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PACIFIC JOURNAL OF MATHEMATICS

Volume 199 No. 2 June 2001

PacificJournalofM

athematics

2001Vol.199,N

o.2

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PACIFIC JOURNAL OF MATHEMATICSVol. 199, No. 2, 2001

ON MIXED PRODUCTS OF COMPLEX CHARACTERS OFTHE DOUBLE COVERS OF THE SYMMETRIC GROUPS

Christine Bessenrodt

In this article, two families of (almost) homogeneous mixedKronecker products of non-faithful and of spin characters ofthe double covers of the symmetric groups are described. Thisis then applied to classify the irreducible mixed products,thus completing the classification of all irreducible Kroneckerproducts of characters of the double covers of the symmetricgroups.

1. Introduction.

Kronecker products of complex characters of the symmetric group Sn havebeen studied in many papers. Information on special products and on thecoefficients of special constituents have been obtained but there is no ef-ficient combinatorial algorithm in sight for computing these products. In[1], products of Sn-characters with few homogeneous components and ho-mogeneous products of characters of the alternating group An have beenclassified. In particular, there are no non-trivial homogeneous Kroneckerproducts for Sn, but there are such products for An, when n is a squarenumber (these are even irreducible).

For the double covers Sn of the symmetric groups, information aboutproducts of characters is even more sparse. Recently, in [2] some resultshave been obtained on products of spin characters of Sn which led to aclassification of homogeneous spin products. Here, homogeneous productsdo occur for all triangular numbers n, but non-trivial irreducible productsoccur only for n = 6.

In this article, we consider mixed products of complex characters for thedouble covers Sn, i.e., products of a non-faithful character of Sn (corre-sponding to a character of Sn) with a spin character. In this situation, thereare some interesting homogeneous or almost homogeneous mixed products;finding such mixed products was greatly helped by the special maple pack-ages SF and QF for dealing with symmetric functions by John Stembridge.Two families of homogeneous resp. almost homogeneous products are de-scribed; one for any composite number and the other one for triangularnumbers. The irreducible mixed products are then classified; they occur foreven numbers and triangular numbers satisfying a congruence condition.

257

http://pjm.math.berkeley.edu/pjm

http://dx.doi.org/10.2140/pjm.2001.199-2

258 CHRISTINE BESSENRODT

2. Preliminaries.

We denote by P (n) the set of partitions of n. For a partition λ ∈ P (n),l(λ) denotes its length, i.e., the number of (non-zero) parts of λ. The setof partitions of n into odd parts only is denoted by O(n), and the set ofpartitions of n into distinct parts is denoted by D(n). We write D+(n) resp.D−(n) for the sets of partitions λ in D(n) with n− l(λ) even resp. odd; thepartition λ is then also called even resp. odd.

We write Sn for the symmetric group on n letters, and Sn for one of itsdouble covers; so Sn is a non-split extension of Sn by a central subgroup〈z〉 of order 2. It is well-known that the representation theory of thesedouble covers is ‘the same’ for all representation theoretical purposes. Thespin characters of Sn are those that do not have z in their kernel. For anintroduction to the properties of spin characters resp. for some results wewill need in the sequel we refer to [5], [10], [11], [13]. Below we collect someof the necessary notation and some results from [13] that are crucial in latersections.

For λ ∈ P (n), we write [λ] for the corresponding irreducible character ofSn; this is identified with the corresponding character of Sn. The associateclasses of spin characters of Sn are labelled canonically by the partitions inD(n). For each λ ∈ D+(n) there is a self-associate spin character 〈λ〉 =sgn 〈λ〉, and for each λ ∈ D−(n) there is a pair of associate spin characters〈λ〉, 〈λ〉′ = sgn 〈λ〉. We write

〈λ〉 =〈λ〉 if λ ∈ D+(n)〈λ〉+ 〈λ〉′ if λ ∈ D−(n)

ελ =

1 if λ ∈ D+(n)√2 if λ ∈ D−(n).

In [13], Stembridge introduces a projective analogue of the outer tensorproduct, called the reduced Clifford product, and proves a shifted analogueof the Littlewood-Richardson rule which we will need in the sequel. To statethis, we first have to define some further combinatorial notions.

Let A′ be the ordered alphabet 1′ < 1 < 2′ < 2 < .... The letters1′, 2′, . . . are said to be marked, the others are unmarked. The notation |a|refers to the unmarked version of a letter a in A′. To a partition λ ∈ D(n)we associate a shifted diagram

Y ′(λ) = (i, j) ∈ N2 | 1 ≤ i ≤ l(λ), i ≤ j ≤ λi + i− 1.

A shifted tableau T of shape λ is a map T : Y ′(λ)→ A′ such that T (i, j) ≤T (i+ 1, j), T (i, j) ≤ T (i, j + 1) for all i, j, and every k ∈ 1, 2, . . . appearsat most once in each column of T , and every k′ ∈ 1′, 2′, . . . appears atmost once in each row of T . For k ∈ 1, 2, . . . , let ck be the number of boxes

MIXED PRODUCTS OF CHARACTERS OF eSn 259

(i, j) in Y ′(λ) such that |T (i, j)| = k: Then we say that the tableau T hascontent (c1, c2, . . . ). Analogously, we define skew shifted diagrams and skewshifted tableaux of skew shape λ \ µ if µ is a partition with Y ′(µ) ⊆ Y ′(λ).For a (possibly skew) shifted tableau S we define its associated word w(S) =w1w2 · · · by reading the rows of S from left to right and from bottom totop. By erasing the marks of w, we obtain the word |w|.

Given a word w = w1w2 . . . , we define

mi(j) = multiplicity of i among wn−j+1, . . . , wn, for 0 ≤ j ≤ nmi(n+ j) = mi(n) + multiplicity of i′ among w1, . . . , wj , for 0 < j ≤ n.

This function mi corresponds to reading the rows of the tableau first fromright to left and from top to bottom, counting the letter i on the way, andthen reading from bottom to top and left to right, counting the letter i′ onthis way.

The word w satisfies the lattice property if, whenever mi(j) = mi−1(j),then

wn−j 6= i, i′, if 0 ≤ j < nwj−n+1 6= i− 1, i′, if n ≤ j < 2n.

For two partitions µ and ν we denote by µ∪ ν the partition which has asits parts all the parts of µ and ν together.

Theorem 2.1 ([13, 8.1 and 8.3]). Let µ ∈ D(k), ν ∈ D(n− k), λ ∈ D(n),and form the reduced Clifford product 〈µ〉 ×c 〈ν〉. Then we have(

(〈µ〉 ×c 〈ν〉) ↑eSn , 〈λ〉) =

1ελεµ∪ν

2(l(µ)+l(ν)−l(λ))/2fλµν ,

unless λ is odd and λ = µ ∪ ν. In that latter case, the multiplicity of 〈λ〉 is0 or 1, according to the choice of associates.

The coefficient fλµν is the number of shifted tableaux S of shape λ \ µ andcontent ν such that the tableau word w = w(S) satisfies the lattice propertyand the leftmost i of |w| is unmarked in w for 1 ≤ i ≤ l(ν).

We will also use the following result from [13] on inner tensor productswith the basic spin character 〈n〉:

Theorem 2.2 ([13, 9.3]). Let λ ∈ D(n), µ a partition of n. We have

(〈n〉[µ], 〈λ〉) =1

ελε(n)2(l(λ)−1)/2gλµ ,

unless λ = (n), n is even, and µ is a hook partition. In that case, themultiplicity of 〈λ〉 is 0 or 1 according to choice of associates.

The coefficient gλµ is the number of “shifted tableaux” S of unshifted shapeµ and content λ such that the tableau word w = w(S) satisfies the latticeproperty and the leftmost i of |w| is unmarked in w for 1 ≤ i ≤ l(λ).

As an interesting consequence, this implies


Corollary 2.3. Let n ∈ N. Then

〈n〉 · 〈n〉 = [n] +n−1∑i=1

[n− i, 1i] .

Finally, we collect some information about certain constituents in squaresresp. in ‘almost’ squares.

Theorem 2.4 ([12], [14], [15]). Let n ∈ N. Let λ ∈ P (n) with λ 6= (n),(1n). Define a, b ∈ N0 by

[λ]2 = [n] + a[n− 2, 2] + b[n− 3, 13] + other constituents.

Denote by hi the number of hooks of λ of length i, for i ∈ 1, 2. Let h21 bethe number of hooks of λ of length 3 and arm length 1. Then we have:

(i) a = h2 + h1(h1 − 2).In particular, a > 0 if n ≥ 4.

(ii) b = h1(h1 − 1)(h1 − 3) + (h1 − 1)(h2 + 1) + h21.In particular, b > 0, unless λ is (n− 1, 1) or (n− 1, 1)′.

Theorem 2.5 ([8, Theorem 4.3]). Let n ∈ N, n ≥ 4. Let µ ∈ D(n) withµ 6= (n) and µ 6= (k, k − 1, . . . , 2, 1). Let s, t ∈ N be defined by

〈µ〉 · 〈µ〉 = [n] + s[n− 1, 1] + t[n− 2, 2] + other constituents.

Then:(i) t ≥ 1.(ii) If n ≥ 5, µ is even and µ is not of the form (k+ r, k− 1+ r, . . . , 1+ r)

for some r, then t ≥ 2.(iii) If µ is not of the form (k + r, k − 1 + r, . . . , 1 + r) for some r, then

s ≥ 1.

We have already considered the exceptional case µ = (n) in Corollary 2.3.Note that for even n it is not clear which hook characters appear in theproduct 〈n〉 · 〈n〉 and which appear in the product 〈n〉 · 〈n〉′, except that eachproduct contains one out of a pair of conjugate hook characters. Clearly,both products do not contain [n − 2, 2] as a constituent, i.e., for the basicspin character we have t = 0 in the notation of the Theorem above.

The exceptional case of a staircase partition µ = (k, k − 1, . . . , 1) will betreated in the next section in Theorem 3.5.

3. Almost homogeneous mixed products.

We now want to study the case of mixed products, i.e., products of the form〈µ〉 · [ν]. Of course, if we know all the constituents in the case of productsof spin characters then we also know all the coefficients in the case of mixedproducts, since

(〈µ〉〈ν〉, [λ]) = (〈µ〉[λ], 〈ν〉)


and 〈ν〉 = 〈ν〉 or 〈ν〉′, depending on n − l(ν) mod 4. But it is not clearhow to obtain a classification of the homogeneous mixed products from theresults we know so far. In the following, we will describe some “combinato-rially homogeneous” mixed products, and we classify the irreducible mixedproducts. We call a character of Sn “homogeneous” if it is of the form c〈λ〉resp. “almost homogeneous” if it is of the form c〈λ〉 for some λ ∈ D(n) andc ∈ N.

We will need the following combinatorial result (see [2], Lemma 4.1 andits proof).

Lemma 3.1 ([2]). Let k ∈ N. Let H(k) denote the product of the hooklengths in ν = (k, k−1, . . . , 2, 1), and let B(k) denote the product of the barlengths in ν. Set n = 1

2k(k + 1). Then:

(i) B(k) = 2n−kH(k).

(ii) B(k + 1) = B(k)k+1∏j=1

(k + j).

First we want to classify all homogeneous resp. almost homogeneous prod-ucts with the basic spin character 〈n〉.

Theorem 3.2. Let µ ∈ P (n), µ 6= (n), (1n). Then the product 〈n〉 · [µ] isalmost homogeneous if and only if µ is a rectangle.

Up to conjugation, we may assume in this case that µ = (ba) with 1 <a ≤ b, and then the product is

〈n〉 · [ba] =

2a−32 〈a+ b− 1, a+ b− 3, · · · , b− a+ 1〉, if a is odd and

b is even

2[a−12 ]〈a+ b− 1, a+ b− 3, · · · , b− a+ 1〉, else.

Proof. First assume that µ is not a rectangle. Let hii, i = 1, . . . , d = d(µ),denote the principal hook lengths in µ. By Theorem 2.2 the product 〈n〉[µ]always has a constituent 〈h11, h22, . . . , hdd〉 as is illustrated by the followingtableau for µ = (73, 62):

1′ 1 1 1 1 1 11′ 2′ 2 2 2 2 21′ 2′ 3′ 3 3 3 31′ 2′ 3′ 4′ 4 41 2 3 4 5 5

.

If µ is not a rectangle, then take j maximal with µj > µj+1 > 0. We can thenreplace the final two entries j′ j in the j th column by the entries j j+1 andstill obtain a tableau of the type counted by the coefficients gλµ occurringin Theorem 2.2, giving a constituent labelled by a partition different from


(h11, h22, . . . , hdd). Thus if the product 〈n〉 · [µ] is almost homogeneous thenµ has to be a rectangle.

So now we consider the case that µ is a rectangle, and we may assume thatµ = (ba) with a ≤ b. In this situation, we have for the partition consideredabove:

λ = (h11, h22, . . . , hdd) = (a+ b− 1, a+ b− 3, . . . , b− a+ 1).

The multiplicity of the constituent 〈a+ b− 1, a+ b− 3, . . . , b− a+ 1〉 inthe product can be calculated by Theorem 2.2; it is easily seen that gλµ = 1,and hence the multiplicity is

1ελε(n)

2a−12 =

2a−32 , if a is odd and b is even

2[a−12 ], else,

as is easily checked. In the first case, the character 〈a+ b− 1, a+ b− 3, . . . ,b− a+ 1〉 is not self-associate, and the associate character appears with thesame multiplicity as µ is not a hook.

We now prove the statement of the Theorem by comparing degrees onboth sides.By the degree formulae for ordinary and spin characters we have for the lefthand side:

〈n〉[ba](1) = 2[n−12 ] n!

H(a, b),

where we denote by H(a, b) the product of the hook lengths in (ba). Forthe right hand side in the statement of the Theorem we obtain by the barformula

2[a−12 ] · 2[n−a2 ] n!

B(a, b)= 2[n−1

2 ] n!B(a, b)

,

where B(a, b) denotes the product of the bar lengths in (a + b − 1, a + b −3, . . . , b− a+ 1).Hence we have to prove that for all a ≤ b we have H(a, b) = B(a, b).

For this, we divide the Young diagram resp. the shifted diagram into threeregions:

b− a b− a

The middle region in both diagrams is of width b− a. It is easy to checkthat the hook lengths in the middle region of (ba) are exactly the same asthe bar lengths in the middle region of (a+ b−1, a+ b−3, . . . , b−a+1); let


Hm(a, b) = Bm(a, b) denote the product of the hook lengths resp. bar lengthsin these middle regions. Let Hl(a, b) and Hr(a, b) denote the product of thehook lengths in the left resp. right region of the diagram for (ba); similarly,let Bl(a, b), Br(a, b) denote the product of the bar lengths in the left resp.right region of the diagram for (a+ b− 1, a+ b− 3, . . . , b− a+ 1). The barlengths occurring in the left region of (a+ b− 1, a+ b− 3, . . . , b− a+ 1) areexactly the same as the hook lengths in the left region of (ba) each multipliedby 2, so

Bl(a, b) = 2(a2)Hl(a, b) .

In the notation of Lemma 3.1 we have Hr(a, b) = B(a) and Br(a, b) = H(a).Since by Lemma 3.1 we know that B(a) = 2(a2)H(a), we obtain

H(a, b) = Hl(a, b)Hm(a, b)Hr(a, b) = Hl(a, b)Bm(a, b) · 2(a2)Br(a, b)

= Bl(a, b)Bm(a, b)Br(a, b) = B(a, b)

and thus the result is proved.

Next we deal with the natural character [n−1, 1] and describe some almosthomogeneous products with this character.

Theorem 3.3. Let n be a triangular number, say n =(k+12

). Then

〈k, k − 1, . . . , 2, 1〉 · [n− 1, 1] = 〈k + 1, k − 1, k − 2, . . . , 3, 2〉.

Proof. First we check that the character given on the right hand side in thestatement above does indeed appear as a constituent:

(〈k, k − 1, . . . , 2, 1〉 · [n− 1, 1], 〈k + 1, k − 1, k − 2, . . . , 3, 2〉)

= (〈k, k − 1, . . . , 2, 1〉 · 〈k + 1, k − 1, k − 2, . . . , 3, 2〉, [n− 1, 1])

= (〈k, k − 1, . . . , 2, 1〉 ↓eSn−1, 〈k + 1, k − 1, k − 2, . . . , 3, 2〉 ↓eSn−1

)

= 1

where the last equality follows from the spin branching theorem. Now toprove the assertion it suffices to check degrees on both sides.

Let again denote B(k) the product of the bar lengths in (k, k−1, . . . , 2, 1),and let B′(k) denote the product of the bar lengths in (k + 1, k − 1, k − 2,. . . , 2). Then by the bar formula we have to check whether the followingequation holds:

2[n−k2 ] n!B(k)

(n− 1) = 2[n−k+22 ] n!

B′(k)or equivalently,

(n− 1)B′(k) = 2B(k).


We want to prove the claim by induction on k, starting with k = 3, wherethe claim is easily checked. By Lemma 3.1

B(k + 1) = B(k)2k+1∏j=k+1

j.

Let N = N(k− 2) be the product of the bar lengths in (k− 1, k− 2, . . . , 2).Then by considering the shifted diagrams one sees that

B′(k) = N · k(k + 1)2k∏

j=k+3

j

B′(k + 1) = N · (k − 1)k

2k−1∏j=k+2

j

(k + 1)(k + 2)2k+2∏i=k+4

i.

Hence

B′(k + 1) = B′(k)(k − 1)(k + 2)(2k + 1)(2k + 2)

k + 3

2k−1∏j=k+2

j.

Since n =(k+12

), we know by induction that

(k2 + k − 2)B′(k) = 4B(k),

and we have to show that

(k2 + 3k)B′(k + 1) = 4B(k + 1).

Using the relations given above this is a straightforward calculation.Hence the assertion of the Theorem is proved.

Theorem 3.4. Let n ∈ N, n ≥ 3, and let µ ∈ D(n). Then the product〈µ〉 · [n − 1, 1] is irreducible if and only if n is a triangular number, sayn =

(k+12

), with k ≡ 2 or 3 mod 4, and µ = (k, k − 1, . . . , 2, 1). In this case,

〈k, k − 1, . . . , 2, 1〉 · [n− 1, 1] = 〈k + 1, k − 1, k − 2, . . . , 3, 2〉.

Proof. If µ and k are as stated, then µ is odd and (k+1, k−1, k−2, . . . , 3, 2)is even, and so by the previous Theorem the stated product is indeed irre-ducible.

Now assume that the product 〈µ〉 · [n − 1, 1] is irreducible. By the clas-sification result for products with the basic spin character, we know thatµ 6= (n).

So assume now µ 6= (k, k − 1, . . . , 1). Then by Theorem 2.5 resp. byTheorem 2.4 both [n− 2, 2] and [n] are constituents of the product 〈µ〉 · 〈µ〉as well as of the square [n− 1, 1]2. Hence

([n− 1, 1] · 〈µ〉, [n− 1, 1] · 〈µ〉) = (〈µ〉 · 〈µ〉, [n− 1, 1] · [n− 1, 1]) ≥ 2,

so the product is not irreducible.


For the classification of the irreducible mixed products, we will need somefurther information in the special case of staircase partitions.

Theorem 3.5. Let k ∈ N, n =(k+12

). Define the coefficients a1, a2, a3, b2,

b3, c3 by

〈k, . . . , 1〉 · 〈k, . . . , 1〉 = [n] + a1[n− 1, 1] + a2[n− 2, 2] + a3[n− 3, 3]

+ b2[n− 2, 12] + b3[n− 3, 13] + c3[n− 3, 2, 1]

+ other constituents.

Then:

(i) a1 = a2 = c3 = 0.

(ii) b2 =

1 if k ≡ 0 or 1 mod 40 if k ≡ 2 or 3 mod 4.

(iii) a3 = b3 = 1.

Proof. Set ρk = (k, . . . , 1) and ϕk = 〈k, . . . , 1〉. Note that ρk ∈ D+ if andonly if k ≡ 0 or 1 mod 4. Also, let πα = 1Sα ↑Sn= 1eSα ↑eSn . So

(ϕk · ϕk, πα) = (ϕk ↓eSα , ϕk ↓eSα),

and for computing the restriction we use the spin branching theorem resp.the shifted Littlewood-Richardson Rule provided by Theorem 2.1.

Since ϕk ↓eS(n−1,1)= 〈k, . . . , 2〉, (ϕk · ϕk, π(n−1,1)) = 1. As [n − 1, 1] =

π(n−1,1) − [n], this yields

a1 = (ϕk · ϕk, [n− 1, 1]) = 0.

Next, ϕk ↓eS(n−2,2)is the irreducible character 〈k, . . . , 3, 1〉×c 〈2〉 (up to the

choice of associates in the case k ≡ 0 or 1 mod 4), hence (ϕk ·ϕk, π(n−2,2)) = 1.As [n− 2, 2] = π(n−2,2) − π(n−1,1), this implies

a2 = (ϕk · ϕk, [n− 2, 2]) = 0.

The restriction ϕk ↓eS(n−3,3)has the two irreducible constituents 〈k, . . . , 4,

2, 1〉×c 〈3〉 and 〈k, . . . , 3〉×c 〈2, 1〉 (up to the choice of associates in the casek ≡ 2 or 3 mod 4). Hence (ϕk · ϕk, π(n−3,3)) = 2, and from the equation[n− 3, 3] = π(n−3,3) − π(n−2,2) we now deduce

a3 = (ϕk · ϕk, [n− 3, 3]) = 1.

Now ϕk ↓eS(n−2,12)= 〈k, . . . , 3, 1〉. Thus,

(ϕk · ϕk, π(n−2,12)) =



As [n− 2, 12] = π(n−2,12) − π(n−2,2) − [n− 1, 1], we obtain

b2 = (ϕk · ϕk, [n− 2, 12]) =


For the restriction ϕk ↓eS(n−3,2,1)we obtain

〈k, . . . , 4, 2, 1〉 ×c 〈2〉+ 〈k, . . . , 4, 3〉 ×c 〈2〉if k ≡ 0 or 1 mod 4, and

〈k, . . . , 4, 2, 1〉 ×c 〈2〉+ 〈k, . . . , 4, 3〉 ×c 〈2〉up to a choice of associates for the second summand, if k ≡ 2 or 3 mod 4.Hence

(ϕk · ϕk, π(n−3,2,1)) =


As

[n− 3, 2, 1] =12

(π(n−3,2,1) − π(n−3,3) − [n− 2, 2]− [n− 1, 1]− [n− 2, 12]),

we obtainc3 = (ϕk · ϕk, [n− 3, 2, 1]) = 0.

Finally, we have

ϕk ↓eS(n−3,13)=

〈k, . . . , 3〉+ 2〈k, . . . , 4, 2, 1〉 if k ≡ 0 or 1 mod 4〈k, . . . , 3〉+ 〈k, . . . , 4, 2, 1〉 if k ≡ 2 or 3 mod 4.

Thus

(ϕk · ϕk, π(n−3,13)) =


As

[n− 3, 13] = π(n−3,13) − π(n−3,2,1) − [n− 1, 1]− [n− 2, 2]− 2[n− 2, 12],

we obtainb3 = (ϕk · ϕk, [n− 3, 13]) = 1.

Theorem 3.6. Let λ ∈ P (n), λ 6= (n), (1n). Let µ ∈ D(n). Then [λ] · 〈µ〉is irreducible if and only if one of the following occurs:

(i) n = 2k, λ = (k, k) or (2k) and µ = (n).Here the products are

[k, k] · 〈n〉 = [2k] · 〈n〉 = 〈k + 1, k − 1〉 .

(ii) n =(k+12

)for some k ∈ N with k ≡ 2 or 3 mod 4, λ = (n − 1, 1) or

(2, 1n−2) and µ = (k, k − 1, . . . , 2, 1).Here the products are

[n− 1, 1] · 〈k, . . . , 1〉 = [2, 1n−2] · 〈k, . . . , 1〉 = 〈k + 1, k − 1, k − 2, . . . , 2〉 .


Proof. In the cases (i) and (ii) described above, the product is irreducibleby Theorem 3.2 resp. Theorem 3.4. In the following we may assume thatn ≥ 4.

Now assume that [λ] · 〈µ〉 is irreducible. Then

1 = ([λ] · 〈µ〉, [λ] · 〈µ〉) = ([λ]2, 〈µ〉 · 〈µ〉).As both [λ]2 and 〈µ〉·〈µ〉 have [n] as a constituent, it suffices to find a furthercommon constituent in all situations not covered by (i) and (ii).

By Theorem 2.4, [λ]2 always contains a constituent [n − 2, 2], and byTheorem 2.5, also 〈µ〉 · 〈µ〉 contains a constituent [n − 2, 2], unless µ = (n)or µ = (k, k−1, . . . , 1). So we only have to consider mixed products [λ] · 〈µ〉with µ of these two exceptional types.

For µ = (n), this was done in Theorem 3.2, giving the products describedin case (i) as the only irreducible mixed products with the basic spin char-acter.

So it remains to deal with the case of a staircase µ = (k, k− 1, . . . , 1). ByTheorem 2.4 and Theorem 3.5 we then find a common constituent [n−3, 13]in [λ]2 and 〈µ〉 · 〈µ〉, unless λ is (n − 1, 1) or (n − 1, 1)′. But in this lattersituation, we can apply Theorem 3.3 which leads exactly to the irreduciblemixed products given in (ii).

Acknowledgements. The financial support by the Deutsche Forschungs-gemeinschaft (grants Be 923/6-1 and JAP-115/169/0) at different stages ofthe work for this article is gratefully acknowledged. Special thanks for thehospitality enjoyed at the University of Osaka where some part of the workfor this article was done go to Professor Uno. The author is also gratefulto John Stembridge for sharing his maple packages SF and QF, that helpedfinding homogeneous mixed products.

References

[1] C. Bessenrodt and A. Kleshchev, On Kronecker products of complex representationsof the symmetric groups, Pacific J. Math., 190(2) (1999), 201-223, MR 2000i:20017.

[2] C. Bessenrodt and A. Kleshchev, On Kronecker products of spin characters of thedouble covers of the symmetric groups, Pacific J. Math., 198 (2001), 295-305.

[3] M. Clausen and H. Meier, Extreme irreduzible Konstituenten in Tensordarstellungensymmetrischer Gruppen, Bayreuther Math. Schriften, 45 (1993), 1-17, MR 94j:20009,Zbl 799.05004.

[4] Y. Dvir, On the Kronecker product of Sn characters, J. Algebra, 154 (1993), 125-140,MR 94a:20023, Zbl 848.20006.

[5] P. Hoffman and J.F. Humphreys, Projective Representations of the SymmetricGroups, Oxford, 1992, MR 94f:20027, Zbl 777.20005.

[6] G. James, The Representation Theory of the Symmetric Groups, Springer LectureNotes Math., 682 (1978), MR 80g:20019, Zbl 393.20009.

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[7] G. James and A. Kerber, The Representation Theory of the Symmetric Group,Addison-Wesley, London, 1981, MR 83k:20003, Zbl 491.20010.

[8] P.B. Kleidman and D.B. Wales, The projective characters of the symmetric groupsthat remain irreducible on subgroups, J. Algebra, 138 (1991), 440-478, MR 92e:20008,Zbl 792.20012.

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Received September 9, 1999.

Fakultat fur MathematikOtto-von-Guericke-Universitat MagdeburgD-39016 MagdeburgGermanyE-mail address: [email protected]

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http://www.ams.org/mathscinet-getitem?mr=94b:20021




CLOSED CONFORMAL VECTOR FIELDS ANDLAGRANGIAN SUBMANIFOLDS IN COMPLEX SPACE

FORMS

Ildefonso Castro, Cristina R. Montealegre, and FranciscoUrbano

We study a wide family of Lagrangian submanifolds in non-flat complex space forms that we will call pseudoumbilical be-cause of their geometric properties. They are determined byadmitting a closed and conformal vector field X such that Xis a principal direction of the shape operator AJX , being Jthe complex structure of the ambient manifold. We empha-size the case X = JH, where H is the mean curvature vectorof the immersion, which are known as Lagrangian submani-folds with conformal Maslov form. In this family we offerdifferent global characterizations of the Whitney spheres inthe complex projective and hyperbolic spaces.

Let Mn be a Kaehler manifold of complex dimension n. The Kaehlerform Ω on M is given by Ω(v, w) = 〈v, Jw〉, being 〈, 〉 the metric and J thecomplex structure on M . An immersion φ : M −→ M of an n-dimensionalmanifold M is called Lagrangian if φ∗Ω ≡ 0. This property involves only thesymplectic structure of M . In this family of Lagrangian submanifolds, onecan study properties of the submanifold involving the Riemannian structureof M . One must take into account the nice property of the second funda-mental form σ of these submanifolds which says that the trilinear form

〈σ(v, w), Jz〉is totally symmetric. Sometimes this property of the second fundamentalform gives obstructions to the existence of examples satisfying classical Rie-mannian properties. So, if one considers like classical property the umbilic-ity, automatically our umbilical Lagrangian submanifold is totally geodesicand only appear trivial examples. Then two natural questions arise: Whatis the Lagrangian version of umbilicity? Which are the corresponding exam-ples?.

In [RU] it was proposed that the Lagrangian version of umbilicity wasthat the second fundamental form satisfies

〈σ(v, w), Jz〉 =n

n+ 2∂v,w,z〈v, w〉〈H,Jz〉,(0.1)

269



270 I. CASTRO, C.R. MONTEALEGRE, AND F. URBANO

where the symbol ∂v,w,z means cyclic sum over v, w, z andH is the mean cur-vature vector of the submanifold. The corresponding classification was alsomade when the ambient space was complex Euclidean space Cn. Besides thelinear Lagrangian subspaces, the Whitney spheres ([RU], see Paragraph 4for the definition) were the only examples, which can be considered like theLagrangian version of the round hyperspheres of Euclidean space. For n = 2this classification was made in [CU1].

In [CU2] when n = 2 and in [Ch1] in arbitrary dimension, the Lagrangiansubmanifolds satisfying (0.1) of the complex projective and hyperbolic spacesCPn and CHn were classified. Again, in the complex projective space onlythe Whitney spheres appeared, but in the complex hyperbolic space, besidesthe Whitney spheres, two new families of noncompact examples appeared,topologically equivalent to S1 × Rn−1 and Rn, which can be considered likethe Lagrangian version of the tubes over hyperplanes and the horospheresin the real hyperbolic space.

Following with the analogy between umbilical hypersurfaces of real spaceforms and our family of Lagrangian submanifolds of complex space forms,the umbilical hypersurfaces are the easiest examples of submanifolds withconstant mean curvature. Our Lagrangian examples, except the totallygeodesic ones, do not have parallel mean curvature vector, property whichis usually taken as a version on higher codimension of the notion of constantmean curvature, but their mean curvature vectors H satisfy that JH areconformal fields on the submanifolds. So, we will take this property likethe Lagrangian version of the concept of hypersurfaces of constant meancurvature. As the dual form of JH is the well-known Maslov form, wewill refer to these submanifolds as Lagrangian submanifolds with conformalMaslov form. In [RU] this family was studied when the ambient space isCn.

The Lagrangian submanifolds satisfying (0.1) also verify that JH is aprincipal direction of AH . Motivated by this fact, in this paper we study awide family of Lagrangian submanifolds, defined by the property that thesubmanifold admits a closed and conformal vector field X (JH is always aclosed field) such that X is a principal direction of AJX , without assumingthat JX is the mean curvature vector H. In this family a minor impor-tant degenerate case appears, which is completely studied and classified inParagraph 3. Motivated by Proposition 1, the nondegenerate submanifoldsof our family are called pseudoumbilical. A particular family of this kind ofsubmanifolds, which was called H-umbilical by B.Y. Chen, was studied in[Ch2].

In Paragraphs 1 and 2 we study deeply pseudoumbilical Lagrangian sub-manifolds of CPn and CHn. In Theorem 1, we describe the pseudoumbilicalLagrangian submanifolds (see Definition 1), showing that they have a simi-lar behavior to the “umbilical” Lagrangian submanifolds. Theorem 1 allows

LAGRANGIAN SUBMANIFOLDS IN COMPLEX SPACE FORMS 271

us to classify pseudoumbilical Lagrangian submanifolds of CPn and CHn

(Corollary 1 and Theorem 2); they are described in terms of planar curvesand Lagrangian (n-1)-submanifolds of the complex projective, hyperbolic orEuclidean spaces depending on the elliptic, hyperbolic or parabolic characterof the submanifold.

In Paragraph 4, we use the above classification in order to study La-grangian submanifolds with conformal Maslov form. Among the most im-portant global results, we show the following Lagrangian version of a Hopftheorem:

The Whitney spheres are the only compact (nonminimal) La-grangian submanifolds of CPn and CHn with conformal Maslovform and null first Betti number.

Another global result is established in terms of the Ricci curvature. In thiscontext, we prove in Corollary 3 that the Whitney spheres are the only com-pact (nonminimal) Lagrangian submanifolds of CPn with conformal Maslovform such that Ric (JH) ≥ (n− 1)|H|2.

Finally, we prove that all the orientable compact Lagrangian submanifoldsof CPn and CHn with nonparallel conformal Maslov form and first Bettinumber one are elliptic pseudoumbilical and then this allows describe them(Corollary 4).

1. Closed conformal fields and Lagrangian submanifolds.

The Lagrangian submanifolds we are going to consider in this paper willhave a closed and conformal vector field. So, in this section, we will describeproperties of this kind of vector fields, as well as properties of Lagrangiansubmanifolds admitting this kind of vector fields.

The following lemma summerizes some of the known results about Rie-mannian manifolds which admit closed and conformal vector fields (see [RU]and references there in).

Lemma 1. Let (M, 〈, 〉) be an n-dimensional Riemannian manifold endowedwith a nontrivial vector field X which is closed and conformal. Then:

(i) The set Z(X) of the zeros of X is a discrete set.(ii) If divX denotes the divergence of X, then

∇VX =divXn

V, |X|2∇ (divX) = −nRic(X)n− 1

X,

for any vector field V on M , where Ric denotes the Ricci curvature ofM .

(iii) The curvature tensor R of M satisfies

|X|2R(v, w)X =Ric(X)n− 1

〈w,X〉v − 〈v,X〉w.


(iv) If M ′ = M −Z(X), then

p ∈M ′ 7→ D(p) = v ∈ TpM / 〈v,X〉 = 0

defines an umbilical foliation on (M ′, 〈, 〉). In particular, |X|2 anddivX are constant on the connected leaves of D.

(v) (M ′, g) with g = |X|−2〈, 〉, is locally isometric to (I × N, dt2 × g′),where I is an open interval in R, t ×N is a leaf of the foliation Dfor any t ∈ R, and X = (∂/∂t, 0). Moreover, if ∆g is the Laplacian ofg then

∆g log |X|+ Ric(X)n− 1

= 0.

In this paper, Mn(c) will denote a complete simply-connected com-plex space form with constant holomorphic sectional curvature c,with c = 4, 0,−4, i.e., complex Euclidean space Cn if c = 0, thecomplex projective space CPn if c = 4 and the complex hyperbolicspace CHn if c = −4.

Let S2n+1 = z ∈ Cn+1 : 〈z, z〉 = 1 be the hypersphere of Cn+1 centeredat the origin with radius 1, where 〈 , 〉 denotes the inner product on Cn+1.We consider the Hopf fibration Π : S2n+1 −→ CPn, which is a Riemanniansubmersion. We can identify Cn with the open subset of CPn defined by

Π(z1, . . . , zn+1) ∈ CPn : zn+1 6= 0.

Then the Fubini-Study metric g is given on Cn by

gp =1

1 + |p|2

〈, 〉 − R(α⊗ α)

1 + |p|2

where 〈, 〉 is the Euclidean metric, R denotes real part and α is the complex1-form on Cn given by

αp(v) = 〈v, p〉+ i〈v, Jp〉.

On the other hand, if c = −4, let H2n+11 = z ∈ Cn+1 : (z, z) = −1 be

the anti-De Sitter space where ( , ) denotes the hermitian form

(z, w) =n∑i=1

ziwi − zn+1wn+1,

for z, w ∈ Cn+1. Then 〈z, w〉 = R(z, w) induces on H2n+11 a Lorentzian

metric of constant curvature −1. If Π : H2n+11 −→ CHn denotes the Hopf

fibration, it is well-known that Π is a Riemannian submersion. Then CHn

can be identified with the unit ball

Bn =

(z1, . . . , zn) ∈ Cn :

n∑i=1

|zi|2 < 1

,


endowed with the Bergmann metric

gp =1

1− |p|2

〈, 〉+ R(α⊗ α)

1− |p|2

.

We recall that CHn has a smooth compactification CHn ∪ S2n−1(∞), whereS2n−1(∞) can be identified with asymptotic classes of geodesic rays in CHn.It is known that S2n−1(∞) can be also identified with

S2n−1(∞) ≡ π(N ),

being

N = z ∈ Cn+1 − 0 : (z, z) = 0and π the projection given by the natural action of C∗ over N . This iden-tification is given by

[β(s)] 7→ π

(β(s) +

β′(s)

|β′(s)|

)being β(s), s ≥ 0, a geodesic ray in CHn and β(s), s ≥ 0, a geodesic ray inH2n+1

1 with Π(β(s)) = β(s).Finally, in order to understand Theorem 1, it is convenient to remember

that although in CPn and CHn there do not exist umbilical real hypersur-faces, M. Kon in [K] and S. Montiel in [M] proved that the geodesic spheresof CPn and the geodesic spheres, the tubes over complex hyperplanes andthe horospheres of CHn are the only η-umbilic real hypersurfaces of CPnand CHn. The horospheres of CHn with infinity point C ∈ S2n−1(∞) andradius λ > 0 are defined by

Π(z) ∈ CHn : |(z, C)|2 = λ2|Cn+1|2

where C = (C1, . . . , Cn+1) is a point in N such that π(C) = C.

Let φ be an isometric immersion of a Riemannian n-manifoldM inMn(c).If the almost complex structure J of Mn(c) carries each tangent space of Minto its corresponding normal space, φ is called Lagrangian. We denote theLevi-Civita connection of M and the connection on the normal bundle by∇ and ∇⊥, respectively. The second fundamental form will be denoted byσ and the shape operator by Aξ. If φ is Lagrangian, the formulas of Gaussand Weingarten lead to

∇⊥Y JZ = J∇Y Z,σ(Y, Z) = JAJY Z = JAJZY,

for tangent vector fields Y and Z. These formulas imply that 〈σ(Y, Z), JW 〉is totally symmetric, where 〈, 〉 denotes the metric in M

n(c) and the in-duced one in M by φ. Using the Codazzi equation, 〈(∇σ)(Y, Z,W ), JU〉 is


also totally symmetric, where ∇σ is the covariant derivative of the secondfundamental form.

If φ : M −→ CPn is a Lagrangian immersion of a simply-connectedmanifold M , then it is well-known that φ has a horizontal lift (with re-spect to the Hopf fibration) to S2n+1 which is unique up to rotations onS2n+1. We will denote by φ this horizontal lift. The horizontality meansthat 〈φ∗(v), Jφ〉 = 0 for any tangent vector v to M , where J is the complexstructure of Cn+1. We remark that only the Lagrangian immersions in CPnhave (locally) horizontal lifts.

In a similar way, if φ : M −→ CHn is a Lagrangian immersion of a simply-connected manifold M , then φ has a horizontal lift to H2n+1

1 , that we willdenote by φ. We also remark that only the Lagrangian immersions in CHn

have (locally) horizontal lifts.

Proposition 1. Let φ : Mn −→ Mn(c) be a Lagrangian isometric im-

mersion of a connected Riemannian manifold M endowed with a closedand conformal field X. Following the notation of Lemma 1, suppose thatσ(X,X) = ρJX, for some function ρ defined on M ′ = M − Z(X). Then ρis constant on the connected leaves of the foliation D and we have only twopossibilities:(A) At any point p of M , AJX has two constant eigenvalues b1 and b2 onD(p), with constant multiplicities n1 and n2. In this case, ρ is constantand the field X is parallel.

(B) At any point p of M ′, AJX has only one eigenvalue b(p) on D(p). Inthis case, b is constant on the connected leaves of the foliation D and

c

4|X|2 +

(divXn

)2

+ b2 = λ ∈ R.

Remark 1. An immersion φ, under the assumptions of Proposition 1, ad-mits a 1-parameter family of closed and conformal vector fields Xµ = µXwith µ a nonnull real number, satisfying σ(Xµ, Xµ) = µρJ(Xµ). So, if (A)is satisfied, we always will consider one of the two parallel fields X suchthat |X| = 1. If condition (B) is satisfied, then σ(V,Xµ) = µbJV for Vorthogonal to Xµ. So

c

4|Xµ|2 +

(divXµ

n

)2

+ (µb)2 = µ2λ.

When λ 6= 0 we always will consider one of the two closed and conformalfields X such that the corresponding constant λ will be 1 or −1 and wewill refer to them to be elliptic or hyperbolic cases. The parabolic case willcorrespond to λ = 0.

Proposition 1 motivates the following definition.


Definition 1. A Lagrangian immersion φ : Mn −→ Mn(c) is said to be

pseudoumbilical if the Riemannian manifold M is endowed with a closedand conformal field X (without zeros) such that σ(X,X) = ρJX for certainfunction ρ on M and condition (B) in Proposition 1 is satisfied.

Remark 2. When c = 4, a pseudoumbilical Lagrangian immersion is al-ways elliptic. When c = 0, it can be either parabolic or elliptic. And whenc = −4, it can be elliptic, parabolic or hyperbolic.

Proof of Proposition 1. We start proving that ∇ρ = hX for a certain func-tion h, which means that ρ is constant on the connected leaves of D. Forthis purpose, derivating σ(X,X) = ρ JX with respect to a tangent vectorfield V and using Lemma 1, (ii) we get

〈(∇σ)(V,X,X), JW 〉+ 2 divXn〈σ(V,X), JW 〉(1.1)

= 〈∇ρ, V 〉〈X,W 〉+ ρdivXn〈V,W 〉,

for any field W . Using properties of φ and the Codazzi equation of φ weobtain that all the terms in the above equation are symmetric on V andW , except possible 〈∇ρ, V 〉〈X,W 〉. So this one must be also symmetric andthis means that ∇ρ = hX for certain function h. In particular ρ is constanton the leaves of D.

On the other hand, as X(p) is an eigenvector of AJX , then AJX can bediagonalized on D(p). Now using Lemma 1, (iii) and the Gauss equation,the eigenvalues λ′s of AJX on D(p) satisfy the equation

λ2 − ρλ+Ric(X)n− 1

− c

4|X|2 = 0.(1.2)

In particular there are at most two. From Lemma 1, (v), Ric(X) is constanton the connected leaves of D. So the two possible eigenvalues of AJX onD(p) are also constant on the leaves of D.

Let

M0 = p ∈M ′ /AJX has two different eigenvalues onD(p).

Claim. M0 is a closed subset of M ′.

If these eigenvalues are denoted by bi, i = 1, 2, and their multiplicitiesby ni, i = 1, 2, then n〈H,JX〉 = ρ + n1b1 + n2b2. Because bi are smoothfunctions on M0, we have that ni, i = 1, 2 are constant on each connectedcomponent of M0. So,

Di(p) = v ∈ D(p) /AJXv = biv, i = 1, 2,

define distributions on the connected components of M0 such that D =D1 ⊕D2.


If Vi are vector fields on Di, from Lemma 1, (iii) and the Gauss equationof φ, we have that

σ(V1, V2) = 0.(1.3)

Also derivating 〈σ(Vi, Vi), JX〉 = bi〈Vi, Vi〉 with respect to Vj , with j 6= i,and using (1.3) and the fact that ∇bi = hiX for certain functions hi, weobtain that

〈(∇σ)(Vj , Vi, Vi), JX〉 = 0,(1.4)

for Vi ∈ Di, Vj ∈ Dj and i 6= j. But derivating with respect to Vi,〈σ(X,Vi), JVj〉 = 0 and using (1.3) and (1.4) we get

(bi − bj)〈∇ViVi, Vj〉 = 0

and then 〈∇ViVi, Vj〉 = 0 when i 6= j. So

∇ViVi = −divXn|X|−2|Vi|2X + (∇ViVi)i,

where i means component onDi. Finally, taking i 6= j, derivating σ(Vi, Vj) =0 with respect to Vi and using the above equation and (1.3) we get

〈(∇σ)(Vi, Vi, Vj), JVj〉 = bj(divX/n)|X|−2|Vi|2|Vj |2

for i 6= j. Changing the roles of i and j and using the symmetry of ∇σ wefinally get

(bi − bj) divX |Vi|2|Vj |2 = 0.

Since i 6= j we get that divX = 0 and then X is a parallel field on M0.Now we see that bi are constant on each connected component of M0.

Taking in Equation (1.1) V = Vi and W = Vj with i 6= j we obtain

〈(∇σ)(Vi, X,X), JVj〉 = 0.

Derivating 〈σ(X,Vi), JVj〉 = 0 with respect to X and using the above equa-tion and (1.3) we obtain

(bi − bj)〈∇XVi, Vj〉 = 0,

and so ∇XVi is a field on Di. Now if we derivate σ(X,Vi) = biJVi withrespect to X, and use the above information and the parallelism of X in(1.1), we obtain that X(bi) = 0, and therefore bi are constant on eachcomponent of M0. So, by continuity, AJX on D has also two eigenvalueson the clousure of M0, and we get that M0 is a closed subset of M ′. Thisproves the claim.

As M ′ is also connected, there are only two possibilities: Either M0 = M ′

or M0 = ∅.If M0 = M ′, then the parallel field X has no zeros and M ′ = M . As

ρ = b1 + b2 then ρ is also constant and we prove (A).


If M0 = ∅, then for any point p of M ′, AJX has only one eigenvalue b(p)on D(p), which is constant on the leaves of D.

Given any field V on D, we have that σ(X,V ) = bJV . Then derivatingwith respect to X and using that ∇XV is a field on D we obtain

(∇σ)(X,X, V ) +divXn

bJV = X(b)JV.

But from (1.1) we have that

(∇σ)(V,X,X) = (ρ− 2b)divXn

JV,

which, joint to the above equation, gives

X(b) = (ρ− b)divXn

.(1.5)

If h : M ′ −→ R is the function defined by

h =c

4|X|2 +

(divXn

)2

+ b2,

then using Lemma 1, (ii), (1.5), the fact that b is constant on the leaves ofD and (1.2), we have

∇h =−2divXn|X|2

((−c/4)|X|2 +

Ric(X)n− 1

+ b2 − ρb)X = 0.

So h is a constant λ. This finishes the proof.

2. Pseudoumbilical Lagrangian submanifolds of complex spaceforms.

In the following result we characterize pseudoumbilical Lagrangian immer-sions in complex space forms (see Definition 1).

Theorem 1. Let φ : Mn −→ Mn(c) be a Lagrangian immersion of a con-

nected manifold M and exp : TM −→M the exponential map of M .

(i) φ is elliptic pseudoumbilical if and only if there exist a point C ∈M , avector field (without zeros) X on M and a nontrivial smooth functionh : M −→ C satisfying c

4 |X|2+|h|2 = 1, such that for any point outside

the zeros of h, p ∈M −Z(h), the geodesic

βp(s) = exp(φ(p), s f(p)h(p)Xp)


with

f(p) =

1 when c = 0

arccos |h||hX| (p) when c = 4

cosh−1 |h||hX| (p) when c = −4

pass through the point C at s = 1. This point C of M will be calledthe center of φ.

(ii) φ : M −→ CHn is hyperbolic pseudoumbilical if and only if there exista complex hyperplane CHn−1, a vector field X on M and a nontrivialsmooth function h : M −→ C satisfying |h|2 − |X|2 = −1, such thatfor any point outside the zeros of h, p ∈M −Z(h), the geodesic

βp(s) = exp(φ(p), s f(p)h(p)Xp)

with f(p) = cosh−1 |X||hX| (p) cuts orthogonally to CHn−1 at s = 1.

(iii) φ : M −→ CHn is parabolic pseudoumbilical if and only if there exist avector field (without zeros) X on M and a smooth function h : M −→C satisfying |h|2 − |X|2 = 0 such that the map

p ∈M 7→ [βp] ∈ S2n−1(∞),

with βp the geodesic ray βp(s) = exp(φ(p), s h(p)Xp), s ≥ 0, is a cons-tant C ∈ S2n−1(∞) and the horosphere with infinity point C, whereφ(p) lies in, has radius δ|Xp|, with δ a positive real number.

Remark 3. If φ : M −→ Mn(c) is an elliptic pseudoumbilical Lagrangian

immersion and d is the distance on Mn(c), then

d(C, φ) =

|X| when c = 0

arccos |h| when c = 4

cosh−1 |h| when c = −4.

So, if N is the leaf of the foliation D passing through a point p ∈M −Z(h),then φ(N) lies on the geodesic sphere of Mn(c) centered at C and radiusd(C, φ(p)).

When c = 0 or c = −4, Z(h) = ∅. When c = 4, h can have zeros. Bycontinuity, Z(h) is given by

Z(h) = p ∈M : d(φ(p), C) = π/2,i.e., φ(Z(h)) is the intersection of φ(M) with the cut locus of the point C.


Also, if Mn = Cn, φ is given by φ = C + hX with |h| = 1. In this case, ifG : Cn − C −→ Cn − C is the inversion centered at the point C, thenG φ is also an elliptic pseudoumbilical Lagrangian immersion with centerC. The corresponding closed and conformal field is X/|X|2.

Remark 4. If φ : M −→ CHn is a hyperbolic pseudoumbilical Lagrangianimmersion and d is the distance on CHn, then

d(CHn−1, φ) = cosh−1 |X|.

So, if N is the leaf of the foliation D passing through a point p ∈ M , thend(CHn−1, φ(N)) = cosh−1 |Xp|, which means that φ(N) lies on the tube overCHn−1 of radius cosh−1 |Xp|. Also

Z(h) = p ∈M : φ(p) ∈ CHn−1.

Finally, if φ : M −→ CHn is a parabolic pseudoumbilical Lagrangian im-mersion, and N the leaf of the foliation D passing through a point p ∈ M ,then φ(N) lies on the (2n-1)-dimensional horosphere of CHn with infinitypoint C and radius δ|Xp|.

These η-umbilic hypersurfaces of CHn carry a contact structure and theleaves of the foliation D of our pseudoumbilical Lagrangian submanifoldsare integral submanifolds of maximal dimension of such structure (see [B]for details).

Proof. Let φ : M −→ Mn(c) a Lagrangian immersion, h : M −→ C a

nontrivial smooth function, X a vector field without zeros on M and f :M − Z(h) −→ R a smooth function. For each p ∈ M − Z(h), βp(s) willdenote the geodesic exp(φ(p), sf(p)h(p)Xp).

Given a vector v tangent to M at p and a curve α : (−ε, ε)→M − Z(h)with α(0) = p and α′(0) = v, we consider a variation G : (−ε, ε)×R −→Mof the geodesic βp given by G(t, s) = γt(s), where γt is the geodesic on M

with γt(0) = φ(α(t)) and γ′t(0) = (fhX)(α(t)). Then K(s) = ∂G∂t (0, s) is a

Jacobi field along the geodesic βp.To determine K(s) we need to control its initial conditions K(0) and

K ′(0). It is clear that K(0) = φ∗(v) ≡ v. Also

K ′(0) =∂2G

∂s∂t(0, 0) =

∂

∂t |0

∂G

∂s(t, 0) = d(fh)p(v)Xp + (fh)(p)∇vX,

where ∇ is the Levi-Civita connection on M .


As β′p(0) = (fhX)(p), we decompose K(0) and K ′(0) in the following way

K(0) =R(h)f |hX|2

〈Xp, v〉β′p(0)− I(h)f |hX|2

〈Xp, v〉Jβ′p(0) + v⊥,(2.1)

K ′(0) = 〈∇ log f |hX|, v〉β′p(0)

+(〈σ(Xp, Xp), Jv〉

|X|2+I(v(h)h)|h|2

)Jβ′p(0) + fh(∇vX)⊥

where ⊥ means the component orthogonal to the complex plane spanned byXp, R (resp. I) denotes real (resp. imaginary) part, and ∇ is the gradientof the induced metric. Now, taking into account (2.1), our Jacobi field Kalong βp is given by

K(s) = (νs+ µ)β′p(s) + K1(s) + K(s),

being

ν = 〈∇ log f |hX|, v〉, µ =<(h)〈v,X〉f |hX|2

and K1(s) and K(s) the Jacobi fields along βp given by

K1(s) =

A1(s) + sB1(s) when c = 0

cos(2√λs)A1(s) + sin(2

√λs)

2√λ

B1(s) when c = 4

cosh(2√−λs)A1(s) + sinh(2

√−λs)

2√−λ B1(s) when c = −4,

K(s) =

A(s) + sB(s) when c = 0

cos(√λs)A(s) + sin(

√λs)√λ

B(s) when c = 4

cosh(√−λs)A(s) + sinh(

√−λs)√−λ B(s) when c = −4,

where λ = c4(f |hX|)2(p) and A1(s), B1(s), A(s), B(s) are, respectively, the

parallel fields along βp(s) with

A1(0) = − I(h)f |hX|2

〈Xp, v〉Jβ′p(0),

B1(0) =(〈σ(Xp, Xp), Jv〉

|X|2+I(v(h)h)|h|2

)Jβ′p(0),

A(0) = v⊥, B(0) = fh(∇vX)⊥.


We remark that K1(s) is a Jacobi field colinear with Jβ′p(s) and K(s) is aJacobi field orthogonal to β′p(s) and Jβ′p(s).

Proof of (i). First, we suppose that φ is elliptic pseudoumbilical. Then (seeDefinition 1 and Proposition 1) there exists a closed and conformal vectorfield X (without zeros) such that σ(X,X) = ρJX and σ(X,V ) = bJV forany field V orthogonal to X. Defining h by

h = −divXn

+ ib,

X can be chosen (see Remark 1) in such a way that |h|2+ c4 |X|

2 = 1. If h ≡ 0,then X is a parallel vector field and hence Ric(X) = 0. As b also vanishes,(1.2) says that c = 0, which contradicts the fact that |h|2 + c

4 |X|2 = 1. So

h is nontrivial. In fact the set of points where h does not vanish is dense inM .

Now we consider the function F : M −Z(h) −→M given by

F (p) = βp(1) = exp(φ(p), f(p)h(p)Xp).

We are going to compute dFp(v) for any p ∈ M − Z(h) and any v ∈ TpM .It is clear that dFp(v) = K(1), being K(s) the Jacobi field associated to v(see the beginning of the proof).

From the definition of f(p) and the fact that |h|2 = 1 − c4 |X|

2 we candirectly get

sin√λ = |X|, cos

√λ = |h|, when c = 4(2.2)

sinh√−λ = |X|, cosh

√−λ = |h|, when c = −4.

From Lemma 1, Proposition 1, (1.2), (1.5) and the expression of f , weget that µ + ν = 0. Also, from Lemma 1, Proposition 1, (1.5) and an easycomputation, it follows that

B1(0) = b

(1|X|2

− c

4|h|2

)〈Xp, v〉Jβ′p(0),

which allows to prove, using (2.2), that K1(1) = 0.Finally, from Lemma 1 we have that B(0) = −f |h|2v⊥, which implies

using again (2.2) that K(1) = 0. So we have got that

dFp(v) = K(1) = 0,

for any tangent vector v ∈ TpM and any p ∈M −Z(h).If c = 0 or c = −4, Z(h) = ∅ and then F : M −→ M is a constant

function C.If c = 4, let M1 and M2 be two different connected components of M −

Z(h) with M1 ∩M2 6= ∅. Then F is constant on each Mi, i.e., F (Mi) = Cifor i = 1, 2. Let p ∈ Z(h) ∩ M1 ∩ M2 . From Lemma 1, (v) we canparameterize M around p like (−δ, δ)×Nn−1 such that φ(0, x0) = p. In this


neighborhood, h is a function of t ∈ (−δ, δ) (see Lemma 1 again), and so0 ×N ⊂ Z(h). So

φ0 : N −→ CPn

defined by φ0(x) = φ(0, x) is a Lagrangian immersion lying in the cut locusof C1 and C2. If C1 6= C2, then the image of φ0 lies in the intersection ofboth, which is a linear (n− 2)-subspace of CPn. This is impossible becauseφ0 is Lagrangian and N is (n − 1)-dimensional. So we have obtained thatC1 = C2 = C and the necessary condition is proved.

Conversely, suppose that there exist a tangent vector field X (withoutzeros) and a nontrivial function h : M −→ C such that F (p) = βp(1) =exp(φ(p), f(p)h(p)Xp) : M − Z(h) −→ M is a constant function. ThendFp(v) = 0 for any p ∈M−Z(h) and for any v ∈ TpM . So the correspondingJacobi fieldK(s) along the geodesic βp satisfiesK(1) = dFp(v) = 0. Lookingat the expression of K(s), this means that

µ+ ν = 0, K1(1) = 0 and K(1) = 0.

But µ+ ν = 0 implies, taking into account the expression of f , that

∇|X|2 = −2R(h)X.(2.3)

Also, from K(1) = 0 and using (2.2), we obtain that

|h|2v⊥ + h(∇vX)⊥ = 0.

Now by decomposing the above equation in tangential and normal compo-nents to φ we get

(∇vX)⊥ = −R(h)v⊥, σ(Xp, v)⊥ = I(h)Jv⊥.(2.4)

Now (2.3) and the first part of Equation (2.4) say that

∇vX = −R(h)v,

for any v ∈ TpM , and for any p ∈M −Z(h). This means that X is a closedand conformal vector field on M −Z(h) with div(X) = −nR(h).

Also, taking v = Xp in the second part of Equation (2.4) we obtain thatσ(X,X) = ρJX for certain function ρ. Finally, taking v orthogonal to Xp inthe same Equation (2.4), we get σ(v,Xp) = I(h)Jv. So (see Proposition 1)φ : M −Z(h) −→M is an elliptic pseudoumbilical Lagrangian immersion.

If c = 0,−4 the proof is finished. If c = 4, φ(Z(h)) is contained in the cutlocus of the center C, which is a linear (n − 1)-subspace of CPn. As φ is aLagrangian immersion, Z(h) has no interior points and M − Z(h) is denseon M . So the whole φ is an elliptic pseudoumbilical immersion.

Proof of (ii). First, we suppose that φ : M −→ CHn is hyperbolic pseu-doumbilical. Then, from Proposition 1 and Definition 1, there exists a closed


and conformal vector field X (without zeros) such that σ(X,X) = ρJX andσ(X,V ) = bJV for any V orthogonal to X. Defining h by

h = −divXn

+ ib,

X can be chosen (see Remark 1) in such a way that |h|2 − |X|2 = −1. Ifh ≡ 0, then X is parallel and so Ric(X) = 0. As b = 0, (1.2) gives acontradiction. So h is nontrivial.

In this case, we consider F : M −Z(h) −→ CHn given by

F (p) = βp(1) = exp(φ(p), f(p)h(p)Xp).

Following the proof of (i), we can prove in this case that µ + ν = 0 andK1(1) = 0 and hence

dFp(v) = K(1),

which means that dFp(v) is orthogonal to the complex plane spanned byβ′p(1). Also, using the expressions of K(s) and f , it is easy to check thatK ′(1) = 0, and then β′p(1) : p ∈ M − Z(h) spans a 1-complex dimen-sional parallel subbundle on CHn along M − Z(h). So, for each connectedcomponent Ci of M − Z(h) there exists a complex hyperplane CHn−1

i ofCHn such that F (Ci) ⊂ CHn−1

i . It is clear that for any p ∈ Ci, βp(s) cutsorthogonally to CHn−1

i at s = 1.Following a similar reasoning as in (i), we can prove that M − Z(h) is

connected and the proof of the necessary condition is finished.Conversely, suppose that there exist a vector field X without zeros on M ,

a nontrivial smooth function h : M −→ C satisfying −|X|2+|h|2 = −1 and alinear hyperplane CHn−1 such that the geodesic βp(s) =exp(φ(p), f |hX|(p))cuts orthogonally to CHn−1 at s = 1. We follow again the proof of (i),and then, if K is the corresponding Jacobi field along βp, dFp(v) = K(1) istangent to CHn−1. Also, β′p(1) spans the normal space to CHn−1 and henceK ′(1) is a normal vector to CHn−1 at βp(1). This means that K ′(1) = 0.From these two facts, we obtain using the expression of K(s) that

µ+ ν = 0, K1(1) = 0, K ′(1) = 0.

Using the above like in the proof of (i), we can prove that X is a closedand conformal vector field such that σ(X,X) = ρX and σ(X,V ) = I(h)JVfor any v orthogonal to X. So φ : M − Z(h) −→ CHn is a hyperbolicpseudoumbilical Lagrangian immersion. By continuity, φ(Z(h)) ⊂ CHn−1,and since φ is Lagrangian, Z(h) has no interior points, and M − Z(h) isdense in M . This means that the whole φ is hyperbolic pseudoumbilical.

Proof of (iii). We denote by F : M −→ S2n−1(∞) the map F (p) = [βp]. Letφ be a local horizontal lift of φ to H2n+1

1 . Using the notation of Paragraph 1,


βp = Π(βp), with βp(s) = cosh(s|Xp|2)φ(p) + sinh(s|Xp|2)h(p)Xp/|Xp|2,where we have identified X and φ∗X. From the identification S2n−1(∞) ≡π(N ), we have that

F ≡ π ϕ, with ϕ = φ+hX

|X|2.

Suppose now that φ is parabolic pseudoumbilical. Then, from Proposition 1and Definition 1 there exists a closed and conformal vector field X (withoutzeros) such that σ(X,X) = ρJX and σ(X,V ) = bJV for any V orthogonalto X. We define

h = −divXn

+ ib.

Then Proposition 1 and Remark 1 say that we can take X in order to|h|2−|X|2 = 0. For any tangent vector v ∈ TpM , using Lemma 1, (1.2) and(1.5), we obtain v(ϕ) = h

|X|2 〈v,X〉ϕ. This means that dFp(v) = 0 and fromthe connection ofM this implies that F is a constant function C ∈ S2n−1(∞).Following the notation of Paragraph 1, we can take C = ϕ. Using once moreLemma 1, (1.2) and (1.5), it is straightforward to prove that

v

(|(φ, C)|2

|X|2|Cn+1|2

)= 0.

This finishes the proof of the necessary condition.Conversely, if v is any tangent vector to M , then dFp(v) = 0, which

implies that

v(ϕ) =h

|X|2〈v,X〉ϕ.(2.5)

On the other hand, since φ(p) lies in the horosphere with infinity pointC and radius δ|Xp|, we deduce that v(|ϕn+1|2|X|2) = 0. Using (2.5) inthis equation we get ∇|X|2 = −2R(h)X. Using this information in (2.5)and following a similar reasoning like in the proof of (i), we arrive at φ isparabolic pseudoumbilical.

Theorem 1 shows that elliptic pseudoumbilical Lagrangian immersionshave a center and that they can be constructed using the exponential map ofthe ambient space. This fact will imply that they are invariant under certaintransformations of the ambient space, what allows us construct them fromthe classification given by A. Ros and the third author for complex Euclideanspace.

Proposition 2. (i) Let φ : Mn −→ Cn be an immersion, 〈, 〉 the Eu-clidean metric on Cn and g the Fubini-Study metric on Cn. Then φ is


pseudoumbilical Lagrangian with center 0 in (Cn, g) if and only if φ iselliptic pseudoumbilical Lagrangian with center 0 in (Cn, 〈, 〉).

(ii) Let φ : Mn −→ Bn ⊂ Cn be an immersion and g the Bergmann metricon Bn. Then φ is elliptic pseudoumbilical Lagrangian with center 0in (Bn, g) if and only if φ is elliptic pseudoumbilical Lagrangian withcenter 0 in (Bn, 〈, 〉).

Remark 5. Proposition 2 is not true when the center of the immersion isa point C ∈ Cn different from 0.

Proof of (i). From the definition of g, if Ωg and Ω denote the Kaehlertwo-forms on (Cn, g) and (Cn, 〈, 〉) respectively, then

φ∗Ωg =1

1 + |p|2

φ∗Ω− φ∗(=(α ∧ α))

1 + |p|2

.(2.6)

Also, the expressions of the exponential maps at 0 ∈ Cn with respect to themetrics 〈, 〉 and g are given respectively by

exp(0, v) = v, expg(0, v) = cos(|v|) sin(|v|) v

|v|,

and it is not difficult to see that the parallel transport along the geodesicexpg(0, tv) is (like in the Euclidean case) the identity.

First, we suppose that φ is an elliptic pseudoumbilical Lagrangian immer-sion with respect to (Cn, 〈, 〉). From Theorem 1 we have that

φ = hX,

with |h|2 = 1.Then, (φ∗α)(v) = h〈v,Xp〉. So from (2.6) we have that φ∗Ωg = 0, which

means that φ is a Lagrangian immersion in (Cn, g).On the other hand, our immersion φ = hX can be rewritten as

φ(p) = expg(0, f(p)(hX)(p)),

with h = h√1+|X|2

, X =√

1 + |X|2X and f(p) the corresponding function

given in Theorem 1. Although (Cn, g) is not complete, it is easy to checkthat Theorem 1 can be used here in order to say that φ is a pseudoumbilicalLagrangian immersion in (Cn, g) with Z(h) = ∅.

Conversely, we suppose that φ is a pseudoumbilical Lagrangian immersionwith respect to (Cn, g). From Theorem 1 and the fact that the cut locus of0 in Cn is empty we have that

φ = expg(0, fhX),

with g(X,X) + |h|2 = 1, f = arccos |h||h|√g(X,X)

and Z(h) = ∅. So using the

expression of expg(0,−) we have that

φ = hX.


So (φ∗α)(v) = h√g(X,X)

〈v,Xp〉 and from (2.6) we have that φ∗Ω = 0, which

means that φ is a Lagrangian immersion in (Cn, 〈, 〉).Now, our immersion φ can be rewritten as φ = exp(0, hX) with h = h

|h|and X = |h|X. This means, using Theorem 1 again, that φ is an ellipticpseudoumbilical Lagrangian immersion in (Cn, 〈, 〉). This finishes the proofof (i).

To prove (ii) we follow the proof of (i) taking into account that in thiscase

expg(0, v) = cosh(|v|) sinh(|v|) v

|v|,

and the parallel transport along the geodesic expg(0, tv) is the identity.

Proposition 2 allows us, using the results got by A. Ros and the thirdauthor for the case of complex Euclidean space, to classify the elliptic pseu-doumbilical Lagrangian submanifolds of CPn and CHn. In fact, we startrecalling the classification for complex Euclidean space.

Theorem ([RU]). Let φ : Mn −→ Cn be an elliptic pseudoumbilical La-grangian immersion with center C. Then locally Mn is a product I × N ,where I is an interval of R and N a simply-connected manifold, and φ isgiven by

φ(t, x)− C = α(t)ψ(x),

where ψ : N −→ S2n−1 ⊂ Cn is a horizontal lift to S2n−1 of a Lagrangianimmersion ψ : N −→ CPn−1 and α : I −→ C∗ a regular curve.

The proof of the statements given in the following families of examples isstraightforward and so will be omitted.

Examples 1. Let α : I −→ CP1 − Π(0, 1) be a regular curve, ψ : N −→CPn−1 a Lagrangian immersion of an (n-1)-dimensional simply-connectedmanifold N , and α = (α1, α2) : I −→ S3 ⊂ C2 and ψ : N −→ S2n−1 ⊂ Cn

horizontal lifts of α and ψ respectively. Then,

α ∗ ψ : I ×N −→ CPn

given by

(α ∗ ψ)(t, x) = Π(α1(t)ψ(x); α2(t)),

is a pseudoumbilical Lagrangian immersion with center C = Π(0, . . . , 0, 1).

Remark 6. Looking at Remark 3, it is clear that this immersion α ∗ ψverifies that

Z(h) = (t, x) ∈ I ×N : α(t) = Π(1, 0).


Examples 2. Let α : I −→ CH1 − Π(0, 1) be a regular curve, ψ : N −→CPn−1 a Lagrangian immersion of an (n−1)-dimensional simply-connectedmanifold N , and α = (α1, α2) : I −→ H3

1 ⊂ C2 and ψ : N −→ S2n−1 ⊂ Cn

horizontal lifts of α and ψ respectively. Then,

α ∗ ψ : I ×N −→ CHn

given by

(α ∗ ψ)(t, x) = Π(α1(t)ψ(x); α2(t)),

is an elliptic pseudoumbilical Lagrangian immersion with center C = Π(0,. . . , 0, 1).

Remark 7. The immersions α∗ψ given in Examples 1 and 2 do not dependon the horizontal lift of the curve α and, up to holomorphic isometries ofthe ambient space, do not depend on the horizontal lift of ψ.

Corollary 1. Let φ : Mn −→ CPn (respectively φ : Mn −→ CHn) be anelliptic pseudoumbilical Lagrangian immersion. Then φ is locally congruentto some of the immersions α ∗ ψ described in Examples 1 (respectively inExamples 2).

Proof. We start by proving the projective case. From Theorem 1, φ has acenter C ∈ CPn, and there is no restriction if we take it as C = Π(0, . . . , 0, 1).Let

CPn−1 = Π(z1, . . . , zn+1) ∈ CPn : zn+1 = 0

be the cut locus of the point C. Then

F : Cn −→ CPn − CPn−1,

given by

F (p) = Π

(1√

1 + |p|2(p, 1)

),

is a diffeomorphism with F (0) = C and F ∗〈, 〉 = g, where 〈, 〉 (respectivelyg) denotes the Fubini-Study metric on CPn (respectively on Cn).

From Remark 2 and Proposition 2 it is easy to see that

F−1 φ : M −B −→ Cn,

with B = p ∈ M : φ(p) ∈ CPn−1 is an elliptic pseudoumbilical La-grangian immersion in (Cn, 〈, 〉) with center 0, which means, using Theorem[RU], that locally

F−1 φ = βψ


with β : I −→ C∗ a regular curve and ψ the horizontal lift to S2n−1 of aLagrangian immersion ψ : N −→ CPn−1, being N an (n−1)-dimensionalsimply-connected manifold. So, φ : M −B −→ CPn − CPn−1 is given by

φ = Π

(1√

1 + |β|2(βψ, 1)

)= Π(α1ψ, α2),

where

(α1, α2) =eiθ√

1 + |β|2(β, 1),(2.7)

being

θ(t) = −∫ t

t0

〈β′, Jβ〉1 + |β|2

dr

with t0 a point of the interval I. From (2.7) it is easy to see that α = (α1, α2)is a horizontal curve in S3 and so it is the horizontal lift to S3 of the regularcurve α = Π α in CP1. This finishes the proof of the projective case.

To prove the hyperbolic case, we proceed in a similar way taking intoaccount that the center C of φ can be taken as C = Π(0, . . . , 0, 1) and thatF : Bn −→ CHn given by

F (p) = Π

(1√

1− |p|2(p, 1)

)is a diffeomorphism with F (0) = C and F ∗〈, 〉 = g, being 〈, 〉 (respectivelyg) the metric of CHn (respectively the Bergmann metric on Bn).

To finish the description of these submanifolds in CHn, it remains the hy-perbolic and parabolic cases. As always, we start describing examples whoseassertions will not be proved and will take again into account Remark 7.

Examples 3. Let α : I −→ CH1 be a regular curve, ψ : N −→ CHn−1 aLagrangian immersion of an (n−1)-dimensional simply-connected manifoldN , and α = (α1, α2) : I −→ H3

1 ⊂ C2 and ψ : N −→ H2n−11 ⊂ Cn horizontal

lifts of α and ψ respectively. Then

α ∗ ψ : I ×N −→ CHn

given by

(α ∗ ψ)(t, x) = Π(α1(t), α2(t)ψ(x)),

is a hyperbolic pseudoumbilical Lagrangian immersion.

To describe the parabolic pseudoumbilical Lagrangian examples in CHn,we need some previous remarks.


First, we will consider the null vectors e1 = 12(0, . . . , 0, 1,−1) and e2 =

12(0, . . . , 0, 1, 1), in such a way that (e1, e2) = 1/2 where (, ) is the Hermitianproduct in Cn+1.

Second, if ψ : Nn−1 −→ Cn−1 is a Lagrangian immersion of a simply-connected manifold N , then the complex 1-form α on Cn−1 defined beforeas

αp(v) = 〈v, p〉+ i〈v, Jp〉verifies that ψ∗α is a closed 1-form and so there exists a complex functionfψ : N −→ C such that dfψ = 2ψ∗α.

Examples 4. Let α : I −→ CH1 be a regular curve, ψ : N −→ Cn−1 aLagrangian immersion of an (n−1)-dimensional simply-connected manifoldN , and α = α1e1 + α2e2 : I −→ H3

1 ⊂ C21 a horizontal lift of α. Then

α ∗ ψ : I ×N −→ CHn

given by

(α ∗ ψ)(t, x) = Π(α1(t)ψ(x); α1(t)e1 + (α2(t)− fψ(x)α1(t))e2)

with <(fψ) = |ψ|2, is a parabolic pseudoumbilical Lagrangian immersion.

Theorem 2. Let φ : Mn −→ CHn be a pseudoumbilical Lagrangian immer-sion. If φ is hyperbolic (respectively parabolic), then φ is locally congruentto some of the immersions described in Examples 3 (respectively in Exam-ples 4).

Proof. We first consider a horizontal lift φ : U −→ H2n+11 of φ, where U can

be identified (see Lemma 1, (v)) to I ×N , where I is an open interval with0 ∈ I and N is a simply-connected (n−1)-manifold. In addition, followingthe same notation of the proof of Theorem 1, we can take X = (∂/∂t, 0)and, by identifying X and φ∗X, the second fundamental form σ of φ satisfies

σ(X,X) = ρJX + |X|2φ, σ(X, v) = bJv,(2.8)

for any vector v orthogonal to X.Suppose now φ is hyperbolic pseudoumbilical. From Theorem 1(ii), up to

holomorphic isometries, we can take the complex hyperplane CHn−1 as

CHn−1 = Π(z1, . . . , zn+1) : z1 = 0.The equation of a tube of radius r over CHn−1 is |z1| = sinh r (cf. [M]). Ifr = cosh−1 |Xp| (see Remark 4), it becomes in

|φ1|2 = |X|2 − 1 = |h|2.Using Theorem 1 again, the geodesic

βp(s) = Π(

cosh(s f |hX|(p))φ+ sinh(s f |hX|(p))(hX)(p)|hX|(p)

)


with f(p) = cosh−1 |Xp||h(p)Xp| cuts orthogonally to CHn−1 at s = 1. In particular,

βp(1) = Π(|X|φ+ hX

|X|

)∈ CHn−1, which means that the first component of

Υ = |X|2φ+ hX is zero.On the other hand, using (2.8) we obtain that v(hX + |h|2φ) = 0 and

so α1 = −(hX + |h|2φ) is a function of t, with |α1|2 = |h|2 and, from (2.8)again, satisfies α′1 = − |X|

2

h α1.Since |X|2 − |h|2 = 1 and 〈Υ,Υ〉 = −|X|2, from above we can write

φ = α1 + Υ as

φ(t, x) = (α1(t),Υ(t, x)).

Using (2.8), we obtain that Υ′ = −hΥ, so that αx(t) = Υ(t, x) =e−

R t0 h(s)dsαx(0) is a plane curve too. If we now put

ψ(x) =αx(0)√

−〈αx(0), αx(0)〉,

we have that 〈ψ, ψ〉 = −1. From the above definition we can write

φ(t, x) = (α1(t), α2(t)ψ(x)),

with α2(t) = |X|(0)e−R t0 h(s)ds.

Since φ is horizontal, we deduce easily that ψ is a horizontal immersionin a certain H2n−1

1 and that α = (α1, α2) is a horizontal curve in H31. Thus

φ : U −→ CHn is the example α ∗ ψ of Examples 3 with α = Π(α) andψ = Π(ψ).

In the parabolic case, from Theorem 1, (iii), we can take up to holomor-phic isometries the infinity point C = π(C) with C = (0, . . . , 0, 1, 1) andidentify, following the notation of Paragraph 1, C with ϕ = φ+ hX

|X|2 .Then (2.8) says that v(ϕ) = 0, for any vector v orthogonal to X. In this

way, ϕ = ϕ(t) is a null plane curve satisfying, from (2.8) again, X(ϕ) = hϕ.We now put α1 = −|X|2ϕ and it is easy to check that α′1 = −hα1.

Now we define

ψ =1α1

(φ− α1e1 − 2(φ, e1)e2),

where e1 = 12(0, . . . , 0, 1,−1) and e2 = 1

2(0, . . . , 0, 1, 1). It is clear that(ψ, e1) = 0 and from the properties of α1 (or ϕ) and the choosing of e1 ande2 it is not difficult to get that (ψ, e2) = 0 and X(ψ) = 0.

On the other hand, for any v orthogonal to X, we have that (φ∗v, e2) = 0and this implies

(ψ∗v, ψ∗w) =1|α1|2

(φ∗v, φ∗w), (ψ∗v, ψ) = −(φ∗v, e1)α1

.


Let us take α2 as a solution to α′2 + hα2 = −e−Rh/h and define f =eα2−2(eφ,e1)eα1

. We compute that X(f) = 0 and v(f) = 2(ψ∗v, ψ).As a summary, we have shown that φ = α1ψ + α1e1 + (α2 − fα1)e2 and

thus φ : U −→ CHn is the example α ∗ψ of Examples 4 with α = Π(α).

3. The degenerate case for nonflat complex space forms.

In this paragraph we will classify the family of Lagrangian submanifoldsof Mn(c) admitting a closed and conformal field X with σ(X,X) = ρJXand satisfying condition (A) of Proposition 1 when M is a nonflat complexspace form. We will refer to this as the degenerate case. In order to makeself contained the paper, we start describing this family when Mn(c) is thecomplex Euclidean space Cn; this description was obtained by A. Ros andthe third author in [RU, Proposition 2].

Proposition ([RU]). Let φ : Mn −→ Cn be Lagrangian immersion of aconnected manifold M endowed with a closed and conformal vector field X(without zeros) such that σ(X,X) = ρJX. If φ is degenerate (i.e., condition(A) in Proposition 1 is satisfied) then locally Mn is a Riemannian productMn1

1 ×Mn22 with n1 ≥ 2, n2 ≥ 1, and φ is the product of two Lagrangian

immersions φi : Mnii −→ Cni, i = 1, 2, being φ1 a spherical immersion (i.e.,

the image of φ1 lies in a hypersphere of Cn1).

The assertions given in the following examples are not proved becausethey are straightforward.

Examples 5. Let φi : (Ni, gi) −→ CPni , i = 1, 2 be Lagrangian immersionsof ni-dimensional simply-connected manifolds Ni, i = 1, 2 and φi : Ni −→S2ni+1 horizontal lifts of φi, i = 1, 2. Given a real number δ ∈ (0, π/4] andbeing n = n1 + n2 + 1,

φδ1,2 : R×N1 ×N2 −→ CPn

given by

φδ1,2(t, p, q) = Π(cos δ eit tan δφ1(p); sin δ e−it cot δφ2(q)),

is a Lagrangian immersion where X = ( ∂∂t , 0, 0) is a parallel field on (R ×N1 ×N2, dt

2 × cos2 δ g1 × sin2 δ g2) with σ(X,X) = −2 cot 2δJX. Moreoverφ is degenerate with b1 = tan δ and b2 = − cot δ.

Examples 6. Let φ1 : (N1, g1) −→ CPn1 and φ2 : (N2, g2) −→ CHn2 beLagrangian immersions of ni-dimensional simply-connected manifolds Ni,i = 1, 2 and φ1 : N1 −→ S2n1+1 and φ2 : N2 −→ H2n2+1

1 horizontal lifts ofφi, i = 1, 2. Given a positive real number δ and being n = n1 + n2 + 1,

φδ1,2 : R×N1 ×N2 −→ CHn


given by

φδ1,2(t, p, q) = Π(sinh δ eit coth δφ1(p); cosh δ eit tanh δφ2(q)),

is a Lagrangian immersion where X = ( ∂∂t , 0, 0) is a parallel field on (R ×N1×N2, dt

2×sinh2 δ g1×cosh2 δ g2) with σ(X,X) = 2 coth 2δJX. Moreoverφ is degenerate with b1 = coth δ and b2 = tanh δ.

Proposition 3. Let φ : Mn −→ CPn (respectively φ : Mn −→ CHn) bea Lagrangian immersion of a connected manifold M endowed with a closedand conformal vector field X (without zeros) such that σ(X,X) = ρJX. Ifφ is degenerate, then φ is locally congruent to some of the immersions φδ1,2described in Examples 5 (respectively in Examples 6).

Proof. We start with the projective case. From Proposition 1, X is paralleland ρ is constant. We assume that |X| = 1. So the two eigenvalues of AJXon D (see Proposition 1) satisfy

b1 + b2 = ρ, b1b2 = −1.

We start taking the distributions Di, i = 1, 2, on M (see the proof of Propo-sition 1) defined by

Di(x) = v ∈ D(x) : AJXv = biv,of dimensions n1 and n2 respectively. Using a similar reasoning as in theproof of Proposition 1, it is not difficult to prove that

∇ZVi ∈ Di,for any Z tangent to M and Vi ∈ Di. So using also Lemma 1, (v), locallyM is the Riemannian product (I × N1 × N2, dt

2 × g1 × g2), where I is aninterval of R, and Ni are simply-connected manifolds of dimension ni.

We consider a horizontal lift φ : I × N1 × N2 −→ S2n+1 ⊂ Cn+1 of φ,and identify X and φ∗X. If h denotes the second fundamental form of φ inCn+1, we have

h(X,X) = ρJX − φ, h(X, vi) = biJvi,(3.1)

for any tangent vector vi to Ni.We can suppose b1 > 0 > b2 and define γi : I×N1×N2 −→ Cn+1, i = 1, 2,

by

γ1 = (b2 − b1)(b2φ+ JX), γ2 = (b1 − b2)(b1φ+ JX).(3.2)

First, it is easy to check that

|γ1|2 = b2(b2 − b1)3, |γ2|2 = b1(b1 − b2)3, 〈γ1, γ2〉 = 〈γ1, Jγ2〉 = 0.(3.3)

Also, using (3.1) we have

v2(γ1) = 0, X(γ1) = b1Jγ1, v1(γ2) = 0, X(γ2) = b2Jγ2,(3.4)


for any vi tangent to Ni, i = 1, 2. So in particular, from (3.2), (3.3) and(3.4), we recuperate φ in terms of γi by

φ(t, p, q) =1

(b1 − b2)2(eib1tγ1(0, p), eib2tγ2(0, q)),(3.5)

for any (t, p, q) ∈ I ×N1 ×N2.Now we define

φ1(p) =γ1(0, p)|γ1(0, p)|

=γ1(0, p)

(b1 − b2)√b2(b2 − b1)

, p ∈ N1,

φ2(q) =γ2(0, q)|γ2(0, q)|

=γ2(0, q)

(b1 − b2)√b1(b1 − b2)

, q ∈ N2.

So we can rewrite (3.5) as

φ(t, p, q) =

(√b2

b2 − b1eib1tφ1(p),

√b1

b1 − b2eib2tφ2(q)

).

Now the properties of the lift φ say that φi are horizontal immersions in thecorresponding S2ni+1 ⊂ Cni+1, which means that φi are horizontal lifts ofLagrangian immersions φi = Π(φi), i = 1, 2, in CPni .

Because b1 = tan δ for some δ ∈ (0, π2 ), if δ ∈ (0, π4 ], our immersion islocally congruent to some of the immersions φδ1,2 described in Examples 5.If δ ∈ (π4 ,

π2 ), then our immersion is locally congruent to some of the immer-

sions φπ2−δ

1,2 described in Examples 5.We follow with the hyperbolic case. As the proof is quite similar to the

above proof, we will omit some details. From Proposition 1, X is paralleland ρ is constant. We assume that |X| = 1. So the two eigenvalues of AJXon D (see Proposition 1) satisfied

b1 + b2 = ρ, b1b2 = 1.

Reasoning as before, locally M is the Riemannian product (I × N1 ×N2, dt

2 × g1 × g2), where I is an interval of R, and Ni are simply-connectedmanifolds of dimension ni.

We consider a horizontal lift φ : I × N1 × N2 −→ H2n+11 ⊂ Cn+1, and

identify X and φ∗X. If h denotes the second fundamental form of φ in Cn+1

we have

h(X,X) = ρJX + φ, h(X, vi) = biJvi,

for any tangent vector vi to Ni.We can suppose b1 > b2 > 0. Otherwise 0 > b1 > b2 and changing X

by −X we will be in the first case. We define γi : I × N1 × N2 −→ Cn+1,i = 1, 2, by

γ1 = (b2 − b1)(b2φ+ JX), γ2 = (b1 − b2)(b1φ+ JX).


Following the same reasoning as in the projective case, we get that φ can bewritten as

φ(t, p, q) =

(√b2

b1 − b2eib1tφ1(p),

√b1

b1 − b2eib2tφ2(q)

).

Now the properties of the lift φ say that φi are horizontal immersions in thecorresponding S2n1+1 and Hn2+1

1 , and then φ1 = Π(φ1) and φ2 = Π(φ2) areLagrangian immersions in CPn1 and CHn2 respectively, with horizontal liftsφ1 and φ2. As b1 ∈ (1,∞), let δ be the positive real number with coth δ =b1. Then our immersion is locally congruent to some of φδ1,2 described inExamples 6.

4. Lagrangian submanifolds with conformalMaslov form in nonflat complex space forms.

In this section, we will deal with the case that X = JH. Since JH is aclosed vector field (because the ambient manifold is a complex space form),we must only assume that JH is a conformal vector field on the submanifold.It is well-known that, up to a constant, JH is the dual vector field of theMaslov form and so this kind of Lagrangian submanifolds will be referredfrom now on as Lagrangian submanifolds with conformal Maslov form. Theywere deeply studied by A. Ros and the third author in complex Euclideanspace (cf. [RU]).

In the following, we will describe a special family of Lagrangian sub-manifolds with conformal Maslov form in nonflat complex space forms interms of (i) minimal Lagrangian immersions in complex projective, complexhyperbolic and complex Euclidean spaces of less dimension and (ii) a twoparameter family of curves of CP1 and CH1.

To introduce this last family, we start to consider the following 2-para-meter (λ, µ 6= 0) family of o.d.e.s:

u′2 +c

4e2u +

(nµ

n+ 2e2u + λe−nu

)2

=

1, if c = 4A ∈ 1, 0,−1, if c = −4 .

(4.1)

We put h = −u′−i(nµn+2e

2u + λe−nu)

and for the solution to (4.1) satisfyingu′(0) = 0, following the notation of Examples 1-4, we define the curve αλ,µin CP1 if c = 4, in CH1 in the other cases, by means of αλ,µ = Π αλ,µ


where

αλ,µ(t) =(eu(0)e−

R t0 h(s)ds, |h(0)|e

c4

R t0e2u(s)

h(s)ds), if c = 4 or c = −4, A = 1,

(4.2)

αλ,µ(t) =(|h(0)|e−

R t0e2u(s)

h(s)ds, eu(0)e−

R t0 h(s)ds

), if c = −4, A = −1,

αλ,µ(t) = e−R t0 h(s)ds

(2e2u(0)e1 −

(e−2u(0)/2 +

∫ t

0ds/h(s)

)e2

),

if c = −4, A = 0.

Corollary 2. Let φ : Mn −→ Mn(c) be a nonminimal Lagrangian immer-

sion. Then φ has conformal Maslov form and JH is a principal direction ofAH if and only if around each point where H does not vanish, φ is congruentto:

(a) Some α ∗ ψ of Examples 1-4, where ψ is a minimal immersion inCPn−1, CHn−1 or Cn−1 and α = αλ,µ is the curve given in (4.2). Inthis case, |H| = µeu.

(b) Some φδ1,2 of Examples 5 with φi minimal immersions in CPni, i = 1, 2,(n1+n2+1 = n) and arbitrary δ ∈ (0, π/4], or some φδ1,2 of Examples 6with φ1 a minimal immersion in CPn1, φ2 a minimal immersion inCHn2, (n1 + n2 + 1 = n) and arbitrary δ > 0.

Remark 8. In [Ch2], B.Y. Chen introduced the notion of Lagrangian H-umbilical submanifolds in Kaehler manifolds. This kind of submanifoldscorrespond to the simplest Lagrangian submanifolds satisfying the two fol-lowing conditions: JH is an eigenvector of the shape operator AH and therestriction of AH to the orthogonal subspace to JH is proportional to theidentity. He classified them in CPn and CHn (cf. [Ch2]) proving that, ex-cept in some exceptional cases, they are obtained from Legendre curves inS3 or in H3

1 via warped products. In Corollary 2, they appear in (a) byconsidering ψ totally geodesic.

Proof. It is a consequence of Proposition 1, Corollary 1, Theorem 2, Propo-sition 3 and the following facts:

First, when we study the geometric properties of the examples α ∗ ψ wearrive at the following expressions for its mean curvature vector:

nH∗ =1|α1|2

((ρ+ (n− 1)b)JX + (n− 1)(α1H

∗ψ; 0)

)(

resp. nH∗ =1|α2|2

((ρ+ (n− 1)b)JX + (n− 1)(0; α2H

∗ψ)))


for Examples 1, 2, 4 (resp. for Examples 3), where

ρ =|α1||α′|3〈α′′, Jα′〉, b =

〈α′1, Jα1〉|α1||α′|(

resp. ρ =|α2||α′|3〈α′′, Jα′〉, b =

〈α′2, Jα2〉|α2||α′|

).

So if we putXµ = µX = JH (whereX is normalized according to Remark 1)and look at the foregoing expressions we deduce that H∗ψ = 0 and hence ψis a minimal immersion, and using that |X|2 = |α1|2 in Examples 1, 2, 4(resp. |X|2 = |α2|2 in Examples 3), we obtain

−µn|α1|2 = ρ+ (n− 1)b(resp. − µn|α2|2 = ρ+ (n− 1)b)

for Examples 1, 2, 4 (resp. for Examples 3). From (1.5) and the aboveequations, we deduce

b′ + nu′b = −nµu′e2u

where e2u = |X|2 = |H|2/µ2, and then

b = −(

nµ

n+ 2e2u + λe−nu

).

Second, the expressions given for α in Corollary 2 can be deduced follow-ing a constructive reasoning from the proof of Corollary 1 and Theorem 2in all the cases.

Finally, a similar technique can be used to get the cases given in (b).

Next we will consider the easiest examples provided in Corollary 2, takingψ = ψ0 totally geodesic and the curves α0,µ, i.e., putting λ = 0 in (4.1).For our purposes, it is enough to consider µ > 0. We will state alongthis paragraph that these examples play the role of “umbilical” Lagrangianimmersions in non flat complex space forms. If λ = 0, (4.1) becomes in

u′2 +c

4e2u + ν2e4u =

1, if c = 4A ∈ 1, 0,−1, if c = −4 ,(4.3)

with ν = nµn+2 > 0, and we must now take into account that h = −u′− iνe2u

for the curves given in (4.2). The solution to (4.3) satisfying u′(0) = 0 is


given by

e2u(t) =

2√1+4ν2 cosh(2t)+c/4

, if c = 4 or c = −4, A = 1,

21−√

1−4ν2 cos(2t), if c = −4, A = −1,

1ν2+t2

, if c = −4, A = 0.

The only constant solution appears in the hyperbolic case c = −4, A = −1,just when ν = 1/2. In this case e2u(t) ≡ 2 and the corresponding immer-sion has parallel mean curvature vector, just like the examples collected inCorollary 2, (b).

In the elliptic case, we define θ = (1/2) cosh−1√

1 + 4ν2 > 0 and abbre-viate chθ = cosh θ and shθ = sinh θ; now (4.3) and its solution are rewrittenas

u′2 +c

4e2u + ch2

θsh2θe

4u = 1

and

e2u(t) =2

ch2θ cosh(2t) + c/4.

From (4.2), after a long straightforward computation, we arrive at

αθ(t) =1

chθ cosh t+ ishθ sinh t(1, shθ cosh t+ ichθ sinh t),

αθ(t) =1

shθ cosh t+ ichθ sinh t(1, chθ cosh t+ ishθ sinh t),

for c = 4 and c = −4 respectively. By identifying Rn − 0 with R × Sn−1

via the conformal transformation w 7→ (log |w|, w/|w|), αθ ∗ ψ0 defines aLagrangian immersion φθ : Rn − 0 −→ CPn, and it is not complicatedto check that φθ extends regularly to 0 and ∞; hence via stereographicprojection we obtain a family of Lagrangian immersions

φθ : Sn −→ CPn, θ > 0,(4.4)

given by

φθ(x1, . . . , xn, xn+1) =[(

(x1, . . . , xn)chθ + ishθxn+1

;shθchθ(1 + x2

n+1) + ixn+1

ch2θ + sh2

θx2n+1

)]that we will call the Whitney spheres of CPn. We notice that φθ areembeddings except in a double point and that if θ → 0 it appears thetotally geodesic immersion of Sn in CPn.


In a similar way we obtain the Whitney spheres of CHn,

Φθ : Sn −→ CHn, θ > 0,(4.5)

given by

Φθ(x1, . . . , xn, xn+1) =[(

(x1, . . . , xn)shθ + ichθxn+1

;shθchθ(1 + x2

n+1)− ixn+1

sh2θ + ch2

θx2n+1

)],

which are also embeddings except in a double point.In the hyperbolic case, we define β = (1/2) cos−1(

√1− 4ν2) ∈ (0, π/4]

and abbreviate cβ = cosβ and sβ = sinβ; now (4.3) and its solution arerewritten as

u′2 − e2u + s2βc2βe

4u = −1

and

e2u(t) =2

1− c2β cos 2t.

From (4.2), after a long straightforward computation, we arrive at

αβ(t) =1

sβ cos t+ icβ sin t(cβ cos t− isβ sin t, 1).

If RHn−1 = y = (y1, . . . , yn) ∈ Rn : y21 + . . . y2

n−1 − y2n = −1 denotes

the (n−1)-dimensional real hyperbolic space, αβ ∗ ψ0 defines a family ofLagrangian embeddings

Ψβ : S1 × RHn−1 −→ CHn, β ∈ (0, π/4],(4.6)

given by

Ψβ(eit, y) =[

1sβ cos t+ icβ sin t

(cβ cos t− isβ sin t; y)].

We note that Ψπ/4(eit, y) = (e−2it,√

2e−ity) is flat.Finally, in the parabolic case, using (4.2) a straightforward computation

leads to

αν(t) =1

ν + it

(2νe1 − ν

(ν2 + t2

2+ iνt

)e2

).

From Examples 4, αν ∗ ψ0 defines a one-parameter family of Lagrangianembeddings

ϕν : Rn ≡ R× Rn−1 −→ CHn, ν > 0,(4.7)

given by

ϕν(t;x) =[

1ν + it

(2νx;

2νe1 −

(ν(ν2 + t2)

2+

2|x|2

ν+ iν2t

)e2

)].


Theorem 3. Let φ : Mn −→ CPn (respectively φ : Mn −→ CHn) be aLagrangian immersion.

(i) The second fundamental form of φ is given by

〈σ(v, w), Jz〉 =n

n+ 2∂v,w,z〈v, w〉〈H,Jz〉(4.8)

for any tangent vectors v, w and z, (where the symbol ∂v,w,z meanscyclic sum over v, w, z) if and only if either φ is totally geodesic orφ(M) is an open set of some of the Lagrangian submanifolds (4.4) inCPn (respectively of the Lagrangian submanifolds (4.5), (4.6) or (4.7)in CHn).

(ii) [B.Y. Chen] The scalar curvature τ of M satisfies

τ ≤ n2(n− 1)n+ 2

|H|2 + n(n− 1)c

4and the equality holds if and only if φ is totally geodesic or φ is an openset of some of the Lagrangian submanifolds (4.4) in CPn (respectivelyof the Lagrangian submanifolds (4.5), (4.6) or (4.7) in CHn).

Remark 9. In [CU2] when n = 2 and in [Ch1] in arbitrary dimension, thesharp inequality of (ii) between the squared mean curvature and the scalarcurvature for a Lagrangian submanifold in a nonflat complex space form wasestablished. By utilising Jacobi’s elliptic functions and warped products,B.Y. Chen introduced in [Ch1] three families of Lagrangian submanifoldsand two exceptional ones characterized by satisfying the equality in (ii)besides the totally geodesic ones. Later, in [ChV], explicit expressions ofthese Lagrangian immersions were found in a different context from ours. Allof them are exactly the Lagrangian submanifolds described in (4.4), (4.5),(4.6) and (4.7).

Proof. For the proof of (i), it is straightforward to verify that the totallygeodesic immersions and the Lagrangian immersions (4.4)-(4.7) satisfy (4.8).

Conversely, exactly the same proof of Theorem 2 in [RU] works also hereto get that JH is a conformal vector field on M . If H ≡ 0 then (4.8) impliesthat φ is totally geodesic. If H is nontrivial, Lemma 1, (i) says that thezeros of H are isolated. Hence the set of points of M where H does notvanish, say M ′, is a connected open dense subset of M . We will work in it.From (4.8) we first deduce that σ(JH, JH) = 3n

n+2 |H|2H. Then Corollary 2

says that φ is locally congruent to some of the examples described there.Second, (4.8) also gives us that σ(v, JH) = − n

n+2 |H|2Jv for v orthogonal

to JH and this implies that necessarily that φ is locally congruent to someα ∗ ψ of Corollary 2, (a) and, in addition, λ = 0 in (4.1). Using again (4.8)we obtain, for v and w orthogonal to JH, that σ(v, w) = n

n+2〈v, w〉H andthis means that σ∗ψ = 0 when we study the second fundamental form of suchan immersion α ∗ ψ. Thus ψ is totally geodesic. As M ′ is connected and


dense in M , a standard argument shows that φ(M) is an open set in someof the Lagrangian submanifolds (4.4)-(4.7).

Our proof of (ii) follows from (i) and a standard interpolation argument.

Theorem 4. Let φ : Mn −→ CPn (respectively φ : Mn −→ CHn) be aLagrangian immersion of a compact manifold M with conformal Maslovform. If the first Betti number of M vanishes, then φ is either minimal(necessarily in CPn) or congruent to some of the Whitney spheres (4.4) inCPn (respectively congruent to some of the Whitney spheres (4.5) in CHn).

Proof. Suppose (only in the projective case) that H does not vanishes iden-tically. Since JH is a closed vector field and the first Betti number of M iszero, there exists a function f on M such that JH = ∇f . So H has zerosat the critical points of f . Hence JH is a closed and conformal vector fieldwith at least a zero. Under these conditions, exactly the same reasoningof Theorem 3 in [RU] proves that M is conformally equivalent to a roundsphere, the leaves of the umbilical foliation D (see Lemma 1) are spheres andY = Jσ(JH, JH) − (3n/(n + 2))|H|2JH is a harmonic vector field. Sincethe first Betti number of M is zero, we deduce that JH is an eigenvectorof AH . If N is one of the leaves of D, using Corollary 2 we have a minimalLagrangian immersion ψ necessarily in CPn−1 because in Cn−1 and CHn−1

there are no compact minimal submanifolds. Then we can use the sameproof in the above mentioned theorem of [RU] to conclude that ψ is totallygeodesic and that σ is given as in (4.8) on the whole of M . The proof finishesthanks to Theorem 3.

Next, we obtain the following corollary using the same argument that inCorollary 5 of [RU], where the Whitney spheres are characterised in termsof the behaviour of the Ricci curvature.

Corollary 3. Let φ : Mn −→ CPn be a Lagrangian nonminimal immersionof a compact manifold M with conformal Maslov form. Then:

(i) Ric(JH) ≥ (n − 1)|H|2 if and only if φ is congruent to some of theWhitney spheres (4.4) in CPn;

(ii) Ric(JH) ≥ 0 if and only if either φ has parallel mean curvature vectoror φ is congruent to some of the Whitney spheres (4.4) in CPn.

To finish this section we state that if our Lagrangian submanifold has firstBetti number equal to one then it belongs to our family.

Corollary 4. Let φ : Mn −→ CPn (respectively φ : Mn −→ CHn) be aLagrangian immersion of an orientable compact manifold M with nonpar-allel conformal Maslov form such that the first Betti number of M is one.


Then the universal covering of φ is congruent to some α ∗ψ as described inCorollary 2,(a).

Proof. From the proof of Theorem 4 we can deduce that H has no zeros onM and so the vector field |H|−nJH is well-defined on M . Using Lemma 1,one can check that it is a harmonic vector field on M and the same happensfor Y = Jσ(JH, JH)− (3n/(n+ 2))|H|2JH. By the hypothesis, there is aconstant a ∈ R such that

σ(JH, JH) =(

3nn+ 2

|H|2 + a|H|−n)H

and we can go to Corollary 2 to get the result.

References

[B] D.E. Blair, Contact Manifolds in Riemannian Geometry, Springer-Verlag, (1976),MR 57 #7444, Zbl 319.53026.

[CU1] I. Castro and F. Urbano, Lagrangian surfaces in the complex Euclidean plane withconformal Maslov form, Tohoku Math. J., 45 (1993), 565-582, MR 94j:53064,Zbl 792.53030.

[CU2] , Twistor holomorphic Lagrangian surfaces in the complex projective andhyperbolic planes, Ann. of Glob. Anal. and Geom., 13 (1995), 59-67, MR 96c:53093,Zbl 827.53059.

[Ch1] B.Y. Chen, Jacobi’s elliptic functions and Lagrangian immersions, Proc. RoyalSoc. Edinburgh, 126 (1996), 687-704, MR 97k:53071, Zbl 866.53045.

[Ch2] , Interaction of Legendre curves and Lagrangian submanifolds, Israel J.Math., 99 (1997), 69-108, MR 98i:53086, Zbl 884.53014.

[ChV] B.Y. Chen and L. Vrancken, Lagrangian submanifolds satisfying a basic equal-ity, Math. Proc. Cambridge Phil. Soc., 120 (1996), 291-307, MR 97e:53101,Zbl 860.53034.

[HL] R. Harvey and H.B. Lawson, Calibrated geometries, Acta Math., 148 (1982), 47-157, MR 85i:53058, Zbl 584.53021.

[K] M. Kon, Pseudo-Einstein real hypersurfaces in complex space forms, J. DifferentialGeometry, 14 (1979), 339-354, MR 81k:53050, Zbl 461.53031.

[M] S. Montiel, Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan,37 (1985), 515-535, MR 96i:53027, Zbl 554.53021.

[RU] A. Ros and F. Urbano, Lagrangian submanifolds of Cn with conformal Maslov formand the Whitney sphere, J. Math. Soc. Japan, 50 (1998), 203-226, MR 98k:53081,Zbl 906.53037.

Received September 13, 1999 and revised March 31, 2000. This research was partiallysupported by a DGICYT grant No. PB97-0785.

Departamento de Matematicas Escuela Politecnica SuperiorUniversidad de Jaen23071 JaenSpain





http://www.ams.org/mathscinet-getitem?mr=96c:53093

















E-mail address: [email protected]


Departamento de Geometrıa y TopologıaUniversidad de Granada18071 GranadaSpainE-mail address: [email protected]





THE SUPPORT OF THE EQUILIBRIUM MEASURE FOR ACLASS OF EXTERNAL FIELDS ON A FINITE INTERVAL

S.B. Damelin, P.D. Dragnev, and A.B.J. Kuijlaars

We investigate the support of the equilibrium measure as-sociated with a class of nonconvex, nonsmooth external fieldson a finite interval. Such equilibrium measures play an im-portant role in various branches of analysis. In this paper weobtain a sufficient condition which ensures that the supportconsists of at most two intervals. This is applied to exter-nal fields of the form −c sign(x)|x|α with c > 0, α ≥ 1 andx ∈ [−1, 1]. If α is an odd integer, these external fields aresmooth, and for this case the support was studied before byDeift, Kriecherbauer and McLaughlin, and by Damelin andKuijlaars.

1. Introduction.

In recent years, equilibrium measures with external fields have found anincreasing number of applications in a variety of areas. We refer to [2,3, 4, 5, 8, 10, 14, 15] for these relations, ranging from classical topics asweighted transfinite diameter and weighted Chebyshev polynomials, to morerecent developments in weighted approximation, orthogonal polynomials,integrable systems, and random matrix theory.

In the present paper we consider equilibrium problems on the interval[−1, 1]. With a continuous function Q : [−1, 1] → R, we associate theweighted energy of a measure µ on [−1, 1] as follows

IQ(µ) =∫∫

log1

|s− t|dµ(s)dµ(t) + 2

∫Q(t)dµ(t).(1.1)

The equilibrium measure in the presence of the external field Q is the uniqueprobability measure µQ on [−1, 1] minimizing the weighted energy amongall probability measures. Thus

IQ(µQ) = minIQ(µ) : µ ∈ P([−1, 1])(1.2)

where P([−1, 1]) denotes the class

P([−1, 1]) = µ : µ is a Borel probability measure on [−1, 1].

303



304 S.B. DAMELIN, P.D. DRAGNEV, AND A.B.J. KUIJLAARS

The determination of the support of the equilibrium measure is a majorstep in obtaining the measure. As described by Deift [2, Chapter 6] the in-formation that the support consists of N disjoint closed intervals, allows oneto set up a system of equations for the endpoints, from which the endpointsmay be calculated. Knowing the endpoints, the equilibrium measure maybe obtained from a Riemann-Hilbert problem or, equivalently, a singularintegral equation.

There are two general useful facts about the equilibrium measure. Thefirst one, due to Mhaskar and Saff [12], says that for a convex externalfield, the support is always one single interval. The other one, due to Deift,Kriecherbauer and McLaughlin [3], says that for a real analytic externalfield, the support always consists of a finite number of intervals. The actualdetermination of this number is a nontrivial problem. To illustrate thedifficulties, Deift, Kriecherbauer and McLaughlin considered explicitly thefamilies of monomial external fields Q(x) = −cxn with c 6= 0, n ∈ N andx ∈ [−1, 1].

In the even case (n = 2m) the external field is convex if c < 0, andtherefore the support is a single interval. For c > 0, the external field isconcave, and the analysis becomes more involved. Independently from [3],this case was considered in [9], and it was shown that for every c > 0, thereare at most three intervals in the support of the equilibrium measure. Thesame result was also found to be valid for the nonsmooth (i.e., not realanalytic) external fields Q(x) = −c|x|α with α ≥ 1 not necessarily an eveninteger.

In the odd case (n = 2m + 1) the external field is an odd function, and,by symmetry, we may restrict attention to c > 0. In this case the resultsof [3] were extended to the full range of parameters in [1]. For all c and allodd integers n, it was shown that the support of the equilibrium measureconsists of at most two intervals.

It is the aim of the present paper to study the nonsmooth analogues of−cx2m+1 given by

Qα,c(x) := −c sign(x)|x|α =

c|x|α for x ∈ [−1, 0],−cxα for x ∈ [0, 1],(1.3)

with a real number α ≥ 1 and c > 0. The functions (1.3) are both non-convex and nonsmooth, and therefore it is of interest to develop methods todetermine the nature of the support of the equilibrium measures associatedwith these external fields.

Our first theorem presents a sufficient condition which ensures that thesupport of the equilibrium measure is the union of at most two intervals.To formulate it, we use C1+ε([−1, 1]) to denote the class of differentiablefunctions on [−1, 1], whose derivative satisfies a Holder condition for some

SUPPORT OF EQUILIBRIUM MEASURE 305

positive exponent. Thus Q ∈ C1+ε([−1, 1]) if and only if

|Q′(x)−Q′(y)| ≤ C|x− y|ε, x, y ∈ [−1, 1]

for some ε > 0 and some positive constant C independent of x and y.

Theorem 1.1. Let Q ∈ C1+ε([−1, 1]). Suppose that there exists a numbera1 ∈ [−1, 1] such that

(a) Q is convex on [−1, a1], and(b) for every a ∈ [−1, a1], there is t0 ∈ [a1, 1], such that the function

t 7→ 1π

–∫ 1

a

Q′(s)s− t

√(1− s)(s− a)ds(1.4)

is nonincreasing on (a1, t0) and nondecreasing on (t0, 1). The integralin (1.4) is a principal value integral.

Then supp (µQ) is the union of at most two intervals.

Remark 1.2. For the special case a1 = −1, Theorem 1.1 was given alreadyin [9, Theorem 2].

In our second main result we show that the conditions of Theorem 1.1 aresatisfied for the external fields (1.3).

Theorem 1.3. For α ≥ 1 and c > 0, let Qα,c be given by (1.3). Then forevery a ∈ [−1, 0], there exists t0 ∈ [0, 1) such that

1π

–∫ 1

a

Q′α,c(s)s− t

√(1− s)(s− a)ds(1.5)

decreases on (0, t0) and increases on (t0, 1). As a result, the support of µQα,cconsists of at most two intervals.

Remark 1.4. For α an odd integer, Theorem 1.3 was established in [1].The proof for this special case differs from the one given here in severalrespects. For example, the function (1.5) is a polynomial in t whenever αis an odd integer. The proof of the decreasing/increasing property of (1.5)was based in [1] on the calculation of the polynomial coefficients and theDescartes’ rule of signs for polynomials.

Another difference between [1] and the present paper is that in [1] theproblem was viewed in terms of the parameter c. Quite complicated per-turbation arguments were used to obtain from the decreasing/increasingproperty of (1.5) the conclusion that the support consists of at most twointervals. Here we use Theorem 1.1 and this simplifies the arguments con-siderably, also in the case where α is an odd integer.

Remark 1.5. To view the problem in terms of the parameter c is quitenatural, since there is a monotonicity with respect to c. To be precise,if Q is fixed then the support supp (µcQ) is decreasing as c increases, see


[1] or [14]. Using this, we can show the following behavior of the supportdepending on the parameter in case α > 1. There exist three critical values0 < c1 < c2 < c3 depending on α such that:

(a) For 0 < c ≤ c1, the support supp (µQα,c) is equal to the full interval[−1, 1].

(b) For c1 < c ≤ c2, there exists a ∈ (−1, 0) such that

supp (µQα,c) = [a, 1].

(c) For c2 < c < c3, there exist a1, b1 and a2 such that −1 < a1 < b1 <a2 < 1, a1 < 0, and

supp (µQα,c) = [a1, b1] ∪ [a2, 1].

(d) For c ≥ c3, there exists a ∈ (0, 1) such that

supp (µQα,c) = [a, 1].

See [1, Theorem 1.1] where this was shown for odd integers α ≥ 3.Note that for α = 1, the external field (1.3) is convex and the support of

µQ1,c is an interval containing 1 for every c > 0.

Acknowledgement. This work would not have been possible without grantsof our institutions that have allowed for mutual visits. We are grateful fortheir support.

2. The Proof of Theorem 1.1.

In this section, we shall prove Theorem 1.1.

2.1. Preliminaries. Let Q ∈ C1+ε([−1, 1]) be fixed. The equilibrium mea-sure µQ is characterized by the Euler-Lagrange variational conditions asso-ciated with the extremal problem (1.2), which are

Uµ(x) +Q(x) = F for x ∈ supp (µ),(2.1)Uµ(x) +Q(x) ≥ F for x ∈ [−1, 1],(2.2)

where F is a constant and

Uµ(x) =∫

log1

|x− t|dµ(t)(2.3)

denotes the logaritmic potential of µ, see [2, 14]. The equilibrium measureµQ is the only measure from P([−1, 1]) satisfying (2.1) and (2.2) for someconstant F .

If supp (µQ) = Σ and if µQ has a density v, then Equation (2.1) yields∫Σ

log |x− t| v(t)dt = Q(x)− F, x ∈ Σ.(2.4)


Then there is a unique constant F , such that the integral equation (2.4) hasa solution v(t) satisfying ∫

Σv(t)dt = 1.(2.5)

If Σ consists of a finite number of nondegenerate closed intervals, then (2.4)may be differentiated for x in the interior of Σ (sinceQ′ is Holder continuous)to give the singular integral equation

–∫

Σ

v(t)x− t

dt = Q′(x), x ∈ intΣ.(2.6)

It is well-known, see [7, §42.3], that the general solution of (2.6) depends onN parameters, where N is the number of intervals in Σ. These parametersare uniquely determined by the normalization (2.5) and the conditions thatthe constant F in (2.4) should be the same on each interval of Σ. We alsorecall that the solutions of (2.6) are Holder continuous on the interior of Σ,and may become unbounded at endpoints of Σ, cf. [7, §5, §42.3].

If we do not know that Σ is the support of µQ, we can still consider thefunction v(t) determined by Equations (2.4) and (2.5). Then in general thefunction v(t) will not be nonnegative on Σ. Thus v(t) is the density of asigned measure η that depends on Σ:

dη(t) = dηΣ(t) = v(t)dt.

The signed measure ηΣ satisfies

supp (ηΣ) ⊂ Σ,∫dηΣ = 1,(2.7)

and it minimizes the weighted energy IQ(η) amongst all signed measuressatisfying (2.7).

For the special case Σ = [a, 1], with a ∈ [−1, 1), we have thatdηΣ

dt= v(t) =

1π√

(1− t)(t− a)[1 +G(t)] , a < t < 1,(2.8)

with

G(t) =1π

–∫ 1

a

Q′(s)s− t

√(1− s)(s− a)ds,(2.9)

see [7, §42.3] or [16, §4.3]. Note that (2.9) is equal to the function from(1.4).

Next, we recall the notion of balayage of a measure. The balayage of anonnegative measure ν with compact support and continuous potential ontoa set Σ of positive capacity, is the unique measure ν such that supp (ν) ⊂ Σ,‖ν‖ = ‖ν‖ and for some constant c,

U ν(x) = Uν(x) + c, for quasi every x ∈ Σ.(2.10)


Here ‘quasi every’ means with the possible exception of a set of capacityzero. We refer the reader to [11, 13, 14] for these and other notions fromlogarithmic potential theory. Instead of ν we also write Bal(ν; Σ). For asigned measure ν with Jordan decomposition ν = ν+ − ν−, the balayage ofν onto Σ is

Bal(ν; Σ) = Bal(ν+; Σ)−Bal(ν−; Σ)

provided the balayages of ν+ and ν− exist.From their defining properties it is then easy to see that the measures ηΣ

are related by balayage. That is, if Σ1 ⊂ Σ2, then

ηΣ1 = Bal(ηΣ2 ; Σ1).(2.11)

The following result will be used in the proof of Theorem 1.1 below. Wesay that two sets A and B are quasi-equal, if A \B and B \A have capacityzero.

Lemma 2.1. Let Σ and Σn, n ∈ N, be closed subsets of [−1, 1] havingpositive capacity such that

Σ =⋂n

⋃k≥n

Σk(2.12)

and Σ is quasi-equal to ⋃n

⋂k≥n

Σk.(2.13)

Then the following hold.(a) For every finite measure ν with compact support and continuous po-

tential, we have

limn→∞

Bal(ν; Σn) = Bal(ν; Σ)

with convergence in the sense of weak∗ convergence of measures on[−1, 1].

(b) If Σ and Σn, n ∈ N, are finite unions of closed intervals, then

limn→∞

ηΣn = ηΣ

in the sense of weak∗ convergence of signed measures.

Proof. (a) Let us write νn = Bal(ν; Σn). Then by (2.10), we have for someconstant cn,

Uνn(x) = Uν(x) + cn for quasi every x ∈ Σn.

By weak∗ compactness, we may assume that (νn) converges, say with weak∗

limit ν∗. Then ‖ν∗‖ = ‖ν‖ and because of (2.12) we have supp (ν∗) ⊂ Σ.The lower envelope theorem [14] says that

Uν∗(x) = lim inf

n→∞Uνn(x) for quasi every x ∈ C.


Since Σ is quasi-equal to (2.13) it then follows that

Uν∗(x) = Uν(x) + lim inf

n→∞cn for quasi every x ∈ Σ.

Then lim inf cn is finite and it follows that ν∗ is the balayage of ν onto Σ.(b) Let η0 = η[−1,1]. The positive and negative parts of η0 in the Jordan

decomposition η0 = η+0 − η−0 are compactly supported. They also have

continuous potentials. Indeed, the function G from (2.9) (with a = −1)is continuous, and therefore it is bounded on [−1, 1]. Then it follows fromthe representation (2.8)–(2.9) for η0, that both η+

0 and η−0 are boundedabove by a constant times the measure 1/(π

√1− t2)dt. This measure has

a continuous potential — in fact its potential is constant on [−1, 1] — andtherefore the potentials of η+

0 and η−0 are continuous as well, see [6, Lemma5.2]. Thus it follows from part (a) that

Bal(η+0 ; Σn)

∗→ Bal(η+0 ; Σ)

and

Bal(η−0 ; Σn)∗→ Bal(η−0 ; Σ).

Then

Bal(η0; Σn)∗→ Bal(η0; Σ).

Since ηΣ is equal to the balayage of η0 onto Σ, and similarly ηΣn is thebalayage of η0 onto Σn, part (b) follows.

2.2. A lemma on convexity. The convexity assumption (a) of Theo-rem 1.1 will be used via the following lemma.

Lemma 2.2. Let Q ∈ C1+ε([−1, 1]). Let Σ ⊂ [−1, 1] be a finite union ofnondegenerate closed intervals. Let η = ηΣ be the signed measure associatedwith Σ, as described in Section 2.1, and let v be the density of η. Supposethat [a, b] ⊂ Σ and that

(a) Q is convex on [a, b],(b) v(a) ≥ 0, and v(b) ≥ 0,(c) v(t) ≥ 0 on Σ \ [a, b].

Then v(t) > 0 for all t ∈ (a, b).

Remark 2.3. The density v is continuous on the interior of Σ, and maybecome unbounded (±∞) at endpoints of Σ. The assumption (b) is alsosatisfied if v(a) = +∞ in case a is an endpoint, and similarly for b.

Proof. First, we reduce the problem to the case Σ = [a, b]. Write η = η1+η2,where η1 is the restriction of η to [a, b] and η2 the restriction to Σ \ [a, b].From (2.4) we get

Uη1(x) +Q1(x) = F for x ∈ [a, b],


where

Q1(x) = Uη2(x) +Q(x), x ∈ [a, b].

The measure η2 is nonnegative by assumption (c). Then it is easy to seefrom (2.3) that the logarithmic potential Uη2 is convex on [a, b]. Thus Q1 isconvex on [a, b] because of assumption (a). The potential Uη2 is real analyticon the open interval (a, b), and therefore Q′1 satisfies a Holder condition on(a, b). At the endpoins a and b, Q′1 could have a singularity of logarithmictype, but this will not affect the arguments that follow. In particular, therepresentation (2.14) below, remains valid, cf. [16, §4.3].

Therefore, we have reduced the proof of the lemma to the case when Σ =[a, b]. Without loss of generality, we may also assume that [a, b] = [−1, 1].Then, as in (2.8), the density v is given by

v(t) =1

π√

1− t2[1 +G(t)](2.14)

and

G(t) =1π

–∫ 1

−1

Q′(s)s− t

√1− s2ds.

In the principal value integral we remove the singular part as follows

G(t) =1π

∫ 1

−1

Q′(s)−Q′(t)s− t

√1− s2ds+

Q′(t)π

–∫ 1

−1

1s− t

√1− s2ds.

The remaining principal value integral we write as

–∫ 1

−1

1− s2

s− tds√

1− s2=∫ 1

−1

(1− s2)− (1− t2)s− t

ds√1− s2

= −∫ 1

−1

s+ t√1− s2

ds,

where we used the fact that

–∫ 1

−1

1s− t

ds√1− s2

= 0.

Thus

G(t) =1π

∫ 1

−1


√1− s2ds− Q′(t)

π

∫ 1

−1

s+ t√1− s2

ds.(2.15)

Next, we have that(1 + t

2

)G(1) +

(1− t

2

)G(−1) = −

(1 + t

2

)1π

∫ 1

−1Q′(s)(1 + s)

ds√1− s2

+(

1− t2

)1π

∫ 1

−1Q′(s)(1− s) ds√

1− s2.

Combining the two integrals, and using

−(

1 + t

2

)(1 + s) +

(1− t

2

)(1− s) = −(s+ t),


we obtain (1 + t

2

)G(1) +

(1− t

2

)G(−1) = − 1

π

∫ 1

−1Q′(s)

s+ t√1− s2

ds.(2.16)

From (2.15) and (2.16) we learn that

G(t)−(

1 + t

2

)G(1)−

(1− t

2

)G(−1)

(2.17)

=1π

∫ 1

−1


√1− s2ds− 1

π

∫ 1

−1(Q′(t)−Q′(s)) s+ t√

1− s2ds

=1π

∫ 1

−1

Q′(t)−Q′(s)t− s

√1− s2ds− 1

π

∫ 1

−1


t2 − s2√1− s2

ds

=1π

∫ 1

−1


[√1− s2 − t2 − s2√

1− s2

]ds

=1π

∫ 1

−1


1− t2√1− s2

ds.

The convexity of Q implies that


≥ 0

for every s and t in (−1, 1). Then for t ∈ (−1, 1), the integral (2.17) isnonnegative and this proves the inequality

G(t) ≥(

1 + t

2

)G(1) +

(1− t

2

)G(−1), −1 < t < 1.(2.18)

Actually, we have strict inequality in (2.18), unless Q′ is a constant. Indeed,if equality holds in (2.18) at a certain t ∈ (−1, 1), then it follows from (2.17)that


= 0

for almost all s ∈ (−1, 1). Since Q′ is continuous, this can only happen ifQ′(s) = Q′(t) for all s, and this means that Q′ is constant.

Thus, if Q′ is not a constant, we see that

G(t) >(

1 + t

2

)G(1) +

(1− t

2

)G(−1), −1 < t < 1,(2.19)

and then it follows from assumption (b) and (2.14) that 1 + G(1) ≥ 0 and1+G(−1) ≥ 0. The right-hand side of (2.19) is a convex combination of G(1)and G(−1). Thus it follows from (2.19) that 1+G(t) > 0 for all t ∈ (−1, 1).In view of (2.14), we then have v(t) > 0 in case Q′ is not a constant.


If Q′ is a constant, say Q′(t) = k, then we obtain from (2.15) that G(t) =−kt. Hence

v(t) =1− kt

π√

1− t2, −1 < t < 1.

Then from v(−1) ≥ 0 and v(1) ≥ 0, we get |k| ≤ 1, and then clearly v(t) > 0on (−1, 1). This completes the proof of Lemma 2.2.

2.3. Proof of Theorem 1.1.

Proof. We write µ = µQ. Let us first assume that supp (µ) ⊂ [a1, 1]. Fromthe assumption (b) of Theorem 1.1 with a = a1, we have that there existst0 ∈ (a1, 1), such that

1π

∫ 1

a1

Q′(s)s− t

√(1− s)(s− a1)ds

is nonincreasing on (a1, t0) and nondecreasing on (t0, 1). As no points ofsupp (µ) lie to the left of a1, we may apply [9, Theorem 2] on the restrictedinterval [a1, 1] and deduce that supp (µ) is either an interval containing a1,or an interval containing 1, or the union of an interval containing a1 withan interval containing 1. This proves the theorem in case the support of µis contained in [a1, 1].

For the rest of the proof, we shall assume that supp (µ) is not containedin [a1, 1]. Let

a := minx : x ∈ supp (µ)(2.20)

so that a < a1.For every pair (p, q) with a < p ≤ q ≤ 1, we let vp,q be the density of the

signed measure ηΣ with Σ = [a, p] ∪ [q, 1] if q < 1 and Σ = [a, p] if q = 1.See Section 2.1 for the definition of ηΣ.

We introduce the set Z consisting of all pairs (p, q) satisfying the followingfour conditions:

(a) a < p ≤ q ≤ 1 and q ≥ a1.(b) supp (µ) ⊂ [a, p] ∪ [q, 1].(c) If q < 1 then π

√(1− t)(t− a)vp,q(t) is nondecreasing for t ∈ (q, 1).

(d) If p > a1 then π√

(1− t)(t− a)vp,q(t) is nonincreasing for t ∈ (a1, p).

We observe first that Z 6= ∅. Indeed, from the assumption (b) of Theo-rem 1.1 it follows that there exists t0 ∈ [a1, 1] such that

1π

–∫ 1

a

Q′(s)s− t

√(1− s)(s− a)ds, a < t < 1


is nonincreasing on (a1, t0) and nondecreasing on (t0, 1). Since for a < t < 1,we have

π√

(1− t)(t− a)vt0,t0(t) = 1 +1π

–∫ 1

a

Q′(s)s− t

√(1− s)(s− a)ds,

by (2.8) and (2.9), we see that properties (c) and (d) are satisfied for thepair (t0, t0). Properties (a) and (b) are trivially satisfied, so that (t0, t0)belongs to Z. Hence Z is nonempty indeed.

Next, we want to show that Z is closed. To this end, we take (p, q) ∈ Zand we choose sequences (pn) and (qn) such that

(pn, qn) ∈ Z, pn → p, qn → q.

We verify that the properties (a)–(d) hold for the pair (p, q). Since (pn, qn)belongs to Z, we have by (b) that [a, pn] ∪ [qn, 1] contains the support of µfor every n. It then follows that [a, p] ∪ [q, 1] contains supp (µ). Thus (b)holds. Since a ∈ supp (µ) and supp (µ) does not have isolated points, wefind that p > a. The other inequalities of (a) are immediate. To establish(c) and (d), we first note that by Lemma 2.1 we have in the sense of weak∗

convergence of signed measures

vpn,qn(t)dt∗→ vp,q(t)dt.(2.21)

Now suppose that (c) does not hold. Then there exist t1 and t2 with q <t1 < t2 < 1 such that

π√

(1− t1)(t1 − a)vp,q(t1) > π√

(1− t2)(t2 − a)vp,q(t2).Since v is continuous, there exists ε > 0 such that

π

∫ t1+ε

t1−ε

√(1− t)(t− a)vp,q(t)dt > π

∫ t2+ε

t2−ε

√(1− t)(t− a)vp,q(t)dt.

We may assume that ε is chosen sufficiently small so that [t1− ε, t1 + ε] and[t2 − ε, t2 + ε] are disjoint intervals that are both contained in (q, 1). Fromthe weak∗ convergence (2.21) it then easily follows that we must have for nlarge enough,

π

∫ t1+ε

t1−ε

√(1− t)(t− a)vpn,qn(t)dt > π

∫ t2+ε

t2−ε

√(1− t)(t− a)vpn,qn(t)dt.

For n large enough, we also have qn < t1 − ε. Then we arrive at a con-tradiction, since (c) holds for the pair (pn, qn). Thus property (c) holds forthe pair (p, q). In a similar way, it follows that (d) holds. Therefore Z is aclosed set.

Since Z is a closed nonempty subset of [a, 1]× [a1, 1], we can find a pairin Z for which the difference q − p is maximal. Such a maximizer may notbe unique (when we have finished the proof, we will see that it is), but wetake any such pair and denote it by (p∗, q∗). Let Σ = [a, p∗] ∪ [q∗, 1] in case


q∗ < 1, and Σ = [a, p∗] in case q∗ = 1. For brevity, we write v∗ insteadof vp∗,q∗ . Our aim is to show that supp (µ) = Σ. Having established that,it will follow from the uniqueness of µ that (p∗, q∗) is the only maximizerfor the difference q − p. We prove that supp (µ) = Σ by showing that v∗ ispositive on the interior of Σ.

We consider several cases. First we assume q∗ < 1 and we consider theinterval (q∗, 1). Suppose that v∗ is nonpositive somewhere on (q∗, 1). Thenby property (c) it follows that there exists ε ∈ (0, 1 − q∗) such that v∗ isnonpositive on [q∗, q∗+ ε]. We claim that (p∗, q∗+ ε) satisfies the conditions(a)–(d). It is clear that (a) is satisfied. For (b), we recall from [9, Lemma3] that

supp (µ) ⊂ x : v∗(x) > 0 ⊂ [a, p∗] ∪ [q∗ + ε, 1].

To see (c) and (d), we note that vp∗,q∗+ε is obtained from v∗ by taking thebalayage of v∗ onto [a, p∗] ∪ [q∗ + ε, 1]. Since v∗ is nonpositive on the gap(p∗, q∗ + ε), we see using [9, Lemma 4 (2)], that this process preserves theproperties (c) and (d). Thus (p∗, q∗ + ε) ∈ Z. However, this contradictsthe maximality of q∗ − p∗. Thus our assumption that v∗ is nonpositivesomewhere in (q∗, 1) is incorrect, and it follows that v∗ is positive on theinterval (q∗, 1).

Now consider the case p∗ > a1. In a similar way as above it follows thatv∗ is positive on (a1, p

∗). Because of property (d) and the continuity of v∗,we find v∗(a1) > 0. Since

supp (µ) ⊂ x : v∗(x) > 0,see [9, Lemma 3], and a ∈ supp (µ), we also have v∗(a) ≥ 0. Since Q isconvex on [a, a1] and v∗ ≥ 0 outside [a, a1], it follows from Lemma 2.2 thatv∗ > 0 on (a, a1). So we have shown that v∗ > 0 on the interval (a, p∗) incase p∗ > a1.

What remains is the case p∗ ≤ a1. If v∗(p∗) < 0, then v∗ is negative on[p∗ − ε, p∗] for some ε > 0 with ε < p∗ − a. Then we may take the balayageof this negative part onto [a, p∗− ε]∪ [q∗, 1] and it follows as above that thepair (p∗ − ε, q∗) belongs to Z. This is a contradiction. Thus v∗(p∗) ≥ 0.Since Q is convex on [a, p∗] with v∗(a) ≥ 0, v∗(p∗) ≥ 0, and v∗ ≥ 0 outsideof [a, p∗], it follows again from Lemma 2.2 that v∗ is positive on (a, p∗).

Thus in both cases, we find that v∗ > 0 on (a, p∗). We also proved thatv∗ > 0 on (q∗, 1) in case q∗ < 1. Thus v∗ is positive on the interior of Σ. Itfollows that supp (µ) = Σ. This completes the proof of Theorem 1.1, sinceΣ is the union of at most two intervals.

3. The Proof of Theorem 1.3.

Proof. We write Q = Qα,c. Clearly Q is convex on [−1, 0]. Let us set fora ∈ [−1, 0] and for t ∈ [0, 1],


Gα(t) :=1cαπ

–∫ 1

a

Q′(s)s− t

√(1− s)(s− a)ds(3.1)

= − 1π

∫ 0

a

|s|α−1√

(1− s)(s− a)s− t

ds

− 1π

–∫ 1

0

sα−1√

(1− s)(s− a)s− t

ds

=: I1 + I2.

Here the second integral I2 is a principal value integral.We have to prove that there exists tα ∈ [0, 1) so that Gα decreases in

(0, tα) and increases in (tα, 1) (if tα = 0 then the first condition is an emptyone). We establish the following properties:

(i) Gα(0) ≤ 0;(ii) Gα(1) > 0;(iii) For every α > 1, there is tα ∈ [0, 1), such that G′α(t) < 0 on (0, tα),

G′α(t) > 0 on (tα, 1), and G′′α(t) ≥ 0 on (tα, 1).

Clearly, then (iii) implies the decreasing/increasing property of Gα.To show (i), we write

Gα(0) = − 1π

[∫ 1

0sα−2

√(1− s)(s− a)ds−

∫ 0

a|s|α−2

√(1− s)(s− a)ds

]and in the second integral we make the change of variables s 7→ as, to find

Gα(0) = − 1π

∫ 1

0sα−2

√1− s

(√s− a− |a|α−

12

√1− as

)ds.

Since√s− a is greater than or equal to |a|α−

12√

1− as for s in the interval[0, 1], we find that Gα(0) ≤ 0, as claimed in (i). Note that Gα(0) = 0 if andonly if a = −1.

Next, it is easy to see from (3.1) that

Gα(1) =1π

∫ 1

a

|s|α−1√s− a√

1− sds > 0,(3.2)

which establishes (ii) for all α ≥ 1.We now prove (iii) by induction on k = [α], where [α] denotes the integer

part of α.For α = 1, we find by explicit calculation

G1(t) = − 1π

∫ 1

a

√(1− s)(s− a)

s− tds = t− 1 + a

2.


Then (iii) is satisfied with t1 = tα = 0. Suppose now 1 < α < 2. Considerthe analytic function

f(z) := zα−1 [(z − 1)(z − a)]1/2

defined for z ∈ C \ (−∞, 1], where that branch of the square root is chosenwhich is positive for real z > 1. Then the principal value integral I2 may bewritten as

I2 = − 12πi

∫γ

f(z)z − t

dz

with the contour γ going from 0 to 1 on the upper side of the cut (−∞, 1]and back from 1 to 0 on the lower side.

Figure 1. The contours γ and ΓR.

We transform γ into the contour ΓR going from 0 to −R on the upperside of the cut, continuing along the big circle CR of radius R going to −Ron the lower side of the cut, and then going from −R to 0 on the lower sideof the cut. We choose R > 1. See Figure 1 for γ and ΓR.

The contribution from the upper and lower sides of the cut comes fromthe imaginary part of f , which is

=f(x+ i0) =

sin(απ)|x|α−1√

(1− x)(a− x) for x < a,

− cos(απ)|x|α−1√

(1− x)(x− a) for a < x < 0,


and =f(x− i0) = −=f(x+ i0). Therefore

I2 = − 12πi

∫CR

f(z)z − t

dz

+sinαππ

∫ a

−R

|x|α−1√

(1− x)(a− x)x− t

dx

−cosαππ

∫ 0

a

|x|α−1√

(1− x)(x− a)x− t

dx.

Thus we have shown that

Gα(t) = − 12πi

∫CR

f(z)z − t

dz(3.3)

+sinαππ

∫ a

−R

|x|α−1√

(1− x)(a− x)x− t

dx

−(

1 + cosαππ

)∫ 0

a

|x|α−1√

(1− x)(x− a)x− t

dx.

From (3.3), we obtain for the second derivative

G′′α(t) = − 1πi

∫CR

f(z)(z − t)3

dz(3.4)

+2 sinαπ

π

∫ a

−R

|x|α−1√

(1− x)(a− x)(x− t)3

dx

−2 (1 + cosαπ)π

∫ 0

a

|x|α−1√

(1− x)(x− a)(x− t)3

dx.

We let R→∞ in (3.4). Then the integral over the circle CR tends to 0, sincethe integrand behaves like zα−3 as |z| → ∞. Then we get the representation

G′′α(t) =2 sinαπ

π

∫ a

−∞

|x|α−1√

(1− x)(a− x)(x− t)3

dx(3.5)

−2 (1 + cosαπ)π

∫ 0

a

|x|α−1√

(1− x)(x− a)(x− t)3

dx.

Note that the improper integral is convergent because α < 2. Since 1 < α <2, we have sinαπ < 0. Also (x − t)3 < 0 whenever x < 0 < t. Thus weconclude that

G′′α(t) > 0, for t ∈ (0, 1),(3.6)

in case 1 < α < 2. Thus Gα is strictly convex on (0, 1). Since Gα(0) < Gα(1)by properties (i) and (ii) the property (iii) follows for α ∈ (1, 2). Thus wehave established (iii) whenever k = [α] = 1.


Now let k ≥ 2 and suppose that (iii) is true for all α with [α] = k − 1.Let α ∈ [k, k + 1). From (3.1) we obtain for 0 < t < 1,

Gα(t) = − 1π

–∫ 1

a

|x|α−1 − t|x|α−2 + t|x|α−2

x− t√

(1− x)(x− a) dx(3.7)

= − 1π

∫ 1

a

|x|α−1 − t|x|α−2

x− t√

(1− x)(x− a) dx+ tGα−1(t)

=: F (t) + tGα−1(t).

We can write that

F (t) := − 1π

∫ 1

a

|x| − tx− t

|x|α−2√

(1− x)(x− a) dx

=1π

∫ 0

a

x+ t

x− t|x|α−2

√(1− x)(x− a) dx

− 1π

∫ 1

0|x|α−2

√(1− x)(x− a) dx,

from which we obtain

F ′(t) =1π

∫ 0

a

2x(x− t)2

|x|α−2√

(1− x)(x− a) dx < 0, 0 < t < 1(3.8)

and

F ′′(t) =1π

∫ 0

a

4x(x− t)3

|x|α−2√

(1− x)(x− a) dx > 0, 0 < t < 1.(3.9)

Differentiating (3.7) we get

G′α(t) = F ′(t) +Gα−1(t) + tG′α−1(t)(3.10)

and

G′′α(t) = F ′′(t) + 2G′α−1(t) + tG′′α−1(t).(3.11)

By the inductive hypothesis, there exists tα−1, such that G′α−1(t) is nega-tive on (0, tα−1) and positive on (tα−1, 1), as well as G′′α−1(t) ≥ 0 on (tα−1, 1).

Suppose first that tα−1 > 0. Since Gα(0) ≤ 0 and G′α−1(t) < 0 on(0, tα−1), we have that Gα−1(t) is strictly decreasing on (0, tα−1), and there-fore is negative there. This, together with (3.8) and (3.10), implies thatG′α(t) < 0 on (0, tα−1]. On the other hand from (3.9), (3.11) and the induc-tive hypothesis, we obtain that G′′α(t) > 0 on [tα−1, 1). This implies that Gαis strictly convex on (tα−1, 1). Since Gα and G′α are negative on (0, tα−1],and Gα(1) > 0, we see that there exists tα ∈ (tα−1, 1), such that G′α(t) isnegative on (0, tα) and positive on (tα, 1). It is clear also that G′′α(t) > 0 on(tα, 1). Thus property (iii) holds in case tα−1 > 0.


If tα−1 = 0, then we still use (3.9) and (3.11) to derive G′′α(t) > 0 on (0, 1),which implies that Gα is strictly convex on [0, 1]. Since Gα(0) < Gα(1), theproperty (iii) follows as well.

The property (iii) is now established whenever [α] = k. By inductionwe derive that it is true for every k ≥ 1, that is it holds for every α ≥ 1.Thus there exists t0 ∈ [0, 1) such that (1.5) decreases on (0, t0) and increaseson (t0, 1). Since Q is convex on [−1, 0], the conditions of Theorem 1.1 aresatisfied with a1 = 0. It follows from Theorem 1.1 that the support of theequilibrium measure consists of at most two intervals.

References

[1] S.B. Damelin and A.B.J. Kuijlaars, The support of the equilibrium measure for mono-mial external fields on [−1, 1], Trans. Amer. Math. Soc., 351 (1999), 4561-4584,MR 2000g:31003, Zbl 943.31001.

[2] P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Ap-proach, Courant Lecture Notes in Mathematics, Courant Institute, New York, 1999,MR 2000g:47048.

[3] P. Deift, T. Kriecherbauer and K.T.-R. McLaughlin, New results on the equilibriummeasure for logarithmic potentials in the presence of an external field, J. Approx.Theory, 95 (1998), 388-475, MR 2000j:31003, Zbl 918.31001.

[4] P. Deift, T. Kriecherbauer, K.T.-R. McLaughlin, S. Venakides and X. Zhou, Uniformasymptotics for polynomials orthogonal with respect to varying exponential weights andapplications to universality questions in random matrix theory, Comm. Pure AppliedMath., 52 (1999), 1335-1425. CMP 1 702 716, Zbl 944.42013.

[5] P. Deift and K.T.-R. McLaughlin, A continuum limit of the Toda lattice, Mem. Amer.Math. Soc., 624 (1998), MR 98h:58076, Zbl 946.37035.

[6] P. Dragnev and E.B. Saff, Constrained energy problems with applications to orthogonalpolynomials of a discrete variable, J. Anal. Math., 72, (1997) 229-265, MR 99f:42048,Zbl 898.31003.

[7] F.D. Gakhov, Boundary Value Problems, Pergamon Press, Oxford, 1966,MR 33 #6311, Zbl 141.08001.

[8] A.B.J. Kuijlaars, Extremal problems for logarithmic potentials, in ‘ApproximationTheory IX’, C.K. Chui and L.L. Schumaker, editors, Vanderbilt University Press,Nashville, (1998), 201-222, CMP 1 743 007, Zbl 924.31001.

[9] A.B.J. Kuijlaars and P.D. Dragnev, Equilibrium problems associated with fast de-creasing polynomials, Proc. Amer. Math. Soc., 127 (1999), 1065-1074, MR 99f:31003.

[10] A.B.J. Kuijlaars and K.T.-R. McLaughlin, Generic behavior of the density of statesin random matrix theory and equilibrium problems in the presence of real analyticexternal fields, Comm. Pure Appl. Math., 53 (2000), 736-785, CMP 1 744 002.

[11] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin,1972, MR 50 #2520, Zbl 253.31001.

[12] H.N. Mhaskar and E.B. Saff, Where does the sup norm of a weighted polynomial live?,Constr. Approx., 1 (1985), 71-91, MR 86a:41004, Zbl 582.41009.






http://www.ams.org/mathscinet-getitem?mr=1702716

















[13] T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press,Cambridge, 1995, MR 96e:31001, Zbl 828.31001.

[14] E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag,New York, 1997, MR 99h:31001, Zbl 881.31001.

[15] V. Totik, Weighted Approximation with Varying Weight, Lect. Notes in Math., 1569,Springer-Verlag, Berlin, 1994, MR 96f:41002, Zbl 808.41001.

[16] F.G. Tricomi, Integral Equations, Interscience Publishers, New York, 1957,MR 20 #1177, Zbl 078.09404.

Received July 16, 1999 and revised October 11, 2000. The research of the first authorbegan while visiting the third author at the Mathematics Department of the City Uni-versity of Hong Kong and was supported partly by grants from the Centre for ApplicableAnalysis and Number Theory, Johannesburg, and Georgia Southern University.The work of the second author was supported by an Indiana University–Purdue UniversityFort Wayne Summer Research Grant.The work of the third author started while at the City University of Hong Kong. Hisresearch is supported in part by FWO research project G.0278.97, by a research grant ofthe Fund for Scientific Research – Flanders, and by a University of the WitwatersrandResearch grant, jointly with S.B. Damelin.

Department of Mathematics and Computer ScienceGeorgia Southern UniversityStatesboro, Georgia, 30460-8093E-mail address: [email protected]

Department of Mathematical SciencesIndiana University–Purdue UniversityFort Wayne, IN 46805E-mail address: [email protected]

Department of MathematicsKatholieke Universiteit LeuvenCelestijnenlaan 200 B3001 LeuvenBelgiumE-mail address: [email protected]













INVARIANTS OF GENERIC IMMERSIONS

Tobias Ekholm

First order invariants of generic immersions of manifolds ofdimension nm − 1 into manifolds of dimension n(m + 1) − 1,m, n > 1 are constructed using the geometry of self-inter-sections. The range of one of these invariants is related toBernoulli numbers.

As by-products some geometrically defined invariants ofregular homotopy are found.

1. Introduction.

An immersion of a smooth manifold M into a smooth manifold W is asmooth map with everywhere injective differential. Two immersions areregularly homotopic if they can be connected by a continuous 1-parameterfamily of immersions.

An immersion is generic if all its self-intersections are transversal. Inthe space F of immersions M → W , generic immersions form an opendense subspace. Its complement is the discriminant hypersurface, Σ ⊂ F .Two generic immersions belong to the same path component of F − Σ ifthey can be connected by a regular homotopy, which at each instance is ageneric immersion. We shall consider the classification of generic immersionsup to regular homotopy through generic immersions. It is similar to theclassification of embeddings up to diffeotopy (knot theory). In both cases,all topological properties of equivalent maps are the same.

An invariant of generic immersions is a function on F−Σ which is locallyconstant. The value of such a function along a path in F jumps at inter-sections with Σ. Invariants may be classified according to the complexity oftheir jumps. The most basic invariants in this classification are called firstorder invariants (see Section 2).

In [2], Arnold studies generic regular plane curves (i.e., generic immer-sions S1 → R2). He finds three first order invariants J+, J−, and St. In [4],the author considers the case S3 → R5. Two first order invariants J and Lare found. In both these cases, the only self-intersections of generic immer-sions are transversal double points and in generic 1-parameter families thereappear isolated instances of self-tangencies and triple points. The values ofthe invariants J± and J change at instances of self-tangency and remain

321



322 TOBIAS EKHOLM

constant at instances of triple points. The invariants St and L change attriple points and remain constant at self-tangencies.

In this paper we shall consider high-dimensional analogs of these in-variants. The most straightforward generalizations arise for immersionsM2n−1 → W 3n−1, where self-tangencies and triple points are the only de-generacies in generic 1-parameter families. The above range of dimensionsis included in a 2-parameter family, Mnm−1 →Wn(m+1)−1, m,n > 1, wheregeneric immersions do not have k-fold self-intersection points if k > m and ingeneric 1-parameter families there appear isolated instances of (m+ 1)-foldself-intersection. Under these circumstances, we find first order invariants ofgeneric immersions. (For precise statements, see Theorem 1 and Theorem 3in Section 2, where the main results of this paper are formulated.)

In particular, if n is even and Mnm−1 is orientable and satisfies a certainhomology condition (see Theorem 1 (b)) then there exists an integer-valuedinvariant L of generic immersionsMnm−1 → Rn(m+1)−1 which is an analog ofArnold’s St: It changes under instances of (m+1)-fold self-intersection, doesnot change under other degeneracies which appear in generic 1-parameterfamilies, and is additive under connected sum. The value of L at a genericimmersion f is the linking number of a copy of the set of m-fold self-intersection points of f , shifted in a special way, with f(M) in Rn(m+1)−1.(See Definition 4.13.)

For codimension two immersions of odd-dimensional spheres into Eu-clidean space there appear restrictions on the possible values of L (see Theo-rem 2). This phenomenon is especially interesting in the case of immersionsS4j−1 → R4j+1.

In general, it is not known if these restrictions are all the restrictionson the range of L. However, in special cases they are. For example, forimmersions S3 → R5 the range of L is Z (see [4] and also Remark 9.3) andfor immersions S67 → R69 it is 35Z (see Section 9.3).

In [5], the author gives a complete classification of generic immersionsMk → R2k−r, r = 0, 1, 2, k ≥ 2r+4 up to regular homotopy through genericimmersions (under some conditions on the lower homotopy groups of M).Here, the class of a generic immersion is determined by its self-intersectionwith induced natural additional structures (e.g. spin structures). The exis-tence of invariants such as Lmentioned above implies that the correspondingclassification in other dimensions is more involved (see Remark 9.2).

The invariants of generic immersions in Theorems 1 and 3 give rise to in-variants of regular homotopy which take values in finite cyclic groups (Sec-tion 7). We construct examples showing that, depending on the sourceand target manifolds, these regular homotopy invariants may or may notbe trivial (Section 9). It would be interesting to relate these invariants toinvariants arising from the cobordism theory of immersions (see for exampleEccles [3]).

INVARIANTS OF GENERIC IMMERSIONS 323

2. Statements of the main results.

Before stating the main results we define the notions of invariants of orderszero and one. They are similar to knot invariants of finite type introducedby Vassiliev in [13].

As in the Introduction, let F denote the space of immersions and letΣ ⊂ F denote the discriminant hypersurface. The set Σ1 ⊂ Σ of non-genericimmersions which appear at isolated instances in generic 1-parameter fami-lies (see Lemmas 3.5 and 3.6 for descriptions of such immersions Mnm−1 →Wn(m+1)−1) is a smooth submanifold of F of codimension one. (F is anopen subspace of the space of smooth maps, thus an infinite dimensionalmanifold and the notion of codimension in F makes sense.)

If f0 ∈ Σ1 then there is a neighborhood U(f0) of f0 in F cut in twoparts by Σ1. A coherent choice of a positive and negative part of U(f0) foreach f0 ∈ Σ1 is a coorientation of Σ. (The coorientation of the discriminanthypersurface in the space of immersions Mnm−1 →Wn(m+1)−1 is consideredin Section 6.1.)

Let a be an invariant of generic immersions and let f0 ∈ Σ1. Define thejump ∇a of a as

∇a(f0) = a(f+)− a(f−),

where f+ and f− are generic immersions in the positive respectively negativepart of U(f0). Then ∇a is a locally constant function on Σ1.

An invariant a of generic immersions is a zero order invariant (or whichis the same, an invariant of regular homotopy) if ∇a ≡ 0.

Let Σ2 be the set of all immersions which appear in generic 2-parameterfamilies but can be avoided in generic 1-parameter families (see Lemma 3.7for the case Mnm−1 →Wn(m+1)−1). Then Σ2 ⊂ Σ is a smooth codimensiontwo submanifold of F .

An invariant a of generic immersions is a first order invariant if ∇a(f0) =∇a(f1) for any immersions f0, f1 ∈ Σ1 which can be joined by a path inΣ1 ∪ Σ2 such that at intersections with Σ2 its tangent vector is transversalto the tangent space of Σ2.

Theorem 1. Let m > 1 and n > 1 be integers and let Mnm−1 be a closedmanifold.

(a) If m is even and Hn−1(M ; Z2) = 0 = Hn(M ; Z2) then there exists aunique (up to addition of zero order invariants) first order Z2-valuedinvariant Λ of generic immersions Mnm−1 → Rn(m+1)−1 satisfying thefollowing conditions: It jumps by 1 on the part of Σ1 which consists ofimmersions with one (m+ 1)-fold self-intersection point and does notjump on other parts of Σ1.

(b) If n is even, M is orientable, and Hn−1(M ; Z) = 0 = Hn(M ; Z) thenthere exists a unique (up to addition of zero order invariants) first

324 TOBIAS EKHOLM

order integer-valued invariant L of generic immersions Mnm−1 →Rn(m+1)−1 satisfying the following conditions: It jumps by m + 1 onthe part of Σ1 which consists of immersions with one (m + 1)-foldself-intersection point and does not jump on other parts of Σ1.

Theorem 1 is proved in Section 6.4. The invariants Λ and L are definedin Definition 4.12 and Definition 4.13, respectively. If appropriately normal-ized, Λ and L are additive under connected summation of generic immersions(see Section 5.1).

Theorem 2. Let f : S2m−1 → R2m+1 be a generic immersion.

(a) If m = 4j + 1 then 2j + 1 divides L(f).(b) If m = 4j + 3 then 4j + 4 divides L(f).(c) If m = 2j then pr divides L(f) for every prime p and integer r such

that pr+k divides 2j + 1 and pk+1 does not divide µj for some integerk. Here µj is the denominator of Bj

4j , where Bj is the jth Bernoullinumber.

We prove Theorem 2 in Section 8.2.

Theorem 3. Let m > 1 and n > 1 be integers and let Mnm−1 be a closedmanifold and let Wn(m+1)−1 be a manifold.

(a) If n is odd then, for each integer 2 ≤ r ≤ m such that m − r iseven, there exist a unique (up to addition of zero order invariants) firstorder integer-valued invariant Jr, of generic immersions Mnm−1 →Wn(m+1)−1 satisfying the following conditions: It jumps by 2 on thepart of Σ1 which consists of immersions with one degenerate r-foldself-intersection point and does not jump on other parts of Σ1.

(b) If n = 2 then there exists a unique (up to addition of zero orderinvariants) first order integer-valued invariant J , of generic immer-sions M2m−1 → W 2m+1 satisfying the following conditions: It jumpsby 1 on the part of Σ1 which consists of immersions with one degener-ate m-fold self-intersection point and does not jump on other parts ofΣ1.

Theorem 3 is proved in Section 6.3. If appropriately normalized, Jr andJ are additive under connected summation of generic immersions (see Sec-tion 5.1). The value of Jr on a generic immersion is the Euler characteristicof its resolved r-fold self-intersection manifold, the value of J is the numberof components in its m-fold self-intersection (which is a closed 1-dimensionalmanifold).


3. Generic immersions and generic regular deformations.

In this section we define generic immersions and describe the immersions cor-responding to non-generic instances in generic 1- and 2-parameter families.We also describe the versal deformations of these non-generic immersions.

3.1. Generic immersions. Let M and W be smooth manifolds and letf : M → W be an immersion. A point q ∈ W is a k-fold self-intersectionpoint of f if f−1(q) consists of exactly k points. Let Γk(f) ⊂W denote theset of k-fold self-intersection points of f and let Γk(f) = f−1(Γk(f)) ⊂ M

denote its preimage. Note that f |Γk(f)→ Γk(f) is a k-fold covering.

Definition 3.1. Let m > 1 and n > 1. An immersion f : Mnm−1 →Wn(m+1)−1 is generic if

g1 Γr(f) is empty for r > m, andg2 if q = f(p1) = · · · = f(pk) ∈ Γk(f) (k ≤ m) then for any i, 1 ≤ i ≤ k

df TpiM + ∩s 6=idf TpsM = TqW.

A standard application of the jet-transversality theorem shows that theset of generic immersions is open and dense in the space of all immersionsF .

We remark that the set of k-fold self-intersection points Γk(f) of a genericimmersion f : Mnm−1 → Wn(m+1)−1 is a smooth submanifold of W of di-mension n(m− k + 1)− 1. Moreover, the deepest self-intersection Γm(f) isa closed manifold.

Lemma 3.2 below gives a local coordinate description of a generic im-mersion close to a k-fold self-intersection point. To state it we introducesome notation: Let xi ∈ Rnm−1, we write xi = (x0

i , x1i , . . . , x

k−1i ), where

x0i ∈ Rn(m−k+1)−1 and xri ∈ Rn, for 1 ≤ r ≤ k − 1. Similarly, we writey ∈ Rn(m+1)−1 as y = (y0, y1, . . . , yk), where y0 ∈ Rn(m−k+1)−1 and yr ∈ Rn,for 1 ≤ r ≤ k.

Lemma 3.2. Let f : Mnm−1 →Wn(m+1)−1 be a generic immersion and letq = f(p1) = · · · = f(pk) be a k-fold self intersection point of f . Then thereare coordinates y in V ⊂ Wn(m+1)−1 centered at q and coordinates xi onUi ⊂Mnm−1 centered at pi, 1 ≤ i ≤ k such that f is given by

f(x1) = (x01, x

11, . . . , x

k−21 , xk−1

1 , 0),

f(x2) = (x02, x

12, . . . , x

k−22 , 0, xk−1

2 ),...

f(xk) = (x0k, 0, x

1k, x

2k, . . . , x

k−1k ).

(That is, if y = f(xi) then y0(xi) = x0i , y

r(xi) = xri for 1 ≤ r ≤ k − i,yk−i+1(xi) = 0, and yr(xi) = xr−1

i for k − i+ 2 ≤ r ≤ k.)

326 TOBIAS EKHOLM

Proof. The proof is straightforward.

When the source and target of a generic immersion are oriented and thecodimension is even then there are induced orientations on the self intersec-tion manifolds:

Proposition 3.3. Let n = 2j > 1, m > 1, 2 ≤ k ≤ m, and let f : M2jm−1

→ W 2j(m+1)−1 be a generic immersion. Orientations on M2jm−1 andW 2j(m+1)−1 induce an orientation on Γk(f).

Proof. Let N denote the normal bundle of the immersion. The decomposi-tion

f∗TW 2j(m+1)−1 = TM2jm−1 ⊕Ninduces an orientation on N . If q ∈ Γk(f), q = f(p1) = · · · = f(pk) then

TqW2j(m+1)−1 = TqΓk(f)⊕ki=1 Npi .

The orientation on N induces an orientation on ⊕ki=1Npi . Since the dimen-sion of the bundle N is 2j which is even, the orientation on the sum isindependent on the ordering of the summands. Hence, the decompositionabove induces a well-defined orientation on TΓk(f).

3.2. The codimension one part of the discriminant hypersurface.Our next result describes (the 1-jets of) immersions in Σ1 (see Section 2).

Lemma 3.4. If f0 : Mnm−1 → Wn(m+1)−1 is an immersion in Σ1 then g1and g2 of Definition 3.1 holds, except at one k-fold (2 ≤ k ≤ m + 1) self-intersection point q = f0(p1) = · · · = f0(pk), where,

(a) if k = 2,

dim(df0 Tp1M + df0 Tp2M) = n(m+ 1)− 2,

or(b) if 2 < k ≤ m+ 1, for i 6= l

dim(df0 TpiM + ∩r 6=i,r 6=ldf0 TprM) = n(m+ 1)− 1

anddim(df0 TpiM + ∩r 6=idf0 TprM) = n(m+ 1)− 2.

Proof. We have to show that degeneracies as above appears at isolated pa-rameter values in generic 1-parameter families, and that further degeneracies(of the 1-jet) may be avoided . This follows from the jet-transversality the-orem applied to maps Mnm−1 × [−δ, δ]→Wn(m+1)−1.

A k-fold self-intersection point q of an immersion f0 : Mnm−1 →Wn(m+1)−1 where g1 and g2 of Definition 3.1 does not hold will be calleda degenerate self-intersection point of f0. If 2 ≤ k ≤ m we say that q is ak-fold self-tangency point of f0 if k = m+ 1 we say that q is a (m+ 1)-foldself-intersection point of f0.


Recall that a deformation F of a map f is called versal if any deformationof f is equivalent (up to left-right action of diffeomorphisms) to one inducedfrom F .

Let f0 be an immersion in Σ1. Then its versal deformation ft is a 1-parameter deformation. In other words, it is a path λ(t) = ft in F whichintersects Σ1 transversally at f0 and thus, ft are generic immersions forsmall t 6= 0.

Next, we shall describe immersions f0 ∈ Σ1 in local coordinates closeto their degenerate self intersection point. Self-tangency points and (m +1)-fold self-intersection points are treated in Lemma 3.5 and Lemma 3.6,respectively.

To accomplish this we need a more detailed description of coordinatesthan that given in Lemma 3.2, we use coordinates as there with one moreingredient: We write (when necessary) xri ∈ Rn and yri ∈ Rn, r ≥ 1 asxri = (ξri , u

ri ) and yr = (ηr, vr), where ξri , η

r ∈ R and uri , vr ∈ Rn−1. (Greek

letters for scalars and Roman for vectors.)

Lemma 3.5. Let f0 : Mnm−1 →Wn(m+1)−1 be an immersion in Σ1 and letq = f0(p1) = · · · = f0(pk) be a point of degenerate k-fold self intersection2 ≤ k ≤ m. Then there are coordinates y on V ⊂ W centered at q andcoordinates xi on Ui ⊂ M centered at pi, 1 ≤ i ≤ k such that in thesecoordinates the versal deformation ft, −δ < t < δ of f0 is constant outsideof ∪iUi and in neighborhoods of pi ∈ Ui it is given by

ft(x1) = (x01, x

11, . . . , x

k−21 , xk−1

1 , 0),(a)

ft(x2) = (x02, x

12, . . . , x

k−22 , 0, xk−1

2 ),...

ft(xk−1) = (x0k−1, x

1k−1, 0, x

2k−1, . . . , x

k−1k−1).

(That is, if y = ft(xi) then for 1 ≤ i ≤ k − 1, y0(xi) = x0i , y

1(xi) = x1i ,

yr(xi) = xri for 1 ≤ r ≤ k − i, yk−i+1(xi) = 0, and yr(xi) = xr−1i for

k − i+ 2 ≤ r ≤ k.)

ft(xk) =(x0k, (ξ1k, 0), (ξ2k, u

1k), . . . , (ξk−1

k , uk−2k ),(b)

(−ξ2k − · · · − ξk−1k , uk−1

k ))

+(0, (0, 0), (0, 0), . . . , (0, 0), (Q(x0

k, ξ1k) + t, 0)

)where Q is a non-degenerate quadratic form in the n(m − k + 1) variables(x0k, ξ

1k).

328 TOBIAS EKHOLM

Proof. It is straightforward to see that we can find coordinates xi on Ui,i = 1, . . . , k, so that up to first order of approximation f0|Ui is given by theexpressions in (a) and the first term in (b) above.

We must consider also second order terms: Let N be a linear subspace inthe coordinates y transversal to Tf0(U1)∩ · · · ∩Tf0(Uk−1). Let φ : f(Uk)→N be orthogonal projection into N . Then ker(dφ) = Tf0(U1)∩· · ·∩Tf0(Uk)and coker(dφ) is 1-dimensional. The second derivative d2φ : ker(φ) →coker(φ) must be a non-degenerate quadratic form, otherwise, we can avoidf0 in generic 1-parameter families. (This is a consequence of the jet transver-sality theorem.) The Morse lemma then implies that, after possibly adjust-ing the coordinates in Uk by adding quadratic expressions in (x0

k, ξ1k) to ξjk,

j > 1, there exists coordinates for f0 as stated.Finally, we must prove that the deformation ft as given above is versal.

A result of Mather [9] says that it is enough to prove that the deformationis infinitesimally versal. This is straightforward.

In Lemma 3.6 below we will use coordinates as at generic m-fold selfintersection points. That is, x = (x0, x1, . . . , xm−1) ∈ Rnm−1 and y =(y0, y1, . . . , ym) ∈ Rn(m+1)−1, where x0, y0 ∈ Rn−1 and xk, yk ∈ Rn fork 6= 0.

Lemma 3.6. Let f0 : Mnm−1 →Wn(m+1)−1 be an immersion in Σ1 and letq = f0(p1) = · · · = f0(pm+1) be a point of (m + 1)-fold self-intersection.Then there are coordinates y on V ⊂ W centered at q and coordinates xion Ui ⊂M centered at pi, 1 ≤ i ≤ m+ 1 such that in these coordinates theversal deformation ft, −δ < t < δ of f0 is constant outside of ∪iUi and inneighborhoods of pi ∈ Ui it is given by

ft(x1) = (x01, x

11, . . . , x

m−21 , xm−1

1 , 0),(a)

ft(x2) = (x02, x

12, . . . , x

m−22 , 0, xm−1

2 ),...

ft(xm) = (x0m, 0, x

1m, x

2m, . . . , x

m−1m ).

(That is, if y = ft(xi) then for 1 ≤ i ≤ m, y0(xi) = x0i , y

r(xi) = xri for1 ≤ r ≤ m− i, ym−i+1(xi) = 0, and yr(xi) = xr−1

i for m− i+ 2 ≤ r ≤ m.)

(b) ft(xk) =(0, (ξ1k, x

0k), (ξ2k, u

1k), . . . , (ξm−1

k , um−2k ),

(−ξ1k − · · · − ξm−1k + t, um−1

k )).

Proof. The proof is similar to the proof of Lemma 3.5, but easier.


3.3. The codimension two part of the discriminant hypersurface.

Lemma 3.7. Let f0,0 : Mnm−1 →Wn(m+1)−1 be an immersion in Σ2. Theneither (a) or (b) below holds.

(a) f0,0 has two distinct degenerate self-intersection points q1 and q2. Lo-cally around qi, i = 1, 2, f0,0 is as in Lemma 3.5 or Lemma 3.6. Theversal deformation of f0,0 is a product of the corresponding 1-parameterversal deformations.

(b) f0,0 has one degenerate k-fold 2 ≤ k ≤ m self-intersection point q.There are coordinates y centered at q and coordinates xi centered atpi, 1 ≤ i ≤ k such that in these coordinates the versal deformation fs,t,−δ < t, s < δ of f0,0 is constant outside of ∪iUi and in a neighborhoodof pi ∈ Ui it is given by

fs,t(x1) = (x01, x

11, . . . , x

k−21 , xk−1

1 , 0),(b1)

fs,t(x2) = (x02, x

12, . . . , x

k−22 , 0, xk−1

2 ),...

fs,t(xk−1) = (x0k−1, x

1k−1, 0, x

2k−1, . . . , x

k−1k−1).

That is, if y = fs,t(xi) then for 1 ≤ i ≤ k−1, y0(xi) = x0i , y

1(xi) = x1i ,

yr(xi) = xri for 1 ≤ r ≤ k − i, yk−i+1(xi) = 0, and yr(xi) = xr−1i for

k − i+ 2 ≤ r ≤ k.

(b2) fs,t(xk) =(x0k, (ξ1k, 0), (ξ2k, u

1k), . . . , (ξk−1

k , uk−2k ), (−ξ2k − · · · − ξ

k−1k , uk−1

k ))

+(0, (0, 0), (0, 0), . . . , (0, 0), (Q(x0

k) + ξ1k((ξ1k)

2 + s) + t, 0))

where Q is a non-degenerate quadratic form in the n(m − k + 1) − 1variables x0

k.

Proof. The jet-transversality theorem applied to maps of M2jm−1× [−δ, δ]2intoW 2j(m+1)−1 shows that immersions with points of k-fold self-intersectionk ≥ m+2, as well as immersions with points of k-fold self-intersection points2 ≤ k ≤ m+ 1 at which the 1-jet has further degenerations than the 1-jetsof Lemma 3.4 can be avoided in generic 2-parameter families.

Assume that f0,0 has a degenerate k-fold self-intersection point. If k =m + 1 it is easy to see that f0,0 has the same local form as the map inLemma 3.6. So, immersions with (m+1)-fold self-intersection points appearsalong 1-parameter subfamilies in generic 2-parameter families. If k < m+ 1we proceed as in the proof of Lemma 3.5 and construct the map φ. Applyingthe jet transversality theorem once again we see that if the rank of d2φ issmaller than n(m−k+1)−1 then f0,0 can be avoided in generic 2-parameterfamilies. If the rank of d2φ equals n(m−k+1)−1 then the third derivative

330 TOBIAS EKHOLM

of φ in the direction of the null-space of d2φ must be non-zero otherwise f0,0

can be avoided in generic 2-parameter families.With this information at hand we can find coordinates as claimed. Finally,

as in the proof of Lemma 3.5, we must check that the deformation given isinfinitesimally versal. This is straightforward.

Σ

Σ

Σ

Σ

(a) (b)

Figure 1. The discriminant intersected with a small generic2-disk.

Pictures (a) and (b) in Figure 1 correspond to cases (a) and (b) inLemma 3.7. The codimension two parts of the discriminant are repre-sented by points. In (a) two branches of the discriminant, which consistof immersions with one degenerate r-fold and one degenerate k-fold self-intersection point respectively, intersect in Σ2. In (b) the smooth pointsof the semi-cubical cusp represents immersions with one degenerate k-foldself-intersection point. The singular point represents Σ2. If an immersion inΣ above the singular point (Σ2) is moved below it then the index of the qua-dratic form Q in local coordinates close to the degenerate self-intersectionpoint (see Lemma 3.5) changes.

4. Definition of the invariants.

In this section we define the invariants Jr, J , L and Λ. To this end, wedescribe resolutions of self-intersections (for the definitions of Jr and J) andwe compute homology of image-complements of and of normal bundles (forthe definitions of L and Λ).

4.1. Resolution of the self intersection. For generic immersions f :Mnm−1 → Wn(m+1)−1 let Γj(f) = Γj(f) ∪ Γj+1(f) ∪ · · · ∪ Γm(f), for 2 ≤j ≤ m and let Γj(f) = f−1(Γj(f)). Resolving Γj(f) we obtain a smoothmanifold ∆j(f):

Lemma 4.1. Let m > 1 and n > 1 and 2 ≤ j ≤ m be integers. Letf : Mnm−1 → Wn(m+1)−1 be a generic immersion. Then there exists closed


(n(m−j+1)−1)-manifolds ∆j(f) and ∆j(f), unique up to diffeomorphisms,and immersions σ : ∆j(f)→M and τ : ∆j(f)→W such that the diagram

∆j(f) σ−−−→ f−1(Γj(f)) ⊂M

p

y yf∆j(f) τ−−−→ Γj(f) ⊂W

,

commutes. The maps σ and τ are surjective, have multiple points only alongΓj+1(f) and Γj+1(f) respectively, and p is a j-fold cover.

Proof. This is immediate from Lemma 3.2: Close to a k-fold self intersectionpoint Γk(f) looks like the intersection of k (nm−1)-planes in general positionin Rn(m+1)−1.

4.2. Definition of the invariants Jr and J.

Definition 4.2. Let m > 1 and n > 1 be integers and assume that n is odd.For a generic immersion f : Mnm−1 →Wn(m+1)−1 and an integer 2 ≤ r ≤ msuch that m− r is even, define

Jr(f) = χ(∆r(f)),

where χ denotes Euler characteristic.

Definition 4.3. Let m > 1 be an integer. For a generic immersion f :M2m−1 →W 2m+1, define

J(f) = The number of components of Γm(f).

Lemma 4.4. The functions Jr and J are invariants of generic immersions.

Proof. Let ft, 0 ≤ t ≤ 1 be a regular homotopy through generic immersions.If F : M × [0, 1] → W × [0, 1] is the immersion F (x, t) = (ft(x), t) thenΓj(F ) ∼= Γj(f0)× I. It follows that ∆j(f0) is diffeomorphic to ∆j(f1).

4.3. Homology of complements of images.

Lemma 4.5. Let f : Mnm−1 → Rn(m+1)−1 be a generic immersion. As-sume that M is closed. Then

(a)

Hn−1(Rn(m+1)−1 − f(M); Z2) ∼= Hnm−1(M ; Z2) ∼= Z2,

and(b) if M is oriented then

Hn−1(Rn(m+1)−1 − f(M); Z) ∼= Hnm−1(M ; Z) ∼= Z.

332 TOBIAS EKHOLM

Proof. Alexander duality implies that

Hn−1(Rn(m+1)−1 − f(M)) ∼= Hnm−1(f(M)).

By Lemma 3.2, we can choose triangulations of M and f(M) such thatthe map f : M → f(M) induces an isomorphism of the associated cellularcochain complexes in dimensions (nm− k), 1 ≤ k ≤ n.

Remark 4.6. In case (a) of Lemma 4.5 we will use the unique Z2-orienta-tion of M and the duality isomorphism to identify Hn−1(Rn(m+1)−1−f(M);Z2) with Z2. In case (b) of Lemma 4.5, a Z-orientation of M determinesan isomorphism Hnm−1(M ; Z) → Z. We shall use this and the dualityisomorphism to identify Hn−1(Rn(m+1)−1 − f(M); Z) with Z.

A small (n − 1)-dimensional sphere going around a fiber in the normalbundle of f(M)− Γ2(f) generates these groups.4.4. Homology of normal bundles of immersions. Consider an im-mersion f : Mnm−1 → Wn(m+1)−1. Let N denote its normal bundle. ThenN is a vector bundle of dimension n over M . Choose a Riemannian metricon N and consider the associated bundle ∂N of unit vectors in N . This isan (n− 1)-sphere bundle over M . Let ∂F ∼= Sn−1 denote a fiber of ∂N .

Lemma 4.7. Let f : Mnm−1 →Wn(m+1)−1 be an immersion. Let i : ∂F →∂N denote the inclusion of the fiber.

(a) If Hn−1(M ; Z2) = 0 = Hn(M ; Z2) then

i∗ : Hn−1(∂F ; Z2)→ Hn−1(∂N ; Z2),

is an isomorphism.(b) If M and W are oriented and Hn−1(M ; Z) = 0 = Hn(M ; Z) then

i∗ : Hn−1(∂F ; Z)→ Hn−1(∂N ; Z),

is an isomorphism.

Proof. This follows from the Leray-Serre spectral sequence.

Remark 4.8. In case (a) of Lemma 4.7 we use the isomorphism i∗ and acanonical generator of Hn−1(∂F ; Z2) to identify Hn−1(∂N ; Z2) with Z2. Incase (b) of Lemma 4.7, orientations of M and W induce an orientation ofthe fiber sphere ∂F . We use this orientation and i∗ in Lemma 4.7 to identifyHn−1(∂N ; Z) with Z.4.5. Shifting the m-fold self-intersection. Let f : Mnm−1→Wn(m+1)−1

be a generic immersion. Recall that the set of m-fold self-intersection pointsof f is a closed submanifold Γm(f) of Wn(m+1)−1 of dimension n− 1 and itspreimage Γm(f) is a closed submanifold of the same dimension in Mnm−1.

Lemma 4.9. If f : Mnm−1 →Wn(m+1)−1 is an immersion then there existsa smooth section s : Γm(f) → ∂N , where ∂N is the unit sphere bundle ofthe normal bundle of f .


Proof. This is immediate: The base space of the n-dimensional vector bundleN |Γm(f) has dimension n− 1.

Let φ denote the canonical bundle map over f : Mnm−1 →Wn(m+1)−1,

φ : N → TWn(m+1)−1,

and let s be as in Lemma 4.9. We define a vector field v : Γm(f) →TWn(m+1)−1 along Γm(f): For q = f(p1) = · · · = f(pm) let

v(q) = φ(s(p1)) + · · ·+ φ(s(pm)).

It is easy to see that v is smooth.We now restrict attention to the case where the target manifold is Eu-

clidean space.

Definition 4.10. For a generic immersion f : Mnm−1 → Rn(m+1)−1, letΓ′m(f, v, ε) be the submanifold of Rn(m+1)−1 obtained by shifting Γm(f) asmall distance ε > 0 along v.

Remark 4.11. Note that for ε > 0 small enough Γ′m(f, v, ε) ∩ f(M) = ∅.Moreover, if a > 0 is such that for all 0 < ε < a the corresponding Γ′m(f, v, ε)is disjoint from f(M) then Γ′m(f, v, ε) and Γ′m(f, v, a) are homotopic inRn(m+1)−1 − f(M).

4.6. Definition of the invariants L and Λ.

Definition 4.12. Let f : Mnm−1 → Rn(m+1)−1 be a generic immersion. As-sume that M satisfies

Hn−1(M ; Z2) = 0 = Hn(M ; Z2).(CΛ)

Define Λ(f) ∈ Z2 as

Λ(f) = [Γ′m(f, v, ε)]− s∗[Γm(f)] ∈ Z2,

where ε > 0 is very small and [Γ′m(f, v, ε)] ∈ Hn−1(Rn(m+1)−1−f(M); Z2) ∼=Z2 (see Remark 4.6) and s∗[Γm(f)] ∈ Hn−1(∂N ; Z2) ∼= Z2 (see Remark 4.8).

Definition 4.13. Let f : Mnm−1 → Rn(m+1)−1 be a generic immersion. As-sume that n is even and that M is oriented and satisfies

Hn−1(M ; Z) = 0 = Hn(M ; Z).(CL)

Define L(f) ∈ Z as

L(f) = [Γ′m(f, v, ε)]− s∗[Γm(f)] ∈ Z,

where ε > 0 is very small and [Γ′m(f, v, ε)] ∈ Hn−1(R2m(j+1)−1− f(M); Z) ∼=Z (see Remark 4.6) and s∗[Γm(f)] ∈ Hn−1(∂N ; Z) ∼= Z (see Remark 4.8).

334 TOBIAS EKHOLM

Remark 4.14. If Tor(Hn−2(M ;Z),Z2) = 0 and M satisfies condition (CL)in Definition 4.13 then by the universal coefficient theorem M also satisfiescondition (CΛ) in Definition 4.12. It follows from Remarks 4.6, 4.8 that inthis case Λ = Lmod2.

Lemma 4.15. Λ and L are well-defined. That is, they do neither dependon the choice of s, nor on the choice of ε > 0.

Remark 4.16. We shall often drop the awkward notation Γ′m(f, v, ε) andsimply write Γ′(f). This is justified by Lemma 4.15.

Proof. The independence of ε > 0 follows immediately from Remark 4.11.We will therefore not write all ε’s out in the sequel of this proof.

Let s0 and s1 be two homotopic sections of ∂N |Γm(f). Let v0 and v1 bethe corresponding normal vector fields along Γm(f).

A homotopy st between s0 and s1 induces a homotopy vt between v0 andv1 and hence between Γ′m(f, v0) and Γ′m(f, v1) in Rn(m+1)−1 − f(M). Thisshows that Λ and L only depend on the homotopy class of s.

To see that Λ and L are independent of the vector field we introduce thenotion of adding a local twist: Let s : Γm(f) → ∂N be a section and letp ∈ Γm(f). Choose a neighborhood U of p in Γm(f) and a trivialization∂N |U ∼= U × Sn−1 such that s(u) = (u,w), where w is a point in Sn−1. LetDn−1 be disk inside U and let σ : D → Sn−1 be a smooth map of degree ±1such that σ ≡ w in a neighborhood of ∂D. Let stw be the section which iss outside of D and σ in D. We say that stw is the result of adding a localtwist to s. (In the case when M is oriented and n is even, the local twist issaid to be positive if the degree of σ is +1 and negative if the degree of σ is−1.)

Let stw be the vector field obtained by adding a twist to s. Let vtw andv be the vector fields along Γm(f) obtained from stw and s respectively.Clearly,

stw∗[Γm(f)] = s∗[Γm(f)]± 1 and [Γ′m(f, vtw)] = [Γ′m(f, v)]± 1.

Hence, Λ and L are invariant under adding local twists.A standard obstruction theory argument shows that if s and s′ are two

sections of ∂N |Γm then by adding local twists to s′ we can obtain a sections′′ which is homotopic to s. Hence, Λ and L are independent of the choiceof section.

Lemma 4.17. Λ and L are invariants of generic immersions.

Proof. Let ft, 0 ≤ t ≤ 1 be a regular homotopy through generic immersions.Consider the induced map

F : M × I → Rn(m+1)−1 × I, F (x, t) = (ft(x), t),


where I is the unit interval [0, 1]. Shifting Γm(F ) ∼= Γm(f0)×I off F (M×I)using a suitable vector field in N |Γm(F ) it is easy to see that L(f0) =L(f1).

5. Additivity properties, connected sum, and reversingorientation.

In this section we study how our invariants behave under two natural oper-ations on generic immersions: Connected sum and reversing orientation.

5.1. Connected summation of generic immersions. For (oriented)manifolds M and V of the same dimension, let M ]V denote the (oriented)connected sum of M and V .

Let f : Mnm−1 → Rn(m+1)−1 and g : V nm−1 → Rn(m+1)−1 be two genericimmersions. We shall define the connected sum f ] g of these. It will be ageneric immersion M ]V → Rn(m+1)−1.

Let (u, x), u ∈ R and x ∈ Rn(m+1)−2 be coordinates on Rn(m+1)−1.Composing the immersions f and g with translations we may assume thatf(M) ⊂ u ≤ −1 and g(V ) ⊂ u ≥ 1. Choose a point p ∈ M anda point q ∈ V such that there is only one point in f−1(f(p)) and ing−1(g(q)). Pick an arc α in Rn(m+1)−1 connecting f(p) to g(q) and suchthat α ∩ (f(M) ∪ g(V )) = f(p), g(q). Moreover, assume that α meetsf(M) and g(V ) transversally at its endpoints. Let N be the normal bun-dle of α. Pick a (oriented) basis of Tf(p)f(M) and an (anti-oriented) oneof Tg(q)(g(V )). These give rise to nm − 1 vectors over ∂a in N . Extendthese vectors to nm − 1 independent normal vector fields along α. Usinga suitable map of N into a tubular neighborhood of α these vector fieldsgive rise to an embedding φ : α ×D → Rn(m+1)−1, where D denotes a diskof dimension nm − 1, such that φ|f(p) × D is an (orientation preserving)embedding into f(M) and φ|g(q) ×D is an (orientation reversing) embed-ding into g(V ). The tube φ(α× ∂D) now joins f(M)− φ(f(p)× int(D)) tog(V )−φ(g(q)× int(D)). Smoothing the corners we get a generic immersionf ] g : M ]V → Rn(m+1)−1.

Lemma 5.1. Let f : Mnm−1 → Rn(m+1)−1 and g : V nm−1 → Rn(m+1)−1 begeneric immersions. The connected sum f ] g is independent of both thechoices of points f(p) and g(q) and the choice of the path α used to connectthem, up to regular homotopy through generic immersions.

Proof. This is straightforward. (Note that the preimages of self-intersectionshas codimension n > 1.)

Lemma 5.2. If f, f ′ :Mnm−1→ Rn(m+1)−1 and g, g′ :V nm−1→ Rn(m+1)−1,m,n > 1 are regularly homotopic through generic immersions then f ] g andf ′ ] g′ are regularly homotopic through generic immersions.

336 TOBIAS EKHOLM

Proof. This is straightforward.

Proposition 5.3. The invariants Jr and J are additive under connectedsummation.

Proof. ∆j(f ] g) = ∆j(f) t∆j(g).

Note that ifMnm−1 and V nm−1 are manifolds which both satisfy condition(CΛ) in Definition 4.12 or condition (CL) in Definition 4.13 then so doesM ]V .

Proposition 5.4. Let f :Mnm−1→ Rn(m+1)−1 and g :V nm−1→ Rn(m+1)−1

be generic immersions.(a) If M and V both satisfy condition (CΛ) then

Λ(f ] g) = Λ(f) + Λ(g).

(b) If n is even and M and V are both oriented and both satisfy condition(CL) then

L(f ] g) = L(f) + L(g).

Proof. Note that Γm(f ] g) = Γm(f) t Γm(g). Consider case (b). Choose2j-chains D and E in u < −1 and u > 1 bounding Γm(f) and Γm(g)respectively and disjoint from the arc α, used in the construction of f ] g.Then

L(f ] g) = (D ∪ E) · f ] g(M ]V ) = D · f(M) + E · g(V ) = L(f) + L(g).

Case (a) is proved in exactly the same way.

5.2. Changing orientation. The invariants Jr, J , and Λ are clearly ori-entation independent. In contrast to this, the invariant L is orientationsensitive.

To have L defined, let n = 2j and consider an oriented closed manifoldM2jm−1 which satisfies condition (CL).

Proposition 5.5. Assume that there exists an orientation reversing diffeo-morphism r : M → M . Let f : M2jm−1 → R2j(m+1)−1 be a generic immer-sion. Then f r is a generic immersion and

L(f r) = (−1)m+1L(f).

Proof. Note that Γm(f r) = Γm(f) = Γm. The orientation of Γm is inducedfrom the decomposition

TqR2j(m+1)−1 = TqΓm(f)⊕N1 ⊕ · · · ⊕Nm.

The immersions f and f r induces opposite orientations on each Ni hencethe orientations induced on Γm agrees if m is even and does not agree if mis odd. Let D be a 2j-chain bounding Γm, with its orientation induced fromf . If m is even then

L(f r) = D · f(r(M)) = D · −f(M) = −L(f).


If m is odd then

L(f r) = −D · f(r(M)) = −D · −f(M) = L(f).

Proposition 5.6. Let R : R2j(m+1)−1 → R2j(m+1)−1 be reflection in a hy-perplane. Let f : M2jm−1 → R2j(m+1)−1 be a generic immersion. Then Rfis a generic immersion and

L(R f) = (−1)mL(f).

Proof. Note that the oriented normal bundle of Rf is −RN .So the correctly oriented 2j-chain bounding Γm(R f) is (−1)m+1RD.

Thus,

L(R f) = (−1)m+1(RD ·Rf(M)) = (−1)m(D · f(M)) = (−1)mL(f).

6. Coorientations and proofs of Theorems 1 and 3.

In this section we prove Theorems 1 and 3. To do that we need a coorien-tation of the discriminant hypersurface in the space of immersions.

6.1. Coorienting the discriminant.

Remark 6.1. Let a be an invariant of generic immersions. If a is Z2-valuedthen∇a (see Section 2) is well-defined without reference to any coorientationof Σ. Moreover, if a is integer-valued and ∆ is a union of path componentsof Σ1 then the notion ∇a|∆ ≡ 0 is well-defined without reference to acoorientation of Σ.

We shall coorient the relevant parts (see Remark 6.1) of the discriminanthypersurface. That is, we shall find coorientations of the parts of the discrim-inant hypersurface in the space of generic immersions where our invariantshave non-zero jumps. These coorientations will be continuous (see [6], Sec-tion 7). That is, the intersection number of any generic small loop in F andΣ1 vanishes and the coorientation extends continuously over Σ2.

Definition 6.2. Let m > 1 and n > 1. Assume that n is odd. Let 2 ≤k ≤ m and assume that m− k is even. Let f0 : Mnm−1 → Wn(m+1)−1 be ageneric immersion in Σ1 with one degenerate k-fold self-intersection point.Let ft be a versal deformation of f0. We say that fδ is on the positive sideof Σ1 and f−δ on the negative side if χ(∆k(fδ)) > χ(∆k(f−δ)).

Remark 6.3. Note that by Lemma 3.5 χ(∆k(fδ)) is obtained fromχ(∆k(f−δ)) by a Morse modification. Since the dimension of χ(∆k(fδ))is n(m − k + 1) − 1 is even the Euler characteristic changes under suchmodifications.

338 TOBIAS EKHOLM

Definition 6.4. Let m > 1. Let f0 : M2m−1 → W 2m+1 be a generic im-mersion in Σ1 with one degenerate m-fold self-intersection point. Let ft bea versal deformation of f0. We say that fδ is on the positive side of Σ1 andf−δ on the negative side if the number of components in Γm(fδ) is largerthan the number of components in Γm(f−δ).

Remark 6.5. Note that by Lemma 3.5 Γm(fδ) is obtained from Γm(f−δ) bya Morse modification. Since these are 1-manifolds the number of componentschange.

The construction of the relevant coorientation for L is more involved.It is based on a high-dimensional counterpart of the notion of over- andunder-crossings in classical knot theory.

Let f0 : M2jm−1 → W 2j(m+1)−1 be an immersion of oriented manifolds.Assume that f0 ∈ Σ1 has an (m+1)-fold self-intersection point, q = f0(p1) =· · · = f0(pm+1). Let ft be a versal deformation of f0. Let Ui be smallneighborhoods of pi and let Sti denote the oriented sheet ft(Ui). Note thatS0i ∩ · · · ∩ S0

m+1 = q and that St1 ∩ · · · ∩ Stm+1 = ∅ if t 6= 0.Let Dt

i = ∩j 6=iStj . Let wi be a line transversal to TS0i + TD0

i at q. Forsmall t 6= 0 both Sti and Dt

i intersects wi. Orienting the line from Sti toDti gives a local orientation (Dt

i , Sti , ~wi) of R2j(m+1)−1. Comparing it with

the standard orientation of R2j(m+1)−1 we get a sign σt(i) = Or(Dti , S

ti , ~wi),

where Or denotes the sign of the orientation. Note that, if we orient the linefrom Dt

i to Sti we get the opposite orientation −~wi of w and

Or(Dti , S

ti , ~wi) = Or(Sti , D

ti ,−~wi),

since Dti and Sti are both odd-dimensional. Note also that σt(i) = −σ−t(i)

(see Lemma 3.6).We next demonstrate that σt(i) = σt(j) for all i, j: Let N t

i be the orientednormal bundle of Sti . Let w be a vector from Dt

1 to Dt2, transversal to both

TS01 + TD0

1 and TS02 + TD0

2. Orient it from Dt1 to Dt

2. Then

σt(1) = Or(St1, Dt1, ~w),

and hence TDt1 + w gives a normal bundle N t

1 and by the convention usedto orient the normal bundle Or(N t

1) = σt(1)Or(Dt1, ~w). In a similar way it

follows that Or(N t2) = σt(2)Or(Dt

2,−~w).Now, by definition

1 = Or(Dt1, N

t2, . . . , N

tm) = σt(2)Or(Dt

1, (Dt2,−~w), N t

3, . . . , Ntm)

= [dim(Dt2) is odd] = −σt(2)Or(Dt

1,−~w,Dt2, N

t3, . . . , N

tm)

= σt(2)σt(1)Or(N t1, D

t2, N

t3, . . . , N

tm) = σt(1)σt(2).

Hence, σt(1)σt(2) = 1 as claimed.


Definition 6.6. Let f0 : M2j(m+1)−1 → W 2j(m+1)−1 be an immersion oforiented manifolds. Assume that f0 has an (m + 1)-fold self-intersectionpoint. Let ft be a versal deformation of f0. We say that fδ is on the positiveside of Σ1 at f0 if

σδ(1) = · · · = σδ(m+ 1) = +1.6.2. First order invariants. The following obvious lemma will be usedbelow.

Lemma 6.7. Any first order invariant of generic immersions is uniquely(up to addition of zero order invariants) determined by its jump.6.3. Proof of Theorem 3. We know from Lemma 4.4 that J and Jr areinvariants of generic immersions. We must calculate their jumps.

We start in case (a): Assume that n is odd. Let ft : Mnm−1 → Rn(m+1)−1,t ∈ [−δ, δ] be a versal deformation of f0 ∈ Σ1 and fix r, 2 ≤ r ≤ m such thatm− r is even.

If f0 has a degenerate k-fold intersection point 2 ≤ k ≤ m + 1 andk 6= r then ∆r(f−δ) is diffeomorphic to ∆r(fδ): If k < r then ∆r(f−δ) isnot affected at all by the versal deformation. If k > r then the immersedsubmanifold τ−1(Γk(f−δ)) of ∆r(f−δ) is changed by surgery (or is deformedby regular homotopy if k = m+ 1) under the versal deformation. This doesnot affect the diffeomorphism class of ∆r(f−δ).

If f0 has a degenerate r-fold self-intersection point then ∆rfδ is obtainedfrom ∆rf−δ by a surgery (see Lemma 3.5). Since the dimension of ∆r(f−δ)is n(m − r + 1) − 1 which is even this changes the Euler characteristic by±2.

According to our coorientation conventions ∇Jr(f0) = 2 if f0 has a de-generate r-fold self-intersection point. Also, ∇Jr(f0) = 0 if f0 has any otherdegeneracy.

By the same argument, the jump of J in case (b) is 1 at degenerate m-foldself intersection points and 0 at other degeneracies

It is evident from Lemma 3.7 (see Figure 1) that J and Jr are first orderinvariants. The theorem now follows from Lemma 6.7.

6.4. Proof of Theorem 1. We start with (b): Let ft : M2jm−1 →R2j(m+1)−1, t ∈ [−δ, δ] be a generic one-parameter family intersecting Σ1

in f0. If the degenerate intersection point of f0 is a k-fold intersection pointwith 2 ≤ k ≤ m − 1 then clearly L(f−δ) = L(fδ), since the m-fold self-intersection is not affected under such deformations.

Assume that f0 has a degenerate m-fold self-intersection point. Withoutloss of generality we may assume that ft is a deformation of the form inLemma 3.5 (we use coordinates as there):

Let F (x, t) = (ft(x), t). Shifting ((Q(y0, v1) = t, 0 . . . , 0), t) we obtaina 2j-chain bounded by Γ′m(fδ)× δ−Γ′m(f−δ)×−δ in R2m(j+1)−1× [−δ, δ]−F (M × [−δ, δ]). It follows that L(fδ) = L(f−δ).

340 TOBIAS EKHOLM

If f0 has an (m + 1)-fold self-intersection then the discussion precedingDefinition 6.6 shows that L(fδ) = L(f−δ) + (m+ 1): At an (m+ 1)-fold selfintersection point m+1 crossings are turned into crossings of opposite sign.

Hence, ∇L(f0) = m + 1 if f0 has an (m + 1)-fold self-intersection pointand ∇L(f0) = 0 if f0 has any other degeneracy.

The calculation of ∇Λ in (a) is analogous. Let us just make a remarkabout the parity of m: At an (m + 1)-fold self-intersection point m + 1crossings are changed. Hence, at instances of (m + 1)-fold self-intersectionthe invariant Λ changes by 1 ∈ Z2 if m + 1 is odd and does not change ifm+ 1 is even.

It is immediate from Lemma 3.7, see Figure 1 that Λ and L are first orderinvariants. The theorem now follows from Lemma 6.7.

7. Invariants of regular homotopy.

A function of immersions M → W which is constant on path componentsof F will be called an invariant of regular homotopy. Our geometricallydefined invariants of generic immersions give rise to torsion invariants ofregular homotopy.

Definition 7.1. Let n be odd. Let f : Mnm−1 →Wn(m+1)−1 be an immer-sion. Let f ′ be a generic immersion regularly homotopic to f . For 2 ≤ r ≤ msuch that m− r is even, define jr(f) ∈ Z2 as

jr(f) = Jr(f ′) mod 2.

Proposition 7.2. The function jr is an invariant of regular homotopy.

Proof. Clearly it is enough to show that for any two regularly homotopicgeneric immersions f0 and f1, jr(f0) = jr(f1). Let ft be a generic regularhomotopy from f0 to f1. Then ft intersects Σ transversally in a finite numberof points in Σ1. It follows from Theorem 3 that jr remains unchanged atsuch intersections. Hence, jr(f0) = jr(f1).

Definition 7.3. Let n be even. Assume that Mnm−1 is a manifold whichsatisfy condition (CL). Let f : Mnm−1 → Rn(m+1)−1 be an immersion. Letf ′ be a generic immersion regularly homotopic to f . Define l(f) ∈ Zm+1 as

l(f) = L(f ′) mod (m+ 1).

Proposition 7.4. The function l is an invariant of regular homotopy.

Proof. The proof is identical to the proof of Proposition 7.2.

Definition 7.5. Let m > 1 be odd. Assume that Mnm−1 is a manifoldwhich satisfy condition (CΛ). Let f : Mnm−1 → Rn(m+1)−1 be an immersion.Let f ′ be a generic immersion regularly homotopic to f . Define λ(f) ∈ Z2

asλ(f) = Λ(f ′).


Proposition 7.6. The function λ is an invariant of regular homotopy.

Proof. This follows from the proof of Theorem 1, where it is noted that whenm is odd Λ remains constant when a regular homotopy intersects Σ1.

8. Sphere-immersions in codimension two.

In this section Theorem 2 will be proved. To do that we will first discussthe classifications of sphere-immersions and sphere-embeddings up to regularhomotopy.

8.1. The Smale invariant. In Smale’s classical work [12] it is proved thatthere is a bijection between the set of regular homotopy classes Imm(k, n)of immersions Sk → Rk+n and the elements of the group πk(Vk+n,k), thekth homotopy group of the Stiefel manifold of k-frames in (k + n)-space. Iff : Sk → Rk+n is an immersion we let Ω(f) ∈ πk(Vk+n,k) denote its Smaleinvariant. Via Ω we can view Imm(k, n) as an Abelian group.

If the codimension is two the groups appearing in Smale’s classificationare easily computed: The exact homotopy sequence of the fibration

SO(2) → SO(2m+ 1)→ V2m+1,2m−1

implies that π2m−1(V2m+1,2m−1) ∼= π2m−1(SO(2m + 1)). Bott-periodicitythen gives:

π2m−1(V2m+1,2m−1) =

Z if m = 2j,Z2 if m = 4j + 1,0 if m = 4j + 3.

Remark 8.1. It is possible to identify the group operations in Imm(k, 2)geometrically. Kervaire [8] proves that the Smale invariant Ω is additiveunder connected sum of immersions. This gives the geometric counterpartof addition. If f : Sn → Rn+2 is an immersion, r : Sn → Sn is an orientationreversing diffeomorphism, and n 6= 2 then Ω(f) = −Ω(f r). (See [4] forthe case S3 → R5, the other cases are analogous.) This gives the geometriccounterpart of the inverse operation in Imm(n, 2) for n 6= 2.

It is interesting to note that for immersions f : S2 → R4, Ω(f) = Ω(f r):The Smale invariant can in this case be computed as the algebraic numberof self-intersection points. This number is clearly invariant under reversingorientation. (The same is true also for immersions S2k → R4k, k ≥ 1.)

Lemma 8.2. The regular homotopy invariant l induces a homomorphism

Imm(2m− 1, 2)→ Zm+1.

Proof. Note that the dimensions are such that L is defined and spherescertainly satisfy condition (CL). The invariant L is additive under connectedsummation and changes sign if an immersion is composed on the left with

342 TOBIAS EKHOLM

an orientation reversing diffeomorphism. Hence, l induces a homomorphism.

Let Emb(n, 2) ⊂ Imm(n, 2) be the set of regular homotopy classes whichcontain embeddings. By Remark 8.1 Emb(n, 2) is a subgroup of Imm(n, 2).A result of Hughes and Melvin [7] states that Emb(4j − 1, 2) ⊂ Imm(4j −1, 2) ∼= Z is a subgroup of index µj , where µj is the order of the imageof the J-homomorphism, J : π4j−1(SO(4j + 1)) → π8j(S4j+1). The Adamsconjecture, formulated in [1] and proved in [11] implies that Bj

4j = νjµj

, where

Bj is the jth Bernoulli number and νj and µj are coprime integers.

8.2. Proof of Theorem 2. By Lemma 8.2, l : Imm(2m− 1, 2)→ Z/(m+1)Z is a homomorphism and clearly, Emb(2m− 1, 2) ⊂ ker(l).

In case (b) Imm(8j + 5, 2) = 0 and hence the image of l is zero, whichproves that L(f) is always divisible by m+ 1 = 4j + 4.

In case (a) Imm(8j+1, 2) ∼= Z/2Z. Hence, for any immersion f , l(f ] f) =0. Thus, L(f ] f) = 2L(f) is divisible by m+ 1 = 4j+ 2, which implies thatL(f) is divisible by 2j + 1.

In case (c) Emb(4j − 1, 2) ∼= µjZ as a subgroup of Imm(4j − 1, 2) ∼= Z.Hence, for any immersion f , m+ 1 = 2j + 1 divides

L(f ] . . . [µj summands] . . . ] f) = µjL(f).

Thus, if if p is a prime and r, k are integers such that pr+k divides 2j + 1and pk+1 does not divide µj then pr divides L(f).

9. Examples and problems.

In this section we construct examples of 1-parameter families of immersionswhich shows that all the first order invariants we have defined are non-trivial.We also discuss some problems in connection with the regular homotopyinvariants defined in Section 7.

9.1. Examples. Using our local coordinate description of immersions withone degenerate self-intersection point we can construct examples showingthat the invariants J , Jr, Λ and L are non-trivial. We start with Λ and L:

Choose m standard spheres S1, . . . , Sm of dimension (nm−1) intersectingin general position in Rn(m+1)−1 so that S1 ∩ · · · ∩ Sm ∼= Sn−1. Pick a pointp ∈ S1 ∩ · · · ∩ Sm and let Sm+1 be another standard (nm− 1)-sphere inter-secting S1∩ · · ·∩Sm at p so that in a neighborhood of p the embeddings aregiven by the expressions in Lemma 3.6 and so that p is the only degenerateintersection point. Let f0 : Snm−1 → Rn(m+1)−1 be the immersions whichis the connected sum of S1, . . . , Sm+1. Let ft be a versal deformation off . Then, after possibly reversing the direction of the versal deformation wehave Λ(fδ) = 1 ∈ Z2, if n is odd or L(fδ) = ±(m+ 1) if n is even.


In the latter case we would get the other sign of L(fδ) if the orientationof Sm+1 in the construction above is reversed. To distinguish these twoimmersions denote one by h+ and the other by h−, so that L(h+) = m+1 =−L(h−). Then it is an immediate consequence of Theorem 1 that:

Any generic regular homotopy from h+ to h− has at least two instancesof (m+ 1)-fold self-intersection.

Theorem 1 has many corollaries similar to the one just mentioned.To see that L and Λ are non-trivial for other source manifolds. We use

connected sum with the immersions Snm−1 → Rn(m+1)−1 just constructedand Proposition 5.4.

The proofs that the invariants J and Jr are non-trivial are along thesame lines: Construct a sphere-immersion into Euclidean space with onedegenerate self-intersection point as in the local models in Lemma 3.5. Thenembed this Euclidean space with immersed sphere into any target manifoldand use connected sum together with Proposition 5.3.

Remark 9.1. Applying connected sum to the sphere-immersion construct-ed above in the case when it is an immersion S2m−1 → R2m+1 shows thatfor any integer k there exists a generic immersions f : S2m−1 → R2m+1 suchthat L(f) = (m+ 1)k.

Remark 9.2. Let f0 : Mnm−1 → Rn(m+1)−1 be an immersion in Σ1 withan (m + 1)-fold self-intersection point and let ft, −δ < t < δ be a versaldeformation of f0.

Let Γ± = ∪mi=1Γi(f±δ) and Γ± = f−1±δ (Γ±). Then if U± are small regular

neighborhoods of Γ± and U± = f−1±δ (U±) then there exist diffeomorphisms

φ and ψ such that the following diagram commutes

(U−, Γ−)ψ−−−→ (U+, Γ+)

f−δ

y yf+δ(U−,Γ−)

φ−−−→ (U+,Γ+)

.

That is, the local properties of a generic immersion close to its self-inter-section does not change at instances of (m+ 1)-fold self-intersection pointsin generic 1-parameter families.

Assume that M fulfills the requirements of Theorem 1. Then Λ or L isdefined and Λ(f−δ) 6= Λ(fδ) or L(f−δ) 6= L(fδ), respectively. This impliesthat f−δ and fδ are not regularly homotopic through generic immersions.Hence, knowledge of the local properties of a generic immersion Mnm−1 →Rn(m+1)−1 close to its self-intersection is not enough to determine it up toregular homotopy through generic immersions.

9.2. Problems. Are the regular homotopy invariants l, λ and jr non-trivial?

344 TOBIAS EKHOLM

A negative answer to this question implies restrictions on self-intersectionmanifold. A positive answer gives non-trivial geometrically defined regularhomotopy invariants.

9.3. Remarks on the invariant l. Theorem 2 gives information on thepossible range of l for sphere-immersions in codimension two. We considerthe most interesting cases of immersions S4j−1 → R4j+1, j ≥ 1 (Theo-rem 2 (c)). In the first case S3 → R5, l is non-trivial. This was shown in[4]. Hence, for any integer b there are generic immersions f : S3 → R5 suchthat L(f) = b.

Remark 9.3. The invariant l is called λ in [4] and is the mod 3 reduction ofan integer-valued invariant called lk. The definition of lk given in [4] differsslightly from the definition of L given here. There is an easy indirect wayto see that, nonetheless, the two invariants are the same: Due to Theorem 1and Theorem 2 in [4], L − lk is an invariant of regular homotopy which is0 on embeddings and additive under connected summation. Assume thatL(f) − L′(f) = a for some immersion. Then, since µ1 = 24, the connectedsum of 24 copies of any immersion is regularly homotopic to an embedding.Hence, 24a = 0 and therefore a = 0. Thus, L ≡ lk.

If 2j + 1 is prime then Theorem 2 does not impose any restrictions on lsince, in this case, 2j + 1 divides µj (see Milnor and Stasheff [10]).

On the other hand, if none of the prime factors of 2j + 1 divides µj ,Theorem 2 implies that l is trivial and in such cases Remark 9.1 allows us todetermine the range of L which is (2j + 1)Z. The first two cases where thishappens are S67 → R69 and S107 → R109, where 2j+1 equals 35 = 5 · 7 and55 = 5 · 11 and µj equals 24 = 23 · 3 and 86184 = 23 · 34 · 7 · 19, respectively.

9.4. Remarks on the invariant j2. The first case in which to considerthe invariant j2 are immersions of a 5-manifold into an 8-manifold.

Theorem 7.30 in [5] shows that the self-intersection surface of a genericimmersion S5 → R8 must have even Euler characteristic (even though it maybe non-orientable). Thus, in this case j2 is trivial. (Note that π5(V8,5) ∼= Z2

so that there are two regular homotopy classes of immersions S5 → R8.Hence, it is non-trivial to see that l is trivial in this case.)

In contrast, the invariant j2 is non-trivial for immersions S5 → RP 8:Clearly, S5 embeds in RP 8. Thus, there is an immersion f with j2(f) = 0.

Consider three hyperplanes H1, H2, H3 in general position in RP 8. Fixa point q ∈ RP 8 − (H1 ∪ H2 ∪ H3). Note that X = RP 8 − q is a non-orientable line bundle over Hi, i = 1, 2, 3. Choose sections vi of X overHi such that vi = 0 ∼= RP 6 meets H1 ∩ H2 ∩ H3 transversally in Hi

for i = 1, 2, 3. Now, H1 ∩ H2 ∩ H3∼= RP 5. Let g : RP 5 → RP 8 be the

corresponding embedding. Restricting vi to RP 5 we get three normal vectorfields v1, v2, v3 along RP 5 ⊂ RP 8 and v1 = v2 = v3 = 0 ∼= RP 2.


Let p : S5 → RP 5 be the universal cover. Then h = g p : S5 → RP 8

is an immersion. Let Ki = h−1(vi = 0) ∼= S4. Choose Morse functionsφi : S5 → R such that φi = 0 = Ki. Let ε > 0 be small. Then f : S5 →RP 8, given by

f(x) = h(x) + ε3∑i=1

φi(x)vi(h(x)) for x ∈ S5,

is a generic immersion with Γ2(f) ∼= RP 2. Hence, j2(f) = 1.

References

[1] J.F. Adams, On the groups J(X) I, Topology, 2 (1963), 181-195, MR 28 #2553.

[2] V.I. Arnold, Plane curves, their invariants, perestroikas, and classifications, in ‘Sin-gularities and Bifurcations’, edited by Arnold, Adv. Sov. Math., 21 (1994), 39-91; 2(1963), 181-195, MR 95m:57009, Zbl 864.57027.

[3] P.J. Eccles, Characteristic numbers of immersions and self-intersection manifolds,Bolayi Soc. Math. Stud., 4, Topology with Applications (Szekszard) (1993), 197-216,MR 97a:57030.

[4] T. Ekholm, Differential 3-knots in 5-space with and without self intersections, Topol-ogy, 40(1) (2001), 157-196, CMP 1 791 271.

[5] , Immersions in the metastable range and spin structures on surfaces, Math.Scand., 83(1) (1998), 5-41, MR 99k:57062.

[6] , Regular homotopy and Vassiliev invariants of generic immersions Sk →R2k−1, k ≥ 4, J. Knot Theor. Ramif., 7(8) (1998), 1041-1064, MR 2000d:57047,Zbl 949.57015.

[7] J.F. Hughes and P.M. Melvin, The Smale invariant of a knot, Comment. Math. Helv.,60(4) (1985), 615-627, MR 87g:57045, Zbl 586.57017.

[8] M.A. Kervaire, Sur le fibre normal a une sphere immergee dans une espace euclidien,Comment. Math. Helv., 33, (1959), 121-131, MR 21 #3863, Zbl 089.18202.

[9] J. Mather, Stability of C∞-mappings I, Ann. of Math., 87 (1968), 89-104,MR 38 #726, Zbl 159.24902.

[10] J.W. Milnor and J.D. Stasheff, Characteristic Classes, Ann. of Math. Stud., 76,Princeton University Press (1974), MR 55 #13428, Zbl 298.57008.

[11] D. Quillen, The Adams conjecture, Topology, 10 (1971), 67-80, MR 43 #5525,Zbl 219.55013.

[12] S. Smale, Classification of immersions of spheres in Euclidean space, Ann. of Math.,69 (1959), 327-344, MR 21 #3862, Zbl 089.18201.

[13] V.A. Vassiliev, Cohomology of knot spaces, in ‘Theory of Singularities and its Ap-plications’, edited by Arnold, Adv. Sov. Math., 1 (1990), 23-69, MR 92a:57016,Zbl 727.57008.

Received May 24, 1999.

Department of MathematicsUppsala UniversityS-751 06 UppsalaSweden


http://www.ams.org/mathscinet-getitem?mr=95m:57009





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346 TOBIAS EKHOLM




APPROXIMATION WITH NORMAL OPERATORS WITHFINITE SPECTRUM, AND AN ELEMENTARY PROOF OF

A BROWN–DOUGLAS–FILLMORE THEOREM

Peter Friis and Mikael Rørdam

We give a short proof of the theorem of Brown, Douglasand Fillmore that an essentially normal operator on a Hilbertspace is of the form “normal plus compact” if and only if it hastrivial index function. The proof is basically a modification ofour short proof of Lin’s theorem on almost commuting self-adjoint matrices that takes into account the index.

Using similar methods we obtain new results, generalizingresults of Lin, on approximating normal operators by oneswith finite spectrum.

1. Introduction.

Let H be an infinite-dimensional separable Hilbert space, let K denote thecompact operators on H, and consider the short-exact sequence

0 −−−→ K −−−→ B(H) π−−−→ Q(H) −−−→ 0where Q(H) is the Calkin algebra B(H)/K. An operator T ∈ B(H) isessentially normal if T ∗T − TT ∗ ∈ K, or equivalently, if π(T ) is normal.

An operator T ∈ B(H) is Fredholm if π(T ) is invertible in Q(H), and it’sFredholm index is denoted by index(T ). The essential spectrum spess(T ) isthe spectrum of π(T ). The index function of T is the map

C \ spess(T )→ Z; λ 7→ index(T − λ·1).

The index function is invariant under compact perturbations. Hence we maydefine the index function of an invertible S ∈ Q(H) to be that of any liftT ∈ B(H) of S.

The index function is continuous and hence constant on each connectedcomponent of its domain. It vanishes on the unbounded connected compo-nent of its domain. We say that an operator has trivial index function if itsindex function is zero everywhere on its domain.

In Section 2 we give a new proof of the following:

Theorem 1.1 ([BDF1, Cor. 11.2]). An essentially normal operator on aHilbert space is a compact perturbation of a normal operator if and only ifit has trivial index function.

347



348 PETER FRIIS AND MIKAEL RØRDAM

The theorem is a special case of the result by Brown-Douglas-Fillmore([BDF1, Theorem 11.1]) that two essentially normal operators are unitarilyequivalent modulo a compact perturbation if and only if they have the sameessential spectrum and index function. In fact the general theorem is a fairlystraightforward consequence of Theorem 1.1, see [BD, Theorem 5.8] and [D,Proposition 4.1].

Our proof of Theorem 1.1 has two steps. The first step is to show that anessentially normal operator with trivial index function is in the closure of theset of compact perturbations of normal operators on H. We prove this byshowing that a normal element in the Calkin algebra can be approximatedby normal elements with finite spectra if it has trivial index function (seealso Lin [L2]). The methods used here follow closely the methods we used in[FR]. The main difference is that we here must keep track of the index (orK1-class) of the invertible operators used in the various approximation steps.Step two is then to show that the set of compact perturbations of normaloperators is norm closed. This step involves quasidiagonal essentially normaloperators, and its proof is a straightforward consequence of Lin’s Theorem:

Theorem 1.2 (H. Lin, [L2]). For every ε > 0 there is a δ > 0 such that forevery finite dimensional C*-algebra A and every element T ∈ A such that

‖T‖ ≤ 1 and ‖T ∗T − TT ∗‖ < δ

there is a normal element N ∈ A such that ‖T −N‖ < ε.

See also the short proof of Lin’s theorem in [FR].

In Section 3 we consider the general problem of approximating normalelements of a C*-algebra by normal elements with finite spectra.

We show that a normal element a of a unital C*-algebra A of real rankzero can be approximated by normal elements with finite spectra if and onlyif all its translates, a − λ ·1, λ ∈ C, belong to the closure of GL0(A), theconnected component of the identity in GL(A) (Theorem 3.2). A key steptowards this end is that a normal element a in any unital C*-algebra A canbe approximated by normal elements b ∈ A, with 1-dimensional spectra andwith b − λ ·1 ∈ GL0(A) for all λ not in the spectrum of b, if and only ifall translates a − λ ·1 belong to the closure of GL0(A). To complete theargument we use a theorem of Lin ([L3, Theorem 5.4]) that every normalelement with 1-dimensional spectrum in a C*-algebra of real rank zero isthe norm-limit of normal elements with finite spectra.

In Corollary 3.12 to Theorem 3.2 it is shown that if A is a unital C*-algebra of real rank zero and stable rank one, then every normal elementa ∈ A, satisfying π(a)− λ·1 ∈ GL0(A/I) for all proper ideals I of A and forall λ ∈ C \ sp(π(a)), can be approximated by normal elements with finitespectra. This result should be compared with the theorem of Lin in [L4]

NORMAL OPERATORS WITH FINITE SPECTRUM 349

which says that in a simple C*-algebra A of real rank zero and with property(IR) — a property, considered in [FR], which is weaker than stable rank one— every normal element a, with a− λ·1 ∈ GL0(A) for every λ ∈ C \ sp(a),is the norm-limit of normal operators with finite spectra.

The first operator theoretic proof of Theorem 1.1 is due to Berg andDavidson [BD]. In fact, their analysis gives a stronger quantitative versionwhich, subject to a natural resolvent condition on the operator T , gives abound on the norm of the compact perturbation in terms of the norm of theself-commutator T ∗T − TT ∗.

Also let us mention that Lin has generalized Theorem 1.1 to essentiallynormal elements of M(A)/A for certain AF-algebras A (see [L5]).

We thank Larry Brown for several suggestions that helped improve ourresults and our exposition.

2. Proof of Theorem 1.1.

The main part of the proof of Theorem 1.1 consists of showing the theorembelow, which — as indicated — has already been proved by Huaxin Lin(noting that Q(H) is purely infinite and simple). The proof presented here,we believe is shorter and more direct than Lin’s proof.

Theorem 2.1 (cf. H. Lin, [L3, Theorem 4.4]). Let T be a normal elementin Q(H). Then T is the norm limit of a sequence of normal elements inQ(H) with finite spectra if and only if T has trivial index function.

The lemmas below serve to prove the “if”-part of the theorem.Define the continuous function fε : R+ → R+ by fε(t) = maxt − ε, 0.

Let B(λ, r) denote the closed disc in the complex plane with center λ andradius r. We let D denote the unit disc B(0, 1), and T will be the unit circle.

Lemma 2.2. Let T be a normal element in Q(H), let λ ∈ sp(T ) and letε > 0. There exists a normal element S ∈ Q(H) with ‖T − S‖ ≤ 2ε,λ /∈ sp(S), index(S − λ·1) = 0, and

sp(S) \B(λ, ε) = sp(T ) \B(λ, ε).

Proof. We may without loss of generality assume that λ = 0. Let R ∈B(H) be any lift of T , and let R = V |R| be the polar decomposition of R.The operator V fε(|R|) and its adjoint, V ∗fε(|R∗|), have infinite-dimensionalkernels (because π(|R|) = |T | = π(|R∗|) is non-invertible). Let Wfε(|R|)be the polar decomposition of V fε(|R|). The argument above shows that1 −WW ∗ and 1 −W ∗W are both infinite-dimensional projections. HenceW extends to a unitary U ∈ B(H) with V fε(|R|) = Ufε(|R|).

Notice that π(V fε(|R|)) = π(V )fε(|T |) = π(U)fε(|T |) is normal (becauseπ(V ) commutes with |T |). This implies that π(U) commutes with fε(|T |),


and therefore S = π(U)(fε(|T |) + ε ·1) is normal. Clearly, S is invertible,and S lifts to the invertible operator U(fε(|R|) + ε·1), and so index(S) = 0.The distance between S and T is estimated by

‖T − S‖ ≤ ‖π(V )|T | − π(V )fε(|T |)‖+ ‖π(U)fε(|T |)− π(U)(fε(|T |) + ε·1)‖

≤ 2ε.

Let E ∈ Q(H)∗∗ be the spectral projection for |T | corresponding to theinterval [0, ε]. Since |S| = fε(|T |) + ε ·1, it follows that E is the spectralprojection of |S| corresponding to ε in Q(H)∗∗. Since T and S are normal,E commutes with T and S (being a Borel function of T and of S). Wetherefore have

sp(T ) ∪ 0 = sp(TE) ∪ sp(T (1− E)), and

sp(S) ∪ 0 = sp(SE) ∪ sp(S(1− E)).

To complete the argument, notice that |T |(1−E) = (fε(|T |)+ε·1)(1−E) =|S|(1 − E), and hence T (1 − E) = S(1 − E). Also, ‖TE‖ = ‖|T |E‖ ≤ εand ‖SE‖ = ‖|S|E‖ ≤ ε, which entails that sp(TF ) and sp(SF ) are bothcontained in B(0, ε).

Lemma 2.3. Let F be a finite subset of C. The set of elements S ∈ Q(H),satisfying sp(S) ∩ F = ∅ and index(S − λ·1) = 0 for all λ ∈ F , is open.

Proof. The set in question is a finite intersection of open sets of the formGL0(Q(H)) + λ·1.

Lemma 2.4. Let T ∈ Q(H) be normal with trivial index function, let Fbe a finite subset of C, and let ε > 0. Then there exists a normal elementS ∈ Q(H) such that ‖T − S‖ ≤ ε, sp(S) ∩ F = ∅, and

index(S − λ·1) = 0,

for all λ ∈ F .

Proof. For some 0 ≤ k ≤ n we can write F = λ1, λ2, . . . , λn so thatF ∩ sp(T ) = λk+1, λk+2, . . . , λn. By assumption, index(T − λj ·1) = 0 foreach j ≤ k.

Using Lemmas 2.2 and 2.3 we find successively normal elements Tk = T ,Tk+1,Tk+2, . . . , Tn in Q(H) satisfying• ‖Tj − Tj−1‖ ≤ ε/(n− k),• λ1, λ2, . . . , λj /∈ sp(Tj),• index(Tj − λi ·1) = 0, for i = 1, 2, . . . , j, and• λj+1, λj+2, . . . , λn ∈ sp(Tj).

Finally, S = Tn will be as desired.


For each ε > 0 consider the ε-grid Γε in C defined by

Γε = x+ iy ∈ C | x ∈ εZ or y ∈ εZ.

Lemma 2.5. Let T ∈ Q(H) be normal with trivial index function, and letε > 0. Then there exits a normal element S ∈ Q(H) with trivial indexfunction, sp(S) ⊆ Γε, and such that ‖S − T‖ ≤ ε.

Proof. Choose N ∈ N such that Nε ≥ ‖T‖+ ε/4. Put

Σε = x+ iy ∈ C | x, y ∈ ε(Z + 12), X = a+ ib : |a| ≤ Nε, |b| ≤ Nε.

Applying Lemma 2.4 to the finite set Σε ∩ X we get a normal operatorS′ ∈ Q(H) satisfying: ‖S′−T‖ ≤ ε/4, sp(S′)∩(Σε∩X) = ∅, and index(S′−λ·1) = 0 for all λ ∈ Σε ∩X. Note that sp(S′) ⊆ X \ Σε (= Y ).

There is a continuous path t 7→ ft, t ∈ [0, 1], of continuous functionsft : Y → Y so that• f0(z) = z for all z ∈ Y ,• f1(Y ) ⊆ Γε,• ft(z) = z for all z ∈ Γε ∩X and for all t,• |f1(z)− z| < (

√2/2)ε for all z ∈ Y .

Put S = f1(S′). Then S is normal, sp(S) ⊆ Γε ∩ X, and ‖T − S‖ ≤ε/4 + (

√2/2)ε ≤ ε.

If λ ∈ C \ sp(S), then λ is in the same connected component of C \ sp(S)as some λ′ ∈ Σε ∩X, or λ is in the unbounded component of C \ sp(S). Inthe latter case, index(S − λ·1) = 0, and in the former case,

index(S − λ·1) = index(S − λ′ ·1) = index(f1(S′)− λ′ ·1)= index(f0(S′)− λ′ ·1) = index(S′ − λ′ ·1) = 0.

The lemma below is a special case of the Alexander Duality theorem fromtopology. For a compact subset X of C it says that π1(X) ∼= H1(X; Z) ∼=H0(C \ X; Z), and the latter is the free Abelian group generated by thebounded connected components of C \X.

Lemma 2.6. Let X be a compact subset of Γε for some ε > 0. Everycontinuous map f : X → C \ 0 is homotopic to a map of the form

z 7→ (z − λ1)n1(z − λ2)n2 · · · (z − λk)nk ,

for some λi ∈ C \X and some ni ∈ Z.

Proof. Choose n ∈ N such that X ⊆ [−nε, nε]2, and put Y = Γε∩[−nε, nε]2.Then f extends to a continuous function g : Y → C \ 0. (Indeed, byconsidering only one line-segment of Y at a time, this follows from theelementary fact that if X0 is a closed subset of the interval [0, 1] and if


f0 : X0 → C \ 0 is continuous, then f0 extends to a continuous functiong0 : [0, 1]→ C \ 0.)

Let C1, C2, . . . , Ck be the bounded connected components of C \Y . EachCi is an open square, k = 4n2, and

[−nε, nε]2 = Y ∪ C1 ∪ C2 ∪ · · · ∪ Ck.Let γi be the closed curve Ci \ Ci oriented in the positive direction, anddenote by ni the winding number of g around γi.

Choose a yi ∈ Ci for each i. Put

h(z) = (z − λ1)−n1(z − λ2)−n2 · · · (z − λk)−nkg(z), z ∈ Y.The winding number of h around γi is zero for each i. Therefore h|γi

extends to a continuous function Ci → C \ 0 for each i. This showsthat h extends to a continuous function h : [−nε, nε]2 → C \ 0. Since[−nε, nε]2 is contractible, h is homotopic to the constant function 1 in-side C([−nε, nε]2,C \ 0). Hence h|X = h|X is homotopic to 1 insideC(X,C \ 0). The claim of the lemma follows immediately from this.

Lemma 2.7 is a natural generalization of [FR, Lemma 2.3] and the proofrequires only a simple modification using Lemma 2.6. For the reader’s con-venience we include the entire proof.

Lemma 2.7. Let T be a normal element in Q(H) with trivial index functionand with sp(T ) ⊆ Γε for some ε > 0. Suppose that I is a relatively opensubset of sp(T ) which is homeomorphic to the open interval (0, 1). Then foreach λ0 ∈ I and for each δ > 0 there is a normal element S in Q(H) withtrivial index function such that ‖S − T‖ ≤ δ, and sp(S) ⊆ sp(T ) \ λ0.

Proof. Choose one point in each bounded connected component of C\sp(T ),and let F be the finite set of all these points. If S ∈ Q(H) satisfies sp(S) ⊆sp(T ), then S has trivial index function if index(S − λ·1) = 0 for all λ ∈ F .It follows from Lemma 2.3 that S has these properties if ‖T − S‖ ≤ δ0 fora sufficiently small δ0 > 0. We may assume that δ ≤ δ0, and the part of thestatement regarding S having trivial index function is then automaticallytaken care of.

Let J be a relatively open subset of I satisfying

λ0 ∈ J ⊆ J ⊆ I, diam(J) ≤ ε.Let f0 : I → T\−1 be a homeomorphism, and extend f0 to f : sp(T )→ Tby setting f(z) = 1 for all z ∈ sp(T ) \ I. Let V be the unitary element f(T )of Q(H).

It follows from Lemma 2.6 that f is homotopic, inside C(sp(T ),C \ 0),to the function g : sp(T )→ C \ 0 given by

g(z) = (z − µ1)n1(z − µ2)n2 · · · (z − µk)nk ,


for appropriate µi ∈ C \ sp(T ) and ni ∈ Z. Hence f(T ) ∼h g(T ) insideGL(Q(H)) and

index(f(T )) = index(g(T )) =k∑i=1

ni ·index(T − µi ·1) = 0.

This shows that V = f(T ) lifts to a unitary U in B(H). We may nowproceed exactly as in [FR, Lemma 2.3] (see also [BDF1, Lemma 6.1]).

Let E ∈ B(H) be the spectral projection for U corresponding to the(relatively open) subset f(J) of sp(U), and put F = π(E). Then F is aprojection in Q(H). We show below that F commutes with T , and that

spFQ(H)F (TF ) ⊆ J, sp(1−F )Q(H)(1−F )(T (1− F )) ⊆ sp(T ) \ J.(♣)

Once this has been established, we can choose any λ1 ∈ J \ λ0 and set

S = λ1F + (1− F )T.

Then S is normal (because T and F commute),

sp(T ) ⊆ λ1 ∪ sp(T ) \ J ⊆ sp(T ) \ λ0,

and ‖S − T‖ = ‖TF − λ1F‖ ≤ diam(J) ≤ ε as desired.Suppose ϕ : sp(T )→ C is a continuous function which is zero on sp(T )\I.

Then

ϕ(z) =

(ϕ (f |I)−1)(z), if z ∈ T \ −10, if z = −1 ,

defines a continuous function T→ C satisfying ϕ = ϕf , and hence ϕ(T ) =ϕ(V ). Since E commutes with U , it follows that F commutes with V andhence with ϕ(V ) = ϕ(T ).

If ϕ in addition is constant equal to 1 on J , then ϕ is constant equal to 1on f(J), which implies that ϕ(T )F = Fϕ(T ) = F . If ϕ instead vanishes onsp(T ) \ J , then ϕ vanishes on T \ f(J), whence ϕ(T )F = Fϕ(T ) = ϕ(T ).

Let h : sp(T )→ [0, 1] be a continuous function with h|J = 1 and h|sp(T )\I =0. By the argument above, h(T )F = Fh(T ) = F , and since the functionz 7→ zh(z) vanishes on sp(T ) \ I, we get

TF = Th(T )F = FTh(T ) = Fh(T )T = FT.

To show (♣) it suffices to show that ϕ(TF ) = 0 and ψ(T (1− F )) = 0 forevery pair of continuous functions ϕ,ψ : sp(T )→ C, where ϕ vanishes on Jand ψ vanishes on sp(T ) \ I (and where the continuous functions operate inthe respective corner algebras). We may assume that ϕ is equal to 1 on theset sp(T ) \ I. From the argument in the previous paragraph we get

ϕ(TF ) = ϕ(T )F = F − (1− ϕ(T ))F = 0, ψ(T (1− F )) = ψ(T )(1− F ) = 0,

as desired.


Proof of Theorem 2.1. Assume that Tn is a sequence of normal elements ofQ(H) so that Tn tends to T in norm, and Tn has finite spectrum for eachn. Let λ ∈ C \ sp(T ). Then λ ∈ C \ sp(Tn) for all sufficiently large n, andindex(Tn−λ·1)→ index(T−λ·1). But index(Tn−λ·1) = 0 because C\sp(Tn)is connected, so index(T − λ ·1) = 0. This shows that T has trivial indexfunction.

Assume now that T is a normal element of Q(H) with trivial index func-tion, and let ε > 0. By Lemma 2.5 there is a normal element T ′ ∈ Q(H)with sp(T ′) ⊆ Γε, ‖T −T ′‖ ≤ ε, and such that T ′ has trivial index function.

We show next that there exists a normal element T ′′ ∈ Q(H) such that

‖T ′ − T ′′‖ ≤ ε, sp(T ′′) ⊆ Γε,

and such that sp(T ′′) contains no entire line segment of Γε, where an entireline segment is a set of the form

n+ iy | mε < y < (m+ 1)ε or x+ im | nε < x < (n+ 1)ε,

for some n,m ∈ Z.Let I1, I2, . . . , In be the set of entire line segments of Γε that are con-

tained in sp(T ′), and choose λj ∈ Ij for each j. Apply Lemma 2.7 suc-cessively to obtain normal elements R0 = T ′, R1, R2, . . . , Rn in Q(H) suchthat

‖Rj+1 −Rj‖ ≤ ε/n, sp(Rj+1) ⊆ sp(Rj) \ λj+1,

and each with trivial index function. The element T ′′ = Rn then has thedesired property.

It now follows that sp(T ′′) can be partitioned into finitely many clopensets C1, C2, . . . , Cm each with diameter less than 2ε. Choose µi ∈ Ci foreach i, and let f : sp(T ′′) → µ1, µ2, . . . , µm be the continuous functionwhich maps Ci to µi. Then |f(z)− z| < 2ε for all z ∈ sp(T ′′). The elementS = f(T ′′) is normal with sp(S) = µ1, µ2, . . . , µm, and ‖S − T ′′‖ ≤ 2ε, sothat ‖S − T‖ ≤ 4ε.

Recall that an operator T is quasidiagonal if there is an increasing se-quence En∞n=1 of finite rank projections converging strongly to 1 suchthat limn→∞ ‖TEn − EnT‖ = 0. The set of quasidiagonal operators is in-variant under compact perturbations and is norm closed. That the set isnorm closed can be seen by using the equivalent “local” definition of T beingquasidiagonal, that for every finite rank projection E ∈ B(H) and for everyε > 0 there exists a finite rank projection F ∈ B(H) such that E ≤ F and‖TF − FT‖ ≤ ε.

Every normal element N with finite spectrum is quasidiagonal (writeN =

∑kj=1 λjPj and put En =

∑kj=1 F

(j)n , where F (j)

n ∞n=1 is an increasingsequence of finite rank projections converging strongly to Pj). Since every


normal operator in B(H) can be approximated by normal operators withfinite spectrum, every normal operator is in fact quasidiagonal.

The statements in the proposition below were consequences of Theo-rem 1.1 in [BDF1]. Here it is used as a step in our proof.

Proposition 2.8 (cf. [BDF1, Cor. 11.4 and Cor. 11.12]). The set of com-pact perturbations of normal operators on a Hilbert space H is equal to theset of quasidiagonal essentially normal operators. In particular, the set ofcompact perturbations of normal operators operators on H is norm-closed.

Proof. Each compact perturbation of a normal operator is clearly essentiallynormal, and — by the remarks above — also quasidiagonal.

It is well-known that every quasidiagonal operator T ∈ B(H) is block-diagonal plus compact, i.e., there exist S ∈ B(H) and an increasing se-quence En∞n=1 of finite rank projections converging strongly to 1 suchthat T − S ∈ K and SEn = EnS for all n. (Indeed, choose En∞n=1 suchthat

∑∞n=1 ‖EnT − TEn‖ < ∞. Put E0 = 0, and put S =

∑∞n=1(En −

En−1)T (En − En−1). Then

T − S =∞∑n=1

(En − En−1)T (1− En) +∞∑n=1

(En − En−1)TEn−1,

and the right-hand side is compact since the two terms are norm-convergentsums of compact operators.)

Assume now that T is essentially normal and quasidiagonal. Notice thatS, being a compact perturbation of T is also essentially normal. Put

Sn = (En − En−1)T (En − En−1).

Since S =∑∞

n=1 Sn, and SS∗ − S∗S is compact, we have

limn→∞

‖SnS∗n − S∗nSn‖ = 0.

Each Sn lies in the finite dimensional C*-algebra B(Hn), where Hn = (En−En−1)(H), and so Lin’s theorem (Theorem 1.2) says that there exist normaloperators Rn ∈ B(Hn) with limn→∞ ‖Rn − Sn‖ = 0. Put R =

∑∞n=1Rn.

Then R is normal, and R− S is compact. This proves that T is a compactperturbation of a normal operator.

The set of essentially normal operators and the set of quasidiagonal op-erators are both closed. Hence so is their intersection.

Proof of Theorem 1.1. A compact perturbation of a normal operator hastrivial index function since this is the case for a normal operator and sincethe index function is invariant under compact perturbation.

Assume now that T is essentially normal with trivial index function. Thenπ(T ) ∈ Q(H) is normal with trivial index function. Hence, by Theorem 2.1,there is a sequence Sn∞n=1 of normal elements of Q(H) with finite spectrasuch that Sn → π(T ) in norm. Lift Sn to Tn ∈ B(H) such that Tn → T in


norm. Every normal operator in Q(H) with finite spectrum has a lift to anormal operator in B(H). Hence we can find normal operators Rn ∈ B(H)with π(Rn) = Sn = π(Tn). It follows that each Tn is a compact perturbationof a normal operator. By Proposition 2.8 this shows that T itself is a compactperturbation of a normal operator.

Corollary 2.9. An essentially normal operator on a Hilbert space has triv-ial index function if and only if it is quasidiagonal.

Proof. Combine Theorem 1.1 with Proposition 2.8.

3. Approximating normal elements with normal elements withfinite spectra.

In this section we prove various generalizations of Theorem 2.1 using moreor less the same methods as in Section 2.

We first consider an obstruction — analogous to the index-obstruction inTheorem 2.1 — for a normal element of a C*-algebra to be a norm-limit ofnormal elements with finite spectra. The natural generalization of the indexfunction of an element of Q(H) to an element a of a unital C*-algebra A isthe map

C \ sp(a)→ K1(A); λ 7→ [a− λ·1]1.

Proposition 3.1 below shows that we must also take into account the indexfunction of a in every quotient of A. For each proper ideal I of A (properideal meaning I 6= A) we must consider the maps

C \ sp(πI(a))→ K1(A/I); λ 7→ [πI(a)− λ·1]1,

where πI denotes the quotient mapping A→ A/I. This additional obstruc-tion was hidden in the case of the simple C*-algebra Q(H).

Proposition 3.1. Let A be a unital C*-algebra and let a ∈ A be a normalelement. If a is the norm limit of normal elements in A with finite spectra,then

πI(a)− λ·1 ∈ GL0(A/I)(♦)

for every proper ideal I of A and every λ ∈ C \ sp(πI(a)).

Lin showed in [L3] that the natural map GL(A)/GL0(A) → K1(A) isinjective if RR(A) = 0. Since the property real rank zero passes to quotients,this shows that (♦) could be replaced by

[πI(a)− λ·1]1 = 0 in K1(A/I)(♥)

if RR(A) = 0.


Proof. Suppose that πI(a) − λ·1 /∈ GL0(A/I) for some proper ideal I of Aand some λ ∈ C \ sp(πI(a)). Then a belongs to the open set

b ∈ A | πI(b)− λ·1 ∈ GL(A/I) \GL0(A/I).This set can contain no element with finite spectrum, because if b was such anelement, πI(b) ∈ A/I would have finite spectrum and then either πI(b)−λ·1is not invertible or belongs to GL0(A/I) since C\sp(πI(b)) is connected.

Next, we investigate to what extent the converse of Proposition 3.1 holds.Recall the definition of the ε-grid Γε given above Lemma 2.5.

Theorem 3.2. Let A be a unital C*-algebra and let a be a normal elementin A. The following conditions are equivalent:

(i) a− λ·1 lies in the closure of GL0(A) for every λ ∈ C,(ii) for every ε > 0 there exists a normal element b ∈ A such that

sp(b) ⊆ Γε, ‖a− b‖ ≤ 2ε, b− λ·1 ∈ GL0(A)

for all λ ∈ C \ sp(b).If the real rank of A is zero, then (i) and (ii) are equivalent to(iii) for every ε > 0 there exists a normal element b ∈ A with finite spectrum

and with ‖a− b‖ ≤ ε.

The implication (ii) ⇒ (iii) of Theorem 3.2 is contained in a theorem ofH. Lin, [L3, Theorem 5.4]. It also follows from [ELP, Theorem 3.1] af-ter realizing that the conditions on a in (i) and (ii) imply that the mapK1(C∗(a, 1)) → K1(A), induced by the inclusion map, is zero. The impli-cation (ii) ⇒ (i) follows easily from the fact that C \ Γε is dense in C. Theimplication (iii) ⇒ (i) is also easy — see also the proof of Proposition 3.1above.

One could alternatively prove (ii) ⇒ (iii) by mimicking the proof ofLemma 2.7. One would for this approach need Lin’s result, [L1], that ifA is a unital C*-algebra of real rank zero, then every unitary u ∈ U0(A)can be approximated by unitaries with finite spectra. To follow the proofof Lemma 2.7 we would need actual spectral projections for u. They willin general not be available. Instead we can find projections, that approxi-mately commute with u and that approximately divide the spectrum of uinto two disjoint subsets. With some care, one can complete the proof ofLemma 2.7 in this fashion.

The proof of (i) ⇒ (ii) is contained in the three lemmas below:

Lemma 3.3 (cf. [Rø1, Theorem 2.2]). Let A be a unital C*-algebra andlet a be an element in the closure of GL0(A). Let a = v|a| be the polardecomposition of a, with v a partial isometry in A∗∗. For each continuousfunction f : R+ → R+, such that f |[0,ε] ≡ 0 for some ε > 0, there exists aunitary u ∈ U0(A) such that vf(|a|) = uf(|a|).


Proof. This follows from [Rø1, Theorem 2.2] except for the part about theunitary u lying in the connected component of the identity (and the as-sumption that a lies in the closure of GL0(A) rather than in the closure ofGL(A)). As we shall indicate below, the proof of [Rø1, Theorem 2.2] —with obvious modifications — yields the claimed lemma.

We shift to the notation of [Rø1], and denote our C*-algebra by A, andthe element a ∈ A will be denoted by T (so that α(T ) = 0 — the dis-tance from T to the invertibles of A is zero), and T = V |T | is the polardecomposition of T .

In the proof of [Rø1, Theorem 2.1] choose the invertible element A to liein GL0(A) (which is possible by the assumption that T lies in the closure ofGL0(A)). Then the element S produced in that theorem will lie in GL0(A).The element S0 constructed in [Rø1, Lemma 2.3] is homotopic to S, and willtherefore also lie in GL0(A). To see this, go into the proof of [Rø1, Lemma2.3], put Dt = S − tg(|T ∗|)S + tSg(|T |) and put Et = S−1 − tg(|T |)S−1 +tS−1g(|T ∗|). Then, following the proof, one finds that DtEt = EtDt = 1,D0 = S, and D1 = S0. Finally, the unitary U found in [Rø1, Theorem2.2], which satisfies Uf(|T |) = V f(|T |), is equal to S0|S0|−1, and so U ishomotopic to S0 in GL0(A), and hence U ∈ U0(A).

The next lemma is an improvement of Lemma 2.2. Recall the definitionof the functions fε : R+ → R+ from above Lemma 2.2. If f : R+ → R+

is a continuous function with f(0) = 0, then define a continuous functionf : C→ C by f(reiθ) = f(r)eiθ.

Lemma 3.4. Let A be a unital C*-algebra, let a be a normal element in A,and let a = v|a| be a polar decomposition of a, where v ∈ A∗∗ is a partialisometry and |a| = (a∗a)1/2.

(i) If f : R+ → R+ is continuous with f(0) = 0, then vf(|a|) = f(a).(ii) If vfε(|a|) = ufε(|a|) for some unitary u ∈ A, and if b = u(fε(|a|) +

ε·1), then b is normal and invertible, and ‖a − b‖ ≤ 2ε. Moreover, ifg : C→ C is a continuous function which is constant on B(0, ε), theng(a) = g(b), and

sp(b) \ εT = sp(a) \B(0, ε).

Proof. (i). This follows easily by approximating f with functions f0 of theform f0(r) = rh(r), where h : R+ → R+ is continuous (not necessarily withh(0) = 0).

(ii). That b is normal, and ‖a − b‖ ≤ 2ε can be seen as in the proof ofLemma 2.2. Notice that |b| = fε(|a|)+ε·1. This shows that |b|— and henceb — are invertible, and that the spectrum of b does not intersect the openball with center 0 and radius ε.


We shall apply the Borel function calculus inside the von Neumann alge-bra A∗∗. Denoting the indicator function of the (Borel) set E by 1E , set eε =1[0,ε](|a|) = 1[0,ε](|b|) = 1ε(|b|). Put ϕ(t) = t·1(ε,∞)(t) = (fε(t)+ε)·1(ε,∞)(t)for t ∈ R+.

|a|(1− eε) = ϕ(|a|) = |b|(1− eε).Since eε commutes with |b|,

a(1− eε) = v|a|(1− eε) = v(1− eε)|b| = u(1− eε)|b| = b(1− eε).

Now using that eε commutes with a and with b, we get

g(a)(1− eε) = g(a(1− eε)) = g(b(1− eε)) = g(b)(1− eε)

for every continuous function g : C→ C.Assume that g is constant on B(0, ε). Put ψ(z) = g(z)1[0,ε](|z|) =

g(0)1[0,ε](|z|). Then

g(a)eε = ψ(a) = g(0)eε = ψ(b) = g(b)eε.

In conclusion, we have shown that g(a) = g(b). The claim about the spectrafollows from the previous statement.

We now show an analogue of Lemma 2.4.

Lemma 3.5. Let A be a unital C*-algebra, let a be a normal element in A,and let F be a finite subset of C. If a−λ·1 lies in the closure of GL0(A) forall λ ∈ F , then for every ε > 0 there exists a normal element b in A with‖a− b‖ ≤ ε and b− λ·1 ∈ GL0(A) for every λ ∈ F .

Proof. Write F = λ1, λ2, . . . , λn. We find successively normal elementsa0 = a, a1, a2, . . . , an in A satisfying• ‖aj+1 − aj‖ ≤ ε/n,• aj − λi ·1 ∈ GL0(A) for i = 1, 2, . . . , j,• aj − λi ·1 lie in the closure of GL0(A) for i = j + 1, j + 2, . . . , n.

The element b = an will then be as desired.Assume aj−1 has been found and that 1 ≤ j ≤ n. Choose δ > 0 such that• δ < ε/2n,• for every c ∈ A with ‖c − aj−1‖ ≤ δ, we have c − λi ·1 ∈ GL0(A) fori = 1, 2, . . . , j − 1.• |λi − λj | > δ for i = j + 1, j + 2, . . . , n,

Write aj−1 − λj ·1 = v|aj−1 − λj ·1|, with v a partial isometry in A∗∗, anduse Lemma 3.3 to find a unitary u ∈ U0(A) so that vfδ(|aj−1 − λj ·1|) =ufδ(|aj−1 − λj ·1|). Put

aj = u(fδ(|aj−1 − λj ·1|) + δ ·1) + λj ·1.


It then follows from Lemma 3.4 (ii) that aj is normal, that aj−λj·1 ∈ GL0(A),and that ‖aj − aj−1‖ ≤ 2δ < ε/n. By the choice of δ this implies thataj − λi ·1 ∈ GL0(A) for i = 1, 2, . . . , j.

Let i ∈ j + 1, j + 2, . . . , n. We show that aj − λi ·1 is in the closure ofGL0(A). By the choice of δ, we have that λi /∈ B(λj , δ). We can thereforefind continuous functions f, g : C→ C such that• f(z)g(z) = z − λi for all z ∈ C,• f |B(λj ,δ) is constant,• g(z) = exp(h(z)) for some continuous function h : C→ C.

The property of g entails that g(b) ∈ GL0(A) for every normal elementb ∈ A. From Lemma 3.4 (ii) we conclude that f(aj−1) = f(aj). Hence

aj − λi ·1 = f(aj)g(aj) = f(aj−1)g(aj) = (aj−1 − λi ·1)g(aj−1)−1g(aj).

By assumption, aj−1 − λi ·1 lies in the closure of GL0(A), and this showsthat aj − λi ·1 lies in the closure of GL0(A).

Proof of (i) ⇒ (ii) in Theorem 3.2. Copy the proof of Lemma 2.5 usingLemma 3.5 instead of Lemma 2.4.

Recall from [FR] that a unital C*-algebra A is said to have property(IN) if every normal element belongs to the closure of GL(A). A non-unitalC*-algebra A has property (IN) if its unitization A has property (IN).

Definition 3.6. We say that a unital C*-algebra A has property (IN0) ifevery normal element that has 0 as an interior point of its spectrum lies inthe closure of GL0(A).

A non-unital C*-algebra A is said to have property (IN0) if A has property(IN0).

It is clear that property (IN0) implies property (IN). Property (IN) doesnot imply (IN0), not even for C*-algebras of real rank zero and stable rankone as Example 3.7 below shows.

Examples of C*-algebras satisfying (IN0) are given in Proposition 3.8. Thereader may prefer to consider the slightly more restrictive condition (IN00)of a unital C*-algebra A, defined by requiring all normal non-invertibleelements of A to belong to the closure of GL0(A). Trivially, (IN00) implies(IN0), but Example 3.9 gives a C*-algebra of real rank zero for which thereverse implication does not hold.

Example 3.7. Property (IN) does not imply property (IN0).Let B1, B2 be two unital C*-algebras of stable rank one, real rank zero,

so that the unitary group of B1 is disconnected, and (B1 and) B2 are non-scattered. (One could for example take B1 = B2 to be an irrational rotationC*-algebra.) Let A be the C*-algebra B1⊕B2. Then A is unital, sr(A) = 1and RR(A) = 0.


Choose normal elements b1 ∈ B1 and b2 ∈ B2 with

sp(b1) = z ∈ C : 1/2 ≤ |z| ≤ 1, sp(b2) = z ∈ C : |z| ≤ 1/2,and with b1 /∈ GL0(B1). Set a = (b1, b2) ∈ A. Then sp(a) = D, the closedunit disc in the complex plane. Hence a−λ·1 ∈ GL0(A) for every λ ∈ C\sp(a)(because C \ sp(a) is connected). But a does not belong to the closure ofGL0(A), since b1 does not belong to the closure of GL0(B1).

Hence A does not have property (IN0), but sr(A) = 1 (so A has property(IN)), and RR(A) = 0.

Proposition 3.8. Every simple unital C*-algebra, which has stable rankone or is purely infinite, has property (IN0).

Proof. The two classes of C*-algebras in the proposition have in commonthat they have property (IN) (see [Rø2, Theorem 4.4]), and if B is a non-zerohereditary sub-C*-algebra of A, then the natural map U(B)→ U(A)/U0(A)is surjective (see [Ri, Theorem 10.10] and [C, Theorem 1.9]).

Let a be a normal non-invertible element of A, and let ε > 0. Let a = v|a|be the polar decomposition for a, with v ∈ A∗∗, and recall from [Rø1,Theorem 2.1] that there exists a unitary u ∈ A so that vfε(|a|) = ufε(|a|).Let g : R+ → R+ be a continuous function that vanishes on [ε,∞), andwith g(0) = 1. Then g(|a|) 6= 0 because a is non-invertible, and we cantherefore find a unitary w ∈ g(|a|)Ag(|a|) + 1 such that uw ∼h 1. Setb = uw(fε(|a|) + ε·1). Then b ∈ GL0(A) (we are not claiming here that b isnormal), and

‖a− b‖ ≤ ‖a− u(fε(|a|) + ε·1)‖+ ‖w(fε(|a|) + ε·1)− (fε(|a|) + ε·1)‖ ≤ 4ε,

where we have used that wfε(|a|) = fε(|a|).This argument shows that A actually has property (IN00).

Villadsen has found an example of a simple C*-algebra with stable rank 2,showing that a specific normal element of the constructed C*-algebra cannotbe approximated by invertible elements (see [V]). This example is thereforea simple, stably finite C*-algebra that does not have property (IN), andhence neither (IN0) nor (IN00).

Example 3.9. Property (IN0) does not imply property (IN00).Let B be any real rank zero, unital C*-algebra that has property (IN0)

and non-connected group of unitary elements, and set A = B ⊕ C. (HereB could be an irrational rotation C*-algebra, cf. Proposition 3.8.) If zero isan interior point of the spectrum of a = (b, λ) ∈ A, then zero is an interiorpoint of the spectrum of b. Hence b lies in the closure of GL0(B), and fromthis we get that a lies in the closure of GL0(A).

The C*-algebra A does not have property (IN00). Indeed, if u ∈ U(B) \U0(B), and if a = (u, 0) ∈ A, then a is normal and non-invertible, but a isnot in the closure of GL0(A).


Corollary 3.10. Let A be a unital C*-algebra. The following two condi-tions are equivalent:

(i) Every normal element a ∈ A, satisfying

a− λ·1 ∈ GL0(A)

for all λ ∈ C \ sp(a), is the norm-limit of normal elements in A withfinite spectrum.

(ii) A has real rank zero and property (IN0).

Proof. (i)⇒ (ii). Assume that (i) holds. Every self-adjoint element a in A isnormal and satisfies a−λ·1 ∈ GL0(A) for all λ ∈ C\sp(a) (because C\sp(a)is connected). Hence a can be approximated within any given tolerance bya normal element b with finite spectrum. It is easily seen, that (b + b∗)/2is a self-adjoint element with finite spectrum whose distance to a is at most‖a− b‖. This shows that A has real rank zero.

We proceed to show that A has property (IN0). Let a be a normal elementin A, and assume that there exists an r > 0 such that B(0, r) ⊆ sp(a). Leta = v|a| be the polar decomposition of a, with v a partial isometry in A∗∗.

Consider the continuous functions f : R+ → R+, g : R+ → R+, givenby f(t) = minr−1t, 1 and g(t) = maxr, t, and h = f : C → C (seeabove Lemma 3.4). Notice that f(t)g(t) = t, and that vf(|a|) = h(a) (byLemma 3.4 (i)).

The element h(a) is therefore normal and sp(h(a)) = h(sp(a)) = D (theclosed unit disc in the complex plane). Consequently, h(a)− λ·1 ∈ GL0(A)for all λ ∈ C \ sp(h(a)) (because C \ sp(h(a)) is connected). Assuming (i),we conclude that h(a) can be approximated by normal elements with finitespectra, and therefore h(a) lies in the closure of GL0(A) (cf. the proof ofProposition 3.1). Since a = vf(|a|)g(|a|) = h(a)g(|a|), and since g(|a|) ∈GL0(A), we conclude that a lies in the closure of GL0(A). It has now beenproved that A has property (IN0).

(ii) ⇒ (i). By Theorem 3.2 it suffices to show that a − λ ·1 lies in theclosure of GL0(A) for all λ ∈ C. This is the case by assumption on a ifλ /∈ sp(a). By continuity, a − λ·1 lies in the closure of GL0(A) for all λ inthe closure of C \ sp(a). The remaining points, λ, are the interior pointsof the spectrum of a, and there a − λ·1 is in the closure of GL0(A) by theassumption that A has property (IN0).

Corollary 3.11. Let A be a unital C*-algebra. Assume that RR(A) = 0,that A has property (IN), and that

(NT) for every hereditary sub-C*-algebra B of A, the map

U(B)→ U(I)/U0(I),

where I is the ideal of A generated by B, is surjective.


Then every normal element a ∈ A, satisfying

πI(a)− λ·1 ∈ GL0(A/I)(♦)

for every proper ideal I of A, and for every λ ∈ C \ sp(πI(a)), is a normlimit of normal elements in A with finite spectra.

Proof. By Theorem 3.2 it suffices to show that every normal element a ∈ A,satisfying

πI(a) ∈ GL(A/I)⇒ πI(a) ∈ GL0(A/I),lies in the closure of GL0(A). Let ε > 0. Write a = v|a| with v a partialisometry in A∗∗. By the assumption that A has property (IN), and byLemma 3.3, there is a unitary u ∈ A so that vfε(|a|) = ufε(|a|). We wish toreplace u by another unitary that belongs to U0(A). Since A is of real rankzero, we can find a projection p in the hereditary subalgebra of A generatedby fε(|a|) so that ‖(1 − p)fε(|a|)‖ ≤ ε. We show that there is a unitaryv ∈ (1 − p)A(1 − p) such that u∗ ∼h p + v in U(A). This will imply thatb = u(p+ v∗)(fε(|a|) + ε·1) ∈ GL0(A), and

‖a− b‖ ≤ ‖a− u(fε(|a|) + ε·1)‖+ ‖(p+ v∗)(fε(|a|) + ε·1)− (fε(|a|) + ε·1)‖

≤ 2ε+ ‖(v∗ − (1− p))(fε(|a|) + ε·1)‖ ≤ 6ε.

Let I be the closed two-sided ideal in A generated by 1 − p. If I = A,then (1 − p)A(1 − p) contains a unitary v with u∗ ∼h p + v in U(A) bythe assumption (NT). Assume that I 6= A. Because p lies in the hereditarysubalgebra generated by fε(|a|), we have p|a|p ≥ εp; this entails πI(|a|) ≥ ε·1,and so πI(a) ∈ GL(A/I), which by the assumption on a implies πI(a) ∈GL0(A/I). It follows that

πI(u) ∼h πI(ufε(|a|)) ∼h πI(a) ∼h 1

in GL(A/I). We can therefore find w ∈ U0(A) with πI(w) = πI(u). Sincew∗u ∈ U(I) it follows from the assumption (NT) that there exists a unitaryv in (1−p)A(1−p) with (w∗u)∗ ∼h p+v. This completes the proof, becausew∗u ∼h u.

Every C*-algebra of stable rank one has property (IN) (for trivial reasons),and also property (NT) (for less trivial reasons). For the latter one can use[Ri, Theorem 10.10]. With these observations we get the following corollaryto Corollary 3.11:

Corollary 3.12. Let A be a unital C*-algebra of real rank zero and stablerank one. Then every normal element a ∈ A, satisfying

πI(a)− λ·1 ∈ GL0(A/I)(♦)

for every proper ideal I of A and every λ ∈ C \ sp(πI(a)), is a norm limitof normal elements in A with finite spectra.


Corollary 3.12 can for example be applied to the real rank zero AT-algebras classified by George Elliott in [E]. (AT-algebras is the class ofC*-algebras obtained from the C*-algebra C(T) with the operations of ten-soring by Mn(C), taking direct sums, and taking inductive limits.)

It would be interesting to know if one can replace the assumption in Corol-lary 3.12 that sr(A) = 1 with the weaker assumption that A has property(IR) (cf. [FR, 3.1] and the main theorem of [L4]). By Corollary 3.11 thatwould be the case if property (IR) (together with RR(A) = 0) implies prop-erty (NT). This is known to be true for simple C*-algebra, because a simpleunital C*-algebra A has property (IR) if and only if either sr(A) = 1 or Ais purely infinite.

Example 3.13. Of the three sufficient conditions in Corollary 3.11, the realrank zero condition is clearly also necessary (cf. the proof of Corollary 3.10).It is possible that the condition (NT) always holds for real rank zero C*-algebras. Larry Brown has informed us that examples of C*-algebras ofreal rank zero, which do not have property (IN), exist. Condition (NT)is necessary in Corollary 3.11, at least when A is simple, as the followingexample shows:

Assume that A is a simple unital C*-algebra of real rank zero whereproperty (NT) of Corollary 3.11 does not hold — if such an example exists.Since every hereditary sub-C*-algebra of A is the inductive limit of corneralgebras pAp, where p is a projection in A, there is a unitary u ∈ A anda projection p ∈ A with the property that there is no unitary v ∈ pApsatisfying u ∼h v+(1−p). Since A is of real rank zero, and since sp(u) = T,there is a non-zero projection q ∈ A such that ‖quq − q‖ ≤ 1/2. Then(1− q)u(1− q) is invertible in (1− q)A(1− q), and z = q+ (1− q)u(1− q) ishomotopic to u in GL0(A). Put u0 = z|z|−1. Then u0 ∈ U(A), u ∼h u0 andqu0 = u0q = q.

Let x be a non-zero element in qAp, and let e be a non-zero projectionin xAx∗. Then e ≤ q and e - p. It follows that eu0e = e and that thereis no unitary v ∈ eAe such that u0 ∼h v + (1 − e). The corner algebraeAe is non-scattered (because A must be infinite-dimensional), and we cantherefore find a normal element c ∈ eAe with sp(c) = D.

Put a = c + (1 − e)u0(1 − e). Then a is normal and sp(a) = D, and soa − λ ·1 ∈ GL0(A) for every λ ∈ C \ sp(a). We claim that a is not in theclosure of GL0(A), and this will show that a cannot be approximated bynormal elements with finite spectra, cf. Theorem 3.2. Indeed, assume thatb ∈ GL0(A) and that ‖a− b‖ < 1. Then

‖(1− e)u0(1− e)− (1− e)b(1− e)‖ < 1,

and (1− e)u0(1− e) is unitary in (1− e)A(1− e). Hence (1− e)b(1− e) isinvertible in (1−e)A(1−e) with an inverse we denote by r. As in a standard


2 × 2 matrix trick,

(1− e)b(1− e) + (ebe− ebrbe) = (1− ebr(1− e))b(1− (1− e)rbe)∼h b

(♠)

in GL(A). Hence d = ebe − ebrbe ∈ GL(eAe) and so d = v∗|d| for someunitary v ∈ eAe. By (♠) we also have

(1− e) + v ∼h (1− e) + d−1 ∼h (1− e)b(1− e) + e

∼h (1− e)u0(1− e) + e = u0

in contradiction with the stipulated properties of u0 and e.

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[BDF1] L.G. Brown, R. Douglas and P. Fillmore, Unitary equivalence modulo the com-pact operators and extensions of C*-algebras, in ‘Proc. Conf. on Operator The-ory’, Lecture notes in Math., 345, Springer-Verlag, Heidelberg, (1973), 58-128,MR 52 #1378, Zbl 277.46053.

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[D] A.M. Davie, Classification of essentially normal operators, in ‘Spaces of AnalyticFunctions’, Lecture notes in Math., 512, Springer-Verlag, Heidelberg, (1976),31-55, MR 58 #12478, Zbl 337.47010.

[ELP] S. Eilers, T. Loring and G.K. Pedersen, Fragility for subhomogeneous C*-algebraswith one-dimensional spectrum, Bull. London. Math. Soc., 31(3) (1999), 337-344,MR 2000a:46092.

[E] G.A. Elliott, On the classification of C*-algebras of real rank zero, J. Reine Angew.Math., 443 (1993), 179-219, MR 94i:46074, Zbl 809.46067.

[FR] P. Friis and M. Rørdam, Almost commuting self-adjoint matrices — A shortproof of Huaxin Lin’s theorem, J. Reine Angew. Math., 479 (1996), 121-131,MR 97i:46097, Zbl 859.47018.

[L1] H. Lin, Exponential rank of C*-algebras with real rank zero and Brown-Pedersen’sconjecture, J. Funct. Anal., 114 (1993), 1-11, MR 95a:46079, Zbl 812.46054.

[L2] , Almost commuting self-adjoint matrices and applications, in OperatorAlgebras and their Applications, Fields Institute Communications, 13, (1995),193-233, MR 98c:46121, Zbl 881.46042.

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[L5] , The generalized Berg theorem and BDF-theorem, Trans. Amer. Math.Soc., 349 (1997), 529-545, MR 98g:46085, Zbl 870.46034.

[Ri] M.A. Rieffel, Dimension and stable rank in the K-theory of C*-algebras, Proc.London Math. Soc., 46(3) (1983), 301-333, MR 84g:46085, Zbl 533.46046.

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Received November 1, 1998. The first author was supported by The Danish ResearchAcademy.

Department of MathematicsUniversity of Toronto100 St. George StreetToronto, Ontario M5S 3G1CanadaE-mail address: [email protected]

Department of MathematicsUniversity of CopenhagenUniversitetsparken 52100 Copenhagen ØDenmarkE-mail address: [email protected]














PARAMETRIZATION OF THE IMAGE OF NORMALIZEDINTERTWINING OPERATORS

Chris Jantzen and Henry H. Kim

In studying residual automorphic representations, we needto parametrize the image of normalized local intertwining op-erators. This has been done by Moeglin in the case of theresidual spectrum attached to the trivial character of the torusfor split classical groups. In this paper, we extend her result tonon-trivial characters of the torus. To do this, we use Roche’sHecke algebra isomorphisms and Barbasch-Moy’s graded al-gebra isomorphisms to reduce to the case of the trivial char-acter. Along the way, we need to show that Roche’s Hecke al-gebra isomorphisms are compatible with induction in stages,construct a generalized Iwahori-Matsumoto involution, andshow that the images of intertwining operators behave wellwith respect to the Hecke algebra and graded algebra isomor-phisms. We note that this also gives a parameterization of thesquare-integrable and tempered representations supported onthe Borel subgroup.

0. Introduction.

Let G = G(n) be a split classical group Sp(2n, F ), SO(2n + 1, F ), orO(2n, F ) over a p-adic field F with odd residual characteristic (this con-dition comes from [R]) and T be a maximal torus. Throughout this paper,we will drop F in the notation. Let χ be a unitary character of T andp = (a1, b1, . . . , as, bs, as+1) be a chain (as+1 is only for the cases Sp(2n, F )and SO(2n+ 1, F ); see Definition 3.1). Then p gives rise to a character

λp =(a1 − 1

2,a1 − 3

2, . . . ,−b1 − 1

2, . . . ,

as − 12

,as − 3

2, . . . ,

− bs − 12

,as+1 − 1

2,as+1 − 3

2, . . . , 1

).

If we ignore the ordering, p gives rise to a unipotent orbit O in the dualgroup G∗ = SO(2n+ 1,C), Sp(2n,C), or O(2n,C). To p, we attach a Weylgroup element wp. In this paper we parametrize the image of the local nor-malized intertwining operator R(wp, λp, χ)I(λp, χ). We need this result in

367



368 CHRIS JANTZEN AND HENRY H. KIM

the calculation of the residual spectrum [Ki3]. In addition to the applica-tion to the residual spectrum, our result has independent interest in that itparametrizes the square integrable representations which have support onBorel subgroups, via the generalized Iwahori-Matsumoto involution; actu-ally it parametrizes tempered representations which have support on Borelsubgroups (see Remark 3.2.4).

In a remarkable paper [M1], Mœglin solved the problem in the case whenχ is trivial and p satisfies a certain condition, that is, p ∈ P (O) (see Sec-tion 2). She showed that R(wp, λp)I(λp) is semi-simple and its summandsare parametrized by certain characters η of A(O), where A(O) is a finiteabelian group generated by the order two elements σ(a1), σ(b1), . . . , σ(as), σ(bs),σ(as+1) (we take only distinct ones). If O is a distinguished unipotent orbit(i.e., ai’s, bj ’s are all distinct), then the characters are those which satisfyη(σ(ai)) = η(σ(bi)), i = 1, . . . , s and η(σ(as+1)) = 1. We denote by A(p)the set of such characters. Let Unip(p) be the set of direct summands ofR(wp, λp)I(λp). To a unipotent orbit O, she considered a certain set of or-dered partitions P (O) so that each chain p′ ∈ P (O) gives rise to a certaincharacter λp′ which is a conjugate of λp. Let Unip(O) be the union of allUnip(p) as p runs through P (O). She showed that Unip(O) is the set of irre-ducible constituents of the principal series I(λp) whose Iwahori-Matsumotoinvolution is tempered. Then Unip(O) is parametrized by Springer (O),which is the union of A(p) as p runs through P (O); recall that the Springercorrespondence is an injective map from the characters of W , the Weylgroup, into the set of (O, η), where O is a unipotent orbit of G∗ and η is acharacter of A(O). Then Springer (O) is the set of characters of A(O) whichare in the image of the Springer correspondence. Thus Mœglin showed thatUnip(O) is the set of the local components of the residual spectrum attachedto the trivial character of the maximal torus.

The basic approach of this paper is to reduce the problem to the trivialcharacter case. However, there are a number of non-trivial obstacles, whichwe now describe.

First, the basic mechanism we use to reduce from the ramified case to theunramified case is the Hecke algebra isomorphisms of Roche [R]. The firstbasic problem we must deal with is that the representations we are interestedin are not, in general, induced off the Borel, but rather are degenerateprincipal series. So, in order to implement our approach, we must establishthat these isomorphisms behave well with respect to induction in stages.Since such results may be of broader use, we do this in the generality of[R], not just for the particular classical groups we deal with in the restof this paper. In addition, we need to generalize the Iwahori-Matsumotoinvolution. (While this has been done in general in [Au1], [Au2], [Sc-St],these results are done in the Grothendieck group setting; we need to know

INTERTWINING OPERATORS 369

that composition series are respected as well.) Such a generalized Iwahori-Matsumoto involution can easily be defined using Roche’s isomorphism. Butin order to verify some of the properties we need, we have to establish howit behaves with respect to induction in stages. Again, this comes down toshowing that Roche’s isomorphisms respect induction in stages. These issuesare addressed in Section 1.

Second, we need to deal with non-trivial unramified quadratic characters.Mœglin depended on Barbasch-Moy’s results [B-Mo1] which use Kazhdan-Lusztig’s parametrization of unramified representations and the Iwahori-Matsumoto involution. However, the technique cannot be extended to non-trivial unramified quadratic characters. In a subsequent work, Barbasch-Moy [B-Mo2] extended their results to non-trivial unramified quadraticcharacters, using graded Hecke algebras. We use their graded Hecke algebraisomorphisms to reduce to the trivial character case. See Section 3.1. Unfor-tunately, Barbasch-Moy’s results [B-Mo2] are stated for connected groupsand we need them for the disconnected group O(2n). We have stated this asAssumption 3.1.1. We have no doubt that it is true. However, we were notable to verify it. Therefore, our results are complete only for odd orthogonalgroups.

Third, we need to remove a restriction on p (see Remark 3.2) for an arbi-trary chain when χ is trivial. Note that an arbitrary chain does not belongto P (O) in general. But it always comes from the global consideration whenχ = χ(µ1, . . . , µ1︸︷︷︸

r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

), where µi’s are non-trivial qua-

dratic grossencharacters such that µiv = 1. (See Remark 3.3.) Mœglin’sargument by induction shows that R(wp, λp, 1)I(λp) is still semi-simple inthe general case and we denote the set of direct summands still by Unip(p).However, Moeglin’s argument does not work in this case, since the normal-ized local intertwining operators could vanish. For this, we use the globalmethod. By considering the iterated residue of the pseudo-Eisenstein se-ries as in Mœglin [M1], we can show that Unip(p) is contained in Unip(O),where O is the unipotent orbit obtained by ignoring the ordering in p. Re-call that Unip(O) is the union of Unip(p) as p runs through P (O) and thisshows that by considering arbitrary chains, we do not get a new component.

Fourth, we need to show that Hecke algebra isomorphisms of [R] andthe graded Hecke algebra isomorphisms of [B-Mo2] commute with the in-tertwining operators. That reduces us to the case of the trivial character.More precisely, let p = (a1, b1, . . . , as, bs, as+1) and χ = χ(µ1, . . . , µ1︸︷︷︸

r1

, . . . ,

µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

), r0 + · · · + rk = n, r1 ≥ · · · ≥ rk, µi’s are distinct qua-

dratic characters. Here k ≤ 3. (Recall that we are dealing with a p-adic field


with odd residual characteristic and hence there are only three non-trivialdistinct quadratic characters.) Set µ0 = 1. Let G′ = G′1 × · · · × G′k × G′0,where, for i = 1, . . . , k,

G′0 = G(r0), G′i =

O(2ri), if G = Sp(2n), O(2n)SO(2ri + 1), if G = SO(2n+ 1).

Here we note that G′ is an endoscopic group of G. Then by the Hecke algebraisomorphisms of [R] and the graded algebra isomorphisms of [B-Mo2], weget an equivalence of categories

R(G, τ(λp, χ)) ' R(G′, τ ′(λp, 1)),

where τ(λp, χ) is the infinitesimal character associated to subquotients ofI(λp, χ) and R(G, τ(λp, χ)) is the category of smooth finite-length represen-tations of G having infinitesimal character τ(λp, χ). We note that λp maybe viewed as a character of the maximal split torus of G′ in the obviousway: If we write λp = λp1 × · · · ×λpk ×λp0 (corresponding to the decompo-sition r1 + · · ·+ rk + r0 = n above), then λpi may be viewed as a characterof the maximal split torus of G′i. We show in Section 3.2 that under thecategory equivalence, the image R(wp, λp, χ)I(λp, χ) corresponds to the im-age R(wp1 , λp1 , 1)I(λp1)⊗ · · · ⊗R(wpk , λpk , 1)I(λpk)⊗R(wp0 , λp0 , 1)I(λp0).This reduces us to the case when χ = 1. Thus, by matching of the imagesof intertwining operators, we see that R(wp, λp, χ)I(λp, χ) is semi-simple.Let Unip(p, χ) be the set of the direct summands. Then under the categoryequivalence, Unip(p, χ) is contained in the set Unip(O1)× · · · ×Unip(Ok)×Unip(O0), where Oi are certain unipotent orbits obtained by ignoring theordering in p. In particular, the generalized Iwahori-Matsumoto involutionof the elements in Unip(p, χ) is tempered.

We let χ = 1 and parametrize Unip(p). To each (ai, bi), we can attacha Weyl group element σ(ai,bi). Then σ(a1,b1) defines a normalized opera-tor R(σ(a1,b1)). It defines a homomorphism from the group id, σ(a1,b1)into the group of the intertwining operators of R(wp, λp)I(λp). This meansthe following: For X ∈ Unip(p), let R(σ(ai,bi))X = ηp

X(σ(ai,bi))X. ThenηpX defines a character of A(O) such that ηp

X(σ(ai)) = ηpX(σ(bi)). Since

Unip(p) ⊂ Unip(O), ηpX ∈ Springer (O). Therefore we have:

Theorem 0.1 (Theorem 3.4.2). Unip(p) is parametrized by

C(p) = η ∈ Springer (O) : η(σ(ai)) = η(σ(bi)),

i = 1, . . . , s, η(σ(as+1)) = 1.

Note that if p satisfies the condition (3.1) (that is, in Mœglin’s situation),then C(p) = A(p) ([M1, Proposition 1.3.3]). In order to apply this theo-rem to the residual spectrum calculation, let O1, O2 be two distinguishedunipotent orbits in G∗1, G

∗2, resp. (If G = Sp(2n), then G∗1 = O(2r1,C)


and G∗2 = SO(2r0 + 1,C). If G = SO(2n + 1), then G∗1 = Sp(2r1,C) andG∗2 = Sp(2r0,C). If G = O(2n), then G∗1 = O(2r1,C) and G∗2 = O(2r0,C).)Then we get a unipotent orbit O in G∗ by combining O1 and O2. Furtherwe have canonical embedding A(Oi) ⊂ A(O). For pi ∈ P (Oi), i = 1, 2, weget a chain p1 × p2 by shuffling the segments in p1 and p2 so that it satis-fies (3.1), and thus we get Unip(p1 × p2). Let Unip(O1, O2) be the unionof Unip(p1 × p2) as pi runs through P (Oi) for i = 1, 2. It is a subset ofUnip(O). Then we have:

Theorem 0.2 (Theorem 3.4.3). Unip(O1, O2) is parametrized by

C(O1, O2) = η ∈ Springer (O) : η|A(O1) ∈ Springer (O1),

η|A(O2) ∈ Springer (O2).

This can be easily generalized to an arbitrary character. Let us statethe result on the local components of the residual spectrum attached to anarbitrary character in order to apply it to [Ki3]. Let χ = χ(µ1, . . . , µ1︸︷︷︸

r1

, . . . ,

µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

), r0 + · · · + rk = n, r1 ≥ · · · ≥ rk ≥ 2, µi’s are distinct

non-trivial quadratic grossencharacters. Let Oi be a distinguished unipotentorbit in G∗i for i = 0, 1, . . . , k, where, for i = 1, . . . , k,

G∗i =

O(2ri,C), if G = Sp(2n), O(2n)Sp(2ri,C), if G = SO(2n+ 1),

G∗0 =

SO(2r0 + 1,C), if G = Sp(2n)Sp(2r0,C), if G = SO(2n+ 1)O(2r0,C), if G = O(2n).

Let pi ∈ P (Oi) for i = 0, . . . , k and p = p1×· · ·×pk×p0. Then we can shufflethe segments in p so that it satisfies the condition (3.1). We still call it p.For a non-archimedean place v, let Unip(O1, . . . , Ok, O0, χv) be the set ofunion of Unip(p1, . . . , pk, p0, χv) as pi runs through P (Oi) for i = 0, . . . , k.

Theorem 0.3 (Theorem 3.4.10). Πresv = Unip(O1, . . . , Ok, O0, χv) isparametrized by

C(O1, . . . , Ok, O0, χv) = [Springer (O1)×· · ·×Springer (Ok)×Springer (O0)],

where [ ] is defined as follows: If µ1v = µ2v 6= µiv for i = 0, 3, . . . , k, thenwe replace Springer (O1)× Springer (O2) by

C(O1, O2, µ1v) = η ∈ Springer (O) : η|A(Oi) ∈ Springer (Oi), for i = 1, 2,


where O is the unipotent orbit of G∗12 obtained by combining O1, O2, where

G∗12 =

O(2(r1 + r2),C), if G = Sp(2n), O(2n)Sp(2(r1 + r2),C), if G = SO(2n+ 1).

Acknowledgments. We would like to thank Prof. Shahidi for his guidanceand encouragement throughout this paper. The second author would liketo thank Prof. Mœglin for patiently answering many of his questions [M6]and for many correspondences. We would also like to thank Alan Rochefor many useful conversations on Hecke algebra isomorphisms. Finally, wewish to thank Professors A. Moy, M. Reeder, A. Silberger, M. Tadic, andD. Vogan for many helpful correspondences.

1. Hecke algebra isomorphisms and the generalizedIwahori-Matsumoto involution.

In order to reduce from the case of ramified characters to the case of unram-ified characters, we use the Hecke algebra isomorphisms of Roche [R]. Wealso use these isomorphisms to construct a generalized Iwahori-Matsumotoinvolution. (The duality results of Aubert and Schneider-Stuhler are in theGrothendieck group setting; we need to deal with the composition serieshere.) We begin this section by reviewing the Iwahori-Matsumoto involu-tion and the Hecke algebra isomorphisms of Roche. We then show that theseisomorphisms behave well with respect to induction in stages. This will alsoallow us to verify certain properties of the generalized Iwahori-Matsumotoinvolution.

For this section, we work in a more general setting. Let G denote theF -rational points of a split connected reductive group defined over F . Inorder to apply Roche’s results, we also assume the residue characteristic ofF satisfies the conditions in [R]. For our applications to Sp(2n), SO(2n +1), this requires odd residue characteristic. (We note that [Go], or moregenerally [Mr], gives similar Hecke algebra isomorphisms which could beused to extend the results of this section to cover characters of level 0 withoutthe constraints on the residue characteristic.) Fix a set of positive roots Φ+

and a subset of simple roots Π. Let

B = Borel subgroup of GI = Iwahori subgroup of G

K = O − points in G (a maximal compact subgroup)

W = Weyl group of GW = affine Weyl group of G

`(·) = length function on WT = maximal split torus in B


δ(·) = modular function on B.

We may view W ⊂ W by identifying W with N(O)/A(O), where N =NormG(T ). We use BG, IG, etc., if there is more than one group around andconfusion is possible. Set

H(G) = C∞c (G).

It is an algebra under convolution. If (π,G, V ) is a smooth representation,we define π on H(G) by

π(h)v =∫Gh(g)π(g)vdg

for all h ∈ H(G), v ∈ V .We begin by reviewing the Iwahori-Matsumoto involution. First, let us

normalize Haar measure so that |I| = 1. Set 1I = charI . Let

H(G, 1I) = 1I · H(G) · 1I = f ∈ H(G)|f(i1gi2) = f(g)

for all i1, i2 ∈ I, g ∈ G.

The Iwahori-Matsumoto involution is an involution of H(G, 1I). In order todescribe the Iwahori-Matsumoto involution, we must first discuss the struc-ture of H(G, 1I). The following description is due to Bernstein-Zelevinsky(cf. [Lu2]); the classical description is due to Iwahori-Matsumoto [I-M]. Asa vector space, H(G, 1I) = H(K, 1I) ⊗ Θ. Now, H(K, 1I) has Tww∈W asa basis, where Tw denotes the characteristic function of IwI. Further, themultiplication is governed by

T 2s = (q − 1)Ts + q for s simple

Tw1Tw2 = Tw1w2 if `(w1) + `(w2) = `(w1w2).

Θ is an abelian subalgebra of H(G, 1I) with basis θt|t ∈ T/T ∩K. For t ∈T , choose t1, t2 ∈ T− = t ∈ T ||α(t)| ≤ 1 for all simple roots α such thatt = t1t

−12 . Then, θt = δ

12 (t)Tt1T

−1t2

. The multiplication between H(K, 1I)and Θ is governed by the following: If s = sα, α a simple root,

θtTs = Tsθsts + (q − 1)θt − θsts

1− θα($−1),

where α denotes the coroot associated to α. The Iwahori-Matsumoto invo-lution j : H(G, 1I) −→ H(G, 1I) is defined by

j :Tw 7−→ (−q)`(w)(Tw−1)−1

j :θt 7−→ θ−1t

for Tw ∈ H(K, 1I), θt ∈ Θ.


We now discuss Roche’s results [R]. The applications to the representa-tion theory of G will be discussed later; for now we focus on the results deal-ing with the structure of Hecke algebras. Fix a character χ : T ∩K −→ C×.To the character χ, he associates an open compact subgroup J and a char-acter ρ : J −→ C× with ρ|T∩K = χ. The pair (J, ρ) is a type in the sense of[Bu-K]. Let eρ be defined by

eρ(g) =

1|J |ρ−1(g), if g ∈ J

0, if not.

Then, e2ρ = eρ. Let

H(G, ρ) = eρ · H(G) · eρ= f ∈ H(G)|f(j1gj2) = ρ−1(j1j2)f(g) for all j1, j2 ∈ J, g ∈ G.

Roche constructs a split connected reductive group H and a finite abeliangroup Cχ, which acts on H, such that

H(G, ρ) ∼= H(H, 1I)⊗C[Cχ].

The notation ⊗ is used to indicate that multiplication is governed by

(Tw1 ⊗ c1) · (Tw2 ⊗ c2) = Tw1Tc1(w2) ⊗ c1c2for w1, w2 ∈ H, c1, c2 ∈ Cχ (cf. Section 8 [R]). He also constructs a dis-connected group H, with H the connected component of the identity in H,which has H(H, 1I)⊗C[Cχ] ∼= H(H, 1IH ). The following example will be ofinterest in §3.2.

Example 1.1. Suppose G = SO(2(r0 + r1) + 1, F ) or Sp(2(r0 + r1), F ), Fof odd residual characteristic. Let µ denote a ramified quadratic characterof F× and set χ = χ(µ, . . . , µ︸︷︷︸

r1

, 1, . . . , 1︸︷︷︸r0

). (More precisely, it is χ|T (O) that is

needed in Roche’s construction.) Then Jχ = I, and we have the following:

(1) If G = SO(2(r0 + r1) + 1), then H(G, ρχ) ∼= H(H, 1), with H =SO(2r1 + 1)× SO(2r0 + 1) = H ′1 ×H ′0.

(2) If G = Sp(2(r0 + r1)), then H(G, ρχ) ∼= H(H, 1), with H = O(2r1) ×Sp(2r0) = H ′1 ×H ′0.

Further, if λ is an unramified character of T , write λ = λ1×λ0 (with λ1 thecharacter of (F×)r1 consisting of the first r1 terms of λ and λ0 the characterof (F×)r0 consisting of the last r0 terms). Then, under the above Hecke alge-bra isomorphisms, IndGB(λχ) is identified with

(IndH

′1

B1(λ1)

)⊗(IndH

′0

B0(λ0)

).

Remark 1.1. Because of the technical difficulties involved in dealing withdisconnected groups, we will not pursue O(2n) in detail in this paper. How-ever, in order that we may at least indicate what is expected in that case,


we briefly discuss how to extend Roche’s isomorphisms to get the analoguefor O(2n) of the example above.

Let us continue to assume F of odd residual characteristic. Let G =O(2n), G = SO(2n), and µ× · · · × µ︸︷︷︸

r1

× 1× · · · × 1︸︷︷︸r0

, where r0 + r1 = n. Let

H = O(2r1) × O(2r0) and H = S(O(2r1) × O(2r0)). By Roche’s results,there is a support-preserving isomorphism of Hecke algebras

Ψ : H(G, ρ) −→ H(H, 1).

We will use Ψ to construct a support-preserving isomorphism of Hecke al-gebras

Ψ : H(G, ρ) −→ H(H, 1).

To this end, let iG : H(G, ρ) −→ H(G, ρ) and iH : H(H) −→ H(H, 1) denotethe obvious embeddings. If we let cn denote the nth sign change (an elementof O(2n)\SO(2n)) and C = 〈1, cn〉, then we have H(G, ρ) ∼= H(G, ρ)⊗C[C].Similarly, if we let H = SO(2r1) × SO(2r0) and C ′ = 〈1, cr1 , cn, cr1cn〉,C ′′ = 〈1, cr1cn〉, thenH(H, 1) ∼= H(H, 1)⊗C[C ′],H(H, 1) ∼= H(H, 1)⊗C[C ′′].We may then define Ψ by

Ψ :h⊗ 1 7−→ iH Ψ i−1G (h)⊗ 1

h⊗ ecn 7−→ iH Ψ i−1G (h)⊗ e′cn .

The definition of Ψ ensures that it is a support-preserving linear isomor-phism. We need to check that it also respects multiplication. For this, it isenough to show that cn(iH Ψ i−1

G (Tw)) = iH Ψ i−1G (Tcn(w)). Since Ψ is

a support-preserving isomorphism, we may write Ψ(Tw) = awT′w. Then,

cn(iH Ψ i−1G (Tw)) = awT

′cn(w)

andiH Ψ i−1

G (Tcn(w)) = acn(w)T′cn(w).

Therefore, it is enough to show that aw = acn(w). For this, it is enough toshow that aw = acn(w) for w ∈ Wχ = w ∈ W G|w · χ = χ and w = y ∈Y = Hom(Gm, T ). Now, since cn · Π = Π and cn · Πχ = Πχ (cf. [R, p.393]), we have `(cn(w)) = `(w) and `χ(cn(w)) = `χ(w) for w ∈ Wχ, where`χ = `H denotes length taken with respect to Πχ. Since Ψ : q−

12`(w)Tw 7−→

q−12`χ(w)T ′w, this tells us that aw = acn(w) for w ∈ Wχ. This also tells us

that δ cn = δ and δH cn = δH . Therefore, q`(y) = q`(cn(y)) for y ∈ Y + =y ∈ Y |〈y, α〉 ≥ 0 for all α ∈ Φ+ (cf. [Ca, Lemma 1.5.1]), and similarly

for `χ = `H . Since Ψ : δ12 (y)Ty 7−→ δ

12H(y)T ′y (cf. proof of Lemma 9.3 [R]),

it follows that ay = acn(y) for y ∈ Y +, as needed.


The following results describe what happens when χ is replaced by χ−1.They will be needed below.

Lemma 1.1. In Roche’s construction, if (J, ρ) is the type associated to χ,then the type associated to χ−1 is (J, ρ−1).

Proof. Write (Jχ, ρχ), (Jχ−1 , ρχ−1) for the types associated to χ, χ−1, re-spectively. The functions fχ and fχ−1 from Definition 3.3 [R] must be thesame. Therefore, Jχ = Jfχ and Jχ−1 = Jfχ−1 must be the same. It is thenimmediate from the definitions that ρχ−1 = (ρχ)−1 (cf. [R, Proposition3.6]).

Corollary 1.2. H(G, ρ) ∼= H(G, ρ−1) via a support-preserving isomorphism.(The isomorphism may also be taken to be ∗-preserving; cf. Section 6 [R].)

Proof. Observe that in the notation of [R], we have Φχ = Φχ−1 , Φχ,af =Φχ−1,af (cf. [R, Definition 6.1]) and Wχ = Wχ−1 (cf. [R, Theorem 4.14]).Thus, Cχ = Cχ−1 , and therefore Πχ,af = Πχ−1,af (cf. [R, Section 6]). It thenfollows that S0

χ = S0χ−1 and Ωχ = Ωχ−1 (also [R, Section 6]). The corollary

is now an immediate consequence of [R, Theorem 6.3].

We now define an involution j on H(H, 1I)⊗C[Cχ]. For h ∈ H(H, 1I),j(h) is defined as above. For C[Cχ], fix a character σ : Cχ −→ C× withσ2 = 1. Then, for c ∈ Cχ, set j(ec) = σ(c)ec. This extends to an involutionof C[Cχ] (trivial if σ = 1). We extend j to H(H, 1I)⊗C[Cχ] bilinearly.

Proposition 1.3. j is an involution of H(H, 1I)⊗C[Cχ].

Proof. We need to check that j respects multiplication. For h1, h2 ∈ H(H, 1I)and c1, c2 ∈ Cχ, we have

j(h1 ⊗ ec1) · j(h2 ⊗ ec2) = σ(c1c2)[j(h1)c1(j(h2))⊗ ec1c2 ]

andj((h1 ⊗ ec1) · (h2 ⊗ ec2)) = σ(c1c2)[j(h1)j(c1(h2))⊗ ec1c2 ].

Thus, it suffices to show j(c(h)) = c(j(h)) for h ∈ H(H, 1I), c ∈ Cχ. Sincethe action of Cχ on H preserves the set of simple roots of H, hence δH and`H , we see that

j(c(Tw)) = (−q)`H(w)(Tc(w−1))−1 = c(j(Tw))

for all w ∈WH . Also, since c(θt) = θc(t), we get

j(c(θt)) = θ−1c(t) = c(j(θt)),

as needed.


We now consider how these restrict to Levi subgroups of standard parabol-ics. First, if M is a standard Levi of G, we have support-preserving isomor-phisms

ΨG : H(G, ρ) −→ H(H, 1I)⊗C[Cχ] = H(H, 1IH )

ΨM : H(M,ρM ) −→ H(L, 1IL)⊗C[Dχ] = H(L, 1IL)

constructed by Roche. In general, these are not unique. Similarly, let jH ,resp. jL, denote the Iwahori-Matsumoto involutions for H(H, 1I)⊗C[Cχ],resp. H(L, 1I)⊗C[Dχ] constructed above. We note the following:

Lemma 1.4. We have Dχ ⊂ Cχ and may take L to be a standard Levisubgroup of H.

Proof. First, we check that Dχ ⊂ Cχ. In fact, we claim that Dχ = Cχ∩WM .Recall that

Dχ = w ∈WM,χ | w · Φ+M,χ = Φ+

M,χ.

Now, it is immediate from the definitions that WM,χ = WM ∩ Wχ andΦ+M,χ = ΦM ∩ Φ+

χ . Therefore, if w ∈ Cχ ∩ WM , we have w ∈ WM,χ.Further, w · Φ+

χ = Φ+χ (since w ∈ Cχ) and w · ΦM = ΦM (since w ∈ WM ),

implying w · Φ+M,χ = Φ+

M,χ. Thus, Cχ ∩WM ⊂ Dχ. On the other hand,suppose d ∈ Dχ. It is automatic that d ∈ WM . To show d ∈ Cχ ∩WM ,it remains to check that d · Φ+

χ = Φ+χ . From the definition of Dχ, we have

d · Φ+M,χ = Φ+

M,χ. Suppose α ∈ Φ+χ \ Φ+

M,χ. Since Φ+M,χ = ΦM ∩ Φ+

χ , thisforces α ∈ Φ+\Φ+

M . As WM ·(Φ+\Φ+M ) = Φ+\Φ+

M , we have d·α ∈ Φ+\Φ+M .

Thus, d ·α ∈ (Φ+\Φ+M )∩Φχ ⊂ Φ+

χ , as needed. This tells us Dχ ⊂ Cχ∩WM .Thus, Dχ = Cχ ∩WM , as claimed.

Now, we identify L as a standard Levi subgroup ofH. First, we claim thatif ΠM,χ denotes the simple roots in ΦM,χ, then ΠM,χ = Πχ∩ΦM,χ. Certainly,if α ∈ Πχ ∩ ΦM,χ, then α ∈ ΠM,χ (minimal with respect to Φ+

χ impliesminimal with respect to Φ+

M,χ = ΦM ∩Φ+χ ). In the other direction, suppose

α ∈ ΠM,χ but α 6∈ Πχ ∩ ΦM,χ. Since ΠM,χ ⊂ ΦM,χ, this requires α 6∈ Πχ.Therefore, we can write α = β + γ with β, γ ∈ Φ+

χ . Write α =∑aiαi,

β =∑biαi, γ =

∑ciαi, where αi runs over the simple roots Π of G. Then,

ai = bi + ci for all i. Further, ai, bi, ci ≥ 0 (since α, β, γ ∈ Φ+χ ⊂ Φ+). Now,

α ∈ ΦM implies ai = 0 for αi 6∈ ΠM . Therefore, bi = ci = 0 for αi 6∈ ΠM .Thus, β, γ ∈ ΦM . In particular, β, γ ∈ Φ+

χ ∩ ΦM = Φ+M,χ. However, the

decomposition α = β + γ with β, γ ∈ Φ+M,χ contradicts the simplicity of α

in Φ+M,χ. Thus, ΠM,χ = Πχ ∩ ΦM,χ, as claimed. We now let L be the Levi

factor of the standard parabolic subgroup of H associated to the subset ofsimple roots ΠM,χ ⊂ Πχ = ΠH . Then, L has the right root data to appearin the isomorphism H(M,ρM ) ∼= H(L, 1IL)⊗C[Dχ].


The next step is to identify H(M,ρM ) as a subalgebra of H(G, ρ). LetP = MU be the standard parabolic subgroup with Levi factor M ; U theunipotent radical opposite U . Set

J` = J ∩ U, JM = J ∩M, Ju = J ∩ U,

which gives J = J`JMJu. Let

I+M = w ∈WM | wJuw−1 ⊂ Ju, w−1J`w ⊂ J`.

We setI+M,χ = I+

M ∩WM,χ.

If we letH+(M,ρM ) denote the space of functions inH(M,ρM ) with supportcontained in JMI+

M,χJM , then there is a support-preserving embedding (ofC-algebras with 1)

H+(M,ρM ) → H(G, ρ)

([Bu-K, Corollary 6.12]). This may be extended uniquely to an embedding

H(M,ρM ) → H(G, ρ)

([Bu-K, Theorem 7.2], noting that the existence of ζ as in the hypothesesof Theorem 7.2 [Bu-K] follows immediately from the support-preservingisomorphism of Roche and the fact that elements of H(H, 1IH ) supported ona single double-coset are invertible). We fix such embeddings H(M,ρM ) →H(G, ρ) and H(L, 1IL) → H(H, 1IH ).

Now, we note that WM,χ = WM,χ · Y , where Y = Hom(Gm, T ) (whichmay be viewed as a set of representatives for T/T (O)). If we let

Y +M,χ = I+

M,χ ∩ Y,

we have the following:

Lemma 1.5.I+M,χ = WM,χ · Y +

M,χ.

Proof. First, we check thatWM,χ ·Y +M,χ ⊂ I

+M,χ. By definition, Y +

M,χ ⊂ I+M,χ.

Thus, we need only check that WM,χ ⊂ I+M,χ. Recall that Roche defines

fχ(α) =

[cα/2], for α ∈ Φ+,

[(cα + 1)/2], for α ∈ Φ−,,

where cα = cond(χ α) (where cond(λ) is defined to be the lowest positiveinteger such that 1 + pn ⊂ ker(λ)). Then,

J` =∏

α∈Φ−\Φ−M

Uα,fχ(α), Ju =∏

α∈Φ+\Φ+M

Uα,fχ(α),


JM =

∏α∈Φ−M

Uα,fχ(α)

·A(O) ·

∏α∈Φ+

M

Uα,fχ(α)

.

Here, if α ∈ Φ+,

Uα,k = φα

(1 pk

0 1

), U−α,k = φα

(1 0pk 1

),

with φα : SL2 −→ G as usual. Now, suppose w ∈WM and α ∈ Φ+. Then,

w · Uα,fχ(α) = φw·α

(1 pfχ(α)

0 1

),

suitably interpreted if w ·α < 0. To show that wJuw−1 ⊂ Ju for w ∈WM,χ,it is (more than) enough to show that if α ∈ Φ+ \ Φ+

M , then (1) w · α ∈Φ+ and (2) fχ(w · α) = fχ(α). Of course, (1) follows immediately fromw · (Φ+ \Φ+

M ) = Φ+ \Φ+M (which holds for any w ∈WM ). (2) follows from

w ·χ = χ: (w−1 ·χ) α = χ α implies cw·α = cα, so that fχ(w ·α) = fχ(α),as needed. The same argument may be used to show w−1J`w ⊂ J`. Thus,we get WM,χ ⊂ I+

M,χ, as needed. In fact, this shows more: for w ∈ WM,χ,we have wJuw−1 = Ju and w−1J`w = J`.

The containment I+M,χ ⊂WM,χ · Y +

M,χ is now easy. Take

x ∈ I+M,χ ⊂WM,χ = WM,χ · Y

and write x = wy, w ∈ WM,χ, y ∈ Y . Then, it suffices to show thaty ∈ Y +

M,χ. Calculate:

(wy)Ju(wy)−1 ⊂ Ju =⇒ yJuy−1 ⊂ w−1Juw = Ju

since w ∈ WM,χ ⊂ I+M,χ. Similarly, y−1J`y ⊂ J`. Thus, y ∈ Y +

M,χ, asneeded.

Now, we can prove the following:

Proposition 1.6. (1) We may take ΨM = ΨG|H(M,ρM ) to get a support-preserving isomorphism

ΨM : H(M,ρM ) −→ H(L, 1IL)⊗C[Dχ].

(2) If we let jH , resp. jL, denote the Iwahori-Matsumoto involutions forH(H, 1I)⊗C[Cχ], resp. H(L, 1IL)⊗C[Dχ], then

jL = jH |H(L,1IL )⊗C[Dχ].

Proof. We first say a few words about what it means for ΨG to be support-preserving. By construction, Y is associated to both G and H; we useYG, YH when we wish to distinguish the context. Note that Wχ = Y oWχ.Also, since ΦH = Φχ, we can identify W

0H and W

0χ. This gives rise to


an identification of Wχ = W0χ o Cχ with W

0H o Cχ. Thus, we have an

identification

ψG : Wχ = Y oWχ ←→ Y o (W 0H o Cχ) = WH o Cχ.

Then, ΨG : H(G, ρ) −→ H(H, 1I)⊗C[Cχ] is support-preserving means thatif T ∈ H(G, ρ) is supported on JwJ , then ΨG(T ) is supported on IHψG(w)IH .

Now, to make matters precise, let tM : H(M,ρM ) → H(G, ρ) and tL :H(L, 1IL) → H(H, 1IH ) denote the fixed embeddings. To verify (1), we needto show that

ΨG(tM (H(M,ρM ))) = tL(H(L, 1IL)).To this end, we first show that

ΨG(tM (H+(M,ρM ))) ⊂ tL(H+(L, 1IL)).

Since the restriction of tM to H+(M,ρM ) is support-preserving, and simi-larly for tL, it is enough to show that

ψG(I+M,χ) ⊂ I

+L

= WL ·Dχ · Y +L .

By Lemma 1.5 and the fact that ψG(WM,χ) = WL ·Dχ, we are reduced tochecking that ψG(Y +

M,χ) ⊂ Y +L . Now, if y ∈ Y and α ∈ Φ, then yUα,ky

−1 =Uα,k+α(y). Therefore, y ∈ Y +

M,χ if and only if α(y) ≥ 0 for all α ∈ Φ+ \ Φ+M

(noting that the corresponding condition for negative roots also reduces tothis). On the other hand, y ∈ Y +

L if and only if α(y) ≥ 0 for all α ∈Φ+H \ Φ+

L = Φ+χ \ Φ+

M,χ ⊂ Φ+ \ Φ+M , so we see that ψG(Y +

M,χ) ⊂ Y +L , as

needed.We now proceed to show that

ΨG(tM (H(M,ρM ))) = tL(H(L, 1IL)).

First, choose ζ and φm as in Proposition 7.1 [Bu-K]. Let y0 ∈ Y +M,χ

be a representative for ζ. Then, (φm)−1 = φ−m ∈ H(M,ρM ). We notethat H+(M,ρM ) and φ−1 are sufficient to generate H(M,ρM ). We alreadyhave ΨG(tM (H+(M,ρM ))) ⊂ tL(H+(L, 1IL)). Since y0 ∈ I+

M,χ, we haveΨG(tM (φ1)) ∈ tL(H+(L, 1IL)). Further, since tM |H+(M,ρM ) and ΨG aresupport-preserving, ΨG(tM (φ1)) is supported on the double-coset IHy0IH .Therefore, ΨG(tM (φ1)) = tL(φ′1), where φ′1 ∈ H(L, 1IL) is supported on thedouble-coset ILy0IL. As a consequence, φ′1 is invertible and ΨG(tM (φ−1

1 )) =tL(φ′1

−1). Since H+(M,ρM ) and φ−11 are enough to generate H(M,ρM ), we

have ΨG(tM (H(M,ρM ))) ⊂ tL(H(L, 1IL)). On the other hand, we knowthat ΨG(tM (H(M,ρM ))) contains the functions tL(Tw)| w ∈ ψG(WM,χ) =WLDχ, tL(Ty)| y ∈ Y +

M,χ, and tL(T−1y0 ). As these are enough to generate

tL(H(L, 1IL)), we see that

ΨG(tM (H(M,ρM ))) = tL(H(L, 1IL)),


as needed.We now know that t−1

L ΨG tM is an isomorphism from H(M,ρM )

to H(L, 1IL). We must show that it is support-preserving. To this end,observe that since ζ ∈ Z(M), we have α(y0) = 0 for all α ∈ ΦM . Therefore,viewing y0 as an element of YH , we have α(y0) = 0 for all α ∈ ΦL = ΦM,χ.In particular, y0Uα,ky

−10 = Uα,k+α(y0) = Uα,k for all α ∈ ΦL allows us to

conclude that ILy0 = y0IL. Therefore, in H(L, 1IL), we have that Ty0Tw issupported on ILy0wIL for any w ∈ WLDχ. In particular, T−1

y0 is supportedon ILy

−10 IL. Thus, we have that t−1

L ΨG tM is support-preserving on

H+(M,ρM ) and φ−11 = φ−1. By writing x ∈ WM,χ as y−m0 w, with w ∈

W+M,χ ([Bu-K, Lemma 6.14]), one can then check that t−1

LΨG tM (Tx) is

supported on ILy−m0 ψG(w)IL, as needed.It is now straightforward to check the second claim. This follows imme-

diately from the following observations:

(1) Since L is a standard Levi of H, the Iwahori-Matsumoto involutionon H(H, 1I) restricts to give the Iwahori-Matsumoto involution onH(L, 1IL).

(2) Since Dχ ⊂ Cχ, we may choose σ|Dχ for the character of order ≤ 2 ofDχ.

This finishes the proof of the proposition.

Suppose H1,H2 are Hecke algebras and φ : H1 −→ H2 is an isomorphism.Suppose π is a representation of H1 with space V . Let φ(π) denote thecorresponding representation of H2 (i.e., H2 acting on V by π φ−1). Wedefine induction as in [Bu-K]: If L1 ⊂ H1 is a subalgebra and τ is arepresentation of L1, then IndH1

L1τ has space HomL1(H1, Vτ ) (where H1 is

an L1-module by left multiplication). The action is right translation. Weclaim that φ respects induction. In particular, if L2 = φ(L1), we have thefollowing:

Lemma 1.7.φ(IndH1

L1τ) ∼= IndH2

L2φ(τ).

Proof. Let X ∈ HomL1(H1, Vτ ). Then, it is straightforward to check thatX φ−1 ∈ HomL2(H2, Vφ(τ)) (noting that as a vector space, Vφ(τ) = Vτ ; itis written as Vφ(τ) to indicate the representation). In particular, E : X −→X φ−1 is a bijective linear map from HomL1(H1, Vτ ) to HomL2(H2, Vφ(τ)).

We claim that E : HomL1(H1, Vτ ) −→ HomL2(H2, Vφ(τ)) gives the desiredequivalence. In particular, if R1 (resp. R2) denotes right translation onHomL1(H1, Vτ ) (resp. HomL2(H2, Vφ(τ))), this requires

ER1(φ−1(h2))X = R2(h2)EX


for all X ∈ HomL1(H1, Vτ ), h2 ∈ H2. Now, it is easy to check that

[ER1(φ−1(h2))X](h′2) = X(φ−1(h′2h2)) = [R2(h2)EX](h′2).

The desired equivalence follows.

We now discuss the implications for the representation theory of G. First,if (π,G, V ) is a representation of G, let (π,H(G, ρ), V ρ) denote the corre-sponding Hecke algebra representation, where V ρ = π(eρ)V (the action isinherited from H(G) acting on V ). Now, fix a character χ extending χ toT . Let Rχ(G) denote the category of smooth representations having theproperty that every irreducible subquotient of π has supercuspidal supportcontained in λ(w · χ)|w ∈ W,λ an unramified character of T. As men-tioned earlier, (J, χ) is a type. This means, among other things, that themap (π,G, V ) 7−→ (π,H(G, ρ), V ρ) gives an equivalence of categories be-tween Rχ(G) and the category of H(G, ρ) modules. This extends the resultsof Borel [Bo] and Casselman [Ca] on unramified principal series.

We now discuss the Iwahori-Matsumoto involution on representations.Let

ΨG,χ : H(G, ρ) −→ H(H, 1I)⊗C[Cχ]

ΨG,χ−1 : H(G, ρ−1) −→ H(H, 1I)⊗C[Cχ−1 ]

denote the isomorphisms above. Since Cχ = Cχ−1 (which follows easily fromthe definition; cf. [R, Section 8]), the Hecke algebras on the right-hand sideare identical. Therefore, we have an isomorphism

φj : Ψ−1G,χ−1 j ΨG,χ : H(G, ρ) −→ H(G, ρ−1).

If (γ, Vγ) is a representation of H(G, ρ), we let φj(γ) denote the representa-tion of H(G, ρ−1) associated to γ by the Hecke algebra isomorphism above.Now, if M is a standard Levi of G, let

EG,χ : Rχ(G) −→ H(G, ρ)−mod

EM,χ : Rχ(M) −→ H(M,ρM )−mod

denote the functors giving the equivalence of categories. If π ∈ Rχ(G), wedefine the Iwahori-Matsumoto involution of π by

j(π) = E−1G,χ−1 φj EG,χ(π).

Similarly, if τ ∈ Rχ(M), then j(τ) = E−1M,χ−1 φj EM,χ(τ) (noting that

φjM = φj |H(L,1L)⊗C[Dχ] from above).

Theorem 1.8.j(IndGP τ) ∼= IndGP (j(τ)).


Proof. First, by Corollary 8.4 of [Bu-K],

EG,χ(IndGP τ) ∼= IndH(G,ρ)H(M,ρM )(EM,χ(τ)).

Therefore,

j(IndGP τ) ∼= E−1G,χ−1 φj EG,χ(IndGP τ)

∼= E−1G,χ−1 φj

(IndH(G,ρ)H(M,ρM )(EM,χ(τ))

)∼= E−1

G,χ−1

(IndH(G,ρ−1)

H(M,ρ−1M )φj(EM,χ(τ))

),

by Lemma 1.7 above. By definition, j(τ) = E−1M,χ−1 φj EM,χ(τ), or

EM,χ−1(j(τ)) = φj(EM,χ(τ)). Thus,

j(IndGP τ) ∼= EG,χ−1

(IndH(G,ρ−1)

H(M,ρ−1M )

EM,χ−1j(τ))

∼= IndGP (j(τ)),

again, by Corollary 8.4 of [Bu-K].

We close by discussing a special case. First, let G = GL(k) and µ aunitary character of F×. Consider the character

τ = | · |α+−k+12 µ⊗ | · |α+−k+1

2+1µ⊗ · · · ⊗ | · |α+ k−1

2 µ

of T . By definition,

j(τ) = | · |−α+ k−12 µ−1 ⊗ | · |−α+ k−1

2−1µ−1 ⊗ · · · ⊗ | · |−α+−k+1

2 µ−1.

Since (µdetk)|detk |α is the unique irreducible subrepresentation of IndGBτ ,we see that j((µ detk)|detk |α) must be the unique irreducible subrepre-sentation of IndGB(j(τ)), i.e.,

j((µ detk)|detk|α) = (µ−1 detk)|detk|−α ⊗ Stk.

With this in hand, we may easily obtain the following:

Corollary 1.9. Let G = SO(2n + 1) (resp., G = Sp(2n)), P = MU astandard parabolic subgroup with M ∼= GL(k)×SO(2(n−k)+1) (resp., M ∼=GL(k)×Sp(2(n−k))), µ a unitary character of F×, and π a representationof SO(2(n− k) + 1) (resp., Sp(2(n− k))). Then,

j(IndGP ((µ detk)|detk|α ⊗ π)) ∼= IndGP ((µ−1 detk)|detk|−1Stk ⊗ j(π)).


2. The trivial character case; summary of Mœglin’s result.

We recall Mœglin’s results [M1]. All theorems in this section are due toMœglin. Let G(n) = Sp(2n), SO(2n + 1), O(2n). Then the dual group isG∗(n) = O(2n + 1,C), Sp(2n,C), O(2n,C). Given a unipotent orbit O ∈O(2n+ 1,C), Sp(2n,C), or O(2n,C), Mœglin formed a set P (O) of orderedpartitions as follows:

p = (p1, . . . , pr; q1, . . . , qs) ∈ P (O) if and only if:(1) (p1, p1, . . . , pr, pr, q1, . . . , qs) is O if we ignore the ordering.(2) qi are distinct and odd integers in the case of O2n+1(C) and O2n(C),

even integers in the case of Sp2n(C).(3) For all 1 ≤ j ≤ [ s+1

2 ], q2j−1 > q2j and there does not exist 1 ≤ k ≤[ s+1

2 ] such that q2j−1 > q2k−1 > q2j > q2k.(4) If there exists a 1 ≤ k ≤ s such that q2j−1 > qk > q2j , then k < 2j− 1.

We set qs+1 = 0 if s is odd. We can put an equivalence relation on P (O) asfollows: For p = (p1, . . . , pr; q1, . . . , qs), p′ = (p′1, . . . , p

′r; q′1, . . . , q

′s) ∈ P (O),

p ' p′ if and only if for all 1 ≤ i ≤ [ s+12 ], there exists 1 ≤ j ≤ [ s+1

2 ] suchthat q2i−1 = q′2j−1, q2i = q′2j . We note that p1, . . . , pr = p′1, . . . , p′r assets.

Remark 2.1. For a distinguished unipotent orbit, we have r = 0. In thatcase, we write p = (; q1, . . . , qs).

Example 2.1. For a unipotent orbit of the form (5, 3, 1) in O9(C), thereare two non-equivalent elements in P (O), namely, (;5,3,1) and (;3,1,5).

Remark 2.2. The conditions (3), (4) in P (O) are such that the condition in[M1, Cor 0.10.2] is satisfied, and hence implies non-vanishing of normalizedintertwining operators. We use this in Proposition 2.5. In that case, x =q1−1

2 , q1−32 , . . . , q2+1

2 and X is a sub-module of

IndM 1× | |q3−q4

4 × · · · × | |q2[ s+1

2 ]−1−q

2[ s+12 ]

4 ,

where M = GL(q2)×GL( q3+q42 )× · · · ×GL

(q2[ s+1

2 ]−1+q

2[ s+12 ]

2

).

For p = (p1, . . . , pr; q1, . . . , qs), we set, for 2 ≤ i ≤ r, p′i = p1 + · · ·+ pi−1

and p′1 = 0 and for 1 ≤ i ≤ [ s+12 ],

T di =r∑j=1

pj +∑

1≤l<i

q2l−1 + q2l2

,

T fi =r∑j=1

pj +∑

1≤l≤i

q2l−1 + q2l2

.


We recall the definition of λp and wp: λp = (λp,1, . . . , λp,n), where

λp,p′i+t=pi + 1

2− t, for 1 ≤ i ≤ r and 1 ≤ t ≤ pi,

λp,T dk+t =q2k−1 + 1

2− t, for 1 ≤ k ≤

[s+ 1

2

]and 1 ≤ t ≤ q2k1 + q2k

2.

wp is an element of the Weyl group given by:

wp(p′i + t) = p′i+1 − t+ 1, for 1 ≤ i ≤ r and 1 ≤ t ≤ pi

wp(t) = −t, for 1 ≤ k ≤[s+ 1

2

]and T dk < t ≤ T dk +

q2k1 − q2k2

,

wp

(T dk +

q2k−1 − q2k2

+ t

)= T fk − t+ 1,

for 1 ≤ k ≤[s+ 1

2

]and 1 ≤ t ≤ q2k.

Remark 2.3. All λp are conjugates and w2p = 1. Let λO be the conjugate

of λp which is in the closure of the positive Weyl chamber.

We also define σpi for 1 ≤ i ≤ r and σk for 1 ≤ k ≤ [ s2 ] and let Stab(λp, ↑p) be the subgroup of Stab(λp) generated by these elements:

σpi(j) = j, if j /∈ [p′i + 1, p′i+1],

σpi(p′i + t) = −(p′i+1 − t+ 1), if t ∈ [1, pi],

σk(j) = j, if j /∈[T dk +

q2k−1 − q2k2

+ 1, T fk

],

σk

(T dk +

q2k−1 − q2k2

+ t

)= −(T fk − t+ 1), if t ∈ [1, q2k].

Let A(O) be a finite abelian group generated by the order two elementsσ(p1), . . . , σ(pr), σ(q1), . . . , σ(qs). (We take only the distinct ones.) LetA(p) = A(O)/Kp, where Kp is generated by σ(q2i−1)σ(q2i)−1 for all 1 ≤ i ≤[ s+1

2 ]. We set σ(qs+1) = 1 if s is odd.

Lemma 2.1. (1) |A(p)| = 2[ s2].

(2) A(p) is isomorphic to the quotient of Stab (λp, ↑ p) by the subgroupgenerated by σ(pi)σ(pj)−1 for pi = pj and σ(pi)σ−1

k for pi = q2k−1

or pi = q2k. The homomorphism Stab (λp, ↑ p) 7−→ A(p) is given byσ(pi) 7−→ σ(pi)Kp for i = 1, . . . , r and σk 7−→ σ(q2k−1)Kp.

Let Springer (O) is the set of characters of A(O) which is in the image ofthe Springer correspondence. We recall that the Springer correspondence isa one to one map from the set of characters of W , the Weyl group of G∗

into the set of pairs (O, η), where O is a unipotent orbit in G∗ and η is acharacter of A(O). Then:


Theorem 2.2. Springer (O) ' ∪p∈P (O)A(p), where A(p) is the character

group of A(p).

2.1. Local theory. Let I(λp) = IndGB exp(〈λp,HB()〉) (normalized induc-tion). The normalized intertwining operator R(wp, λ) is not holomorphic atλp in general. Mœglin defined R(wp, λp) as composition of several operators.(See [M1] or Section 3) Then we have:

Theorem 2.3. (1) R(wp, λp)I(λp) is a direct sum of |A(p)| irreduciblerepresentations with multiplicity 1. Let Unip(p) be the set of the ir-reducible direct summands and Unip(O) = ∪p∈P (O)Unip(p). Then theIwahori-Matsumoto involution of elements in Unip(O) is tempered.

(2) Unip(O) is exactly the set of irreducible representations of G(n) whoseinfinitesimal character is λO and whose Iwahori-Matsumoto involutionis tempered.

(3) If r = 0, i.e., O is a distinguished unipotent orbit, then the Iwahori-Matsumoto involution of elements in Unip(O) is square integrable.

Mœglin proves Theorem 2.3 by induction: Let p = (p1, . . . , pr; q1, . . . , qs).Suppose r 6= 0. Set p′ = (p2, . . . , pr; q1, . . . , qs). We have

R(wp, λp)I(λp) = IndGL(p1)×G(n−p1)1×R(wp′ , λp′)I(λp′).

We denote by j the Iwahori-Matsumoto involution. Then

jR(wp, λp)I(λp) = IndGL(p1)×G(n−p1)St(p1)× jR(wp′ , λp′)I(λp′),

where St(p1) is the Steinberg representation of GL(p1). By induction,jR(wp′ , λp′)I(λp′) is a semi-simple tempered representation of length |A(p′)|.Let X ′ ∈ Unip(p′).

Proposition 2.4. The induced representation

IndGL(p1)×G(n−p1)St(p1)× jX ′,

i.e., IndGL(p1)×G(n−p1)1 ×X ′ is irreducible if and only if p1 = pj or qk forsome j = 2, . . . , r or k = 1, . . . , s. If it is reducible, then it is a sum of twoirreducible representations.

For x small, the normalized intertwining operator associated to σp1 is

IndGL(1)×···×GL(1)×G(n−p1)| |p1−1

2+x × · · · × | |−

p1−12

+x ×X ′

−→ IndGL(1)×···×GL(1)×G(n−p1)| |−p1−1

2−x × · · · × | |

p1−12−x ×X ′.

This operator is a product of the operator

IndGL(1)×···×GL(1)×G(n−p1)| |p1−1

2+x × · · · × | |−

p1−12

+x ×X ′

−→ IndGL(p1)×G(n−p1)|det |x ×X ′,


and the operator, RX′(σp1 , x):

IndGL(p1)×G(n−p1)|det |x ×X ′ 7−→ IndGL(p1)×G(n−p1)|det |−x ×X ′.

Proposition 2.5. RX′(σp1 , x) is holomorphic at x = 0. Let RX′(σp1 , 0) =RX′(σp1). Then RX′(σp1)

2 = id and RX′(σp1) is the identity if and only ifIndGL(p1)×G(n−p1)1 × X ′ is irreducible. Let R(σp1) be the sum of RX′(σp1)as X ′ runs through Unip(p′). Then it defines an intertwining operator forR(wp, λp)I(λp).

In a similar way, we can define R(σpi).

Suppose r = 0. From [M1, 676] or Section 3, we know that

R(wp, λp)I(λp) ⊂ IndGL(

q1+q22

)×G(n−T d2 )|det |−

q1−q24 ×R(wp, λp≥3

)I(λp≥3).

We use induction: Let Y ∈ Unip(p≥3) and consider

WY = IndGL(

q1+q22

)×G(n−T d2 )|det |

q1−q24 × Y.

Let W ∗Y = IndGL(

q1+q22

)×G(n−T d2 )|det |−

q1−q24 × Y . Then we have

jW ∗Y = IndGL(

q1+q22

)×G(n−T d2 )St

(q1 + q2

2

)⊗ |det |

q1−q24 × jY.

Proposition 2.6. jW ∗Y is reducible and its subrepresentations are tempered.

Consider the following commutative diagram:

ZY = IndGL(1)×···×GL(1)×GL(q2)×G(n− q1+q2

2)| |

q1−12

×| |q1−3

2 × · · · × | |q2+1

2 × 1× Y−−−→ WY

R′Y

y yZ∗Y = Ind

GL(1)×···×GL(1)×GL(q1)×G(n− q1+q22

)| |−

q1−12

×| |−q1−3

2 × · · · × | |−q2+1

2 × 1× Y←−−−←

W ∗Y

where R′Y is the normalized intertwining operator. Its image isR(wp, λp)I(λp)∩Z∗Y . By Proposition 2.4, Ind

GL(q2)×G(n− q1+q22

)1×Y is semi-

simple with length 2.

Proposition 2.7. The image of R′Y is semi-simple with length ≤ 2.

Proof. Let X be a subrepresentation of Z∗Y . Let ξ = | |−q1−1

2 × | |−q1−3

2 ×· · · × | |−

q2+12 . Consider the subrepresentation of the Jacquet module of X

with respect to M = GL(1)× · · · ×GL(1)︸︷︷︸q1−q2

2

×G(n− q1−q22 ) such that the q1−q2

2


copies of GL(1) acts, after semi-simplification, according to ξ, i.e., the spaceof generalized weight ξ. We denote the space by Jacξ X. Then we have

Jacξ X ⊂ Jacξ Z∗Y = ξ × IndGL(q2)×G(n− q1+q2

2)1× Y.

Therefore our assertion follows.

Let IndGL(q2)×G(n− q1+q2

2)1 × Y = X1 + X2. Then by [M1, Cor. 0.10.2],

the normalized intertwining operators

IndGL(1)×···×GL(1)×G(n− q1−q2

2)| |

q1−12 × | |

q1−32 × · · · × | |

q2+12 ×Xi

−→ IndGL(1)×···×GL(1)×G(n− q1−q2

2)| |−

q1−12 × | |−

q1−32 × · · · × | |−

q2+12 ×Xi

are non-vanishing. Therefore:

Proposition 2.8. The image of R′Y is a sum of two irreducible representa-tions.

Proposition 2.9. Let RY (σ1, x) be the operator:

IndGL(q2)×G(n− q1+q2

2)|det |x × Y −→ Ind

GL(q2)×G(n− q1+q22

)|det |−x × Y.

RY (σ1, x) is holomorphic at x = 0 and let RY (σ1, 0) = RY (σ1). ThenRY (σ1)2 = id and RY (σ1) is not the identity since Ind

GL(q2)×G(n− q1+q22

)1×Y

is reducible. Let R(σ1) be the sum of RY (σ1) as Y runs through Unip(p≥2).Then it defines an intertwining operator for R(wp, λp)I(λp).

In the same way, we can define R(σk). For each σ ∈ Stab(λp, ↑ p), wedefine R(σ) in a canonical way ([M1, 686]): If σ = σp1 · · ·σprσ1 · · ·σk,then R(σ) = R(σp1) · · ·R(σpr)R(σ1) · · ·R(σk). We note that R(σwp, λp) =R(σ)R(wp, λp) for σ ∈ Stab(λp, ↑ p).

Theorem 2.10.(1) σ 7−→ R(σ) is a homomorphism of the group Stab (λp, ↑ p) into the

group of the intertwining operators of R(wp, λp)I(λp).(2) For X ∈ Unip(p), let R(σ)X = ηp

X(σ)X. Then ηpX defines a character

of Stab (λp, ↑ p).(3) If pi = pj, then ηp

X(σ(pi)) = ηpX(σ(pj)). If pi = q2k−1 or pi = q2k,

then ηpX(σ(pi)) = ηp

X(σk). Recall in Proposition 2.4 that these happenprecisely when IndGL(pi)×G(m)1 ×X ′ is irreducible for an appropriatem. Therefore by Lemma 2.1, ηp

X defines a character of A(p).(4) By Proposition 2.5 and 2.9, passing to quotient, X 7−→ ηp

X gives rise

to an isomorphism Unip(p) ' A(p) which is extended canonically to

Unip(O) ' Springer (O),


by X 7−→ ηX in the sense that |Unip(p) ∩ Unip(p′)| = | A(p) ∩ A(p′)|and for X ∈ Unip(p) ∩Unip(p′), ηp

X = ηp′

X .

(5) If p ' p′, then Unip(p) ' A(p) = A(p′) ' Unip(p′). In other words,up to isomorphism, Unip(p) depends only on the equivalence class ofp.

Especially, if p = (; q1, . . . , qs) ∈ P (O), O a distinguished unipotent orbit,then we have:

Corollary 2.11. (1) A(p) is isomorphic to Stab(λp, ↑ p).(2) σ 7−→ R(σ) is an isomorphism of the group Stab (λp, ↑ p) onto the

group of the intertwining operators of R(wp, λp)I(λp).

We think of η ∈ A(p) as an element of A(O) such that η|Kp = 1.2.2. Global Theory. Let O be a unipotent orbit and p = (p1, . . . , pr;q1, q2, . . . , qs) ∈ P (O). Let Sp be the set of positive roots defined as follows:

ej − ej+1, for∑k

i=1 pi < j <∑k+1

i=1 pi,

ej − ej+1, for T di < j ≤ T fi − 1 andeT di +

q2i−q2i−12

+ eT fi

, where 1 ≤ i ≤ [ s2 ],

ej − ej+1, for T ds+12

< j < n

2en, if G = Sp(2n), s is odd and qs > 1;en, if G = SO(2n+ 1), s is odd.

We note that Sp ⊂ α > 0|wpα < 0, 〈λp, α∨〉 = 1 and Sp has exactly n− r

elements. We will take the iterated residue of the Eisenstein series along then singular hyperplanes 〈λp, α

∨〉 = 1 for α ∈ Sp.

Definition 2.2.1. For p = (p1, . . . , pr; q1, q2, . . . , qs) ∈ P (O), we define

Mp = GL(p1)× · · · ×GL(pr)×G(n− T d1 ),

M ′p = GL(p1)× · · · ×GL(pr)×GL(q1 + q2

2

)× · · ·

×GL(q2[ s+1

2]−1 + q2[ s+1

2]

2

).

If s is odd, we put the convention thatq2[ s+1

2 ]−1+q

2[ s+12 ]

2 is qs−12 if G = Sp(2n),

and qs2 if G = SO(2n+ 1).

Definition 2.2.2. Let V (p) (resp. V ′(p)) be the set of elements of theform λp + η, where η is a character of Mp(A) (resp. M ′p(A)). Note that ifr = 0, V (p) = λp. We note that V ′(p) is the intersection of the singularhyperplanes 〈λ, α∨〉 = 1, where α ∈ ej − ej+1 for

∑ki=1 pi < j <

∑k+1i=1 pi,

T di < j ≤ T fi − 1, i = 1, . . . , [ s2 ] and T fs < j < n.


We denote the element in V ′(p) as

λp(x1, . . . , xr, z1, . . . , z[ s+12

])

= λp + (x1, . . . , x1︸︷︷︸p1

, . . . , xr, . . . , xr︸︷︷︸pr

, z1, . . . , z1︸︷︷︸q1+q2

2

, . . . , z[ s+12

], . . . , z[ s+12

]︸︷︷︸q2[ s+1

2 ]−1+q

2[ s+12 ]

2

).

Definition 2.2.3. For 1 ≤ k ≤ [ s+12 ], we define

V ′k(p) = λp(x1, . . . , xr, z1, . . . , z[ s+12

]) ∈ V′(p),

such that zi = 0 for all i > k.

In particular, V ′0(p) = V (p) and V ′[ s+1

2]= V ′(p).

Definition 2.2.4. We define W (↑, p) to be the set of the Weyl group ele-ments which send the positive roots of M ′p to the positive roots of M ′p.

Letd(p, λ) =

∏α∈Sp

(〈λp, α∨〉 − 1).

Let Unip be the submodule of ⊗′vRv(wp, λp)Iv(λp) which is the sum of irre-ducible subrepresentations of type ⊗′vXv, where Xv ∈ Unip(p) for all v andthere does not exist p′ > p and Xv ∈ Unip(p′) for all v.

Let proj[p] be the G(A)-projection ⊗′vRv(wp, λp)Iv(λp) 7−→ Unip. Forφ ∈ PW , the set of Paley-Wiener type functions, let

lp(φ, λ) =∑w∈W

r(w,−λ)R(wpw−1, wλ)φ(wλ).

Then we have:

(1) r(wp, λ)d(p, λ) is holomorphic at λ = λp and its value is non-zero.(2) The poles of lp(φ, λ) in a neighborhood of λp are contained in the local

intertwining operators.(3) r(w,−λ) is identically zero on V ′(p) if w /∈W (↑, p). So the restriction

of lp(φ, λ) to V ′(p) is given by

lp(φ, λ) =∑

w∈W (↑,p)


(4) lp(φ, λ)V (p) can be defined inductively by restricting it to V ′k(p) fromk = [ s+1

2 ]− 1 to k = 0.(5) lp(φ, λ)V (p) is holomorphic at λp and lp(φ, λp) ∈ ⊗′vRv(wp, λp)Iv(λp).

This depends only on φ and the equivalence class of p. Let l[p](φ, λp) =proj[p](φ, λp).


(6) Let 〈, 〉 be the inner product in L2(G(F )\G(A)). Then

〈θφ′ , θφ〉 = limT→∞

∑O⊂G∗(n)

∑p∈P (O)

cp

∫λ∈V (p),Reλ=λp,||Imλ||2≤T

〈〈l′[p](φ′, λ), l[p](φ, λ)〉〉V (p)(r(wp, λ)d(p, λ))V (p),

whereO runs through the unipotent orbits inG∗(n) and p runs throughthe set of representatives in each equivalence classes in P (O).

(7) For φ ∈ PW , suppose l[p](φ, λp) generates an irreducible representa-tion. Then for all v finite places, let Xv be the local representation ofGv generated by l[p](φ, λp). Then Xv ∈ Unip(p) and

∏v ηXv = 1.

(8) Conversely, suppose p = (p1, . . . , pr; q1, . . . , qs) ∈ P (O) such thatpi 6= pj for i, j ∈ [1, r] and π = ⊗′vXv is an irreducible automor-phic representation which satisfies the following: (a) Xv ∈ Unip(p)for all v, (b) Xv is spherical almost everywhere and at archimedeanplaces, and (c)

∏v ηXv = 1. Then there exists φ ∈ PW such that the

representation generated by l[p](φ, λp) is isomorphic to π.(9) In fact, for an appropriate φ ∈ PW ,

lp(φ, λp) =∑

τ∈Stab (λp,↑p)

R(τ−1)R(wp, λp)φ(λp).

3. Arbitrary character case.

By conjugation, we can assume χ = χ(µ1, . . . , µ1︸︷︷︸r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

),

r0 + · · ·+ rk = n, r1 ≥ · · · ≥ rk, µi’s are distinct local quadratic characters.Here k ≤ 3 (Recall that we are dealing with a p-adic field with odd residualcharacteristic and hence there are only three non-trivial distinct quadraticcharacters.) Set µ0 = 1. We use the following notation throughout thissection:

(1) If G = Sp(2n), G′ = G′1 × · · · × G′k × G′0, where G′i = O(2ri) fori = 1, . . . , k, G′0 = Sp(2r0). Also we denote G∗i = O(2ri,C) for i =1, . . . , k, G∗0 = O(2r0 + 1,C).

(2) If G = SO(2n+ 1), G′ = G′1× · · · ×G′k ×G′0, where G′i = SO(2ri + 1)for i = 1, . . . , k, G′0 = SO(2r0 + 1). Also we denote G∗i = Sp(2ri,C)for i = 1, . . . , k, G∗0 = Sp(2r0,C).

(3) If G = O(2n), G′ = G′1 × · · · × G′k × G′0, where G′i = O(2ri) fori = 1, . . . , k, G′0 = O(2r0). Also we denote G∗i = O(2ri,C) for i =1, . . . , k, G∗0 = O(2r0,C).

We need to first generalize Mœglin’s results to an arbitrary chain.


Definition 3.1. By a segment attached to (a, b), we mean a descendingsequence of characters

λ(a,b) = | |λ1 × | |λ2 × · · · × | |λn ,

where λt = a+12 −t, n = a+b

2 . We sometimes write it as λ(a,b) = (λ1, . . . , λn).We put the convention that a > b > 0 (a, b are odd in the case of Sp(2n)and O(2n), even in the case of SO(2n + 1)). To (a), a ≥ 3, we attach asegment

λ(a) = | |a−12 × | |

a−32 × · · · × | |1.

We write it as λ(a) = (a−12 , a−3

2 , . . . , 1). By a chain we mean an orderedunion of segments. We put a convention that the segment attached to (a)appears at the end. It can appear only in the case of Sp(2n) and SO(2n+1).A chain is denoted by p = (a1, b1, a2, b2, . . . , as, bs, as+1), where (ai, bi) is asegment and λp = λ(a1,b1) · · ·λ(as,bs)λ(as+1) with an obvious meaning.

Remark 3.1. We note that λ(a,b) is the intersection of the n singular hy-perplanes e1 − e2 = 1, e2 − e3 = 1, . . . ,en−1 − en = 1, ea−b

2+ en = 1.

Remark 3.2. Suppose we ignore the ordering in p. Then it corresponds toa unipotent orbit O in G∗(n). When O is a distinguished orbit, Mœglin’scase is that ai, bi are all distinct and satisfy two additional conditions:

(1) there does not exist 1 ≤ k ≤ s such that ai > ak > bi > bk.(2) if there exists ak or bk such that aj > ak > bj or aj > bk > bj , then

k < j.

Let p = (a1, b1, . . . , as, bs, as+1). If ai, bi ∩ ai+1, bi+1 6= ∅, then wecould permute (ai, bi) and (ai+1, bi+1) by [M1, Proposition 0.9.1]. But ifai, bi ∩ ai+1, bi+1 = ∅, then we cannot permute ai, bi and ai+1, bi+1in general. From now on we assume that in a chain p,

if ai, bi ∩ aj , bj = ∅ for i < j, it does not happen that ai > aj > bj > bi,(3.1)

in order to use [M1, Cor. 0.10.2] on non-vanishing of normalized intertwiningoperators. We note that the condition (3.1) is just a rephrasing of thecondition (2) in Remark 3.2.

Remark 3.3. Let O be a unipotent orbit obtained by ignoring the orderingin a chain p = (a1, b1, . . . , as, bs, as+1). Suppose either O is not distinguishedor p does not satisfy condition (1) in Remark 3.2. Then p can be written asp = p1 × · · · pk × p0, where pi ∈ P (Oi) and Oi is a distinguished unipotentorbit in G∗i for i = 0, 1, . . . , k. It means that it comes from global considera-tion when χ = χ(µ1, . . . , µ1︸︷︷︸

r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

), where µi, i = 1, . . . , k

are non-trivial quadratic grossencharacters such that µiv = 1 for i = 1, . . . , kfor a given non-archimedean place v.


For a segment (a, b), we define the Weyl group elements w(a,b), σ(a,b) asfollows (see [M1, p. 660]):

w(a,b)(t) = −t, if 1 ≤ t ≤ a− b2

,

w(a,b)

(t+

a− b2

)=a+ b

2+ 1− t, if 1 ≤ t ≤ b,

σ(a,b)(t) = t, if 1 ≤ t ≤ a− b2

,

σ(a,b)

(t+

a− b2

)= −

(a+ b

2+ 1− t

), if 1 ≤ t ≤ b.

We note that(1) σ(a,b)w(a,b) = w(a,b)σ(a,b) is the longest element w0 of the Weyl group

of Sp2n, n = a+b2 , i.e., w0 = c1c2 · · · cn, where ci’s are sign changes.

(2) σ(a,b)λ(a,b) = λ(a,b).(3) l(σ(a,b)w(a,b)) = l(w(a,b)) + l(σ(a,b)).(4) If q = 1, σ(a,1) = cn and w(a,1) = c1 · · · cn−1.For a segment attached to (a), we define w(a) = c1c2 · · · ca−1

2and σ(a) = 1.

λ(a) is the intersection of the a−12 singular hyperplanes e1−e2 = 1, e2−e3 = 1,

. . . ,ea−32− ea−1

2= 1.

For a chain p = (a1, b1, . . . , as, bs, as+1), we define wp as wp =w(a1,b1) · · ·w(as,bs)w(as+1) with an obvious meaning and Stab(λp, ↑ p) as thegroup generated by σ(ai,bi) for i = 1, . . . , s.

In order to apply induction, we define p≥i = (ai, bi, . . . , as, bs, as+1). Let,for 1 ≤ i ≤ s, T di =

∑i−1k=1

ak+bk2 and T fi =

∑ik=1

ak+bk2 .

Definition 3.2. For a chain p = (a1, b1, . . . , as, bs, as+1), we define a Levisubgroup M ′p = GL(a1+b1

2 ) × · · · ×GL(as+bs2 ) ×GL([as+1

2 ]) and degenerateprincipal series

I(λp, χ) = IndM ′pχ⊗ |det |

a1−b14 × · · · × |det |

as−bs4 × |det |

12[as+1+1

2],

I(−λp, χ) = IndM ′pχ⊗ |det |−

a1−b14 × · · · × |det |−

as−bs4 × |det |−

12[as+1+1

2],

where χ is the character of M ′p induced by χ.

If we set w′p to be the longest Weyl group element of M ′p, then I(λp, χ)is the image of the normalized intertwining operator R(w′p, λp, χ).

The normalized intertwining operator R(wp, λ, χ) is not holomorphic atλp. In order to define R(wp, λp, χ), we need:

Proposition 3.1. For each segment (ai, bi), R(w(ai,bi), λ(a1,b1) · · ·λ(ai,bi), χ)defines a holomorphic intertwining operator from

IndGL(1)×···×GL(1)×G(n−T fi )

χ(a1,b1)λ(a1,b1) · · ·χ(ai,bi)λ(ai,bi) × I(−λp≥i+1, χ),


into

IndGL(1)×···×GL(1)×G(n−T fi )

χ(a1,b1)λ(a1,b1) · · ·χ(ai−1,bi−1)λ(ai−1,bi−1)

× w(ai,bi)χ(ai,bi)λ(ai,bi) × I(−λp≥i+1, χ).

Its image is included in

IndGL(1)×···×GL(1)×G(n−T fi+1)

χ(a1,b1)λ(a1,b1) · · ·

χ(ai−1,bi−1)λ(ai−1,bi−1) × I(−λp≥i , χ).

Proof. The argument is like that in [M1, 0.13]; the introduction of quadraticcharacters does not create any new complications.

We define the normalized intertwining operator R(wp, λp, χ) as the com-position of the above operators. Then

R(wp, λp, χ) ⊂ I(−λp, χ)

R(wp, λp, χ) ⊂ IndGL(

a1+b12

)×G(n−T f1 )µ1|det |−

a1−b14

×R(wp≥2, λp≥2

, χ)I(λp≥2, χ).

Here χ in R(wp≥2, λp≥2

, χ)I(λp≥2, χ) should be interpreted appropriately.

Lemma 3.2. The normalized intertwining operator R(wp, λp, χ) does notvanish identically.

Proof. Let λO be the conjugate of λp which is in the closure of the positiveWeyl chamber. Let w1 be a Weyl group element such that λp = w1λO.Consider the following commutative diagram.

I(λO, w−11 χ)

R(wO,λO,w−11 χ)

−−−−−−−−−−→ I(−λO, w−11 χ)yR(w1,λO,w

−11 χ)

xR(w−11 ,−λp,χ)

I(λp, χ)R(wp,λp,χ)−−−−−−−→ I(−λp, χ).

Here R(wO, λO, w−11 χ) is the intertwining operator on IndGP λO⊗IndMB w−1

1 χ,where P = MN is the parabolic subgroup such that λO is in the positiveWeyl chamber with respect to P . Then it is non-vanishing. Note that all thenormalized intertwining operators are holomorphic. Therefore, R(wp, λp, χ)is non-vanishing.

We first reduce to the case χ = 1.


3.1. Review of the results of Barbasch-Moy [B-Mo2]. We now con-sider the case when χ is unramified. The following discussion is based on[B-Mo2].

Let G = SO(2n+1), θ = | |a1µ×· · ·×| |an1µ×| |b1×· · ·×| |bn0 , a characterof T ⊂ G (n0 + n1 = n), where µ is the non-trivial unramified quadraticcharacter and a1, . . . , an1 , b1, . . . , bn0 ∈ R. Let H ′ = H ′1 × H ′0, with H ′1 =SO(2n1 +1), H ′0 = SO(2n0 +1). Set θ′ = θ′1×θ′0 with θ′1 = | |a1×· · ·×| |an1 ,θ′0 = | |b1 × · · · × | |bn0 characters of T ′1 ⊂ H ′1, T

′0 ⊂ H ′0. If τ (resp. τ ′, τ ′1,

τ ′0) denotes the infinitesimal character associated to subquotients of IndGB θ

(resp. IndH′

B′ θ′, IndH

′1

B′1θ′1, IndH

′0

B′0θ′0), then the results of [B-Mo2] tell us

that there is an equivalence of categories

R(G, τ) ' R(H ′, τ ′) ' R(H ′1 ×H ′0, τ ′1 × τ ′0),where R(G, τ) denotes the category of smooth finite-length representationsof G with infinitesimal character τ .

Suppose G = Sp(2n). Then the same discussion as above applies, exceptthat in this case H ′1 = O(2n1) and H ′0 = Sp(2n0).

Ultimately, we are going to apply the results of [B-Mo2] to those of [R] todeal with representations of Sp(2n) and SO(2n+1). Therefore, we also needto discuss [B-Mo2] for O(2n) (cf. Example 1.1). Unfortunately, the resultsof [B-Mo2] do not apply to disconnected groups, nor does there appearto be any obvious way to extend the results of [B-Mo2] from SO(2n) toO(2n). Thus, we are forced to assume the results of [B-Mo2] hold for O(2n)as well (cf. Assumption 3.1.1 below). The above discussion then applies toG = O(2n) as well. In this case, H ′1 = O(2n1) and H ′0 = O(2n0).

Assumption 3.1.1. The results in Sections 1-6 of [B-Mo2] hold for O(2n).

Next, we note that this equivalence respects induction, in a suitable sense.Let G be one of the groups above (SO(2n+1), Sp(2n), O(2n)) andM a stan-dard Levi subgroup of G. Let M ′,H ′ be the groups corresponding to M,Gunder [B-Mo2]. Let τM , τ ′M ′ be infinitesimal characters for M,M ′ whichcorrespond under [B-Mo2], τ, τ ′ the infinitesimal characters for G,H ′ ob-tained by induction. Then by [B-Mo2, Theorem 6.2], the following diagramcommutes:

R(G, τ) −−−→ R(H ′, τ ′)x⊗-Ind

x⊗-Ind

R(M, τM ) −−−→ R(M ′, τ ′M ′)where ⊗-Ind denotes induction defined via tensor product at the Heckealgebra level.

Example 3.1.1. Let α1, . . . , αk ∈ R and

π = IndGP (|detm1 |α1µ detm1 × · · ·


× |detmj |αjµ detmj × |detmj+1 |αj+1 × · · · × |detmk |αk)

π′ = IndH′

P ′ (|detm1 |α1 × · · · × |detmj |αj × |detmj+1 |αj+1 × · · · × |detmk |αk),

where P, P ′ are standard parabolics of G,H ′ whose Levi factors both havethe form GL(m1)× · · ·×GL(mk). Observe that by [Ja1, Proposition 2.1.2](which may be extended to cover O(2n); we may choose w0 = c1c2 · · · cn,and similarly for smaller rank orthogonal groups appearing in standard Levisubgroups), we have

π ∼= ⊗− IndGP (|detm1 |−α1µ detm1 × · · ·× |detmj |−αjµ detmj × |detmj+1 |−αj+1 × · · · × |detmk |

−αk)

π′ ∼= ⊗− IndH′

P ′ (|detm1 |−α1 × · · ·× |detmj |−αj × |detmj+1 |−αj+1 × · · · × |detmk |

−αk).

Since the inducing representation for π is the unique irreducible subrepre-sentation of

IndMBM

((| |α1+

−m1+12 µ× · · · × | |α1+

m1−12 µ

)× · · ·

×(| |αj+

−mj+1

2 µ× · · · × | |αj+mj−1

2 µ

)×(| |αj+1+

−mj+1+1

2 × · · · × | |αj+1+mj+1−1

2

)× · · ·

×(| |αk+

−mk+1

2 × · · · × | |αk+mk−1

2

)),

and similarly for π′, we see that the inducing representations correspondunder R(M, τM ) ' R(M ′, τ ′M ′). By the discussion above, this means that πand π′ correspond under R(G, τ) ' R(H ′, τ ′).

3.2. Matching of images of intertwining operators under Hecke al-gebra isomorphisms. Let χ = χ(µ1, . . . , µ1︸︷︷︸

r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

), where

r0 + · · · + rk = n, r1 ≥ · · · ≥ rk, and the µi’s are distinct local qua-dratic characters. Let G′ = G′1 × · · · × G′k × G′0. Combining the Heckealgebra isomorphisms of Roche and the graded algebra isomorphisms ofBarbasch-Moy, we get an equivalence of categories, which gives rise to a cor-respondence between subquotients of the induced representation IG(λ, χ)and subquotients of the induced representation IG

′(λ, 1) (cf. Example 1.1

and previous section). Note that if we write λ = λ1 × · · · × λk × λ0

with λ1 = | |a1 × · · · × | |ar , a1, . . . , ar ∈ R, etc. we have IG′(λ, 1) ∼=

IG′1(λ1, 1) × · · · × IG

′k(λk, 1) × IG

′0(λ0, 1). The correspondence preserves

temperedness, square-integrability, etc. The correspondence behaves wellwith respect to intertwining operators:


Proposition 3.2.1. The above correspondence behaves well with respect tointertwining operators, i.e., R(wp, λp, χ)I(λp, χ) corresponds to

R(wp1 , λp1)IG′1(λp1)× · · · ×R(wpk , λpk)I

G′k(λpk)×R(wp0 , λp0)IG′0(λp0),

where wp = wp1 × · · · × wpk × wp0 and λp = λp1 × · · · × λpk × λp0.

We prove this in this section. The arguments used in this section arebased largely on [Ca], [Ca2], [Re2] (with their presentation also influencedby [Re1]).

Since we are assuming F has odd residual characteristic here, we havethat F admits three non-trivial quadratic characters. Let µ, resp. µnr, de-note a ramified, resp. unramified, non-trivial quadratic character (so thatµ, µnr, µµnr are the three non-trivial quadratic characters). If we fix a uni-formizer $, we may assume µ is the ramified quadratic character satisfyingµ($) = 1. For convenience, assume χ (cf. Section 1) has the form

χ = µ× · · · × µ︸︷︷︸n1

× 1× · · · × 1︸︷︷︸n0

.

(As in Example 1.1, it is actually χ|T (O) that is needed in Roche’s construc-tion.) Then we have Jχ = I = Iwahori subgroup and

H(G, ρ) ' H(H, 1IH ).

Here ρ = ρχ as in Section 1. It will be convenient to write H ′ (resp.,I ′, B′) for H (resp., IH , BH). Recall the decomposition H ′ = H ′1 ×H ′0 fromExample 1.1.

Now, H(G, ρ) has linear basis Tww∈Wχ , where Tw is supported onIwI. If we identify Wχ with representatives in G chosen as in the proofof Lemma 9.3 [R] (which in turn is based on [Mr]), we can normalize Twso that it is 1 at w. Also, observe that Wχ = W

′, which may be identifiedwith W (H ′1) ×W (H ′0). We let w0 denote the longest element of W , andnote that w0 ∈Wχ.

If π = IndGB (λχ) with λ unramified, then V I,χπ has as basis fww∈Wχ

,where

fw(g) =

δ

12 (t)λ(t)χ(t)χ(i), if g = (tu)wi ∈ BwI

0, if not.

Similarly, if π′ = IndH′

B′ λ, then V I′π′ has basis f ′ww∈W ′ , where

f ′w(g′) =

δ′

12 (t′)λ(t′), if g′ = (t′u′)wi′ ∈ B′wI ′

0, if not.

If we let T ′w, w ∈ W ′, denote the characteristic function of I ′wI ′, we havethe following:


Lemma 3.2.2. Let s ∈W ′ be the reflection associated to the χ-simple rootα ∈ Πχ. For w ∈W ′, we have

π′(T ′s)f′w =

f ′ws, if wα > 0,qf ′ws + (q − 1)f ′w, if wα < 0.

Further, for c ∈ Cχ,π′(Tc)f ′w = f ′wc−1 .

Proof. Since it is well-known how to do such calculations (and straightfor-ward), we omit the details.

Corollary 3.2.3. Let H(K, ρ) ⊂ H(G, ρ) denote the subalgebra consistingof functions supported on K. Then, fw0 generates V I,χ

π under the action ofH(K, ρ).

Proof. Observe that the corresponding result for π′ is straightforward: Fromthe preceding lemma, f ′w = π′(T ′w−1)f ′1. Therefore, f ′1 = π′(T ′

w−10

−1)f ′w0.

Thus, f ′w = π′(T ′w−1)π′(T ′w−10

−1)f ′w0. To extend this to cover π, observe

that f ′w|K′ = T ′w and fw|K = Tw. Therefore, for w1, w2 ∈ Wχ, if T ′w2∗

T ′w−1

1

=∑cwT

′w, then π′(T ′w1

)f ′w2=∑cwf

′w, and similarly for π. Suppose

Ψ : Tw 7→ awT′w. If T ′

w−11

∗ T ′w2=∑cwT

′w, one can then conclude that

π(Tw1)fw2 =∑cwaw1aw2a

−1w fw. The corollary follows.

If B = TU is the Levi factorization of B, we use πU to denote the (un-normalized) Jacquet module of π with respect to U . It has

space: (Vπ)U = Vπ/Vπ(U),

where Vπ(U) = span π(u)v − v| u ∈ U, v ∈ Vπ,action: πU (t)(v + Vπ(U)) = π(t)v + Vπ(U).

We have the following:

Lemma 3.2.4. The restriction of the quotient map Vπ −→ (Vπ)U to V I,ρπ

gives rise to an isomorphism V I,ρπ ' (Vπ)

T0,χU as vector spaces.

Proof. The proof is essentially the same as that done by Casselman forunramified principal series, so we just give a sketch here.

First, one checks that if v ∈ V U−1 T0,χπ , then v−π(eρ)v ∈ Vπ(U) (note that

eρ = T1). Consequently, (V I,ρπ )U = (V U−1 T0,χ

π )U . Next, we choose a finite-dimensional subspace X ⊂ V T0,χ

π which maps onto (V T0,χπ )U . If we take a

compact subgroup U−k ⊂ U− which acts trivially on X and t ∈ T such that

t−1U−1 t ⊂ U−k , we get π(t)X ⊂ V T0U

−1 ,χ

π . Therefore, (V T0,χπ )U = πU (t)XU ⊂

(V T0U−1 ,χ

π )U = (V I,ρπ )U , giving surjectivity. Injectivity then follows from a

comparison of dimensions.


Lemma 3.2.5. For t ∈ T−, π(Tt)fw0 = δ−12 (t)(w0λ)(t)(w0χ)(t)fw0. (This

result also holds when G = O(2n) and w0 = c1c2 · · · cn.)

Proof. Recall that we have a B-stable filtration Vπ =⋃w∈W (Vπ)w, where

(Vπ)w =f ∈ Vπ

∣∣∣∣ supp f ⊂⋃x≥w

BxB

,

(where ≥ denotes the Bruhat order). This gives rise to a T -filtration (Vπ)U =⋃w∈W ((Vπ)w)U . By Lemma 6.3.5 [Ca] (also, see the proof of Lemma 2.12

[B-Z]), T acts on ((Vπ)w0)U by the character δ12w0(λχ). Since Bw0I ⊂

Bw0B, we see that fw0 is a basis for V I,ρπ ∩ (Vπ)w0 . Thus, by Lemma 3.2.4,

fw0 + Vπ(U) is a basis for ((Vπ)w0)U . In particular,

π(t)(fw0 + Vπ(U)) = δ12 (t)(w0λ)(t)(w0χ)(t)fw0 + Vπ(U).

Next, the same basic argument used in [Ca2, Proposition 2.5] tells us that|ItI|−1π(Tt)fw0 and π(t)fw0 have the same image in (Vπ)U . Since |ItI| =δ−1(t) ([Ca, Lemma 1.5.1]), we see that

π(Tt)fw0 + Vπ(U) = δ−1(t)π(t)fw0 + Vπ(U)

= δ−12 (t)(w0λ)(t)(w0χ)(t)fw0 + Vπ(U).

Finally, if one writes π(Tt)fw0 =∑

w∈Wχcwfw, taking Jacquet modules gives

π(Tt)fw0+Vπ(U) =∑

w∈Wχcwfw+Vπ(U). It is now immediate from the pre-

ceding lemma and the equality above that cw0 = δ−12 (t)(w0λ)(t)(w0χ)(t)fw0

and cw = 0 for w 6= w0.For G = O(2n), let λ, χ be as above and w0 = c1c2 · · · cn. Let G = SO(2n)

and w0 ∈ W (G) of maximal length. The preceding argument then tells usthat for π = IndGBλχ, π(Tt)fw0 = δ−

12 (t)(w0λ)(t)(w0χ)(t)fw0 when t ∈ T−.

If w0 = w0 (n even), the O(2n) result is immediate. If w0 = w0cn (n odd),it follows easily from the fact that π(ecn)fw0 = fw0 and δ cn = δ.

For λ, χ as above, let π = IndGB(λχ) and π′ = IndH′

B′ (λ). If λ = | |x1×· · ·×| |xn1×| |y1×· · ·×| |yn0 , we set λ1 = | |x1×· · ·×| |xn1 and λ0 = | |y1×· · ·×| |yn0

(so that λ = λ1 × λ0). We note that π′ ' π′1 ⊗ π′0, where π′1 = IndH′1

B′1λ1

and π′0 = IndH′0

B′0λ0. We define a map M : V I,χ

π 7−→ V I′π′ as follows: Let

M : fw0 7−→ f ′w0. (Note that under the identification Wχ = W

′ = W (H ′1)×W (H ′0), w0 corresponds to w0,1·w0,0, with w0,i ∈W (H ′i) consisting of all signchanges. Under the identification π′ ' π′1⊗π′0, we have f ′w0

7−→ f ′w0,1⊗f ′w0,0

.)Then by Corollary 3.2.3, we can extend M to get a linear isomorphismsatisfying π′(Ψ(h))(Mf) = M(π(h)f) for all h ∈ H(K, ρ), f ∈ V I,χ

π . Weclaim that the equivalence of categories Rχ(G) ' R1(H ′) comes from themap M. More precisely, we have the following:


Proposition 3.2.6. The pair (H(G, ρ), π) is equivalent to (H(H ′, 1), π′)under (Ψ,M). In particular,

π′(Ψ(h))(Mf) = M(π(h)f)

for all h ∈ H(G, ρ), f ∈ V I,χπ .

Proof. By definition, π′(Ψ(h))(Mf) = M(π(h)f) holds for h ∈ H(K, ρ).Take y ∈ Y +. Then Ψ : δ

12 (y)Ty 7−→ δ′

12 (y)T ′y (cf. proof of Lemma 9.3 [R]).

We note that by definition, Ty = Ty(w). Since the extension of χ from T (O)to T satisfies χ(y(w)) = 1 for all y ∈ Y , it follows from Lemma 3.2.5 thatπ′(Ψ(Ty))(Mfw0) = M(π(Ty)fw0) for y ∈ Y +. Therefore, π′(Ψ(Ty))(Mf) =M(π(Ty)f) for all y ∈ Y, f ∈ V I,χ

π . The proposition follows.

We now give a technical lemma which we will need below. In the lemma,we use lG to denote length for W (G), lH for length for W (H).

Lemma 3.2.7. Let G,χ be as above, M the Levi factor of a standard par-abolic subgroup of G. Then, there exists a set W TM

χ ⊂ Wχ such that thefollowing all hold:

(1) W TMχ is a set of representatives for Wχ(M)\Wχ.

(2) For x ∈Wχ(M), w ∈W TMχ , we have lG(xw) = lG(x) + lG(w).

(3) For x ∈Wχ(M), w ∈W TMχ , we have lH(xw) = lH(x) + lH(w).

Proof. For explicitness, let G = Sp(2n). Write χ = µ× · · · × µ︸︷︷︸n1

× 1× · · · × 1︸︷︷︸n0

and M ' GL(m1) × · · · × GL(ml) × Sp(2m0), with n0 + n1 = m0 +m1 +· · · + ml = n. Observe that H = H o Cχ with Cχ = 1, cn1, where cn1

denotes the n1th sign change. Let L be the subgroup of H correspondingto M (cf. Lemma 1.4). We define

WTL(H) = w ∈W (H)|w−1α > 0 for all α ∈ Φ+

L

WTL(H) = w ∈W (H)|w−1α > 0 for all α ∈ Φ+

L.

It is known that if x ∈W (L), w ∈W TL(H), then lH(xw) = lH(x) + lH(w);since lH(cn1) = 0, this result clearly extends to w ∈ W TL(H). We considertwo cases.

Case 1: n1 ≤ m1 + · · ·+ml.In this case, L = L is connected. Set W TM

χ = WTL(H). Since Wχ(M) =

W (L), property (3) is immediate. The first property follows easily fromthe fact that W

TL(H) is a set of representatives for W (L)\W (H) andW

TL(H) = WTL(H) ∪W TL(H)cn1 .


Finally, for the second property, one can directly check that there is astandard Levi M ′ of G such that Φ+

M,χ = Φ+L = Φ+

M ′ (if n0 = m1 + · · ·+mi

for some i, M ′ = M ; otherwise, M ′ < M). Then Wχ(M) = W (M ′) and

WTMχ = W

TL(H) ⊂W TM ′. The result follows.

Case 2: n1 > m1 + · · ·+ml.In this case, L = L o Cχ is disconnected. Set W TM

χ = WTM ∩Wχ =

w ∈ Wχ|w−1α > 0 for all α ∈ Φ+M. As a first step, we check that

WTL(H) = W

TMχ ∪ cn1W

TMχ . It follows easily from the definitions that

WTMχ ∪ cn1W

TMχ ⊂ W

TL(H). For the reverse containment, observe that

w ∈ WTL(H) has w ∈ Wχ. Further, w−1Φ+

M,χ = w−1Φ+L ⊂ Φ+

H = Φ+χ .

Now, the only simple root in ΠM which is not in Φ+M,χ is

αn1 =

en1 − en1+1, if n1 < n

2en, if n1 = n.

There are two possibilities: w−1αn1 > 0 or w−1αn1 < 0. If w−1αn1 > 0,then w−1ΠM ⊂ Φ+, implying w ∈ W

TM , as needed. If w−1αn1 < 0, weclaim w−1cn1αn1 > 0. (This is clear for n1 = n. Suppose n1 < n. Observethat w ∈ Wχ implies wen1 = ±ej with j ≤ n1 and wen1+1 = ±ek withk > n1. Further, since w ∈ W

TL, w(2en1+1) > 0, so wen1+1 = ek. Theclaim follows.) Also cn1(ΠM − αn1) ⊂ cn1Φ

+M,χ ⊂ Φ+

M,χ (noting that

cn1 ∈ W (L) = Wχ(M)). Thus, w−1cn1ΠM ⊂ Φ+, implying cn1w ∈ WTM ,

as needed. Therefore, W TL(H) = WTMχ ∪ cn1W

TMχ .

Since |W TMχ | = |W (H)|/|W (L)|, the first property is equivalent to

W (L)W TMχ = W (H). We calculate:

W (L)W TMχ = (W (L) ∪W (L)cn1)W

TMχ

= W (L)(W TMχ ∪ cn1W

TMχ )

= W (L)W TL(H).

That W (L)W TL(H) = W (H) follows easily from W (L)W TL(H) = W (H).The second property follows immediately from Wχ(M) ⊂ W (M) and

WTMχ ⊂W TM .To check the third property, write x = xHcx, w = cwwH with cx, cw ∈ Cχ

and xH , wH ∈ W (H). Then since lH(cn1) = 0, we have lH(x) = lH(xH),lH(w) = lH(wH). Set x′H = (cxcw)−1xH(cxcw). Then lH(x′H) = lH(x).Since xH ∈ W (L) and wH ∈ W

TL(H), we have lH(x′H) + lH(wH) =lH(x′HwH) = lH(cxcwx′HwH) = lH(xw). The result follows.


The case of G = SO(2n + 1) is much easier; since H = H is connected,the argument is just that of Case 1 above.

Lemma 3.2.8. Suppose M is a standard Levi for G, L the correspondingsubgroup of H. If ν → IndMBMλχ has space V IM ,ρM

ν ⊂ V IM ,ρMIndMBM

λχ, let V I′L

ν′ ⊂

VI′L

IndLBLλ

be its image under MM (with ν ′ denoting the restriction of IndLBLλ

to V I′Lν′ ). Then the image of V I,ρ

IndGPMν⊂ V I,ρ

π under M is V I′

IndHPLν′⊂ V I′

π′ .

Remark 3.2.1. Certainly, the restriction of π′ to MV I,ρ

IndGPMν

is equivalent

to IndHPLν′. But for our purposes, it is necessary to know that the subspaces

actually match up.

Proof of Lemma 3.2.8. First we need to identify the subspaces for IndGPM ν

and IndHPLν′. Suppose fMi i=1,... ,k =

∑x∈Wχ(M) b

ixf

Mx i=1,... ,k is a basis

for V IM ,ρMν ⊂ V IM ,ρM

IndMBMλχ

. Let fi =∑

x∈Wχ(M) bixfx ∈ V

I,ρπ . Then a basis for

V I,ρ

IndGPMλ⊂ V I,ρ

π is

π(Tw−1)fi i=1,... ,k

w∈WTMχ

=

∑x∈Wχ(M)

bixfxw

i=1,... ,k

w∈WTMχ

,

with WTMχ as in the preceding lemma. The proof of this is essentially the

same as that of [Ja1, Lemma 2.1.4] (noting that W TMχ is a set of representa-

tives forW (M)\W (G)). Similarly, if f ′iLi=1,... ,j =

∑x∈W (L) c

ixf′xLi=1,... ,j

is a basis for V I′Lν′ ⊂ V

I′L

IndLBLλ, then IndHPLν

′ has basis

π′(T ′w−1)f ′i

i=1,... ,j

w∈WTL

=

∑x∈W (L)

cixf′xw

i=1,... ,j

w∈WTL

.

Here we are taking W T L = WTMχ . (By the preceding lemma, W TM

χ has theproperties we need; we change notation only for appearance’s sake.)

We now check how subspaces match up. First observe that M : fw 7−→π′(Ψ(Tw−1)Ψ(T−1

w−10

))f ′w0. Therefore, if Ψ : Tw 7−→ awT

′w, we see that Mfw =

m(w)f ′w, where m(w) = aw−1a−1

w−10

. Similarly, using Proposition 1.6, we get

MMfMw = mM (w)f ′w

L with mM (w) = aw−1a−1

w−10,M

(w0,M the longest element


of Wχ(M)). If fMi i=1,... ,k as above is a basis for V IM ,ρMν , then V I,ρ

IndGPMλ

has basis∑

x∈Wχ(M) bixfxw

i=1,... ,k

w∈WTMχ

. Therefore, MV I,ρ

IndGPMλ

has basis (using

Wχ(M) = W (L), W TMχ = W

T L) ∑x∈W (L)

bixπ′(Ψ(Tw−1x−1)Ψ(T−1

w−10

))f ′w0

i=1,... ,k

w∈WTL

.

On the other hand, V I′Lν′ has basis

MMf

Mi

i=1,... ,k

=

∑x∈W (L)

bixmM (x)f ′xL

i=1,... ,k

.

Therefore, V I′

IndHPLν′

has basis ∑x∈W (L)

bixmM (x)f ′xw

i=1,... ,k

w∈WTL

.

Finally, by Lemma 3.2.7, for x ∈Wχ(M) and w ∈W TMχ , we have Tw−1x−1 =

Tw−1Tx−1 and T ′w−1x−1 = T ′w−1T′x−1 . From this, one sees that aw−1x−1 =

aw−1ax−1 . Therefore, m(xw) =(aw−1

aw−1

0,M

aw−1

0

)mM (x). The conclusion fol-

lows.

Let R(wp, λp, χ) denote the normalized standard intertwining operatordefined earlier. Since wp ∈ Wχ, we may identify wp ∈ W

′ with wp1 ·wp0 ∈ W (H ′1) ×W (H ′0) (p1, p0 are ordered partitions for H ′1,H

′0). If we

use H ′ = H ′1×H ′0 to identify the degenerate principal series IH′(λp, 1) withIH′

1(λp1 , 1)⊗ IH′

0(λp0 , 1), we have

R′(wp, λp, 1) = R′1(wp1 , λp1 , 1)⊗R′0(wp0 , λp0 , 1),

for the corresponding intertwining operators.

Proposition 3.2.9. R(wp, λp, χ) is a non-zero multiple of

M−1 R′1(wp1 , λp1 , 1)⊗R′0(wp0 , λp0 , 1) M.

Proof. We argue as in [Re2]. Let x = (x1, . . . , xs), y = (y1, . . . , yt+1) andset

λp1 + x = |det |a1−b1

4+x1 × · · · × |det |

as−bs4

+xs


λp0 + y = |det |c1−d1

4+y1 × · · · × |det |

ct−dt4

+yt × |det |ct+1

4+yt+1

and λp +(x, y) = (λp1 +x)× (λp0 +y). Since I(λp +(x, y), χ) and I ′H′1(λp1 +

x, 1), I ′H′0(λp0 + y, 1) are irreducible representations for (x, y) 6= (0, 0) near

zero, Schur’s lemma for p-adic groups tells us

R(wp, λp + (x, y), χ)(∗)= c(x, y)M−1 R′1(wp1 , λp1 + x, 1)⊗R′0(wp0 , λp0 + y, 1) M

for such (x, y) (c(x, y) scalar). Now, observe that M is independent of(x, y). By Proposition 3.1, Lemma 3.2 and [M1], R(wp, λp + (x, y), χ) andM−1 R′1(wp1 , λp1 +x, 1)⊗R′0(wp0 , λp0 +y, 1)M are both holomorphic andnon-zero at (x, y) = (0, 0). By analytic continuation, (∗) holds at (x, y) =(0, 0). Further, we see that c(x, y) must be holomorphic and non-vanishingat (x, y) = (0, 0). The proposition follows.

The results above begin the process of separating the effects of the fourcharacters of order≤ 2, essentially allowing us to deal with 1, µnr separatelyfrom µ, µµnr (with µ, µnr associated to 1, µnr for H ′1). To finish this pro-cess, we need to separate the effects of 1 and µnr for both symplectic andorthogonal groups. To do this, we use an argument similar to that above,only with [B-Mo2] playing the role that [R] played above. We give theargument for the symplectic case; the orthogonal case is similar.

Let χ = (µnr, . . . , µnr︸︷︷︸n1

, 1, . . . , 1︸︷︷︸n0

) and λ = | |x1×· · ·×| |xn1×| |y1×· · ·×| |yn0

with x1, . . . , xn1 , y1, . . . , yn0 ∈ R. Again, set H ′ = H ′1 × H ′0. Let π =IndGB (λχ) and π′ = IndH

′B′ (λ) = (IndH

′1

B′1λ1) ⊗ (IndH

′0

B′0λ0) = π′1 ⊗ π′0 (with

λ1 = | |x1×· · ·× | |xn1 and λ0 = | |y1×· · ·× | |yn0 ). We let τ, τ ′, τ ′1, τ′0 denote

the infinitesimal characters of π, π′, π′1, π′0, resp.

We define a map Mτ : V Iπ −→ V I′

π′ (unlike the situation above, Mτ isnot independent of λ). Let Mτ : fw0 7−→ f ′w0

. If q, q′ denote the quo-tient maps q : H(G) −→ Hτ (G), q′ : H(H ′) −→ Hτ ′(H ′) (cf. [B-Mo2,p. 619]), then we certainly have fw0 , f

′w0

generating V Iπ , V

I′π′ , under the ac-

tion of π(q(H(K))), π′(q′(H(K ′))). Thus, as above, we may extend Mτ toget a linear isomorphism satisfying π′(Ψτ (h))(Mτf) = Mτ (π(h)f) for allh ∈ q(H(K)), f ∈ V I

π . Here Ψ : Hτ (G) −→ Hτ ′(H ′) is the isomorphismof quotient algebras obtained by composing the isomorphisms in [B-Mo2].With notation as above, we have the following:

Lemma 3.2.10. Mτ has the following properties:

(1) π′(Ψτ (h))(Mτf) = Mτ (π(h)f) for all h ∈ H(G), f ∈ V Iπ .

(2) Mτ is real analytic in x1, . . . , xn1 , y1, . . . , yn0.


Proof. The proof of (1) is similar to Proposition 3.2.6 above. For h ∈q(H(K)), it holds by definition. For t, t′ corresponding to the same elementof Y , we have that Ψτ : δ

12 (t)q(Tt) 7−→ χ(t)δ

12 (t′)q′(Tt′) (a consequence of

[B-Mo2, (4.4)] and the construction of Ψτ ). With this observation, the restof (1) follows as in the proof of Proposition 3.2.6.

For (2), if w ∈Wχ = W (H ′), we have

Mτ (fw) = π′(Ψτ (q(Tw)))f ′w0.

It is sufficient to show Ψτ (q(Tw)) is analytic, which follows from [B-Mo2,Theorem 4.3].

Let R(wp, λp, χ) be the normalized standard intertwining operator definedearlier. As before, we can write wp = wp1wp0 ∈ W (H ′1) ×W (H ′0), and piis an ordered partition for H ′i. We decompose λp = λp1 × λp0 as above.Let R′(wp, λp, 1) = R′(wp1 , λp1 , 1) ⊗ R′(wp0 , λp0 , 1) be the correspondingintertwining operator for H ′.

Proposition 3.2.11. R(wp, λp, χ) = Mτ R′(wp1 , λp1 , 1)⊗R′(wp0 , λp0 , 1)M−1

τ .

Proof. We again argue as in [Re2]. With notation as in the proof of Propo-sition 3.2.9, we again have that if (x, y) 6= 0 near 0,

R(wp, λp + (x, y), χ)(∗∗)= Mτ(x,y) R′(wp1 , λp1 + x, 1)⊗R′(wp0 , λp0 + y, 1) M−1

τ(x,y),

where τ(x, y) is the infinitesimal character associated to IndGB (λp +(x, y))χ.In this case, there is no need to introduce a scalar c(x, y)–the action of (∗∗)on K-fixed vectors tells us we actually have equality. By Proposition 3.1 andthe preceding lemma, both sides of (∗∗) are analytic in (x, y). Therefore byanalytic continuation, (∗∗) holds at (0,0), as needed.

Remark 3.2.2. Hecke algebras are not available for archimedean places.Also Roche’s results are not available for the place v, v|2.

Remark 3.2.3. Above, we have used the results of Barbasch-Moy andRoche to identify IndGB λχ with IndG

′1

B′1λ1⊗ IndG

′2

B′2λ2⊗ IndG

′3

B′3λ3⊗ IndG

′0

B′0λ0,

where χ = (µ, . . . , µ︸︷︷︸r1

, µµnr, . . . , µµnr︸︷︷︸r2

, µnr, . . . , µnr︸︷︷︸r3

, 1, . . . , 1︸︷︷︸r0

) and λ = λ1 ×

λ2 × λ3 × λ0. The same isomorphisms allow us to identify IndG′1

B′1λ1 with

IndG1B1

λ1χ1, where χ1 = (µ, . . . , µ︸︷︷︸r1

) and G1 = G(r1) and similarly for G′2, G′3.

Thus, we may also identify IndGB λχ with IndG1B1

λ1χ1 ⊗ IndG2B2

λ2χ2 ⊗IndG3

B3λ3χ3 ⊗ IndG

′0

B′0λ0 (with G0 = G′0). This correspondence is done in


general in [Ja3]. However, there are two basic obstacles to using [Ja3] here.The first is that, as with [Au1], [Au2], [Sc-St], the results in [Ja3] aredone in the Grothendieck group setting, hence do not deal with compositionseries. The second is that we deal with IndG

′1

B′1λ1, etc., by using Mœglin’s

results. To work with IndG1B1

λ1χ1, etc., we would have to establish thecorresponding results ourselves.

To reduce our problem to that covered by Mœglin’s results, we will alsoneed the following proposition. (We continue to use j for the Iwahori-Matsumoto involution.)

Proposition 3.2.12. Suppose π and π′ = π′1 × π′2 × π′3 × π′0 are corre-sponding irreducible representations. Then, j(π) is tempered (resp., square-integrable) if and only if j(π′1), j(π

′2), j(π

′3), j(π

′0) are all tempered (resp.,

square-integrable).

Proof. Note that j(π′) is tempered (resp., square-integrable) if and only ifj(π′1), j(π

′2), j(π

′3), j(π

′0) are all tempered (resp., square-integrable).

Let us write θ ∈ Jac(π) if θ appears in the normalized Jacquet moduleof π (with respect to the Borel subgroup) with multiplicity at least one.Observe that by the abelianness of T and Frobenius reciprocity, we haveθ ∈ Jac(π) if and only if π → IndGBθ. Therefore, by Theorem 1.8 andFrobenius reciprocity, we see that θ ∈ Jac(π) if and only if θ−1 ∈ Jac(j(π)).Similarly, θi ∈ Jac(π′i) if and only if θ−1

i ∈ Jac(j(π′i)).For notational convenience, let µ(k) = µ× · · · × µ︸︷︷︸

k

. Let µ1 = µ, µ2 = µµnr,

µ3 = µnr, µ0 = 1, as above. We claim that λ1µ(r1)1 × λ2µ

(r2)2 × λ3µ

(r3)3 ×

λ0µ(r0)0 ∈ Jac(π) if and only if λi ∈ Jac(π′i) for i = 0, 1, 2, 3. First, let

G′′0 = G(r3 + r0) and

G′′1 =

O(2(r1 + r2)), if G = Sp(2n), O(2n)SO(2(r1 + r2) + 1), if G = SO(2n+ 1).

Suppose that π corresponds to π′′1 × π′′0 under Roche’s isomorphism. Wethen argue as follows: (λ1µ

(r1)1 × λ2µ

(r2)2 ) × (λ3µ

(r3)3 × λ0µ

(r0)0 ) ∈ Jac(π) if

and only if π → IndGB((λ1µ(r1)1 ×λ2µ

(r2)2 )× (λ3µ

(r3)3 ×λ0µ

(r0)0 )) if and only if

π′′1 → IndG′′1

B′′1(λ1 × λ2µ

(r2)nr ) and π′′0 → IndG

′′0

B′′0(λ3µ

(r3)nr × λ0) (cf. [R, Theorem

9.5]) if and only if λ1×λ2µ(r2)nr ∈ Jac(π′′1) and λ3µ

(r3)nr ×λ0 ∈ Jac(π′′0). We use

the same basic argument in conjunction with the results of Barbasch-Moy,making a few minor modifications to cover induction via tensor product.We argue as follows for π′′1 : λ1 × λ2µ

(r2)nr ∈ Jac(π′′1) if and only if π′′1 →

IndG′′1

B′′1(λ1×λ2µ

(r2)nr ) if and only if π′′1 → ⊗-IndG

′′1

B′′1(λ−1

1 ×λ−12 µ

(r2)nr ) if and only


if π′1 → ⊗-IndG′1

B′1λ−1

1 and π′2 → ⊗-IndG′2

B′2λ−1

2 (cf. [B-Mo2, Theorem 6.2]) ifand only if λ1 ∈ Jac(π′1) and λ2 ∈ Jac(π′2). The argument for π′′0 is similar.This verifies our claim.

Next we claim that any θ ∈ Jac(π) has the form sh(λ1µ(r1)1 , λ2µ

(r2)2 ,

λ3µ(r3)3 , λ0µ

(r0)0 ) for some λ1, λ2, λ3, λ0 with λi ∈ Jac(π′i), i = 0, 1, 2, 3. Here,

sh denotes a shuffle, used in the usual sense (e.g., see [Ja3, Definition 3.1]).This claim follows immediately from the discussion above and [Ja3, Lemma5.4].

Since θ ∈ Jac(π) if and only if θ−1 ∈ Jac(j(π)), the inequalities requiredby the Casselman criteria for j(π) to be tempered (resp., square-integrable)have the same form as those in [M1, Remarque 1.3.5] (π′1, π

′2, π′3, π′0 are al-

ready covered by [M1, Remarque 1.3.5]). Further, observe that λ1, λ2, λ3, λ0

each satisfy the inequalities of [M1, Remarque 1.3.5] if and only ifsh(λ1µ

(r1)1 , λ2µ

(r2)2 , λ3µ

(r3)3 , λ0µ

(r0)0 ) satisfies the inequalities of [M1, Remar-

que 1.3.5] for every shuffle sh. The proof of this is straightforward; essentiallyidentical to that of [Ja3, Corollary 8.3]. Thus, j(π) is tempered (resp.,square-integrable) if and only if j(π′1), j(π

′2), j(π

′3), j(π

′0) are all tempered

(resp., square-integrable), as needed.

Remark 3.2.4. We may also use the above result to classify the square-integrable (resp., tempered) representations of Sp(2n, F ), SO(2n+1, F ) sup-ported on the Borel subgroup (at least for F having odd residual character-istic). By [Ta3, Theorem 6.2], such a square-integrable (resp., tempered)representation has cuspidal support contained in | |αµα∈R,µ2=1. Now,to classify such representations, it suffices to classify the representations πwith cuspidal support in | |αµα∈R,µ2=1 such that j(π) is square-integrable(resp., tempered). By the preceding proposition, it suffices to classify thecorresponding representations π′1, π

′2, π′3, π′0 of G′1, G

′2, G

′3, G

′0. This is done

in [M1].3.3. The definition of Unip(p, χ). We prove:

Proposition 3.3.1.(1) R(wp, λp, χ)I(λp, χ) is semi-simple and the generalized Iwahori-Matsu-

moto involution of its direct summands is tempered. Let Unip(p, χ) bethe set of direct summands of R(wp, λp, χ)I(λp, χ).

(2) Under the Hecke algebra isomorphism, Unip(p, χ) corresponds to a sub-set of Unip(O1)× · · · ×Unip(Ok)×Unip(O0).

Proof. By Proposition 3.2.1, it is enough to prove for χ = 1. For simplicity,we denote Unip(p, 1) = Unip(p). Let O be the unipotent orbit obtainedfrom p by ignoring the ordering: We will prove Unip(p) ⊂ Unip(O), whereO is a unipotent orbit obtained from p by ignoring the ordering.

If p = (a1, b1, . . . , as, bs, as+1) satisfies the two conditions in Remark 3.2,then we are in Mœglin’s situation. So it is clear by Mœglin’s result.


Otherwise, by Remark 3.3, such chain can be written as p = p1×· · ·×pk×p0, where pi ∈ P (Oi) and Oi is a distinguished unipotent orbit in G∗i for i =0, 1, . . . , k. Let µi, i = 1, . . . , k be non-trivial quadratic grossencharacterssuch that µiv = 1 for i = 1, . . . , k for a given non-archimedean place v. Letχ = χ(µ1, . . . , µ1︸︷︷︸

r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

).

Consider the pseudo-Eisenstein series attached to χ from [Ki3]:

lp(φ, λ, χ) =k∏i=0

∑wi∈Wi

r(wi,−λi,Φi)R(wpw−1k · · ·w

−10 , w0 · · ·wkλ, χ)

∑d∈D

r(dw1 · · ·wkw0,−λ,ΦD)R(d−1, dw1 · · ·wkw0λ, dχ)φ(dw1 · · ·wkw0λ),

where Φi’s are given by:

Φ1 = ei ± ej , 1 ≤ i < j ≤ r1,Φ2 = er1+i ± er1+j , 1 ≤ i < j ≤ r2,

...

Φk = er1+···+rk−1+i ± er1+···+rk−1+j , 1 ≤ i < j ≤ rk,Φ0 = er1+···+rk+i ± er1+···+rk+j ,

1 ≤ i < j ≤ r0, 2er1+···+rk+i, i = 1, . . . , r0,

ΦD = Φ+ −k⋃i=0

Φk.

We note that the above is for G = Sp(2n). If G = SO(2n + 1), we needto add, to Φi, er1+···+ri−1+j , j = 1, . . . , ri, for i = 1, . . . , k and in Φ0,2er1+···+rk+i should be er1+···+rk+i. If G = O(2n), then Φ0 does not havethe roots 2er1+···+rk+i, i = 1, . . . , r0.

Also D is the set of distinguished coset representatives for θ = ∆ −er1 − er1+1, er1+r2 − er1+r2+1, . . . , er1+···+rk − er1+···+rk+1 ⊂ ∆ = e1 −e2, . . . , en−1 − en and Wi is the Weyl group of G′i for i = 0, 1, . . . , k.Let λ = λ1 + · · · + λk + λ0, where λi = ar1+···+ri−1+1er1+···+ri−1+1 + · · · +ar1+···+rier1+···+ri for i = 1, . . . , k and λ0 = ar1+···+rk+1er1+···+rk+1 + · · · +anen.

Now we substitute χ = 1 and we show that lp(φ, λp, χ = 1) is well-defined.Mœglin showed that r(wi,−λi,Φi) is identically zero on V ′(pi) if wi /∈W (↑, pi). Since the local intertwining operators R(wp, λp) are well-defined byProposition 3.1, the only thing we need to show is that r(dw1 · · ·wkw0,−λ,ΦD) is holomorphic at λp for wi ∈ W (↑, pi) even if χ = 1. Recall that fornon-trivial χ, χ α is non-trivial for α ∈ ΦD and so it is holomorphic.


For this, we need to show that if for wl ∈ W (↑, pl), wl(er1+···+rl−1+i) < 0for some i = 1, . . . , rl, then wl(er1+···+rl−1+i) = −er1+···+rl+1−i. Then thepoles and zeros in r(dw1 · · ·wkw0,−λ,ΦD) cancel each other. We show itfor w1 ∈ W (↑, p1). The remaining cases are similar. Suppose w1(ei) = −ejfor 1 ≤ i, j ≤ r1. Recall the definition of W (↑, p1). It is the set of the Weylgroup elements of G′1 which send ei − ej , 1 ≤ i < j ≤ r1 to itself. Thenconsider w1(ep− ei) = w1(ep)+ ej . So w1(ep) = −eq, where q > j. Considerw1(ei− ep) = −ej +w1(ep). Then w1(ep) = −eq, where q < j. Therefore wecan see that w1(ep) = −er1+1−p.

Since lp(φ, λp, χ = 1) is well-defined, by Mœglin’s inner product for-mula (Section 2.2), it belongs to the residual spectrum L2(G(F )\G(A))(T,1).Its local components are precisely the image of intertwining operatorR(wp, λp, χv = 1)I(λp, χv = 1). Even though Proposition 2.8 is no longertrue since the normalized intertwining operators could vanish, we can showthat the image of intertwining operator R(wp, λp, χv = 1)I(λp, χv = 1) issemi-simple in the same way as in Proposition 2.7. Then by Mœglin [M1,p. 734], it is included in Unip(O), where O is the unipotent orbit obtainedby ignoring the ordering in p.

3.4. Parametrization of Unip(p, χ). Next we parametrize Unip(p, χ).

3.4.1. The case χ = 1. Because of Proposition 3.3.1, we can still defineR(σ(ai,bi)) by Proposition 2.9 and the following still holds.

Proposition 3.4.1. σ(ai,bi) 7−→ R(σ(ai,bi)) is a homomorphism of the groupid, σ(ai,bi) into the group of the intertwining operators of R(wp, λp)I(λp).

This means the following: For X ∈ Unip(p), let R(σ(ai,bi))X =ηpX(σ(ai,bi))X. Then ηp

X defines a character of A(O) such that ηpX(σ(ai)) =

ηpX(σ(bi)).Since Unip(p) ⊂ Unip(O), ηp

X ∈ Springer (O). Therefore we have:

Theorem 3.4.2. Unip(p) is parametrized by

C(p) = η ∈ Springer (O) : η(σ(ai)) = η(σ(bi)),

i = 1, . . . , s, η(σ(as+1)) = 1.

Example 3.4.1. Let G = Sp(8) and p = (5, 1, 3). Then λp = (2, 1, 0, 1)and wp = c1c2c4. O = (5, 3, 1) and Springer (O) has 3 elements, namely,η ∈ Springer (O) if and only if η(σ(5)) = η(σ(3)), η(σ(1)) = 1 or η(σ(3)) =η(σ(1)), η(σ(5)) = 1. Therefore, η ∈ Springer (O) : η(σ(5)) = η(σ(1)),η(σ(3)) = 1 has only the trivial character.

In order to apply the above theorem to the global situation, let O1, O2

be two distinguished unipotent orbits in G∗1 and G∗2, resp. (If G = Sp(2n),then G∗1 = O(2r1,C) and G∗2 = O(2r2 + 1,C). If G = SO(2n + 1), then


G∗1 = Sp(2r1,C) and G∗2 = Sp(2r2,C). If G = O(2n), then G∗1 = O(2r1,C)and G∗2 = O(2r2,C).) Then we get a unipotent orbit O in G∗ by combiningO1 and O2. Further we have canonical embedding A(Oi) ⊂ A(O).

For pi ∈ P (Oi), i = 1, 2, we get a chain p1 × p2 by shuffling the segmentsin p1 and p2 so that it satisfies (3.1), and thus we get Unip(p1 × p2). LetUnip(O1, O2) be the union of Unip(p1 × p2) as pi runs through P (Oi) fori = 1, 2. It is a subset of Unip(O). Then we have:

Theorem 3.4.3. Unip(O1, O2) is parametrized by

C(O1, O2) = η ∈ Springer (O) : η|A(O1) ∈ Springer (O1),

η|A(O2) ∈ Springer (O2).

This can be generalized easily. Let Oi be distinguished unipotent orbitsin G∗i for i = 0, 1, . . . , k. Let O be the unipotent orbit of G∗, obtained bycombining Oi’s. Then we can define Unip(O1, · · · , Ok, O0) and it is a subsetof Unip(O) and:

Theorem 3.4.4. Unip(O1, · · · , Ok, O0) is parametrized by

C(O1, . . . , Ok, O0) = η ∈ Springer (O) : η|A(Oi) ∈ Springer (Oi),

for i = 0, . . . , k.

Corollary 3.4.5. Let G = Sp(2n) and let O1 = (q1, . . . , qs−1, 1), s even, bea distinguished unipotent orbit of O(2n,C) and O0 = (1). Then Unip(O1, O0)is parametrized by Springer (O′), where O′ = (q1, . . . , qs−1).

3.4.2. The case χ = χ(µ, . . . , µ), µ non-trivial quadratic. Let p =(a1, b1, . . . , as, bs). In this case the above theorems for χ = 1 case hold. Weneed to use the generalized Iwahori-Matsumoto involution in Section 1. Wecan define R(σ(ai,bi), µ) in the similar way as in Proposition 2.9.

Let Unip(p, µ) be the set of components of R(wp, λp, χ)I(λp, χ). Let Obe a unipotent orbit obtained from p by ignoring the ordering. Then:

Theorem 3.4.6. Unip(p, µ) is parametrized by

C(p) = η ∈ Springer (O) : η(σ(ai)) = η(σ(bi)), i = 1, . . . , s.

Let Oi be distinguished unipotent orbits in G∗i for i = 1, . . . , k. Let O bethe unipotent orbit of G∗, obtained by combining Oi’s. Then we can defineUnip(O1, · · · , Ok, µ) and:

Theorem 3.4.7. Unip(O1, · · · , Ok, µ) is parametrized by

C(O1, . . . , Ok, µ) = η ∈ Springer (O) : η|A(Oi) ∈ Springer (Oi),

for i = 1, . . . , k.


3.4.3. The general case. It is enough to consider the case, χ =χ(µ, . . . , µ︸︷︷︸

r1

, 1, . . . , 1︸︷︷︸r0

), where µ is a non-trivial quadratic character. Let pi,

i = 1, 2, be chains in G∗1, G∗2, resp, from which we get unipotent orbits O1,

O2, by ignoring the ordering in pi’s. (If G = Sp(2n), then G∗1 = O(2r1,C)and G∗2 = O(2r0 + 1,C). If G = SO(2n + 1), then G∗1 = Sp(2r1,C) andG∗2 = Sp(2r0,C). If G = O(2n), then G∗1 = O(2r1,C) and G∗2 = O(2r0,C).)Let p = p1 × p2.

We obtain R(σi)’s and R(σj , µ)’s. Let Unip(p, χ) be the set of componentsof R(wp, λp, χ)I(λp, χv). Then:

Theorem 3.4.8. Unip(p, χ) is parametrized by C(p1)× C(p2).

Let χ = χ(µ1, . . . , µ1︸︷︷︸r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

), r0 + · · · + rk = n, r1 ≥

· · · ≥ rk, µi’s are distinct non-trivial quadratic characters. Set µ0 = 1. Letpi, i = 0, 1, . . . , k, be chains in G∗i , from which we get unipotent orbitsO1, . . . , Ok, O0, by ignoring the ordering in pi’s. Let p = p1 × · · · pk × p0.Then:

Theorem 3.4.9. Unip(p, χ) is parametrized by C(p1)×· · ·×C(pk)×C(p0).

In order to apply the above theorem to the global situation, let χ =χ(µ1, . . . , µ1︸︷︷︸

r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

), r0 + · · · + rk = n, r1 ≥ · · · ≥ rk ≥

2, µi’s are distinct non-trivial quadratic grossencharacters. Let Oi be adistinguished unipotent orbit in G∗i for i = 0, 1, . . . , k. Let pi ∈ P (Oi) fori = 0, . . . , k and p = p1 × · · · × pk × p0. Then we can shuffle the segmentsin p so that it satisfies the condition (3.1). We still call it p. For a non-archimedean place v, let Unip(O1, . . . , Ok, O0, χv) be the set of union ofUnip(p1, . . . , pk, p0, χv) as pi runs through P (Oi) for i = 0, . . . , k.

Theorem 3.4.10. Πresv = Unip(O1, . . . , Ok, O0, χv) is parametrized by

(3.2) C(O1, . . . , Ok, O0, χv) = [Springer (O1)× · · ·× Springer (Ok)× Springer (O0)],




G∗12 =



Example 3.4.2. Let G = Sp(28) and χ = χ(µ, . . . , µ︸︷︷︸10

, 1, . . . , 1︸︷︷︸4

), µ is a

quadratic grossencharacter. Let O1 = (9, 7, 3, 1) be a unipotent orbit inO20(C) and O2 = (5, 3, 1) a unipotent orbit in O9(C). Then for a non-archimedean place v, if µv 6= 1, then Πresv is parametrized by Springer (O1)×Springer (O2). It has 18 elements. Let µv = 1. Let O = (9, 7, 5, 3, 3, 1, 1).Then A(O) is an abelian group generated by order 2 elements σ(1), σ(3),σ(5), σ(7), σ(9). Consequently

Springer (O) = η ∈ A(O) : η(σ(9)) = η(σ(7)), η(σ(5)) = 1

or η(σ(7)) = η(σ(5)), η(σ(9)) = 1.Then C(O1, O2) = η ∈ Springer (O) : η|A(Oi) ∈ Springer (Oi), i = 1, 2.C(O1, O2) has 5 elements.

σ(9) 1 -1 1 1 -1

σ(7) 1 -1 -1 1 -1

σ(5) 1 1 -1 1 1

σ(3) 1 1 -1 -1 -1

σ(1) 1 1 1 -1 -1

Remark 3.4.1. Let λO be the conjugate of λp which is the closure ofthe positive Weyl chamber as in Lemma 3.2. Then by inducing in stages,I(λO, w−1

1 χv) = IndGP λO ⊗ IndMB w−11 χv, where P = MN is the parabolic

subgroup such that λO is in the positive Weyl chamber with respect to P .Then we can consider the Knapp-Stein R-group of IndMB w−1

1 χv. It is asubset of (3.2). In fact, the Knapp-Stein R-group is spanned by the or-der 2 elements cr1+···+ri for µiv 6= 1. Therefore, we can think of (3.2) as ageneralization of the Knapp-Stein R-group.

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Received June 15, 1999. The second author was partially supported by NSF grantDMS9610387.

Dept. of MathematicsOhio State UniversityColumbus, OH 43210E-mail address: [email protected]

Dept. of MathematicsEast Carolina UniversityGreenville, NC 27858

Dept. of MathematicsSouthern Illinois UniversityCarbondale, IL 62901E-mail address: [email protected]















RESIDUAL SPECTRUM OF SPLIT CLASSICAL GROUPS;CONTRIBUTION FROM BOREL SUBGROUPS

Henry H. Kim

We completely determine the residual automorphic repre-sentations coming from the torus of odd orthogonal groups.Under certain technical assumption on disconnected groups,our results are valid for symplectic groups and even orthog-onal groups. Moeglin has determined the residual spectrumattached to the trivial character of the torus for split classicalgroups. However, non-trivial characters present some non-trivial difficulties, both local and global. Local problems areresolved in a companion paper with C. Jantzen, where manyother useful results are obtained.

Introduction.

Let F be a number field and A its ring of adeles. Let G be a reductivegroup defined over F . For simplicity we assume that the center of G isanisotropic over F . A central problem in the theory of automorphic forms isto decompose the right regular representation of G(A) acting on the Hilbertspace L2(G(F )\G(A)). It has continuous spectrum and discrete spectrum.

L2(G(F )\G(A)) = L2dis (G(F )\G(A))⊕ L2

cont (G(F )\G(A)).

We are mainly interested in the discrete spectrum. Main contributions havebeen made by Langlands and Arthur. First of all Langlands described, usinghis theory of Eisenstein series, an orthogonal decomposition of this space ofthe form:

L2dis (G(F )\G(A)) =

⊕(M,π)

L2dis (G(F )\G(A))(M,π),

where (M,π) is a Levi subgroup with a cuspidal automorphic representa-tion π taken modulo conjugacy (here we normalize π so that the actionof the maximal split torus in the center of G at the archimedean places istrivial) and L2

dis (G(F )\G(A))(M,π) is a space of residues of Eisenstein seriesassociated to (M,π). Here we note that the subspace

⊕(G,π)L2dis (G(F )\G(A))(G,π),

is the space of cuspidal representations L2cusp (G(F )\G(A)). Its orthogonal

complement in L2dis (G(F )\G(A)) is called the residual spectrum and we

417



418 HENRY H. KIM

denote it by L2res (G(F )\G(A)). Therefore we have an orthogonal decompo-

sition

L2dis (G(F )\G(A)) = L2

cusp (G(F )\G(A))⊕ L2res (G(F )\G(A)).

Arthur described, motivated by his trace formula, described a conjecturaldecomposition of this space as follows:

L2dis (G(F )\G(A)) = ⊕ψL2(G(F )\G(A))ψ,

where ψ runs, modulo conjugacy, through the set of morphism ψ : LF ×SL2(C) 7−→ G∗, where LF is the conjectural tannakian group, G∗ is theLanglands’ L-group and ψ satisfies certain conditions; in particular, ψ re-stricted to SL2(C) is algebraic, ψ restricted to LF parametrizes a cuspidaltempered representation of a Levi subgroup and the image of ψ is not in-cluded in proper Levi subgroups. The space L2(G(F )\G(A))ψ is defined bylocal data (3.3).

Let Πres v be the set of the local components of the residual spectrumL2

dis (G(F )\G(A))(M,π). It is of great importance to prove that Πres v ⊂Πψv for some ψv and the multiplicity formula (3.1) holds: In other words,the Arthur parameters parametrize the residual spectrum. For this, it isnecessary to construct a set of characters Cres v ⊂ Cψv attached to Πres v .

We should note that Arthur’s trace formula will not separate the cuspi-dal part from the residual one and therefore one needs to study the residualspectrum by computing the residues of Eisenstein series. It is noted that thefirst exotic example of the residual spectrum of the split group G2, whichwas discovered by Langlands and further studied later by Moeglin and Wald-spurger, has been of significance to Arthur in formulating his conjectures.

There are two problems in calculating the residual spectrum. The firstproblem is global. It is concerned with calculating the residues of theEisenstein series via the constant term of the Eisenstein series. Computingresidues of Eisenstein series is a very difficult problem and requires both theknowledge of the poles of corresponding automorphic L-functions, as well ashandling very difficult computational combinatorics. These computationsare already highly non-trivial even for rank 2 split groups. For example, forthe split group G2, the Eisenstein series, built out of the maximal parabolicsubgroup attached to the long simple root, contains the third symmetricpower L-function of GL2. The precise location of the poles of the third sym-metric power L-function was resolved only in the recent work of [Ki-Sh2](see the introduction of the paper for the long history). The Eisenstein seriesbuilt out of the minimal parabolic subgroups do not present a problem withregards to poles of automorphic L-functions since the L-functions are HeckeL-functions. However, in this case, cancelation of poles of the normalizedintertwining operators presents a problem. Even for Sp4 [Ki1], one has touse a non-trivial fact on the subtle analysis of the normalized intertwining

RESIDUAL SPECTRUM OF SPLIT CLASSICAL GROUPS 419

operators for SL2 due to Labesse-Langlands [L-La]. The second problemis local. It is concerned with calculating the image of the normalized localintertwining operators of the generalized principal series. While the unitaryprincipal series are well understood thanks to the Knapp-Stein R-group the-ory, the generalized principal series are not. Moeglin’s result [M1] on theunramified principal series of the split classical groups seems to be mostsatisfactory so far.

At this moment, the most satisfying result about the residual spectrumis due to Moeglin-Waldspurger [M-W2] who completely determined theresidual spectrum for GLn and Arthur [A2] gave the Arthur parameters forthem. Their result is that the residual spectrum for GLn is parametrizedby the cuspidal representations of GLm and the principal unipotent orbitof GL n

m(C), where m|n. As we observed, Arthur’s parameters require the

introduction of the hypothetical group LF . However, there are parts of itthat can be parametrized by our existing knowledge. This is the case inparticular, if the Eisenstein series is built out of minimal parabolics. Inthis case, LF can be replaced by the global Weil group WF . Therefore, wewill restrict ourselves to the study of L2

dis (G(F )\G(A))(T,χ), where T is amaximal split torus and χ is a unitary character of T .

WhenG is a split classical group and χ is trivial, Moeglin [M1] completelysolved the problem with the restriction that archimedean components arespherical. In this case the Arthur parameter is given by

ψ : WF × SL2(C) 7−→ G∗,

where ψ|WF= 1 and ψ|SL2(C) is determined by a distinguished unipotent

orbit of G∗. Recall that the Springer correspondence is an injective mapfrom the set of irreducible characters of W , the Weyl group, into the setof pairs (O, η), where O is a unipotent orbit and η is an irreducible char-acter of A(u) = C(u)/C(u)0, u ∈ O, where C(u) is the centralizer of u inG∗. Let Springer (O) be the set of characters of A(u) which are in the im-age of the Springer correspondence. Then Moeglin’s result is that the localcomponents of the residual spectrum are parametrized by the distinguishedunipotent orbits O of G∗ and Springer (O). Let Unip(O) be the set of thelocal components of the residual spectrum. Then there is a pairing betweenUnip(O) and Springer (O) which gives a desired Arthur’s multiplicity for-mula (3.2).

Moeglin [M4] also constructed the local representations associated to theremaining characters of A(u) which are NOT in Springer (O). These shouldbe local components of cuspidal representations. She [M5] obtained a partialresult on how to determine when π ∈ Πψ is a cuspidal representation. Itwould be an important but a difficult problem. We note that Lusztig [Lu]

420 HENRY H. KIM

has a theory of the generalized Springer correspondence which gives theremaining characters of A(u).

In this paper, we will extend Moeglin’s result when χ is an arbitraryunitary character of T . For simplicity, we will describe the result for G =Sp2n. Because the result in [Ja-Ki] for Sp2n, O2n is not complete, our resultfor Sp2n is not complete. This is due to the fact that Barbasch-Moy’s result[B-Mo2] is not available for disconnected groups like O2n. However, weobtain a complete result for SO2n+1. Nevertheless, the symplectic groupcase will illustrate the technique and the heart of the matter.

As in [M1], we use G∗ = O2n+1(C) to denote its dual group. Letµ1, . . . , µk be k distinct non-trivial quadratic grossencharacters of F . Fixintegers r1 ≥ · · · ≥ rk ≥ 2 and r0 so that r0 + r1 + · · · + rk = n. Thenχ = χ(µ1, . . . , µ1︸︷︷︸

r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

). We showed in [Ki-Sh] that only

the character of the above type contributes to the residual spectrum. Thecharacter χ defines a homomorphism of the Weil group WF into a Cartansubgroup of SO2n+1(C). Composing this homomorphism with the stan-dard action of O2n+1(C) on C2n+1 will then give a completely reduciblerepresentation of WF on C2n+1 which decomposes according to eigenval-ues µ1, . . . , µk, and 1, with multiplicities 2r1, . . . , 2rk, and 2r0 + 1, respec-tively. Write C2n+1 = V0 ⊕ V1 ⊕ . . . ⊕ Vk, where each Vi,dimVi = 2ri, isthe eigenspace attached to eigenvalue µi, 1 ≤ i ≤ k, and V0 is the triv-ial eigenspace of dimension 2r0 + 1. In this way we get an embedding ofk∏i=0

O(Vi) ⊂ O2n+1(C). For each i, 0 ≤ i ≤ k, let Oi be the distinguished

unipotent orbit of O(Vi). Then Arthur parameter of interest to us is ahomomorphism

ψ : WF × SL2(C) 7−→k∏i=0

O(Vi) ⊂ O2n+1(C),

satisfying,

(1) ψ|WF : w 7−→ 1 × µ1(w) × · · · × µk(w) ∈ ±1 × ±1 × · · · × ±1,where ±1 is the center of O(Vi) for i = 0, . . . , k.

(2) By Jacobson-Morozov theorem, ψ|SL2(C) defines the unipotent orbit∏ki=0Oi of G∗.

The unipotent orbits Oi determine, through the Jacobson–Morozov’s the-orem, certain conjugacy class of unramified characters of T . Let λ0 =λ1,0 + . . . + λk,0 + λ0,0 be the one in the positive Weyl chamber. To ψ,Arthur associates a Langlands’ parameter φψ:

φψ : WF −−−→ O2n+1(C),


where φψ(w) = ψ

(w,

(|w|

12 0

0 |w|−12

)). It is given by φψ = χ ⊗ exp〈λ0,

HB( )〉. Its non-tempered part is φ+ψ = exp〈λ0,HB( )〉.

In [Ki-Sh], we studied the special case when Oi is the unipotent orbitof the form (2ri − 1, 1) for i = 1, . . . , k and O0 is of the form (2r0 + 1).This is the opposite of the trivial character case. In this case, we showedthat the local components of the residual spectrum in Πφψ are parametrizedby the Knapp-Stein R-group of the unitary principal series Iv = IndM

B0χv,

where M is the Levi subgroup whose L-group is M∗ = Cent(imφ+ψ , G

∗). Wenote that M is of the form GLn1 ×· · ·×GLnk ×Sp2k. Keys-Shahidi pairing[Ke-Sh] gives a desired multiplicity formula (3.2). We note that the methodof [Ki-Sh] can only prove “if” part in (3.2). We need the inner product ofpseudo-Eisenstein series to prove “if and only if.”

In general case, the Knapp-Stein R-groups are not enough. Our result isthat the local component of the residual spectrum Cres v is parametrized bythe unipotent orbits Oi, i = 0, . . . , k and [Springer (O0)×· · ·×Springer (Ok)],where [ ] is defined as follows (see Theorem 6.2.2): If µ1v = µ2v 6= µiv fori = 3, . . . , k, then we replace Springer (O1)× Springer (O2) by


where O is a unipotent orbit of Sp(2(r1 + r2),C) by combining O1, O2. InSection 6, we show that the calculation of the residues of the Eisenstein seriesis reduced to that of the Eisenstein series of O2r1×· · ·×O2rk×Sp2r0 attachedto the trivial character. There remains a local problem of calculating theimage of the normalized local intertwining operators. This has been takencare of by [Ja-Ki], under some assumptions that Barbasch-Moy’s result[B-Mo2] holds for disconnected groupO2n. Therefore for symplectic groups,the result is not complete. We review the result in Section 6.

In Section 3, we review quadratic unipotent Arthur parameters andMoeglin’s reformulation of Arthur’s conjecture on the multiplicity formula.In Section 4, we review Eisenstein series and pseudo-Eisenstein series inthe sense of [M-W1] attached to a character of a Borel subgroup. In Sec-tion 7, we show how automorphic representations obtained by Kudla-Rallis[Ku-Ra] by considering degenerate principal series, are fit as a special case.In Section 8, we explain how the result in [Ki-Sh] is fit as a special case.

Acknowledgments. We would like to thank Prof. Shahidi. Without hisguidance and encouragement this paper could not have been finished. Wewould like to thank Prof. Moeglin for patiently answering many of ourquestions [M5] and for many correspondences. We also thank Dr. Jantzenfor many discussions. Without his help on local questions [Ja-Ki], thispaper could not have been finished.

422 HENRY H. KIM

1. Preliminaries.

Let F be a field and let G = SO2n+1, Sp2n or SO2n over F . Let Jn be then× n matrix given by

Jn =

1

1.

..

1

.

Let J ′2n =(

Jn−Jn

). Then

Sp(2n) =g ∈ GL(2n)

∣∣ tgJ ′2ng = J ′2n,

andSO(n) =

g ∈ GL(n)

∣∣ tgJng = Jn; det(g) = 1.

In each case we let T be the maximal split torus consisting of diagonalmatrices in G. Then

T (F ) =

t(l1, . . . , ln) =

l1l2

. . .ln

l−1n

. . .l−12

l−11

∣∣∣∣ li ∈ F ∗

,

if G = Sp(2n) or SO(2n), and

T (F ) =

t(l1, ..., ln) =

l1l2

. . .ln

1l−1n

. . .l−12

l−11

∣∣∣∣ li ∈ F ∗

if G = SO(2n+ 1).

Let Φ(G,T ) be the roots of G with respect to T . We choose the orderingon the roots so that the Borel subgroup B is the subgroup of upper triangular


matrices in G. Let ∆ be the simple roots in Φ(G,T ) given by ∆ = αjnj=1 ,

with αj = ej − ej+1 for 1 ≤ j ≤ n− 1, and

αn =

en G = SO(2n+ 1),2en G = Sp(2n),en−1 + en G = SO(2n).

We let 〈, 〉 be the standard Euclidean inner product on Φ(G,T ). If Φ is aroot system of type Bn, Cn, or Dn, then we denote by G(Φ) the split groupwith root system Φ.

For G = SO(2n+ 1) or Sp(2n), the Weyl group W (G/T ) ' Sn n Zn2 . Snacts by permutations on the λi, i = 1, . . . , n. We will use standard cyclenotation for the elements of Sn. Thus (ij) interchanges λi and λj . If ci isthe non-trivial element in the i-th copy of Z2 then ci takes λi to λ−1

i . Theelement ci is called a sign change because its action on Φ(G,T ) takes ei to−ei. For G = SO(2n), the Weyl group is given by W (G/T ) ' Sn n Zn−1

2 .Sn acts by permutations on the λi, and Zn−1

2 acts by even numbers of signchanges. The requirement that the number of sign changes be even comesfrom the determinant condition in SO(2n). Note that the sign change ciis an element of O(2n) and normalizes T (F ). Each ci acts on SO(2n) byconjugation, and cn induces the non-trivial graph automorphism on theDynkin diagram of Φ(G,T ).

2. Unipotent orbits of classical groups over C.

Theory of Jordan normal forms implies that a unipotent matrix in GLN isconjugate to J(p1)⊕J(p2)⊕· · ·⊕J(ps), p1 ≥ p2 ≥ · · · ≥ ps, p1+p2+· · ·+ps =N , where J(p) is the p × p Jordan matrix with entries 1 just above thediagonal and the diagonal and zero everywhere else. Therefore unipotentclasses in GLN are in 1 to 1 correspondence with partitions λ of N . We usethe following standard notation for λ: λ = (1r1 , 2r2 , 3r3 , . . . ), where rj is thenumber of pi equal to j.

Let G be a classical group, of type Bn (O2n+1(C)), Cn (Sp2n(C)) or Dn

(O2n(C)). We start with the following facts:(1) X,X ′ ∈ G are conjugate in G if and only if they are conjugate in

GLN , N = 2n+ 1 or 2n.(2) Let X ∈ GLN be unipotent. Then X is conjugate to an element of G

if and only if ri is even for even i in the orthogonal case and for odd iin the symplectic case.

Therefore for G = O2n+1(C), unipotent classes are in 1 to 1 correspondencewith partitions λ of 2n+ 1 such that ri is even for even i.

Let u be a unipotent element in G and let Su be its centralizer in G. Thenwe have:

424 HENRY H. KIM

(3) In the orthogonal case (resp. symplectic) case, Su/Su is k product ofZ/2Z, where k is the number of odd (resp. even) i such that ri > 0.

Here we note that for G = GLN (C), the centralizer ZG(S) is connectedfor any subset S of G.

We say that a unipotent element u is distinguished if all maximal tori ofCent (u,G) are contained in the center of G, the connected component ofthe identity. This is equivalent to the fact that the unipotent orbit O of udoes not meet any proper Levi subgroup of G (Spaltenstein [Sp, p. 67]).(I.e., if L is a Levi subgroup of a parabolic subgroup of G and u ∈ L for au ∈ O, then L = G.) If G = O2n+1(C), then G = SO2n+1(C) and G

has trivial center. By Carter [C], for G = O2n+1(C) or O2n(C), if u is aunipotent element with Jordan blocks (1r1 , 2r2 , . . . ), then the reductive partof the connected centralizer Cent(u,G) is of type∏

i even

Cri/2 ×∏

i odd , ri even

Dri/2 ×∏

i odd , ri odd

B(ri−1)/2.

Therefore, O is a distinguished unipotent class if and only if it has Jordanblocks (1r1 , 3r3 , 5r5 , . . . ), where ri = 0 or 1.

Jacobson-Morozov Theorem. Suppose u is a unipotent element in asemi-simple algebraic group G. Then there exists a homomorphism φ :

SL2 7−→ G such that φ(

1 10 1

)= u.

Here, replacing φ by a conjugate underG, we can assume that φ(a 00 a−1

)is in the closure of the positive Weyl chamber in the maximal torus. Infact, by the theory of weighted Dynkin diagrams (cf. Section 5.6 of [C]),

φ

(a 00 a−1

)is uniquely determined by the unipotent orbit of u as follows

(Carter [C, p. 395]):Suppose O has Jordan blocks (d1, d2, d3, . . . ). For each di, we take the set

of integers di − 1, di − 3, . . . , 3− di, 1− di. We then take the union of thesesets for all di and write this union as (ξ1, ξ2, ξ3, . . . ) with ξ1 ≥ ξ2 ≥ ξ3 ≥ . . . .Then

(2.1) φ

(a 00 a−1

)= diag(aξ1 , aξ2 , aξ3 , . . . ).

Lemma ([B-V, Prop. 2.4]). Let u be a unipotent element and φ : SL2 7−→

G be a homomorphism such that φ(

1 10 1

)= u. Let Sφ = Cent(imφ,G) ⊂

Su = Cent(u,G) and Uu be the unipotent radical of Su. Then:

(1) Su = Sφ · Uu, a semi-direct product. Sφ is reductive.


(2) The inclusion Sφ ⊂ Su induces an isomorphism between Sφ/SφZG andSu/S

uZG.

3. Quadratic unipotent Arthur parameters.

We follow Moeglin [M3]. Let F be a number field and let WF be the globalWeil group of F . Let G = Sp2n, SO2n+1, O2n and we can take the dual groupG∗ = O2n+1(C), Sp2n(C), O2n(C). An Arthur parameter is a homomorphism

ψ : WF × SL2(C) 7−→ G∗,

with the following properties: (The usual definition of Arthur parameteruses Langlands’ hypothetical group LF instead of WF . But since we areonly dealing with Langlands’ quotients which come from principal series,WF is enough.)

(1) ψ(WF ) is bounded and included in the set of semi-simple elements ofG∗.

(2) The restriction of ψ to SL2(C) is algebraic.(3) Composing ψ|WF

with the determinant of G∗ gives a quadratic char-acter of WF , denoted by detψ. We want detψ = 1.

We call an Arthur parameter quadratic unipotent if the followingcondition is satisfied:

(4) ψ|WFis trivial on the intersection of the kernels of the quadratic char-

acters of WF .Because of conditions (1), and (4), the action of ψ(WF ) gives an orthog-

onal decomposition:

C2n+1,C2n = V1 ⊕ · · · ⊕ Vk ⊕ V0,

where dim V0 = 2r0 + 1 or 2r0, dim Vi = 2ri, 2r0 + 1 + 2r1 + · · · + 2rk =2n + 1, 2n, r1 ≥ · · · ≥ rk and Vi is the eigenspace with eigenvalue µi. Hereµ1, . . . , µk are non-trivial distinct quadratic grossencharacters of F , viewedas characters of WF , and dim Vi, i = 1, . . . , k, being even comes from con-dition (3).

The parameter ψ factors through∏ki=0G

∗i , where

G∗i =

O(Vi), if G = Sp2n, O2n

Sp(Vi), if G = SO2n+1:

ψ : WF × SL2(C) 7−→k∏i=0

G∗i .

(1) WF is mapped into the product of centers of G∗iψ|WF : w 7−→ 1× µ1(w)× · · · × µk(w) ∈ ±1 × ±1 × · · · × ±1,

where ±1 is the center of G∗i , for i = 0, . . . , k.

426 HENRY H. KIM

(2) By Jacobson-Morozov theorem, ψ|SL2(C) defines a unipotent orbit ofG∗ of the form

k∏i=0

Oi,

where Oi is a unipotent orbit of G∗i . Inside Oi we fix an element uisuch that

ψ

(1 10 1

)=

k∏i=0

ui.

Let Sψ = Cent(imψ,G∗) and

Cψ = Sψ/SψZG∗ .

We know that Sψ is a maximal reductive subgroup of∏ki=0 Cent (ui, G∗i ).

Therefore Sψ = 1, i.e., Sψ is finite if and only if each ui is a distinguishedunipotent element inG∗i . Especially, since O2(C) has no distinguished unipo-tent element, we have:

Lemma. Let ψ be a quadratic unipotent Arthur parameter. Suppose Sψ = 1.Then rk ≥ 2.

Now it is clear that Sψ/SψZG∗ is equal to

Cent (u0, G∗0)/Cent (u0, G

∗0)ZG∗0

k∏i=1

Cent (ui, G∗i )/Cent (ui, G∗i ).

Here Cent (ui, G∗i )/Cent (ui, G∗i ) is t product of Z/2Z, where t is the number

of i with ri > 0 in Jordan blocks.For each place v of F , we have a map ψv = ψ|WFv×SL2(C). As in global

case, we can then define Sψv . But in the local case, µiv may not be distinct.Suppose µ1v = µ2v 6= µiv for i = 3, . . . , k and µ1v 6= 1. Then in the aboveformula,

Cent (u1, G∗1)/Cent (u1, G

∗1) × Cent (u2, G

∗2)/Cent (u2, G

∗2)

must be replaced by

Cent (u1 × u2, G∗12)/Cent (u1 × u2, G

∗12),

where G∗12 = O(V1⊕V2) or Sp(V1⊕V2). Now we recall Moeglin’s reformula-tion of Arthur’s conjecture ([M3]): It is a part of local Arthur’s conjecturethat for each irreducible character ηv of Cψv , there exists an irreduciblerepresentation π(ψv, ηv). For each v, let Πψv be the set of π(ψv, ηv).

We define the global Arthur packet Πψ to be the set of irreducible rep-resentations π = ⊗vπv of G(A) such that for each v, πv belongs to Πψv .


Arthur’s Conjecture (Global).(1) The representations in the packet corresponding to ψ may occur in the

discrete spectrum if and only if Sψ is finite, i.e., Sψ = 1. We call suchan Arthur parameter elliptic.

(2) For an elliptic Arthur parameter ψ, any π ∈ Πψ occurs in L2dis (G(F )\

G(A)) if and only if

(3.1)∑x∈Cψ

∏v

ηv(xv) 6= 0,

where π = ⊗vπ(ψv, ηv), x = (xv). Note that, if Cψ is abelian, (3.1) isequivalent to

(3.2)∏v

ηv|Cψ = 1,

when ψ|WFis trivial.

We define

(3.3) L2(G(F )\G(A))ψ = Πψ ∩ L2dis (G(F )\G(A)).

Remark 3.1. For split classical groups, Cψ is abelian and Moeglin [M2]proved the multiplicity formula (3.2). However, in general, Cψ is not abelian.For example, in the case of split exceptional group of type G2, Cψ can beS3, the symmetric group on 3 letters. This fact leads to a “bizarre” multi-plicity formula in Moeglin-Waldspurger’s work on the residual spectrum ofG2 [M-W1, Ki2].

Let Πres v be the subset of Πψv which consists of the local components ofthe residual spectrum. It is of great importance to parametrize the elementsin Πres v and prove the multiplicity formula (3.1), i.e., we will construct aset of characters Cres v ⊂ Cψv and each character of Cres v gives an elementin Πres v .

Remark 3.2. To any Arthur parameter ψ, Arthur associates a Langlands’parameter φψ : WF 7−→ G∗ as follows:

φψ(w) = ψ

(w,

(|w|

12 0

0 |w|−12

)).

Let Sφψ = Cent (imφψ, G∗) and Cφψ = Sφψ/S

φψZG∗ . For each place v, we

have Sφψv , Cφψv . For each v, there is a natural surjection Cψv → Cφψv . Theparameter φψv gives a L-packet Πφψv

which consists of Langlands’ quotients.It is a part of Arthur’s original local conjecture that for each place v, thereis a pairing 〈 , 〉 on Cφψv × Πφψv

and an enlargement Πψv of Πφψvwhich

allows an extension of 〈 , 〉 to Cψv × Πψv such that π ∈ Πφψv⊂ Πψv if and

only if the function 〈 ,π〉 lies in the image of Cφψv in Cψv . Since Cψv is

428 HENRY H. KIM

abelian in our case, giving a pairing between Cψv and Πψv is the same asgiving a character of Cψv .

Note that in the unipotent case (Moeglin’s case) where ψ|WFis trivial,

Cφψ = 1. Therefore, Πφψvconsists of only one element, that is, the spherical

one. So there is no non-trivial extension to Πψv .

4. Eisenstein series attached to Borel subgroups.

Let α∨ be the coroot corresponding to α ∈ Φ+(G,T ). Explicitly, for α =ei − ej , α∨(l) = t(1, . . . , l

i, . . . , l−1

j, . . . , 1) ∈ T (F ) for 1 ≤ i < j ≤ n. For

α = ei+ ej , α∨(l) = t(1, . . . , li, . . . , l

j, . . . , 1), for 1 ≤ i < j ≤ n. For α = 2ei,

α∨(l) = t(1, . . . , li, . . . , 1) for 1 ≤ i ≤ n. Here dots represent 1.

Let X(T ) (resp. X∗(T )) be the character (resp. cocharacter) group of T .There is a natural pairing 〈, 〉 : X(T ) ×X∗(T ) 7−→ Z. For α, β ∈ Φ(G,T ),〈β, α∨〉 = 2(β, α)/(α, α), where ( , ) is the standard inner product inΦ(G,T ). Let ωi = e1 + · · · + ei. Then ω1, . . . , ωn are the fundamentalweights of G with respect to (G,T ). (If G = Sp2n, since G is simplyconnected, X(T ) = Zω1 + · · · + Zωn and X∗(T ) = Zσ∨1 + · · ·Zσ∨n .) Seta∗ = X(T ) ⊗ R, a∗C = X(T ) ⊗ C, and a = X∗(T ) ⊗ R = Hom (X(T ),R),aC = X∗(T )⊗ C. The positive Weyl chamber in a∗ is

C+ = λ ∈ a∗|〈λ, α∨〉 > 0, for all α positive roots

=

n∑i=1

aiωi| ai > 0

.

We fix a non-trivial additive character ψF = ⊗vψFv of A/F and letL(z, µ) be the Hecke L-function with the ordinary Γ-factor so that it satis-fies the functional equation L(z, µ) = ε(z, µ)L(1 − z, µ−1), where ε(z, µ) =∏v ε(z, µv, ψFv) is the usual ε-factor. If µ is the trivial character µ0, then

we write simply L(z) for L(z, µ0). We have the Laurent expansion of L(z)at z = 1:

L(z) =c(F )z − 1

+ a+ · · · .

4.1. Definition of Eisenstein series. For µ1, . . . , µn grossencharacters ofF , we define a character χ = χ(µ1, . . . , µn) of T (A) by

χ(µ1, . . . , µn)(t(l1, . . . , ln)) = µ1(l1) · · ·µn(ln).Let B = TU , where U is the unipotent radical. Let I(χ) be the spaceof functions Φ on G(A) satisfying Φ(utg) = χ(t)Φ(g) for any u ∈ U(A),t ∈ T (A) and g ∈ G(A). Then for each λ ∈ a∗C, the representation of G(A)on the space of functions of the form

g 7−→ Φ(g) exp〈λ+ ρB,HB(g)〉, Φ ∈ I(χ),


is equivalent to I(λ, χ) = IndB↑G

χ ⊗ exp(λ,HB( )). We form the Eisenstein

series:E(g, f, λ) =

∑γ∈B(F )\G(F )

f(γg),

where f = Φ e〈λ+ρB ,HB( )〉 ∈ I(λ, χ) and ρB is the half-sum of positive roots,i.e., ρB = ω1 + · · · + ωn. It converges absolutely for Reλ ∈ C+ + ρB andextends to a meromorphic function of λ. It is an automorphic form and theconstant term of E(g, f, λ) along B is given by

E0(g, f, λ) =∫U(F )\U(A)

E(ug, f, λ) du =∑w∈W

M(w, λ, χ)f(g),

where W is the Weyl group of T and

M(w, λ, χ)f(g) =∫wU(A)w−1∩U(A)\U(A)

f(w−1ug) du.

Then M(w, λ, χ) defines an intertwining map from I(λ, χ) to I(wλ,wχ) andsatisfies a functional equation of the form

M(w1w2, λ, χ) = M(w1, w2λ,w2χ)M(w2, λ, χ).

Let S be a finite set of places of F , including all the archimedean placessuch that for every v /∈ S, χv, ψFv are unramified and if f = ⊗fv, for v /∈ S,fv is the unique Kv-fixed function normalized by fv(ev) = 1. We have

M(w, λ, χ) = ⊗vA(w, λ, χv).

Then by applying Gindikin-Karpelevic method, we can see that for v /∈ S,

A(w, λ, χv)fv =∏

α>0,wα<0

L(〈λ, α∨〉, χv α∨)L(〈λ, α∨〉+ 1, χv α∨)

fv,

where fv is the Kv-fixed function in the space of I(wλ,wχ). For any v, let

rv(w) =∏

α>0,wα<0

L(〈λ, α∨〉, χv α∨)L(〈λ, α∨〉+ 1, χv α∨)ε(〈λ, α∨〉, χv α∨, ψFv)

.

We normalize the intertwining operators A(w, λ, χv) for all v by

A(w, λ, χv) = rv(w)R(w, λ, χv).

Let R(w, λ, χ) = ⊗vR(w, λ, χv) and

r(w) = Πvrv(w) =∏

α>0,wα<0

L(〈λ, α∨〉, χ α∨)L(〈λ, α∨〉+ 1, χ α∨)ε(〈λ, α∨〉, χ α∨)

.

Here R(w, λ, χ) satisfies the functional equation

R(w1w2, λ, χ) = R(w1, w2λ,w2χ)R(w2, λ, χ),

430 HENRY H. KIM

for any w1, w2. We know, by Winarsky [W] for p-adic cases and Shahidi[Sh3, p. 110] for real and complex cases, that

(4.1) A(w, λ, χv)∏

α>0,wα<0

Lv(〈λ, α∨〉, χv α∨)−1

is holomorphic for any v. So for any v, R(w, λ, χv) is holomorphic for λ withRe(〈λ, α∨〉) > −1, for all positive α with wα < 0. For χ = χ(µ1, . . . , µn),

χ α∨ =

µiµ−1j , for α = ei − ej

µiµj , for α = ei + ej and i < j

µi, for α = 2eiµ2i , for α = ei.

For α ∈ Φ+, let Sα = λ ∈ a∗C|〈λ, α∨〉 = 1. We call Sα a singularhyperplane. We say that E(g, f, λ) has a pole of order n at λ0 if λ0 isthe intersection of n singular hyperplanes in general position on which theEisenstein series has a simple pole.

For Ψ ⊂ Φ+, we define r(w, λ,Ψ) by

r(w, λ,Ψ) =∏

α∈Ψ,wα<0

L(〈λ, α∨〉, χ α∨)L(〈λ, α∨〉+ 1, χ α∨)ε(〈λ, α∨〉, χ α∨)

.

Observe that we have suppressed the dependence of r(w, λ,Ψ) on χ.

4.2. Definition of pseudo-Eisenstein series. We follow Moeglin [M1]and introduce pseudo-Eisenstein series. For T a maximal torus, a characterχ of T (A)/T (F ) defines a cuspidal representation of T . For any w ∈ W ,wTw−1 = T and so (T,wχ) is conjugate to (T, χ). Let I(χ) be the set ofentire functions φ of Paley-Wiener type such that φ(λ) ∈ I(λ, χ) for each λ.Let

θφ(g) =(

12πi

)n ∫Reλ=λ0

E(g, φ(λ), λ) dλ,

where λ0 ∈ ρB + C+. Let

L2(G(F )\G(A))(T,χ),

be the space spanned by θφ for all φ ∈ I(wχ) as wχ runs through all dis-tinct conjugates of χ. Let L2

dis (G(F )\G(A))(T,χ) be the discrete part ofL2(G(F )\G(A))(T,χ). It is the set of iterated residues of E(g, φ(λ), λ) of or-der n and the residual spectrum attached to (T, χ). In order to decomposeL2

dis (G(F )\G(A))(T,χ), we use the inner product formula of two pseudo-Eisenstein series: Let χ and χ′ be conjugate characters and φ ∈ I(χ),


φ′ ∈ I(χ′). Then

〈θφ, θφ′〉 =1

(2πi)n

∫Reλ=λ0

∑w∈W (χ,χ′)

(M(w−1,−wλ,wχ)φ′(−wλ), φ(λ)) dλ

=1

(2πi)n

∫Reλ=λ0

∑w∈W (χ,χ′)

(M(w, λ, χ)φ(λ), φ′(−wλ)) dλ,

where W (χ, χ′) = w ∈ W |wχ = χ′. We can assume, after conjugation,χ = χ(µ1, . . . , µ1︸︷︷︸

r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

), r0 + · · · + rk = n, r1 ≥ · · · ≥ rk,

where µi’s are non-trivial distinct quadratic grossencharacters. Let D bethe set of distinguished coset representatives in Proposition 6.2.2. Thendχ| d ∈ D is the set of distinct conjugates of χ.

Following Moeglin [M1], we consider the constant term of the pseudo-Eisenstein series:

EPS0 (φ, λ, χ) =∑w∈W

M(w−1, wλ,wχ)φ(wλ).

Here∑

w∈W M(w−1, wλ,wχ)φ(wλ) signifies

∑d∈D

∑w∈W (χ,dχ)

M(w−1, wλ,wχ)φd(wλ)

,

where φd ∈ I(dχ). Here M(w−1, wλ,wχ) = r(w−1, wλ,wχ)R(w−1, wλ,wχ).We show the following:

Lemma 4.1. (1) r(w, λ, χ) = r(w, λ, χ),(2) r(w−1, wλ,wχ) = r(w,−λ, χ−1).

Proof. (1) is immediate. We note that if χ = χ(µ1, . . . , µn), then χ−1 =χ(µ−1

1 , . . . , µ−1n ). Then

r(w−1, wλ,wχ) =∏

α>0,w−1α<0

L(〈wλ, α∨〉, (wχ) α∨)L(〈wλ, α∨〉+ 1, (wχ) α∨)

.

Let γ = −w−1α. Then α > 0|w−1α < 0 = −γ > 0|wγ < 0.If α = ei ± ej , 〈wλ, α∨〉 = (wλ, α) = (λ,w−1α) = −(λ, γ) = −〈λ, γ∨〉.

If α = 2ei, then 〈wλ, α∨〉 = 12(wλ, α) = 1

2(λ,w−1α) = −〈λ, γ∨〉. Also wecan see that wχ (wα)∨ = χ α∨. Therefore, (wχ) α∨ = χ (w−1α)∨ =χ (−γ)∨ = χ−1 γ∨. Hence (2) follows.

If χ is a quadratic character, i.e., χ−1 = χ, then


r(w,−λ, χ)R(w−1, wλ,wχ)φ(wλ).

432 HENRY H. KIM

Even though Moeglin completely analyzed the continuous spectrum aswell, we will only be interested in the discrete spectrum. Let 〈, 〉dis be the in-ner product on L2

dis (G(F )\G(A))(T,χ). The discrete spectrum is spanned bythe square integrable iterated residues of order n of EPS0 (φ, λ, χ). More pre-cisely, EPS0 (φ, λ, χ) has a simple pole on n singular hyperplanes H1, . . . , Hn

which are the poles of the L-functions in the numerator of the normalizingfactor and whose intersection is λ0. Then we have to take iterated residueat H1,H1 ∩H2, . . . , H1 ∩ · · · ∩Hn.

Let us summarize Moeglin’s arguments. Here χ = 1 and we suppress it:Moeglin showed that only the points λp, where p ∈ P (O) and O is a dis-tinguished unipotent orbit (see Section 5), contribute the square integrableresidues. She also found n singular hyperplanes H1, . . . , Hn whose intersec-tion is λp. In order to take the iterated residue at H1,H1 ∩ H2, . . . , H1 ∩· · · ∩ Hn, Moeglin defined the following function, for a certain Weyl groupelement wp,

lp(φ, λ) =∑w∈W


Poles of lp(φ, λ) in a neighborhood of λp are contained in the local inter-twining operators. She showed that lp(φ, λp) can be defined inductively byrestricting lp(φ, λ) to H1,H1∩H2, . . . , H1∩· · ·∩Hn. lp(φ, λp) is the iteratedresidue of EPS0 (φ, λ) and spans the residual spectrum L2

dis (G(F )\G(A))(T,1).lp(φ, λp) belongs to ⊗′vRv(wp, λp)Iv(λp). She analyzed the image of the lo-cal intertwining operator Rv(wp, λp)Iv(λp) and verified Arthur’s multiplicityformula. We will review her result in detail in Section 5.

In Section 6, we calculate the constant term of the pseudo-Eisensteinseries for a general χ and show that we can reduce the calculation of thepoles to that of the trivial character case and apply Moeglin’s result.

5. The trivial character case; summary of Moeglin’s result.

Moeglin [M1] completely analyzed the residual spectrum of Sp2n, SO2n+1

and split O2n, attached to the trivial character of the maximal torus. Herresults also completely describe continuous spectrum. Her results are thatthe residual spectrum of split classical groups attached to the trivial char-acter of the maximal torus is parametrized by distinguished unipotent or-bits O in G∗ and Springer (O), which is a set of characters of A(u) whichare in the image of the Springer correspondence and A(u) = C(u)/C(u)0,C(u) = Cent (u,G∗), u ∈ O. Recall that the Springer correspondence is aninjective map from the set of irreducible characters of W , the Weyl group,into the set of pairs (O, η), where O is a unipotent orbit and η is an irre-ducible character of A(u). We refer [Ja-Ki] for the review of her results.We just remark that since we are only interested in discrete spectrum, we


need to consider distinguished unipotent orbits, i.e., r = 0. In that case weget an explicit order of the set Springer (O).

Lemma 5.1. If O = (q1, ..., qs) is a distinguished unipotent orbit in O2n(C)or O2n+1(C) (thus qi odd ), then

|Springer (O)| = sC[ s2].

Proof. Let Symb (O) = (a1, . . . , as) (see [M1, p. 663]), where ai = [ qi2 ] + i.We note that ai ≤ ai+1− 2 for all i. (If we set qi = 2ri+1, qi+1 = 2ri+1 +1,ri < ri+1, then ai = ri + i and ai+1 = ri+1 + i+ 1.)

Let S = (a′1, . . . , a′s) be a class of Symb (O). Then it satisfies: (a′1, . . . , a

′s)

coincides with Symb (O) and a′i ≤ a′i+2−2 for i = 1, . . . , s−2. For such classS, define η, a character of A(O) by: η(σ(qi)) = 1 if and only if j : a′j = aicontains an element which has the same parity as i. Then it defines abijection between classes of Symb (O) and Springer (O). It is clear that thenumber of classes of Symb (O) is sC[ s

2].

If s is odd, let s = 2r + 1. Then sC[ s2] = 2r+1Cr ∼ 2 22r

√πr

.We interpret Moeglin’s results in terms of Arthur’s parameters:

Theorem 5.2 (Moeglin). Let G = Sp2n, SO2n+1, O2n. Then the residualspectrum of G attached to the trivial character of the torus is parametrizedby unipotent Arthur parameters

ψ : SL2(C) 7−→ G∗,

which are given by distinguished unipotent orbits in G∗= O2n+1(C), Sp2n(C),O2n(C). More specifically, for O a distinguished unipotent orbit, let Cres =Springer (O) ⊂ Cψv and Πres = Unipv(O) ⊂ Πψv for all finite places. Foreach X ∈ Πres , there is a character ηX ∈ Springer (O) and it satisfiesArthur’s conjecture. i.e.,

L2dis (G(F )\G(A))(T,1) ∩ L2(G(F )\G(A))ψ,

is the set of π = ⊗′vXv, where Xv satisfies the following conditions:

(1) There exists p ∈ P (O) such that Xv ∈ Unipv(p) for all v, i.e., ηXfactors through A(p).

(2) Xv is spherical for almost all v and archimedean places.(3)

∏v ηXv is trivial on A(O).

6. Arbitrary character case.

We first do the simple case to explain our method.

434 HENRY H. KIM

6.1. The case G = Sp2n and χ = χ(µ, . . . , µ), µ non-trivial and qua-dratic grossencharacter. Let Φ1 = ei ± ej , 1 ≤ i < j ≤ n andΦ2 = 2ei, i = 1, . . . , n. Then for α ∈ Φ1, χ α∨ = 1 and for α ∈ Φ2,χα∨ is non-trivial. Then the constant term of the pseudo-Eisenstein seriesis given by

EPS0 (λ, φ, χ) =∑w∈W

r(w,−λ, χ)R(w−1, wλ,wχ)φ(wλ)

=∑w∈W

r(w,−λ,Φ1)r(w,−λ,Φ2)R(w−1, wλ,wχ)φ(wλ).

We note that for f ∈ IndO2nB exp(λ,HB( )),

EPS0 (λ, f) =∑w∈W

r(w,−λ,Φ1)R(w−1, wλ)f(wλ),

is a pseudo-Eisenstein series for O2n attached to the trivial character ofthe maximal torus. Since χ α∨ is non-trivial for α ∈ Φ2, r(w,−λ,Φ2) isholomorphic.

Let O be a distinguished unipotent orbit in O2n(C) and p ∈ P (O). Let,for φ ∈ PW ,

lp(λ, φ, χ) =∑w∈W

r(w,−λ, χ)R(wpw−1, wλ,wχ)φ(wλ).

Since r(w,−λ,Φ1) is identically zero on V ′(p) if w /∈W (↑, p), the restric-tion of lp(λ, φ, χ) to V ′(p) is given by

lp(φ, λ, χ)|V ′(p) =∑

w∈W (↑,p)

r(w,−λ,Φ1)r(w,−λ,Φ2)R(wpw−1, wλ,wχ)φ(wλ).

By the definition of W (↑, p), the poles of lp(φ, λ, χ)|V ′(p) is contained inr(w,−λ,Φ1). Therefore we apply Moeglin’s global result for O2n in thetrivial character case, i.e.,

〈θφ′ , θφ〉dis =∑

O⊂O2n(C)

∑p∈P (O)

cp〈〈l′[p](φ′, λp, χ), l[p](φ, λp, χ)〉〉.

We havelp(φ, λp, χ) ∈ ⊗′vRv(wp, λp, χ)I(λp, χv).

From [Ja-Ki], we summarize the result on the image of the local intertwiningoperator Rv(wp, λp, χ)I(λp, χv).

Theorem 6.1.1 (Theorem 3.4.7 of [Ja-Ki]). Let O be a distinguished uni-potent orbit in O2n(C) and p ∈ P (O). Then Rv(wp, λp, χ)I(λp, χv) is semi-simple. Let Unip(p, µv) be the set of direct summands and Unip(O,µv) bethe set of union of Unip(p, µv). Then for each X ∈ Unip(O,µv), there is acharacter ηX ∈ A(O) such that:

(1) If µv 6= 1, then C(O,µv) = ηX |X ∈ Unip(O,µv) = Springer (O).


(2) If µv = 1, then

C(O,µv) = ηX |X ∈ Unip(O,µv)

=

Springer (O), if O does not contain 1Springer (O′), if O contains 1,

where O′ = O − 1.

So we have:

Theorem 6.1.2. Let χ = χ(µ, . . . , µ), µ non-trivial quadratic grossencha-racter. Then the residual spectrum attached to χ is parametrized by the dis-tinguished unipotent orbits in O2n. More specifically, a distinguished unipo-tent orbit O ∈ O2n(C) and χ give a quadratic unipotent Arthur parameter ψand let Cres v = C(O,µv) ⊂ Cψv and Πres v = Unip(O,µv) ⊂ Πψv for all non-archimedean places. For each X ∈ Πres v , there is a character ηX ∈ Cres v

and it satisfies Arthur’s conjecture, i.e.,

L2dis (G(F )\G(A))(T,χ) ∩ L2(G(F )\G(A))ψ,


(1) There exists p ∈ P (O) such that Xv ∈ Unipv(p, µv) for all v.(2) Xv is spherical for almost all v and archimedean places.(3)

∏v ηXv is trivial on Cψ.

6.2. General case. For χ a non-trivial character, we can assume, afterconjugation, that χ = χ(µ1, . . . , µ1︸︷︷︸

r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

), r0+· · ·+rk = n,

r1 ≥ · · · ≥ rk. We use the following notation throughout this section:

(1) If G = Sp(2n), G′ = G′1 × · · · × G′k × G′0, where G′i = O(2ri) fori = 1, . . . , k, G′0 = Sp(2r0). Also we denote G∗i = O(2ri,C) for i =1, . . . , k, G∗0 = O(2r0 + 1,C).

(2) If G = SO(2n+ 1), G′ = G′1× · · · ×G′k ×G′0, where G′i = SO(2ri + 1)for i = 1, . . . , k, G′0 = SO(2r0 + 1). Also we denote G∗i = Sp(2ri,C)for i = 1, . . . , k, G∗0 = Sp(2r0,C).

(3) If G = O(2n), G′ = G′1 × · · · × G′k × G′0, where G′i = O(2ri) fori = 1, . . . , k, G′0 = O(2r0). Also we denote G∗i = O(2ri,C) for i =1, . . . , k, G∗0 = O(2r0,C).

We recall some basic facts from [Ki-Sh]. Let E(g, φ, λ) be the Eisensteinseries associated to the character χ.

Proposition 6.2.1. The Eisenstein series has a pole of order n only ifrk ≥ 2 and µi is a quadratic grossencharacter for i = 1, . . . , k.

436 HENRY H. KIM

We divide the set of positive roots Φ+ as follows:

Φ1 = ei ± ej , 1 ≤ i < j ≤ r1,Φ2 = er1+i ± er1+j , 1 ≤ i < j ≤ r2,

...

Φk = er1+···+rk−1+i ± er1+···+rk−1+j , 1 ≤ i < j ≤ rk,Φ0 = er1+···+rk+i ± er1+···+rk+j , 1 ≤ i < j ≤ r0,

2er1+···+rk+i, i = 1, . . . , r0,

ΦD = Φ+ −k⋃i=0

Φk.

We note that the above is for G = Sp2n. If G = SO2n+1, we need toadd, to Φi, er1+···+ri−1+j , j = 1, . . . , ri, for i = 1, . . . , k and in Φ0,2er1+···+rk+i should be er1+···+rk+i. If G = O2n, then Φ0 does not havethe roots 2er1+···+rk+i, i = 1, . . . , r0.

Φ1, . . . ,Φk are root systems of type Dn and Φ0 is a root system of typeCn. This corresponds to the decomposition O(V1)× · · · ×O(Vk)×O(V0) ⊂O2n+1(C).

Let Wi be the Weyl group corresponding to Φi for i = 1, . . . , k and Wi

be the Weyl group of OVi for i = 0, . . . , k. Then Wi = Wicr1+···+ri fori = 1, . . . , k. Then we have W (χ, χ) = W1 × · · · ×Wk ×W0. Let λ = λ1 +· · ·+λk+λ0, where λi = ar1+···+ri−1+1er1+···+ri−1+1 + · · ·+ar1+···+rier1+···+rifor i = 1, . . . , k and λ0 = ar1+···+rk+1er1+···+rk+1 + · · ·+ anen.

We recall the following well-known result. (Carter [C].)

Proposition 6.2.2. Let ∆ be a set of simple roots and W be the associatedWeyl group. Let wα be the simple reflection with respect to α ∈ ∆. Then Wis generated by the wα, α ∈ ∆. Let θ be a subset of ∆ and Wθ be the subgroupof W generated by the wα, α ∈ θ. Then each coset wWθ has a unique elementdθ characterized by any of the following equivalent properties:

(1) dθθ > 0.(2) dθ is of minimal length in wWθ.(3) For any x ∈Wθ, l(dθx) = l(dθ) + l(x).

We apply Proposition 6.2.2 to ∆ = e1 − e2, . . . , en−1 − en and θ =∆ − er1 − er1+1, er1+r2 − er1+r2+1, . . . , er1+···+rk − er1+···+rk+1. Let D bethe set of such distinguished coset representatives. Then for d ∈ D,wi ∈Wi,i = 0, . . . , k, we have

α > 0 : dw1 · · ·wkw0α < 0

= ∪ki=0α ∈ Φi : wiα < 0 ∪ α ∈ ΦD : dw1 · · ·wkw0α < 0.


Then the constant term of pseudo-Eisenstein series is given by


r(w,−λ, χ)R(w−1, wλ,wχ)φ(wλ)

=k∏i=0

∑wi∈Wi

r(wi,−λi,Φi)

·

(∑d∈D

r(dw1 . . . wkw0,−λ,ΦD)R(w−10 w−1

k · · ·w−11 d−1,

dw1 · · ·wkw0λ, dχ)φ(dw1 · · ·wkw0λ)

).

We note that w1 · · ·wkχ = χ. Recall that


r(w,−λ, χ)R(w−1, wλ,wχ)φ(wλ),

actually means

∑d∈D

∑w∈W (χ,dχ)

r(w,−λ, χ)R(w−1, wλ,wχ)φd(wλ)

,

where φd ∈ I(dχ). By the cocycle relation, we have

R(w−10 w−1

k · · ·w−11 d−1, dw1 · · ·wkw0λ, dχ)

= R(w−10 w−1

k · · ·w−11 , w1 · · ·wkw0λ, χ)R(d−1, dw1 · · ·wkw0λ, dχ).

Let

f(w1 · · ·wkw0λ) =∑d∈D

r(dw1 · · ·wkw0,−λ,ΦD)

R(d−1, dw1 · · ·wkw0λ, dχ)φ(dw1 · · ·wkw0λ).

Then we have

EPS0 (φ, λ, χ) =k∏i=0

∑wi∈Wi

r(wi,−λi,Φi)

R(w−10 w−1

k · · ·w−11 d−1, dw1 · · ·wkw0λ, dχ)f(w1 · · ·wkw0λ).

We note that it has the same normalizing factors as the Eisenstein series ofO2r1 , . . . , O2rk and Sp2r0 attached to the trivial character. Let Oi’s be dis-tinguished unipotent orbits in O2ri(C) for i = 1, . . . , k and O0 in O2r0+1(C).

438 HENRY H. KIM

Let pi ∈ P (Oi) for i = 0, . . . , k and p = p0 × · · · × pk. Let

lp(φ, λ, χ) =k∏i=0

∑wi∈Wi

r(wi,−λi,Φi)

·R(wpw−1k · · ·w

−10 , w0 · · ·wkλ, χ)f(w0 · · ·wkλ).

Here wp = wp0 · · ·wpk . Since r(wi,−λi,Φi) is identically zero on V ′(pi) ifwi /∈ W (↑, pi), the restriction of lp(φ, λ, χ) to V ′(p) = V ′(p0)× · · · × V ′(pk)is given by

lp(φ, λ, χ)|V ′(p) =k∏i=0

∑wi∈W (↑,pi)

r(wi,−λi,Φi)

·R(wpw−1k · · ·w

−10 , w0 · · ·wkλ, χ)f(w0 · · ·wkλ).

We note that f(w0 · · ·wkλ) is holomorphic on V ′(p). We apply Moeglin’sresults and define lp(φ, λp, χ) inductively. But the order of induction willmatter. Among V ′(pi)’s, we can shuffle the segments. By shuffling, we meanthe following:

Let p1 = (; q1, . . . , qs) and p2 = (; q′1, . . . , q′t) be two chains. By shuffling

of p1 × p2, we mean any permutation on segments so that (1) (q1, q2), . . . ,(q2[ s−1

2]−1, q2[ s−1

2]) appear in that order and (2) (q′1, q

′2), . . . , (q

′2[ t−1

2]−1, q′

2[ t−12

])

appear in that order.Take a shuffling of segments in such a way that it satisfies a certain condi-

tion (see [Ja-Ki, (3.1)]) to produce a maximum number of components, i.e.,a condition on the non-vanishing of the normalized intertwining operators.

This amounts to starting with a conjugate of χ. If there is no confusion,we will still write it as χ. Then

〈θφ′ , θφ〉dis =k∑i=0

∑Oi

∑p

cp〈〈l′[p](φ′, λp, χ), l[p](φ, λp, χ)〉〉,

where Oi runs through distinguished unipotent orbits in O2ri(C) for i =1, . . . , k and O2r0+1(C) for i = 0. p = p0 × · · · × pk ∈ P (O0)× · · · × P (Ok).We have

l[p](φ, λp, χ) ∈ ⊗′vRv(wp, λp, χ)Iv(λp, χv).We recall the local result on the image of intertwining operatorsRv(wp, λp,

χ)Iv(λp, χv) from [Ja-Ki]: Let Unip(p, χv) be the set of direct summandsof Rv(wp, λp, χ)Iv(λp, χv) and Unip(O1, . . . , Ok, O0, χv) be the set of unionof Unip(p, χv) as pi runs through P (Oi) for i = 0, . . . , k.

Theorem 6.2.1 (Theorem 3.4.10 of [Ja-Ki]). Πres v = Unip(O1, . . . , Ok,O0, χv) is parametrized by

C(O1, . . . , Ok, O0, χv) = [Springer (O1)×· · ·×Springer (Ok)×Springer (O0)],





G∗12 =


Therefore we have:

Theorem 6.2.2. Let χ = χ(µ1, . . . , µ1︸︷︷︸r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

), r0 + · · · +

rk = n, r1 ≥ · · · ≥ rk, where µi’s are non-trivial distinct quadratic grossen-characters. Then the residual spectrum attached to the conjugacy classof (T, χ) is parametrized by the distinguished unipotent orbits in O2ri(C),i = 1, . . . , k and O2r0+1(C). More specifically, distinguished unipotent or-bits Oi ∈ O2ri(C), i = 1, . . . , k and O0 ∈ O2r0+1(C) and χ give a qua-dratic unipotent Arthur parameter ψ and let Cres v = [Springer (O1)× · · · ×Springer (Ok)×Springer(O0)] ⊂ Cψv and Πres v = Unip(O1, . . . , Ok, O0, χv)⊂ Πψv for all non-archimedean places. For each X ∈ Πres v , there is acharacter ηX ∈ Cresv and it satisfies Arthur’s conjecture. i.e.,

L2dis (G(F )\G(A))(T,χ) ∩ L2(G(F )\G(A))ψ,


(1) There exists pi ∈ P (Oi), i = 0, . . . , k such that Xv ∈ Unipv(p1, . . . , pk,p0, χv) for all v.

(2) Xv is spherical for almost all v and archimedean places.(3)

∏v ηXv is trivial on Cψ.

7. Relation to Kudla-Rallis example [Ku-Ra].

7.1. Trivial character case. Let G = Sp2n and P = MN be the Siegelparabolic subgroup so that M = GLn. Consider the degenerate principalseries I(s, χ) = IndG

P χ ⊗ exp(s,HP ( )), where χ is a trivial character ofGLn. Let E(s, f, P ) be the Eisenstein series attached f ∈ I(s, χ).

Then Kudla-Rallis [Ku-Ra] showed the following:

Theorem ([Ku-Ra]).

(1) E(s, f, P ) has a simple pole at s = 0, . . . , ρn − 2, ρn − 1, ρn, whereρn = n+1

2 and 0 means that 0 is omitted in the case when n is odd.(2) The residue at s = ρn is constant.(3) The residue at s = ρn − 1 is not square integrable.

440 HENRY H. KIM

(4) The residue at si = ρn − i is the direct sum ⊕Rn(Q), where the directsum ranges over all classes of quadratic forms Q with dim Q = n +1− 2si and χQ = 1. Here Rn(Q) is obtained by theta correspondence,first by local construction and then by being put together. In this casethe Hasse invariant χQ = 1 controls the obstruction to the existence ofthe global quadratic space.

Let O = (q1, q2, 1), q1 > q2 > 1, q1+q2 = 2n, be a distinguished unipotentorbit in O2n+1(C). There are two inequivalent ordered partitions in P (O),namely, p1 = (; q1, q2, 1) and p2 = (; q2, 1, q1). We show:

Proposition 7.1. The representations ⊕Rn(Q) are exactly those correspo-nding to p1, i.e., l[p1](φ, λp1) for φ ∈ PW . (See Section 5).

Proof. In this case, in Moeglin’s notation in Section 5, λp1 = (λ1, . . . , λn),where λt = q1+1

2 − t, 1 ≤ t ≤ q1+q22 . wp1(t) = −t for 1 ≤ t ≤ q1−q2

2 andwp1(

q1−q22 + t) = n + 1 − t for 1 ≤ t ≤ q2. We know that Sp1 = α >

0|wp1α < 0, 〈λp1 , α∨〉 = 1 has n elements. But we can see easily that Sp1

contains ei − ei+1 for i = 1, . . . , n − 1. This means that since the simpleroots ei − ei+1 for i = 1, . . . , n− 1, generate the Siegel parabolic subgroup,the iterated residue of the Eisenstein series on the singular hyperplanesei − ei+1 = 1, i = 1, . . . , n − 1, is exactly the Eisenstein series attached tothe trivial character of the Siegel parabolic subgroup, which Kudla-Rallisconsidered.

The square integrable residues at s = 0, . . . , ρn − 2 exactly correspondto the partitions (q1, q2), q1 + q2 = 2n, q1 > q2 > 1 (si = ρn − i correspondsto (2n− 2i+ 1, 2i− 1)).

Since l[p1](φ, λp1) ∈ ⊗′vRv(wp1 , λp1)Iv(λp1), the local component of Rn(Q)is the direct summand of the image of the intertwining operator Rv(wp1 ,λp1)Iv(λp1) which is a sum of two irreducible representations. We note thatRv(wp1 , λp1)Iv(λp1) is the subrepresentation of IndG

GLn|det |−

q1−q24 . Jantzen

[Ja3] gave their Langlands’ data.

Theorem ([Ja3, Prop 3.10]). Rv(wp1 , λp1)Iv(λp1) = π1 ⊕ π2. Here π1 is aspherical representation which is the Langlands’ quotient of IndG

M λO × π,where λO is the conjugate of λp which is in the closure of the positive Weylchamber and M = Cent (λO, G∗) = F×× · · ·×F××GL2× · · ·×GL2×SL2

and π = Ind SL2B 1 is an irreducible representation of SL2. And π2 is the

Langlands’ quotient of

IndGM | |

q1−12 × · · · × | |

q2+12 × | |

q2−12 × | |

q2−12 × · · · × | |2 × | |2 × | |1 × T ,

where M = F× × · · · ×F×× Sp4 and T is the unique (irreducible) commoncomponent of Ind Sp4

GL2|det |

12 and Ind Sp4

F××SL21 × StSL2. Here StSL2 is the

Steinberg representation of SL2. We note that T is tempered.


Let Unip(p1) = π1, π2. Then Proposition 7.1 can be rephrased as fol-lows:

Proposition 7.2. The representations ⊕Rn(Q) are the set of π = ⊗′vXv

which satisfies: (a) Xv ∈ Unip(p1) for all v (b) Xv is spherical almosteverywhere (c)

∏v χXv = 1, where χXv is defined in Section 5.

But the non-degenerate Eisenstein series attached to the trivial characterof the maximal torus has more residues. We need to consider p2 ∈ P (O).In this case λp2 = (λ1, . . . , λn), where λt = q2+1

2 − t for 1 ≤ t ≤ q2+12 and

λt+

q2+12

= q1+12 − t for 1 ≤ t ≤ q1

2 . wp2(t) = −t for all t except t = q2+12 .

Then

Rv(wp2 , λp2)Iv(λp2) ⊂ IndGL q2+12

×Spq1−1 |det |−q2−1

4 × tr.

We have Rv(wp2 , λp2)Iv(λp2) = π1 ⊕ π3.

Theorem ([Ja3]). π3 is the Langlands’ quotient of

Ind F×···×F××GL2×···×GL2| |

q1−12 × · · · × | |

q2+32 × | |

q22 σ × · · · × | |

12σ,

where σ is the unique quotient of IndGL2

F××F× | |− 1

2 × | |12 which is square

integrable.

Let Unip(p2) = π1, π3. In this special case, we have:

Theorem 7.1. The residual automorphic representations attached to thetrivial character of the maximal torus and the distinguished unipotent orbitO = (q1, q2, 1) are the set of π = ⊗′vXv which satisfies:

(1) Xv ∈ Unip(p1) for all v or Xv ∈ Unip(p2) for all v.(2) Xv is spherical for almost all v.(3)

∏v χXv = 1.

7.2. χ = χ(µ, . . . , µ) case, µ a quadratic grossencharacter. In thiscase, Kudla-Rallis showed:

Theorem ([Ku-Ra]). The degenerate Eisenstein series attached to thecharacter µ of GLn has a pole at s = 0, . . . , ρn − 2, ρn − 1 and theresidue at si = ρn − i is square integrable and given by the direct sum⊕Rn(Q), where the direct sum ranges over all classes of quadratic formsQ with dim Q = (n + 1) − 2si and χQ = µ. Again for this case, Hasseinvariant gives obstruction to the existence of the global quadratic space.

These correspond to the distinguished unipotent orbits O = (q1, q2) ofO2n(C), where q1 > q2 ≥ 1. There is only one element in P (O), namely,p = (; q1, q2). The local components of Rn(Q) are the direct summandsof Rv(wp, λp, χv)I(λp, χv). Here Rv(wp, λp, χv)I(λp, χv) is a sum of twoirreducible representations unless µv = 1 and q2 = 1, in which case it is

442 HENRY H. KIM

irreducible. If µv 6= 1, then they are the Langlands’ quotients of IndGM λO×

πi, where λO is the conjugate of λp which is in the closure of the positive Weylchamber and M = Cent (λO, G∗) = F××· · ·×F××GL2×· · ·×GL2×SL2

and IndMB χv = π1 ⊕ π2. If µv = 1, q2 6= 1, then they are the ones in the

case when χv = 1.

Remark 7.1. We can show that the residual spectrum corresponding tothe unipotent orbits of the form p = (; q1, q2, q3), q3 > 1, can be obtained byconsidering the degenerate principal series of GLl×Sp2(n−l), where l = n−q3−1

2 . In this case, λp = (λ1, . . . , λn), where λt = q1+12 − t, for 1 ≤ t ≤ q1+q2

2

and λt+

q1+q22

= q3+12 − t, for 1 ≤ t ≤ q3−1

2 . wp(t) = −t for 1 ≤ t ≤ q1−q22 and

q1+q22 ≤ t ≤ n, wp( q1−q22 + t) = q1+q2

2 − t+ 1 for 1 ≤ t ≤ q2.We can show that Sp = α > 0|wpα < 0, 〈λp, α

∨〉 = 1 contains allsimple roots except e

n− q3−12

−en− q3+1

2

. This means that the iterated residueof the Eisenstein series on the singular hyperplanes in Sp is the residue of thedegenerate Eisenstein series attached to the trivial character of the parabolicsubgroup P = MN , M = GLl × Sp2(n−l).

We note that the degenerate Eisenstein series attached to GLn−1 × SL2

parabolic subgroups has been studied by Jiang [Ji].

8. Relation to our previous work [Ki-Sh].

Let χ = χ(µ1, . . . , µ1︸︷︷︸r1

, . . . , µk, . . . , µk︸︷︷︸rk

, 1, . . . , 1︸︷︷︸r0

), r0 + · · ·+ rk = n, r1 ≥ · · · ≥

rk ≥ 2. In our previous work [Ki-Sh], we are concerned only with unipotentorbits of the form Oi = (2ri−1, 1) of O2ri(C), i = 1, . . . , k and O0 = (2r0+1)of O2r0+1(C), which correspond to the half-sum of positive roots of O2ri andSp2r0 , respectively. We showed:

Theorem ([Ki-Sh]). Cφψv is the Knapp-Stein R-group of the unitary prin-cipal series IndM

B χv, where M is conjugate to GLn1 × · · · × GLnr × Sp2k.Here n1, . . . , nr are determined by Oi’s. Πφψv

is the set of the Langlands’quotients of IndG

M πiv × λp, where IndMB χv = ⊕i πiv. λp is in the positive

Weyl chamber of the split component of M .

Let µi1,v, . . . , µil,v be the set of non-trivial distinct quadratic characters.Then Cφψv is spanned by the order two elements cr1+···+ri1 , . . . , cr1+···+ril .On the other hand, by Theorem 6.2.1 in Section 6, Cres v = C(µi1) × · · · ×C(µil)× C0, where C(µij ) ' Z/2Z and C0 is a set determined by Oi’s. Wenote that Cφψv ' C(µi1,v)× · · · × C(µil,v).

Example 8.1. Let χ = χ(µ, µ, 1, 1), where µ is a non-trivial quadraticgrossencharacter. Let O1 = (3, 1) be a distinguished unipotent orbit ofO4(C) and O0 = (5) of O5(C). Then if µv 6= 1, Cres v = Cφψv ' Z/2Z.


But if µv = 1, Cφψv = 1 and Cres v ' Z/2Z. Here Πres v is Unip(p2) inTheorem 7.1 for p2 = (; 3, 1, 5).

9. Correction to our paper [Ki-Sh].

p. 401: In the Abstract, the Introduction, and also in Theorem 5.1, bythe technique used in that paper, we can only claim that Π ∈ Πφψ appears

with multiplicity, not equal to dψ|Cφψ |

∑x∈Cφψ

〈x,Π〉, but greater than or equal

to dψ|Cφψ |

∑x∈Cφψ

〈x,Π〉. We need to use the technique of pseudo-Eisenstein

series in this paper to claim the assertion.p. 415: In Theorem 4.5, w0 is different from w0 in Proposition 4.4. In

fact, w0 in Theorem 4.5 is given by w0 = w1,0 · · ·wk,0w0,0, where wi,0 is thelongest element in Wi for i = 0, . . . , k.

p. 415: The local intertwining operator R(w0,Λ, χv) is not holomorphicat Λ0. We need to define R(w0,Λ0, χv) as the intertwining operator onIndG

P Λ0 ⊗ IndMB χv. We assumed this implicitly there.

p. 418: In Lemma 4.8, wΛ1,0 should be w1,0, the longest element in W1.p. 418: In the proof of Lemma 4.8 and in Lemma 4.9, w1 should be w1,0.p. 420: The sentence “The parameter Λ0 may not be in the positive Weyl

chamber of the split component of M ,” is false. The parameter Λ0 is in thepositive Weyl chamber of the split component of M .

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Received June 15, 1999. This author was partially supported by NSF grant DMS-9610387.

Dept of Math.Southern Illinois UniversityCarbondale, IL 62901E-mail address: [email protected]

















QUADRATIC BASE CHANGE FOR p-ADIC SL(2) AS ATHETA CORRESPONDENCE II: JACQUET MODULES

David Manderscheid

Let F be a p-adic field and let O be the orthogonal groupattached to a quaternary quadratic form with coefficients inF and of Witt rank one over F . We determine, up to onepossible exception, which nonsupercuspidal representationsof O (F ) occur in the theta correspondences attached to(SL2 (F ) , O (F )).

This paper is the second in a series of papers examining in detail the lo-cal theta correspondences attached to reductive dual pairs (SL2(F ),O(F ))where F is a p-adic field of characteristic zero and O is the orthogonal groupattached to a quaternary quadratic form with coefficients in F and of Wittrank one over F . In this paper we determine, up to one possible exception,which nonsupercuspidal representations of O(F ) occur in the correspon-dences. The determination is explicit and in terms of parabolic inducingdata.

The results we obtain are consistent with the first occurrence in tow-ers conjecture [KR1] and Prasad’s conjectures [P2]. They are sharper andmore explicit than the corresponding results in Cognet’s thesis [C] and com-plement the results of Roberts [Ro2] in the case of p odd, which are statedin terms of distinguished representations. In future papers in this series,we will examine which supercuspidal representations of O(F ) occur in thecorrespondence and the explicit correspondence.

To explain our method, we first recall the general setting of theta corre-spondences for symplectic and orthogonal groups; see, e.g., [MVW], [H].For i = 1, 2, let Vi be a finite-dimensional vector space over F equipped witha nondegenerate bilinear form 〈 , 〉i; assume that 〈 , 〉1 is skew-symmetricwhile 〈 , 〉2 is symmetric. Equip W = V1 ⊗ V2 with the skew-symmetricform 〈 , 〉 coming from tensoring the 〈 , 〉i. Let G1, G2 and G be the isom-etry groups of 〈 , 〉1, 〈 , 〉2 and 〈 , 〉, respectively, and identify G1 and G2

with subgroups of G via their usual actions on W ; then (G1, G2) is calleda reductive dual pair in G. Let χ be a nontrivial additive character of Fand let ω∞χ denote the (smooth) oscillator representation of G attached toχ where G is the (unique) nontrivial two-fold cover of G. For H a closedsubgroup of G, let H denote the inverse image of H in G and let Rχ(H)

447



448 DAVID MANDERSCHEID

denote the set of irreducible admissible representations of H which occuras quotients of ω∞χ |H . Then G1 and G2 commute and Rχ(G1G2) givesrise to a correspondence between Rχ(G1) and Rχ(G2). These correspon-dences are called theta correspondences. We denote these correspondencesby θ : Rχ(G1)→ Rχ(G2) and θ : Rχ(G2)→ Rχ(G1); the direction of θ willbe clear from context. Theta correspondences are known in general to bebijections for p odd [Wa] and for all p in the cases considered in this paper[R2]. Furthermore, in all cases considered here, the space V2 will be even-dimensional and thus the G1 and G2 are trivial covers so that we write, inan abuse of notation, Rχ(G1) and Rχ(G2) instead of Rχ(G1) and Rχ(G2).Elements of these sets will be considered as representations of G1 and G2,respectively.

Then our argument and organization for this paper are as follows. In thefirst section, we establish notation and recall briefly known results that willbe necessary in what follows. These results include the parameterizationsof the admissible duals of GL2(F ) and SL2(F ), the results of the first paperin this series [M] on which representations of SL2(F ) occur in Rχ(SL2(F ))for the pair (SL2(F ),O(F )), quadratic base change for GL2(F ) and SL2(F )and finally the results of Cognet’s thesis [C] that will be necessary.

We begin the second section by showing that the representations of O(F )that occur in Rχ(O(F )) must restrict irreducibly to SO(F ). This is a stan-dard seesaw duality [K1] argument. We then parameterize the irreducibleadmissible nonsupercuspidal parameterizations of O(F ) with this property.The parameterization is explicit and in terms of inducing data from the(unique up to conjugacy) maximal parabolic subgroup of SO(F ). This par-abolic has Levi, M say, isomorphic to F× × E1, where F× denotes themultiplicative group of F and E1 is the kernel of the norm map from E× toF× where E/F is the quadratic extension of F attached to the anisotropicpart of the quadratic form, Q say, giving rise to O.

In the third section, we consider ω∞χ in the Schroedinger model. In thismodel, O(F ) acts linearly on S(V ) where V is the space on whichQ is definedand S(V ) is the space of locally constant compactly supported function onV . We use this to determine necessary conditions on the representationsof M that can be used for inducing data of representations occurring inRχ(O(F )). The argument is by calculation of Jacquet modules.

In the fourth and final section, we show that the necessary conditionsof the third section are also sufficient with one possible exception. Theargument involves our previous results, Cognet’s results and results on basechange. It also involves determining which representation of O(F ) pairs withthe trivial representation of SL2(F ). The pairing representation is also one-dimensional and the argument involves the determination of which orbits inV under O(F ) can support this representation. We plan on returning to the

QUADRATIC BASE CHANGE FOR p-ADIC SL(2) 449

possible exception, a generalized Steinberg representation, in a future paperin this series.

Finally, we would like to thank the referee of this paper for pointing outto us that we needed to modify the proof of Lemma 4.2.

1. Notation and known results.

In this section, we establish notation, recall the parameterization of the ad-missible dual of G1 = SL2(F ) and recall some other known results necessaryfor this paper. We will be brief in our discussion.

Let F be a nonarchimedean local field of characteristic 0. Let p denotethe residual characteristic of F and let O = OF , P = PF , ω = ωF , k = kF ,q = qF and | |=| |F denote, respectively, the ring of integers, the primeideal, a uniformizing parameter, the residue field and the absolute valueon F normalized so that |x| = q−ν(x) where ν = νF denotes the orderfunction on F . Let U = UF = O×F and Un = UnF = 1 + PnF for n apositive integer. Further, for K/F a Galois extension of fields, let Γ(K/F )denote the associated Galois group and if, in addition, [K : F ] < ∞, letNK/F = N denote the norm map and let K1 = K1

F , the norm one elementsin K. Finally, fix an algebraic closure F of F and a Weil group WF ; let theassociated Weil group notation be as in [T].

For G a group and σ a representation (all representations assumed smoothunless stated otherwise) of a subgroup H of G, let Ind(G,H;σ) denote therepresentation of G induced by σ (form of induction determined by context)and for g in G, let σg denote the representation of Hg = gHg−1 definedby σg(h) = σ(g−1hg) for h in Hg. If J is a subgroup of H, we let σ|Jdenote the restriction of σ to J . Further, if J C H and σ is a representationof H/J , then we also view σ as a representation of H via inflation. If σand τ representations of G, then we let HomG(σ, τ) denote the set of G-intertwining operators from σ to τ with the category, once again, specifiedby context. Finally, we let G∧ denote the admissible dual of G.

By a character, we mean a (not necessarily unitary) one-dimensional rep-resentation. If χ is a character of F×, we also view χ as a character of WF

via local class field theory and as a character of GL2(F ) by composition withdet, the determinant map. Further, if K/F is a finite-dimensional Galoisextension, we view χ as a character χK of K× via composition with NK/F .If χ is a character of F and a is an element of F , we let χa denote thecharacter of F defined by χa(y) = χ(ay). Finally, we say representations π1

and π2 of GL2(F ) are twist equivalent if there exists a character η of F×

such that π1∼= π2 ⊗ η.

We now briefly recall the parameterization of the admissible dual ofG1(F ) = SL2(F ) in [LL]. To do this we first recall the parameterizationof the admissible dual of G′1(F ) = GL2(F ) in [JL] in a form suitable for our


purposes. If µ and ν are characters of F× such that µ(x)ν−1(x) 6= |x| or|x|−1, let π(µ, ν) denote the irreducibly induced (normalized induction) prin-cipal series representation of G′1 attached to µ and ν. Note that π(µ, ν) ∼=π(ν, µ). If µ(x)ν−1(x) = |x|, write µ = χ | |1/2 and ν = χ | |−1/2 and letσ(µ, ν) denote the special representation corresponding to the unique invari-ant subspace of the space of the associated induced representation from theBorel subgroup of G

′1 and let π(µ, ν)(∼= χ) denote the corresponding quo-

tient. Similarly, if µ(x)ν−1(x) = |x|−1, let σ(µ, ν) denote the correspond-ing special representation (now the quotient) and π(µ, ν) the correspondingone-dimensional. Note that σ(µ, ν) ∼= σ(ν, µ) and π(µ, ν) ∼= π(ν, µ). Fur-ther, if K/F is quadratic and θ is a character of K×, let π(θ) = π(ρ)denote the corresponding irreducible representation of G′1 associated toρ = Ind(WF ,WK ; θ); note that π(θ) ∼= π(θ−1) and note also that π issupercuspidal if and only if θ does not factor through NK/F which in turnhappens if and only if ρ is irreducible. We call representations of the formπ(θ) Weil representations. The irreducible representations of G′1 not of oneof the above forms are called exceptional and occur only if p = 2. These rep-resentations are supercuspidal and can be parameterized naturally in termsof the primitive (i.e., not induced from a proper subgroup) two-dimensionalrepresentations of WF [Ku]; for σ such a representation of WF , we writeπ(σ) for the corresponding exceptional representation. Finally, we note thatthe representations enjoy no other equivalences with the exception that if µand ν are characters of F× with µν−1 of order two, then π(µ, ν) ∼= π(µK)where K/F is the quadratic extension of F associated to µν−1 by local classfield theory.

Now let G1 = G1(F ) = SL2(F ) viewed as a subgroup of G′1. Then wehave:

Theorem 1.1 ([LL]). Let π1 be an irreducible representation of G1; thenthere exists an irreducible representation π of G′1, unique up to twist equiv-alence, which contains π1 upon restriction to G1. The L-packet of π1 is ofthe form π1, . . . , πs where the πi are distinct irreducible representationsof G1 and the restriction of π to G1 decomposes as

⊕si=1 πi. Further, given

1 ≤ i, j ≤ s there exists g in G′1 such that πgi ∼= πj. Moreover, with χ acharacter of F×:

(i) If π is not a Weil representation, then s = 1. Further, if π ⊗ χ ∼= π,then χ is trivial.

(ii) If π = π(θ) with θ a character of K× such that θ|K1 is not of ordertwo, then s = 2 and πgi

∼= πi if and only det g is a norm from K×.Further, π⊗χ ∼= π if and only if χ is trivial or χ = ωK/F , the characterof F× associated to K by local class field theory. If π is supercuspidalin this setting, then ρ is singly imprimitive (i.e., can only be inducednontrivially from WK).


(iii) If π = π(θ) with θ a character of K× such that θ|K1 is of order two,then s = 4. In this case, ρ is triply imprimitive and if Ki, i = 1, 2, 3,are the fields such that ρ may be induced from WKi and L is theircomposite, then Γ(L/F ) ∼= Z/2Z ⊗ Z/2Z and πgi

∼= πi if and only ifdet g is a norm from L×. Further, π ∼= π⊗χ if and only if χ is trivialor χ = ωKi/F for some i.

(iv) The collection of distinct L-packets partitions G∧1 . Further, anotherrepresentation π′ in (G′1)

∧ gives rise to the same L-packet as π if andonly if π and π′ can be realized as follows:

(a) π = π(µ, ν) and π′ = π(µ′, ν ′) with µν−1 = (µ′)(ν ′)−1;(b) π = σ(µ, ν) and π′ = σ(µ, ν ′) with µν−1 = (µ′)(ν ′)−1;(c) π = π(θ) and π = π′(θ′) with θ and θ′ on K× with θ(θ′)−1|K1 = 1;(d) π = π(σ) and π′ = π(σ′) with σ and σ′ primitive projectively equiv-

alent representations.

In what follows we will distinguish among the πi by their Whittaker mod-els. In particular, recall that if π is an infinite-dimensional irreducible repre-sentation of G′ and η is a nontrivial character of F , then π has, up to scaling,a unique Whittaker model with respect to η and, of course, if π is finite-dimensional, then it has no Whittaker models. Thus, for infinite-dimensionalπ, we let π(µ, ν; η) denote the component of π(µ, ν) with η-Whittaker modeland similarly for σ(µ, ν), π(θ) and π(σ). The only remaining representationis the trivial representation which we denote by 1. Finally, for a in F× andπ an irreducible representation of G1, let πa = πg where g is an element ofG′1 with det g = a. Then one checks that if π has an η-Whittaker modelthen πa has an ηa-Whittaker model.

We continue by recalling the result of [M] that will be necessary forthis paper. To this end, let E/F be a quadratic extension and set V =A ∈ M2(E) | At = A where A denotes the matrix obtained from A byapplying σ to each entry where Γ(E/F ) = 〈σ〉. Now the negative of thedeterminant map det : M2(E) → E when restricted to V maps to F anddefines a quadratic form, Q say, on V viewed as an F vector space. Let H1

denote the isometry group of this form. Further, for χ a nontrivial additivecharacter of F , let Rχ(G1) denote the representations in the admissible dualof G1 that occur in the theta correspondence attached to χ and the reductivedual pair (G1,H1); see [M] and [MVW] for more details concerning thetacorrespondences.

Theorem 1.2 ([M]). If π is an irreducible representation of G1 such thateither, for some b in NE/F (E×), πb has a Whittaker model with respect toχ, or π is trivial, then π is in Rχ(G1).

Theorem 1.1 and Theorem 1.2 have the following consequences: If π′ isan irreducible representation of G′1 that cannot be realized as a π(θ) with θa character of E×, then the entire L-packet for G1 associated to π occurs in


Rχ(G1). On the other hand if π can be realized as a π(θ) with θ a characterof E×, then at least half of the representations in the associated L-packetoccur.

We now recall the results on base change from SL2(F ) to SL2(E) [LL]that will be necessary in what follows. To begin, we first recall base changefrom G′1(F ) = GL2(F ) to G′1(E) = GL2(E) [L]. In particular, if π isan irreducible representation of G′1(F ), let Π denote its (Langlands-Saito-Shintani) base change to G′1(E).

Theorem 1.3. Let π be an irreducible representation of G′1(F ).(i) If π ∼= π(µ, ν) with µ and ν characters of F×, then Π ∼= π(µE , νE).(ii) If π ∼= σ(µ, ν) with µ and ν characters of F×, then Π ∼= σ(µE , νE).(iii) If π ∼= π(θ) = π(ρ) with θ a character of K×, K/F quadratic, then

Π ∼= Π(ρ|WE). In particular, if K 6= E, then Π ∼= π(θKE), and if

K = E, then Π ∼= π(θ, θσ) where 〈σ〉 = Γ(E/F ).(iv) If π is exceptional, then so is Π.

Proof. (i) through (iii) follow directly from [L] (see also [GL] for a convenientsummary).

(iv) Suppose Π is not exceptional. Then there exists a nontrivial char-acter, η say, of E× such that Π ⊗ η ∼= Π, by Theorem 1.1. First assume ηfactors through NE/F , i.e., that η = χE for some nontrivial character χ ofF×. Then the base change of π ⊗ χ to G′1(E) is Π⊗ χE ∼= Π [L] and thus,also by [L], π ∼= π ⊗ χ or π ∼= π ⊗ χ ⊗ ωE/F where ωE/F is the characterof F× associated to E/F by local class field theory. But then since π isexceptional, either χ is trivial or χ⊗ωE/F is. By assumption χ is nontrivialso χ = ωE/F but then η is trivial, a contradiction. Thus, assume η does notfactor through NE/F , i.e., that ησ 6= η where 〈σ〉 = Γ(E/F ) and ησ is thecharacter of E× defined by ησ(x) = η(xσ). Now define the representationΠσ of G′1(E) by Πσ(g) = Π(gσ) where σ acts coordinatewise. Then, by [L]since Π is in the image of base change, Π ∼= Πσ. Thus, Π ⊗ η ∼= Π impliesthat Π⊗ ησ ∼= Π whence Π⊗ ηησ ∼= Π. Now, ηησ does factor through NE/F

so it follows from the first part of this argument that ηησ = 1. But η is oforder two (see Theorem 1.1) and thus η = ησ, a contradiction.

Remark 1.4. We only include a proof of (iv) above since we know of noplace where it occurs in the literature. We, however, make no claim to theresult.

Quadratic base change for SL2 is then at the level of L-packets and canbe summarized by the following theorem. Note that, with notation as in thetheorem, the representations ΠiSi=1 actually factor to PSL2(E) since thecentral character of Π is that of π composed with NE/F [L].

Theorem 1.5. If πisi=1 is an L-packet for SL2(F ), then the base change ofπisi=1 to SL2(E) is the packet ΠiSi=1 obtained by restricting Π to SL2(E)


where Π is the base change of π, a representation of GL2(F ) restricting toSL2(F ) to give the L-packet πisi=1.

Finally, we need to recall a result of Cognet. To this end, we first recallthe structure of the orthogonal group H1. Let H ′1 denote the generalizedorthogonal group attached to Q and V . Then (see [D] for further details ofthis discussion) the map Ψ : G′1(E)× F× → EndF (V ) defined by

Ψ(g, u)(A) =(u 00 u

)gAgt,

where denotes Galois conjugation coordinatewise, is a homomorphism intoH ′1. It has kernel ((

a 00 a

), N(a)−1

)∣∣∣∣ a ∈ E×and image of index two. Further, H ′1 ∼= Im Ψ o 〈σ〉 where σ is the elementof V corresponding to the isometry of V given by conjugation and Im Ψconsists of those elements in H ′1 whose determinant is the square of theirsimilitude factor. Now consider the restriction of Ψ to those elements ofthe form (g, u) with N(det g)u2 = 1; call this group H. Then Ψ(H) is thesubgroup, H0

1 say, of H1 consisting of those elements of determinant one andH1 = H0

1 o 〈σ〉. We map G1(E) to H1 via the map k(g) = Ψ((g, 1)) for gin G1. The kernel of k is ±I and thus, in a slight abuse of notation, we canuse k to identify PSL2(E) with a subgroup of H1. Then k(PSL2(E)) is thecommutator subgroup of H1 and has index 2n+3 where n = 0 unless p = 2in which case n = [F : Q2]. Indeed identifying F× with a subgroup of H0

1

via the map i : F× → H01 defined by

i(a) = Ψ((

a 00 1

), a−1

)for a in F× we get that i(F×)k(PSL2(E)) ∼= H0

1 and H01/k(PSL2(E)) ∼=

i(F×)/i((F×)2).

Theorem 1.6 ([C]). If π is an irreducible infinite-dimensional representa-tion of G1 = SL2(F ), then there exists π′ in the L-packet of π such thatπ′ occurs in the theta correspondence attached to χ and (G1,H1) and suchthat the corresponding representation of H1, upon restriction to PSL2(E),decomposes as a sum of representations in the L-packet for SL2(E) obtainedfrom that of π by base change.

Proof. Cognet’s statements are at the level of the similitude groups G′1 andH ′1. However, it is a straightforward argument using results relating simili-tude theta correspondences to regular theta correspondences (see, e.g., [B]or [Ro1]) to obtain the result above from [C].


2. Nonsupercuspidal Galois-invariant representations of H1.

Although the emphasis in this section will be on representations which arenot supercuspidal, our first result will apply to all representations inRχ(H1).To begin, let det : H1 → C× be the representation of H1 defined by thedeterminant map. Then our first result is fairly standard but we provide acomplete proof since we know of no good reference to the literature.

Lemma 2.1. If π is an irreducible admissible representation of H1 whichis in Rχ(H1), then π ⊗ det is not in Rχ(H1). In particular, π and π ⊗ detare not isomorphic and π|H0

1is irreducible with (π|H0

1)σ ∼= π|H0

1.

Proof. It suffices to show that π⊗det does not occur. Suppose the contrary.Let W ′ be a four-dimensional symplectic vector space over F with form 〈 , 〉′and identify W ′ with two transverse copies of the space giving rise to G1.Then in the language of [K1] we have the following seesaw reductive dualpair:

G1 ×G1 H1.

G(W ′) H1 ×H1

""

""b

bbb

Then, since both π and π ⊗ det are in Rχ(H1) for the reductive dual pair(G1,H1), it follows that π⊗(π⊗det) occurs in Rχ(H1×H1) (defined relativeto the pair (G1 ×G1,H1 ×H1)). But then since irreducible representationsof orthogonal groups are self-contragredient [N], it follows that π⊗ (π⊗det)restricted to H1 has det as a quotient. Then from the reciprocity formulafor seesaw reductive dual pairs (see, e.g., [P1] or [M]), it follows that detis in Rχ(H1) relative to the pair (G(W ′),H1). But this contradicts [R1,Appendix] since dim (W ′)/2 ≥ dimV1 does not hold.

An immediate consequence of the above lemma is that representationsin Rχ(H1) are parameterized by their restriction, which is Galois invari-ant, to H0

1 . Our next step is to parameterize such representations in thenonsupercuspidal case.

Recall that we have i(F×)k(PSL2(E)) ∼= H01 . Let j : E1 → H0

1 be theimbedding of E1 in H0

1 defined by

j(a) = Ψ((

a 00 1

), 1)

with Ψ also as in the previous section. We note that i(−1) 6= j(−1) andthat i(−1)j(−1) = −I, the nontrivial element of the center of H0

1 . We notethat for a in E×

i(N(a))j(a/a) = k

(a 00 a−1

),(2.1)


as is easily checked.Now let V ′ denote the subspace of V consisting of those matrices which

are zero with the possible exception of the (1, 1) entry. Let P denote theparabolic subgroup of H0

1 which stabilizes V ′. Then one checks that P =MN where

N =k

(1 a0 1

)∣∣∣∣ a ∈ Eand M = i(F×)j(E1). Moreover, all proper parabolic subgroups of H0

1 areconjugate to P and thus all irreducible nonsupercuspidal representations ofH0

1 may be realized as subrepresentations (or subquotients) of representa-tions induced from P . We parametrize these representations below. Sincewe use normalized induction, we note that the modulus function of P , δPsay, is given by δP (i(a)j(b)) = |a|2F , as is easily checked. Also, note that

δP

(k

(a 00 a−1

))= δP (i(N(a))) = |N(a)|2F = |a|2E .

Let T denote the subgroup of PSL2(E) obtained by considering the diagonalmatrices in SL2(E) modulo ±I. We identify E×/ 〈−1〉 with T via the mapj′ : E× → T defined by

j′(a) =(a 00 a−1

).

We view characters of M as characters of P , as usual, by inflation. Wealso note that M/k(T ) ∼= H0

1/k(PSL2(E)) ∼= F×/(F×)2 where the firstisomorphism is via the map induced by inclusion as can be checked using(2.1) and the second isomorphism is as was noted above. We will thus viewcharacters of these groups interchangeably. Further, for any character of M ,we view λ|k(T ) as a character of E× via pullback along j′ k.

Lemma 2.2. Let λ be a character of M .(i) If λ|k(T ) is not | |E or | |−1

E and is not of order two, then Ind(H01 , P ;λ)

is irreducible. It is Galois invariant if and only if λ2 is trivial uponrestriction to j(E1) or i(F×); in these cases, set π(λ) = Ind(H0

1 , P ;λ).(ii) If λ|k(T ) = | |, then Ind(H0

1 , P ;λ) has a unique irreducible subrepre-sentation, σ(λ) say, and unique irreducible quotient, π(λ) say, both ofwhich are Galois invariant. Further, π(λ) = λ | |−1.

(iii) Similarly, if λ|k(T ) = | |−1, then Ind(H01 , P ;λ) has a unique irreducible

subrepresentation, π(λ) say, and unique irreducible quotient, σ(λ) say,both of which are Galois invariant. Further, π(λ) = λ · | |.

(iv) Assume λ|k(T ) is of order two. Let ωλ be the associated character ofE× and let E(λ)/E be the quadratic extension associated to ωλ by localclass field theory. Then if E(λ)/F is biquadratic, then Ind(H0

1 , P ;λ)


is the direct sum of two distinct irreducible Galois-invariant represen-tations, π+(λ) and π−(λ) say, each of which remains irreducible uponrestriction to k(PSL2(E)) with the sign being determined by requiringthat π+(λ) ∼= π(λ, 1;χ trE/F ) as a representation of PSL2(E). Fur-thermore, in this case λ itself is of order two. If E(λ)/F is cyclic, thenπ(λ) = Ind(H0

1 , P ;λ) is irreducible and Galois invariant. In this caseλ is not of order two but λ2(j(E1)) = λ2(i(NE/F (E×))) = 1. Finally,if E(λ)/F is not Galois, then Ind(H0

1 , P ;λ) is either irreducible or thedirect sum of two distinct irreducible representations. In either case,none of the irreducible representations obtained is Galois invariant.

(v) The π(λ), π±(λ) and σ(λ) constructed above exhaust the nonsupercus-pidal Galois-invariant portion of the admissible dual of H0

1 . Further,π(λ) ∼= π(λ′) if and only if λ′ = λ or λ′ = λ−1 and similarly for σ(λ).Finally, the representations enjoy no other equivalences.

Proof. The composition series statements in (i), (ii) and (iii) follow readilyfrom the background material of the first section, in particular, Theorem 1.1,since i(F×)k(PSL2(E)) = H0

1 and

i(a)k(g)i(a−1) = k

((a 00 1

)g

(a 00 1

)−1)

for g in PSL2(E) and a in F×. Now consider the composition series whenλ|K(T ) is of order two. In this case, it follows from the material of the firstsection that Ind(H0

1 , P ;λ) is irreducible or the direct sum of two irreduciblerepresentations with the latter occurring if and only if F× is contained inNE(λ)/E(E(λ)×). If E(λ)/F is biquadratic, then the decomposition followssince NE(λ)/E(E(λ)×) = x ∈ E× | NE/F (x) ∈ NE′/F ((E′)×) where E′ isany quadratic extension of F such that EE′ = E(λ), see, e.g., [I, Theorem7.6]. On the other hand if E(λ)/F is cyclic, then F×/NE(λ)/F (E(λ)×) iscyclic of order four by local class field theory and thus NE(λ)/F (E(λ)×)cannot contain (F×)2 whence NE(λ)/E(E(λ)×) cannot contain F×. Finally,if E(λ)/F is not Galois, then the composition series statement follows fromTheorem 1.1.

We now consider equivalences amongst the π(λ), π±(λ) and σ(λ). Weconsider here only the infinite-dimensional representations, the other casesbeing trivial. Restricting to k(PSL2(E)), we see that π(λ) can only beisomorphic to some π(λ′) and similarly for π+(λ), π−(λ) and σ(λ). Supposeπ(λ) ∼= π(λ′). Then, also by restricting to k(PSL2(E)), we get that λ′|k(T ) =λ|k(T ) or λ′|k(T ) = λ−1|k(T ). Suppose the former. Write λ′ = λλ′′ with λ′′ acharacter of M trivial on k(T ). Now it follows from Frobenius reciprocitythat π(λ)⊗λ′′ ∼= π(λλ′′) with λ′′ on the left-hand side viewed as a character ofH0

1 . Thus, π(λ)⊗λ′′ ∼= π(λ). This either implies λ′′ = 1 or, by Theorem 1.1,


λ|k(T ) is of order two and λ′|i(F×) i = ωE(λ)/E |F× . If λ′′ = 1, we are done.Thus, assume the latter. In this case, λ′′ is completely determined by (2.1)since λ′′|k(T ) is trivial. Now, also by (2.1), since λ|k(T ) = λ−1|k(T ), it sufficesto show that λ|i(F×) = (λ′)−1|i(F×). But for a in F×, we have

λ2(i(a)) = λ

(k

(a 00 a−1

))as is easily checked and thus

λλ′(i(a)) = λ2λ′′(i(a))

= λ2(i(a))λ′′(i(a))

= λ

(k

(a 00 a−1

))ωE(λ)/E(a)

= ωE(λ)/E(a)ωE(λ)/E(a)= 1.

Now in general considering

π(λ)Ψ(( 01

10),1)

one sees from Frobenius Reciprocity that π(λ) ∼= π(λ−1). Thus, the caseλ′|k(T )

∼= λ−1|k(T ) follows from the previous case. The arguments for theremaining equivalences are similar, using that the relevant composition seriesare multiplicity free.

Finally, we consider Galois invariance. Let π be a nonsupercuspidal ir-reducible representation of H0

1 . Then HomH01(π, Ind(H0

1 , P ;λ)) 6= 0 forsome λ, a one-dimensional representation of M . Now one checks thatInd(H0

1 , P ;λ)σ ∼= Ind(H01 , P ;λσ) since the modulus character is invariant.

Then it follows from a similar argument to that for the equivalences (invari-ance was not used) that λ = λσ or λ = (λ−1)σ. Now one checks that λ = λσ

if and only if λ2(j(E1)) = 1 and λ = (λ−1)σ if and only if λ2(i(F×)) = 1.The invariance portions of (i), (ii), (iii) then follow. Thus, assume λ|k(T )

is of order two. Now by local class field theory E(λ)/F is Galois if andonly if E(λ) = E(λσ). Then since λ|k(T ) is of order two, it follows thatE (λ) = E(λσ) if and only if λ|k(T ) = λσ|k(T ) and λ|k(T ) = (λ−1)σ|k(T ). Butthese are equivalent to λ2(j(E1)) = 1 and λ2(i(NE/F (E×))) = 1. Finally,since

λ2(i(a)) = λ

(k

(a 00 a−1

))for all a in F×, our determination of the order of λ follows from our discus-sion of reducibility.


3. Jacquet modules.

We now turn to Jacquet modules to find restrictions, in addition to Galoisinvariance, on the nonsupercuspidal representations of H1 that can occur inRχ(H1). We realize ωχ on S(V ), the space of locally constant compactlysupported functions on V with the action of H1 being the natural linearaction, i.e., a Schrodinger model. Set S(V )N = S(V )/〈f − ωχ(n)f | f ∈S(V ), n ∈ N〉 and view S(V )N as an M -module (unnormalized) as usual.Finally, for f in S(V ), let f denote the image of f in S(V )N .

For l an integer, let Ul denote the neighborhood of 0 in V consisting ofthose v such that vii is in P lF for i = 1, 2 and v12 is in OEP lF where vijdenotes the (i, j)-entry of v. Then the Ul form a neighborhood basis of 0 inV . For A in V and l an integer, define fA,l in S(V ) by setting

fA,l(v) =

1 if A− v is in Ul,0 otherwise.

Then the fA,l span S(V ).

Theorem 3.1. Suppose λ in M∧ occurs as a quotient of (ωχ)N . Set λ′ =λ · | |−1. Then either λ′|i(F×) is trivial or λ′|j(E1) is trivial. Further, ifλ′|i(F×) is trivial, then λ′ is determined by λ′|k(T ). Finally, if λ′|j(E1) istrivial, then λ′ is determined by λ′|k(T ) up to (possibly) twisting by ωE/F .

Proof. Let T : S(V )N → C be a nonzero element of HomM ((ωχ)N , λ). Thenthere exist A and l such that T fA,l 6= 0.

We proceed by cases. First, suppose νF (a22) < l and that νE(a12a−122 ) ≥ 0.

Let

n =(

1 −a12a−122

0 1

).

Then one checks that ωχ(k(n))fA,l = fA1,l with

A1 =(a11 00 a22

).

It follows that T fA1,l 6= 0 and thus λ(j(E1)) = 1 since

Tωχ(j(α))N fA1,l = λ(j(α))T fA1,l

for all α in E1 as can be checked. Then by (2.1),

λ(i(N(a))) = λ

(k

(a 00 a−1

))for a in F× and the result follows.

Now suppose that ν(a22) < l but νE(a12a−122 ) < 0. Choose m such that

νE((a12a−122 )ωmF ) ≥ 0.


Let Ul,m =A ∈ Ul | a12 ∈ OEP l+mF , a22 ∈ P l+2m

F

. For B in Ul, set

fA,l,B(v) =

1 if A+B − v is in Ul,m0 otherwise.

Then since T fA,l 6= 0, it follows that there exists B in Ul such that T fA,l,B 6=0. Let

n =(

1 −(a12 + b12)(a22 + b22)−1

0 1

).

Then one checks that ωχ(k(n))fA,l,B = fA1,l,B1 where (A1 + B1)12 = 0 and(A1 + B1)ii = (A + B)ii. Then as in the previous case, one checks thatλ(j(E1)) = 1 and

λ(i(N(a))) = λ

(k

(a 00 a−1

))for a in F×, whence the result.

Finally, suppose that ν(a22) ≥ l. In this case, we may assume a22 = 0.Now if a12 is in OEP lF , then arguing as above we obtain that λ(j(E1)) = 1and

λ(i(N(a))) = λ

(k

(a 00 a−1

))for a in F× and we are done. Thus, suppose a12 is not in OEP lF . Nowsuppose a11 is in P lF . Then since E/F is separable, there exists an integerl′ such that trE/F (P l

′Ea12) = P lF . Similarly, if a11 is not in P lF , then there

exists l′ such that trE/F (P l′Ea12) = P

ν(a11)F . Let m′ ≥ l be an integer such

that N(P l′E)ωm

′F ⊆ P l+1

F and tr(P l′EP

m′F ) ⊂ P l+1

F . Further, let m ≥ m′ be aninteger such that P l

′EP

mF ⊆ OEPm

′F . Finally, let U ′ denote the neighborhood

of 0 in V consisting of all v such that νF (v11) ≥ l, νF (v22) > m and v12 is inOEPm

′F . Then by an argument similar to the above in the case a22 = 0, we

may assume that T f ′A is nonzero where A′ is in V such that A−A′ is in Ul,and for any B in V , f ′B in S(V ) is defined by

f ′B(v) =

1 if v −B is in U ′,0 otherwise.

Without loss of generality, we assume A′ = A so that T f ′A 6= 0.Now, by our choice of l′, there exists x in P l

′E such that trE/F (xa12) = a11.

Let

n =(

1 x0 1

).

Then one checks that, by virtue of our choices ofm andm′, ωχ(k(n))f ′A = f ′Bwith b11 = 0, b12 = a12 and b22 = a22 = 0. Thus, we may assume that a11 =


0 in considering T f ′A 6= 0. It then follows that, for b in O×F , ωχ(i(b))f ′A = f ′Aand thus λ(i(b)) = λ′(i(b)) = 1. Hence to complete the proof, it suffices toshow that λ(i(ω−1

F )) = q.For B in V , define f ′′B in S (V ) by

f ′′B(v) =

1 if v −A is in i(ω−1F )U ′,

0 otherwise.

Then it is clear that ωχ(i(ω−1F ))f ′A = f ′′A. Now

f ′′A =∑B∈R

gA+B

where R is a set of coset representatives for U ′′/i(ω−1F )U ′ with U ′′ =

v ∈ U ′ : νF (v22) ≥ m+ 1 and gA+B is defined as follows:

gA+B(v) =

1 if v −A−B is in U ′′,0 otherwise.

Then arguing as above in the case a11 6= 0, one shows that gA+B = gA forall B in R. Thus,

ωχ(i(ω−1

F ))Nf ′A = qgA

so that

Tωχ(i(ω−1

F ))Nf ′A = qT gA

with both sides nonzero. But now

gA =∑C∈S

f ′A+C

with S a set of coset representatives for U ′/U ′′. Then by an argumentsimilar to that in case a22 6= 0, the result of the theorem follows if T f ′A+C isnonzero for any C not in U ′′. Thus, we may assume T gA = T f ′A and thenthe theorem follows.

As an immediate consequence of the above results, the exactness of theJacquet functor and the adjointness of the Jacquet functor and induction,we have the following.

Corollary 3.2. Let π be an irreducible nonsupercuspidal representation ofH1 which is in Rχ(H1). Then π0 = π|H0

1is irreducible and π, as an element

of Rχ(H1), is determined by π0. Moreover, π0 is of the form π(λ), σ(λ), orπ±(λ) for some λ with λ|i(F×) trivial or λ|j(E1) trivial. In particular, withλ as above, λ is determined by λ|k(T ) if λ|i(F×) is trivial and is determinedby λ|k(T ) up to a (possible) twist of ωE/F if λ|j(E1) is trivial.


To close this section we prove an elementary lemma that will be useful inwhat follows. The statements on restriction in the lemma can easily be mademore precise but we leave that to the reader since the following suffices, forour purposes.

Lemma 3.3. Let π be an irreducible representation of H1 such that π0 =π|H0

1is irreducible. Moreover, assume π0 is of the form π(λ), σ(λ) or π±(λ)

for some λ with λ|i(F ) or λ|j(E1) trivial.

(i) Suppose λ|i(F×) is trivial. Then π0 is of the form π(λ) or π±(λ) andwhen restricted to k(PSL2(E)) decomposes as a sum of representationsin the L-packet associated to π(ρ, ρσ) where ρ is any character of E×

such that ρ(a/a) = λ(j(a/a)) for a in E×.(ii) Suppose λ|j(E1) is trivial. Then if π0 is of the form σ(λ), then π0

restricts to σ(λ i N) on k(PSL2(E)). If π0 is of the form π(λ) orπ±(λ), then, when restricted to PSL2(E), π0 decomposes as a sum ofrepresentations in the L-packet attached to π(λ i N, 1).

Proof. Let π1 be an irreducible representation of PSL2(E) that appears inthe restriction of π to k(PSL2(E)). Then since i(F×)k(PSL2(E)) = H0

1 , anyother representation appearing in the restriction of π to k(PSL2(E)) mustbe of the form π

i(a)1 for a in F×. But then

i(a)k(g)i(a−1) = k

((a 00 1

)g

(a 00 1

)−1)

and

k

(b 00 b−1

)= i(N(b))j(b/b)

imply the result.

4. More on occurrence.

In the previous section, we found some necessary conditions, Corollary 3.2,for nonsupercuspidal representations of H1 to occur in the correspondence.In this section, we will show these conditions are also sufficient, with onepossible exception. The one possible exception is a generalized Steinbergrepresentation as is explained in (ii) and (iii) of the following theorem.

Theorem 4.1. Let π0 be an irreducible representation of H01 .

(i) If π0 is of the form π(λ) or π±(λ) with λ|i(F×) trivial or λ|j(E1) trivial,then π0 has a unique extension to H1 which occurs in Rχ(H1).

(ii) If π0 = σ (| |) or σ(ωE/F | |

)then at most one extension of π0 to H1

occurs in Rχ(H1).


(iii) At least one extension of σ (| |) or σ(ωE/F | |

)occurs in Rχ(H1) and

pairs with the Steinberg representation σ (| |) of G1.(iv) No other nonsupercuspidal representations of H1 can occur.

Proof. It suffices to show (i) through (iii) since then (iv) would follow fromCorollary 3.2.

First consider (i) and suppose that λ|i(F×) is trivial. Let θ be a characterof E× such that θ|E1 = λ|j(E1) j. Now suppose further that λ is not oforder two. Then it follows from Theorem 1.2 that π(θ;χ) occurs in Rχ(G1).Further, by Kudla’s perseverence result [K2], it must pair with π(λ). Like-wise, if λ|i(F×) is trivial but λ is of order two, then π(θ;χ) and π(θ;χb) occurin Rχ(G1) and pair with π±(λ) where b is in NE/F (E×) such that π(θ;χ)and π(θ;χb) are distinct. Therefore, in proving the theorem, we may assumethat λ|i(F×) is nontrivial.

Now consider λ with λ|i(F×) nontrivial. Let µ be a character of F×.Assume for the moment that µE 6= | |±1 and that µ2

E is not trivial. Thenµ2 is also nontrivial and thus π(µ, 1) and π(µE , 1) restrict irreducibly toG1(F ) and G1(E), respectively, with π(µE , 1) the base change of π(µ, 1).Now by Theorem 1.2, π(µ, 1) is in Rχ(G1). Then since µ 6= | |±1 , π(µ, 1) isinfinite-dimensional and thus, by Theorem 1.6, the image of π(µ, 1) underthe theta correspondence, θ(π(µ, 1)) say, must restrict to k(PSL2(E)) asa sum of copies of π(µE , 1). Now since µ2 is nontrivial, π(µ, 1) does notoccur in the theta correspondence attached to χ and (G1,H0) where H0 isthe orthogonal group attached to the anisotropic part of (V,Q). Further,since the correspondence attached to χ and (G1,H1) is a bijection, even inp = 2 [R2], it follows from the argument of the previous paragraph thata λ giving rise to θ(π(µ, 1)) must satisfy λ|j(E1) is trivial. Then it followsfrom Lemma 3.3 that we may assume λ must satisfy µE = λ i N , whenceλ = λ1 or λ2 where

λl(i(a)j(b/b)) = µ(a)ωlE/F (a)

for l = 1, 2, b ∈ E× and a in F×. It follows that both π(µ, 1) and π(µωE/F , 1)occur in Rχ(G1) and pair with either π(λ1) or π(λ2) in Rχ(H1). Then, onceagain, since the correspondence is a bijection, we get that the theorem holdsfor all π(λ) with (λ|i(N(E×)))2 nontrivial and λ|i(N(E×)) 6= | |.

If λ is trivial on j(E1) and λ(i(a)) = |a|, then π(λ) is the trivial rep-resentation and, as is well-known, π(λ) occurs in Rχ(H1) and pairs withπ(ωE/F | | , 1

)in Rχ(G1), see [KR2]. Further, by Theorem 1.6 and Lem-

ma 3.3, σ (| |) is in Rχ(G1) and pairs with σ (| |) or σ(ωE/F | |

)in Rχ(H1).

Now let µ be a character of F× of order two with µE nontrivial. Then theL-packets for G1(F ) and G1(E) associated to π(µ, 1) and π(µE,1), respec-tively, each have two components as does the L-packet for G1(F ) attached


to π(ωE/Fµ, 1). By Theorem 1.2, the four representations in the L-packetsassociated to π(µ, 1) and π(µωE/F , 1) occur. Then, by arguments similarto those above, they must pair with the four representations of H1, as inLemma 3.3, attached to λ and ωE/Fλ where λ is the character of M definedby

λ(i(a)j(b/b)) = µ(a).

Further, consider the representation π(1, 1) of G′1(F ). It restricts irreduciblyto G1(F ) and by arguments also similar to those above it occurs and pairswith the representation π(λ) of H1 with

λ(i(a)j(b/b)) = ωE/F (a).

Finally, consider those nontrivial characters of µ on F× such that µ2 isnontrivial while µ2

E is trivial. Such a character has order four and by localclass field theory is associated to a cyclic extension of degree four of Fwith the quadratic subfield being E. Further, if µ is such a character, thenso is µωE/F . Then arguments such as those above show that π(µ, 1) andπ(µωE/F , 1) are in Rχ(G1) and pair with π(λ1) and π(λ2) in Rχ(H1) where

λl(i(a)j(b/b)) = µ(a)ωlE/F (a),

l = 1, 2.To summarize, at this point we have shown that the theorem holds for all

representations ofH01 with the possible exception of the one-dimensional rep-

resentation π(ωE/F | |

). Let π+

(ωE/F | |

)denote the extension of

π(ωE/F | |

)to H1 with σ acting as the identity. Then the following lemma

completes the proof of the theorem.

Lemma 4.2. The one-dimensional representation π+

(ωE/F | |

)occurs in

Rχ(H1) and pairs with the trivial representation of G1.

Proof. It suffices to show that HomH1

(ω∞χ , π+

(ωE/F | |

))is one-dimen-

sional. Write π+

(| |ωE/F

)= ω+

E/F . Let

x =(

1 00 0

)in V . Then one checks that the stabilizer of x inH1 isHx = j(E1)k(N)o〈σ〉.One checks further that Hx is unimodular and that ω+

E/F |Hx is trivial sincethe image of the spinor norm restricted to Hx ∩H0

1 is N(E×) because

k

(a 00 a−1

)= i(N(a))j(a/a).


Thus, it follows from [W] that there exists a unique, up to scalar, linearmap T : S(H1/Hx)→ C such that

Tfg = ω+E/F (g)Tf(4.1)

where fg(x) = f(gx). Now let Y = Q−1(0) and Y ′ = Y − 0. Then byWitt’s Theorem, Y ′ can be identified with H1/Hx and thus we can view Tas a distribution on Y ′.

We claim further that T can be extended to a distribution on S(Y ) sat-isfying (4.1) for all f in S(Y ). To see this we recall that, up to a nonzeroscalar, T can be written

Tf =∫H1/Hx

ω+E/F (y)f(y) dy

for all f in S(H1/Hx) where∫H1/Hx

dy is a left-invariant positive regularBorel measure on Cc(H1/Hx), the space of compactly supported functionsonH1/Hx and ω+

E/F (y) is well-defined since ω+E/F is trivial onHx. Moreover,

the measure is finite on compact sets and may be normalized so that∫H1/Hx

∫Hx

f(yh) dhdy =∫H1

f(g) dg(4.2)

where∫H1dg and

∫Hxdh denote Haar measure on H1 and Hx respectively

and f is any function in Cc(H1). We claim that T can be extended to S(Y )by setting Tf = Tf |Y , for all f in S(Y ). To show this it suffices to show ifU is a compact neighborhood of 0 in Y then U ∩ Y ′ is of finite volume withrespect to

∫H1/Hx

dy. To this end, using a Bruhat decomposition, H1/Hx

can be identified with

i(F×) ∪ k(N)k(w)i(F×)

where w is the standard Weyl element(

0 1−1 0

)for SL2(F ). Now consider the

effect of conjugation by i(a), a ∈ F×, on (4.2). Since H1 is unimodular wehave ∫

H1

f(y) dg =∫H1

f(i(a)g(i(a))−1) dg(4.3)

=∫H1/Hx

∫Hx

f(i(a)y(i(a))−1i(a)h(i(a))−1) dhdy

=∫H1/Hx

|a|−1∫Hx

f(y(i(a))−1h) dhdy

where the last equality follows from explicit realization of the measure∫Hxdh

and left-invariance of∫H1/Hx

dy. Finite volume then follows from explicitly

realizing∫H1/Hx

dy taking into account the |a|−1. It is immediate that theextension satisfies (4.1) since Y ′ is H1 invariant.


Now let T : S(Y ) → C denote the extension constructed above. Thensince Y is closed in V we can extend T to S(V ) by setting Tf = T (f |Y ).Further since Y is H1 invariant T still satisfies (4.1), whence T is a nonzerointertwining map. Furthermore, T is the unique, up to scalar, nonzerointertwining map with image in D(V − Y ), the space of distributions onS(V − Y ), equal to zero since 0 only supports the trivial representation.

Now suppose S : S(V ) → C is an arbitrary nonzero element inHomH1(ω

∞χ , ω

+E/F ). Then it suffices to show that the image of S in D(V −Y )

is zero. Suppose that the image is nonzero. Then by [BZ], there exists anorbit Q−1(a) in V − Y, a ∈ F× which supports a nontrivial intertwiningoperator. Let X = Q−1(a), a ∈ F×, be such an orbit and let y ∈ X. Thensince Q(X) 6= 0, we can write V = 〈y〉 ⊕W with W = 〈y〉⊥. Now let O(W )be the orthogonal group associated to W and Q|W . Then X = H1/O(W ).Now since W is three-dimensional, the spinor norm restricted to O(W ) isonto F×. Further, both H1 and O(W ) are unimodular and thus any inter-twining operator which is trivial when restricted to O(W ) must have beentrivial on H1 [W]—contradicting the surjectivity of the spinor norm.

References

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[GL] P. Gerardin and J.-P. Labesse, The solution of a base change problem for GL(2)(following Langlands, Saito, Shintani), in ‘Automorphic forms, representationsand L-functions’, Proc. Sympos. Pure Math., 33(2), Amer. Math. Soc., Provi-dence, 1979, 115-133, MR 82e:10047, Zbl 412.10018.

[H] R. Howe, θ-series and invariant theory, in ‘Automorphic Forms, Representationsand L-functions’, Proc. Sympos. Pure Math., 33(1), Amer. Math. Soc., Provi-dence (1979), 275-285, MR 81f:22034, Zbl 423.22016.

[I] K. Iwasawa, Local Class Field Theory, Oxford University Press, New York, 1986,MR 88b:11080, Zbl 604.12014.

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[K2] , On the local theta-correspondence, Invent. Math., 83 (1986), 229-255,MR 87e:22037, Zbl 583.22010.




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[KR1] S. Kudla and S. Rallis, On first occurrence in the local theta correspondence, inpreparation.

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[LL] J.-P. Labesse and R.P. Langlands, L-indistinguishability for SL(2), Canad. J.Math., 31 (1979), 726-785, MR 81b:22017, Zbl 421.12014.

[L] R.P. Langlands, Base Change for GL(2), Ann. of Math. Stud., 96, PrincetonUniv. Press, Princeton, 1980, MR 82a:10032, Zbl 444.22007.

[M] D. Manderscheid, Quadratic base change for p-adic SL(2) as a theta corre-spondence I: Occurrence, Proc. Amer. Math. Soc., 127(5) (1999), 1281-1288,MR 99j:22026.

[MVW] C. Moeglin, M.-F. Vigneras and J.-L. Waldspurger, Correspondances de Howesur un Corps p-adique, Lecture Notes in Math., 1291, Springer-Verlag, Berlin,1987, MR 91f:11040.

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[R1] S. Rallis, On the Howe duality conjecture, Compositio Math., 51 (1984), 333-399,MR 85g:22034, Zbl 624.22011.

[R2] , L-functions and the Oscillator Representation, Lecture Notes in Math.,1245, Springer-Verlag, Berlin, 1987, MR 89b:11046, Zbl 605.10016.

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[Ro2] , The non-Archimedean theta correspondence for GSp(2) and GO(4),Trans. Amer. Math. Soc., 351 (1999), 781-811, MR 99d:11056.

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[W] A. Weil, L’Integration dans les Groupes Topologiques et ses Applications, Her-mann, Paris, 1951.

[Wa] J.-L. Waldspurger, Demonstration d’une Conjecture de Dualite de Howe dans lecas p-adique, p 6= 2, Festschrift in honor of I.I. Piatetski-Shapiro on the occasion ofhis sixtieth birthday, Israel Math. Conf. Proc., 2, Weizmann, Jerusalem, (1990),267-324, MR 93h:22035, Zbl 722.22009.

Received June 1, 1999 and revised July 11, 2000. This research was supported in part byNSF grant DMS-9003213 and NSA grant MDA904-97-1-0046.

Mathematics DepartmentUniversity of IowaIowa City, IA 52242-1419E-mail address: [email protected]






























UNIFORM INTEGRABILITY OF APPROXIMATE GREENFUNCTIONS OF SOME DEGENERATE ELLIPTIC

OPERATORS

Ahmed Mohammed

We prove the uniform integrability of the approximateGreen functions of some degenerate elliptic operators in di-vergence form with lower order term coefficients satisfying aKato type condition. Some further properties of the approx-imate Green functions of such operators are also established.

1. Introduction.

In this paper, we study the approximate Green functions of certain degen-erate elliptic operators L on balls in Rn, n > 2, when L has the divergenceform

L := −n∑

i,j=1

∂

∂xi

(aij(x)

∂

∂xj

)+

n∑i=1

bi(x)∂

∂xi+ V (x).

The coefficients aij are real-valued measurable functions whose coefficientmatrix A(x) := (aij(x)) is symmetric and satisfies

ω(x)|ξ|2 ≤ 〈A(x)ξ, ξ〉 ≤ υ(x)|ξ|2.(1.1)

Here 〈., .〉 denotes the usual Euclidean inner product, and υ, ω are weightfunctions that will be stipulated below.

Throughout, we will use the following notations. For functions f and g,we shall write f . g to indicate that f ≤ Cg for some positive constantC. We write f ≈ g if f . g and g . f . We shall use Bt(x) to designatea ball of radius t centered at x. Also, tB will be used to represent the ballconcentric with the ball B, but with radius t times as big. Given a locallyintegrable function f , we shall let f(B) denote the Lebesgue integral of fover the set B. If f ∈ Lloc(dµ), where dµ := γ(x) dx is a weighted measure,then we denote by

−∫Bf(x)γ(x) dx :=

1γ(B)

∫Bf(x)γ(x) dx,

the µ-average of f over B. This average shall also be denoted by fB, γ.

467



468 AHMED MOHAMMED

A non-negative locally integrable function ω on Rn is said to be in theclass A2 if 1/ω is also locally integrable and there is a constant C such thatfor all balls B, (

−∫Bω(x) dx

)(−∫B

1ω(x)

dx

)≤ C.

A non-negative locally integrable function υ on Rn is said to satisfy adoubling condition if there is a constant C such that υ(2B) ≤ Cυ(B) for allballs B. Here C is independent of the center and radius of B. We denotethis by writing υ ∈ D∞. It is known that A2 ⊂ D∞.

It is also known (see [9]) that if υ satisfies a doubling condition, then itsatisfies

υ(tB) ≤ C1tkυ(B), and υ(B) ≤ C2t

−m(tB), t > 1,

for some positive constants C1, C2, k , and m. The second condition is calleda reverse doubling condition.

The following assumptions will be made on ω, and υ.ω and υ are non-negative locally integrable functions on Rn that satisfy

the following conditions:

ω ∈ A2, υ ∈ D∞;(1.2)

ω and υ are related by the existence of some q > 2 such that(1.3)

s

t

[υ(Bs(x))υ(Bt(x))

] 1q

≤ C[ω(Bs(x))ω(Bt(x))

] 12

, 0 < s < t, x ∈ Rn,

for some constant C independent of x, s and t.We shall use the notation σ = q/2 so that σ > 1. Note that when υ and ω

are positive constants, as in the strongly elliptic case, the value of q in (1.3)is q = 2n/(n− 2), so that σ = n/(n− 2).

Let now L0 be the principal part of L; that is

L0 := −n∑

i,j=1

∂

∂xi

(aij(x)

∂

∂xj

).

Let B0 be a ball of radius R that will be fixed in the sequel. Under theconditions (1.2) and (1.3), Chanillo and Wheeden have established, in [3] theexistence and integrability properties of the Green function of L0. Amongother important properties, they have shown that if G(x, y) is the Greenfunction of L0 on 2B0, then for 0 < p < σ,

supy∈B0

∫2B0

G(x, y)pυ(x) dx <∞.(1.4)

UNIFORM INTEGRABILITY OF APPROXIMATE GREEN FUNCTIONS 469

Let B ⊂ B0. In analogy with the way the usual Kato class is defined, weintroduce a class of functions Kn(B) as

Kn(B) :=h ∈ L1

loc(B) : limr→0+

η(h)(r) = 0,

whereη(h)(r) := sup

x∈B

∫Br(x)∩B

G(y, x)|h(y)| dy.

If Lpµ(B) denotes the usual Lp space with respect to the measure µ, then forB ⊂ B0, and p > σ/(σ − 1), the following inclusion holds:

Lpυ1−p(B) ⊂ Kn(B).

To see this let h ∈ Lpυ1−p(B), and x ∈ B. We pick σ/(σ−1) < s < p. Define

s′ by 1/s+1/s′ = 1 (we will use this notation throughout). Then, by Holderinequality∫

Br(x)∩BG(y, x)|h(y)| dy ≤

(∫2B0

G(y, x)s′υ(y) dy

) 1s′

·

(∫Br(x)∩B

|h(y)|sυ1−s dy

) 1s

.

Since υ satisfies a reverse doubling condition, there exist positive constants Cand d such that υ(BR(x)) ≥ C(R/r)dυ(Br(x)) for any 0 < r < R. Therefore∫

Br(x)∩B

(|h(y)|υ

)sυ dx

≤

(∫Br(x)∩B

(|h(y)|υ

)pυ dx

) sp(∫

Br(x)υ dx

) p−sp

≤(∫

B

(|h(y)|υ

)pυ dx

) sp[C( rR

)d] p−sp[υ(2B0)]

p−sp .

Thus, from this last inequality and (1.4), we get the desired conclusion.For notational simplicity, we shall use K for the function space Kn(B0).

Remark 1.1. We should remark that when υ and ω are identically equalto positive constants, as in the strongly elliptic case, the class of functionsK coincides with the usual Kato class (see [1], or [4] for definition). Also, ifυ and ω are constant multiples of each other, then again K is the same asthe one introduced in [6].

The following assumptions will be made of the lower order coefficientsb := (b1, b2, · · · , bn), and V of the degenerate elliptic operator L.

|b|2ω−1, V ∈ K.(1.5)

470 AHMED MOHAMMED

The paper is organized as follows. As the work here relies heavily onthe results of the important works of S. Chanillo and R. Wheeden in theirpapers [2], and [3], we will recall several of their results that are relevantto our discussion in Section 2. We start Section 3 by proving the bounded-ness of certain linear functionals on some Hilbert spaces. These functionalsare associated with elements of the Kato type class defined above. Someproperties related to the approximate Green function of L0 will also be ob-tained. The main result in this paper is Theorem 3.2 which establishes theuniform integrability of approximate Green functions of L on balls. Uniformintegrability of approximate Green functions is a useful tool in proving exis-tence and size estimates of the Green function. See [3], [5] and [8] for suchapplications. In a forthcoming paper, we will use this uniform integrabilityresult to derive Harnack’s inequality for functions naturally associated withnon-negative solutions of the operator L.

2. Preliminaries and background.

Let Ω ⊂ Rn be a bounded open set. Using a standard notation, let Lip(Ω)denote the class of Lipschitz continuous functions on the closure Ω. We saythat φ ∈ Lip0(Ω) if φ ∈ Lip(Ω) and φ has compact support contained in Ω.The following two-weight Sobolev inequality has been proved in [2].

Let ω, υ be non-negative locally integrable functions that satisfy (1.2),(1.3), and q be the constant that appears in (1.3). Then, for any ball B,(

−∫B|f |qυ dx

) 1q

≤ C|B|1n

(−∫B|∇f |2ω dx

) 12

, f ∈ Lip0(B).(2.1)

The constant C is independent of both the ball B and f .Now let us consider the inner product

a0(u, ϕ) :=∫

Ω〈A∇u,∇ϕ〉, u, ϕ ∈ Lip0(Ω).

The completion of Lip0(Ω) with respect to the norm a0(u, u)1/2 is denotedby H0(Ω). An element of H0(Ω) is thus an equivalence class of Cauchysequences uk, uk ∈ Lip0(Ω). If u, ϕ ∈ H0(Ω), with u = uk, ϕ =ϕk, uk, ϕk ∈ Lip0(Ω), then a0(uk, ϕk) is convergent, and we define

a0(u, ϕ) = limka0(uk, ϕk).

In this way, ‖u‖0 := a0(u, u)1/2 defines a norm on the Hilbert space H0(Ω).Lip0(B) is included in H0(B) by considering ϕk with all ϕk = ϕ ∈Lip0(B). As a consequence of the Sobolev inequality (2.1), it is possibleto associate with each ϕ ∈ H0(Ω) a unique pair (ϕ,∇ϕ) so that if ϕ = ϕk,then ϕk → ϕ in L2σ

υ (Ω), and ∇ϕk → ∇ϕ in L2ω(Ω). We shall refer to (ϕ,∇ϕ)


as the pair of functions associated with ϕ. This pair is independent of the par-ticular representation ϕk of ϕ. If ϕ ∈ Lip0(Ω), then ϕ = ϕ, and ∇ϕ = ∇ϕ.Furthermore, it can be shown that given ϕ ∈ H0(Ω), ∇ϕ is the distribu-tional gradient of ϕ. See [3] for proofs of these assertions.

For future reference, we record the following inequality that can be easilyverified using the Cauchy-Schwarz inequality.

‖ϕφ‖0 ≤ ‖ϕ‖∞‖φ‖0 + ‖φ‖∞‖ϕ‖0, ϕ, φ ∈ Lip0(Ω).(2.2)

We will also consider the Hilbert space H(Ω) which is the completion ofLip(Ω) under the inner product

a(u, ϕ) := a0(u, ϕ) +∫

Ωuϕυ, u, ϕ ∈ Lip(Ω).

If u ∈ H(Ω), u = uk, uk ∈ Lip(Ω), then uk converges in L2υ(Ω) to

a function u, and ∇uk converges in L2ω(Ω) to a vector ∇u . If ϕ =

ϕk, ϕk ∈ Lip(Ω), then the limits a(u, ϕ) = limk a(uk, ϕk) and a0(u, ϕ) =limk a0(uk, ϕk) exist, and satisfy

a(u, ϕ) := a0(u, ϕ) +∫

Ωuϕυ.

In this way, a(u, ϕ) defines an inner product on H(Ω), and ‖u‖ := a(u, u)1/2

defines a norm. By the Sobolev inequality (2.1), H0(Ω) is continuouslyembedded in H(Ω).

For u ∈ H(Ω) we say that u ≥ 0 on Ω, if uk ≥ 0 for all k and some ukrepresenting u. If u ≥ 0 on Ω, then u ≥ 0 a.e. on Ω. The following, provedin [3], will be useful to us.

Let u, ϕ ∈ H(Ω), and ∇u, ∇ϕ be the associated gradients respectively.If u = uk, ϕ = ϕk, then as k →∞∫

Ω|〈A∇uk,∇ϕk〉 − 〈A∇u,∇ϕ〉| → 0.(2.3)

In particular

a0(u, ϕ) =∫

Ω〈A∇u,∇ϕ〉, and a(u, ϕ) =

∫Ω〈A∇u,∇ϕ〉+

∫Ωuϕυ.

Before we proceed further, we should perhaps make two remarks. LetB ⊂ B0 be a ball.

Remark 2.1. If fυ−1 ∈ L(2σ)′

υ (B), then

ϕ 7→∫Bfϕ(2.4)

defines a continuous linear functional on H0(B). This follows from Holder’sinequality and the Sobolev inequality (2.1). Therefore, by the Lax-Milgram

472 AHMED MOHAMMED

theorem there is a unique u ∈ H0(B) such that

a0(u, ϕ) =∫Bfϕ.

We shall refer to u as the Lax-Milgram solution of L0u = f in B and u =0 on ∂B.

By the above Remark, given x ∈ B, and ρ > 0 with Bρ(x) ⊂ B there is aunique Gρ ∈ H0(B) such that

a0(Gρ, ϕ) = −∫Bρ(x)

ϕυ, ϕ ∈ H0(B).

Gρ is called an approximate Green’s function of L0 on B with pole x.

Remark 2.2. Let f ∈ L1(B) such that the map in (2.4) is a continu-ous linear functional on H0(B). Suppose ϕk is a bounded sequence inH0(B). Then ϕk contains a subsequence ϕkj that converges weakly tosome element ϕ ∈ H0(B). Now, if u = uk ∈ H(B) is fixed, then since|a0(u, ϕ)| ≤ ‖u‖‖ϕ‖0, we have

a0(u, ϕ) = lim a0(u, ϕkj ) = lim[a0(ukj , ϕkj ) + a0(u− ukj , ϕkj )

]= lim a0(ukj , ϕkj ).

Therefore

a0(u, ϕ)−∫Bfϕ = lim

j→∞

[a0(ukj , ϕkj )−

∫Bfϕkj

].

We shall need several lemmas from [3], and we will state them below forthe readers’ convenience.

Lemma 2.1. Suppose u is a supersolution in H0(Ω); that is u ∈ H0(Ω),and a0(u, ϕ) ≥ 0 whenever 0 ≤ ϕ ∈ Lip0(B). Then u ≥ 0.

Proof. The proof that the approximate Green function Gρ of L0 is non-negative is given on page 323 of [3]. It depends on properties of the innerproduct a0(., .) and the fact that a0(Gρ, ϕ) ≥ 0 for 0 ≤ ϕ ∈ H0(Ω). Exactlythe same proof applies in our case. In fact, if u := uk, then also u = |uk|.See Lemma 3.6 below for a detailed proof.

Lemma 2.2. Let G(x, y) be the Green function of L0 on 2B and fυ−1 ∈Lt

′

υ (2B) for some t < σ. If u is the Lax-Milgram solution of L0u = f in 2B,then

u(y) =∫

2BG(x, y)f(x) dx, for a.e. y ∈ B.

Another useful Lemma is the following weak maximum principle (cf.Lemma 2.1 above).


Lemma 2.3 (Weak Maximum Principle). Let u ∈ H(Ω) satisfy a0(u, ϕ) ≥0 if ϕ ∈ Lip0(Ω), ϕ ≥ 0. Let u = uk, uk ∈ Lip(Ω) and assume that uk ≥ 0in some neighborhood (depending on k) of ∂Ω. Then u ≥ 0 a.e in Ω.

Let Gρ be the approximate Green function of L0 on a ball B with polex ∈ B, and G be the corresponding Green function. In [3], it was shownthat for an appropriate subsequence, Gρk(y)→ G(x, y) pointwise a.e. on Bfor a.e. x ∈ 1

2B. If B ⊂ B∗, then the weak maximum principle, Lemma 2.3shows that Gρ ≤ Gρ∗ a.e on B if Gρ∗ is the approximate Green function ofL0 on B∗ with pole x. Consequently, the inequality G ≤ G∗ holds a.e. on12B ×

12B, where G∗ is the Green function of L0 on B∗.

We also need Lemma (2.7) of [3] in the following slightly modified form.To accomodate this change, we shall indicate the minor alterations neededin the proof of Lemma (2.7) of [3].

Lemma 2.4. Let Bj := Bj(x0) be balls of radius rj for j = 1, 2, 3, withrj < rj+1. If ϕ ∈ H(B3) and ϕ ≤ m a.e. in B2, then given any M > m,and r1 < r < r2, there exist ϕk ∈ Lip(B3) such that ϕk → ϕ in H(B3) andϕk ≤M a.e. on B∗(x0), a ball of radius r.

Proof. As in [3], we pick hk ∈ Lip(B3) with hk → ϕ in H(B3). Thus hk → ϕin L2

υ(B3), and by using a subsequence, we may assume that hk → ϕ a.e. onB3. By hypothesis, ϕ ≤ m a.e. on B2. By Egorov’s theorem, given M > m,and δ > 0, there exist E ⊂ B2, and k0 such that |B2 r E| < δ and hk ≤Mon E, if k ≥ k0. Let χ ∈ C∞c (B2), 0 ≤ χ ≤ 1, and χ ≡ 1 on B∗, whereB∗ := B∗(x0) is a ball of radius r. We now define ϕk := hkχ∧M+hk(1−χ).Clearly ϕk ∈ Lip(B3), and ϕk ≤ M a.e. on B∗. It now remains to showthat ϕk → ϕ in H(B3). Noting that ϕk − hk is supported on B2, the rest ofthe proof proceeds in the same way as that of Lemma (2.7) of [3].

Remark 2.3. If ϕ ≥ 0 in the sense of H(B3), then the ϕk in Lemma 2.4can be taken to be non-negative, as can be seen from the definition of ϕk inthe proof.

Remark 2.4. Let 0 ≤ m, and ϕ ∈ H0(B) such that ϕ ≤ m a.e. on B.Then given M > m, we can choose ϕk ∈ Lip0(B) such that ϕk → ϕ inH0(B), and ϕk ≤ M a.e. on B. This follows from the proof of Lemma 2.4by extending ϕ to be zero outside B.

3. Approximate Green functions.

The following embedding lemma is useful in the subsequent development.In proving the Lemma, we adapt a method used in [6], in the case of equalweights.

474 AHMED MOHAMMED

Lemma 3.1. If f ∈ K, and B ⊂⊂ B0 is a ball of radius r, then for anyu ∈ H0(B) the following holds.∫

B|f |u2 dx . η(f)(3r)

∫B〈A∇u,∇u〉.

Proof. Let G∗ and G be the Green functions of L0 on 2B∗ and 2B0, respec-tively, where B∗ is a concentric slight enlargement of B. As pointed outin the remark following Lemma 2.3, we first observe that G∗ ≤ G a.e. onB∗×B∗. Let fk = |f | ∧ (kυ), k = 1, 2, . . . , and note that fkυ−1 ∈ Ltυ(2B)for any t. Since ω ∈ A2, and ω ≤ υ, we see that υ can not vanish on a setof positive Lebesgue measure. Therefore fk → |f | a.e. on B. Thus once theinequality in the Lemma is shown to hold for fk, then by Fatou’s Lemma,it will also hold for f . So there is no loss of generality in assuming thatfυ−1 ∈ Lt′υ (2B) for some t < σ.

First, let us suppose that u ∈ Lip0(B). Let ζ := ζk ∈ H0(2B∗) be theLax-Milgram solution of L0ζ = |f |χB in 2B∗ and ζ = 0 on ∂(2B∗). Let ζ bethe associated function. Then by the representation theorem in Lemma 2.2,we know that for a.e. x ∈ B∗

ζ(x) =∫BG∗(x, y)|f(y)| dy ≤

∫BG(x, y)|f(y)| dy

≤∫

3Br(x)∩2B0

G(x, y)|f(y)| dy.

Therefore, ζ(x) ≤ η(f)(3r) for a.e x ∈ B∗. By Lemma 2.1, and Lemma 2.4we pick a sequence ζk ∈ Lip0(2B∗) such that 0 ≤ ζk → ζ in H0(2B∗) andζk . η(f)(3r) a.e. on B. By extending u to be zero outside B, we considerthe element ϕ = u2 ∈ H0(2B∗). Then, we write

δk +∫

2B∗|f |u2 =

∫2B∗〈A∇ζk,∇u2〉,(3.1)

whereδk =

∫2B∗〈A∇ζk,∇u2〉 −

∫2B∗|f |u2.

By Cauchy-Schwarz inequality, we have

〈A∇ζk,∇u2〉 = 2〈A(u∇ζk),∇u〉 ≤ 4η〈A∇u,∇u〉+ 14η〈A(u∇ζk), u∇ζk〉,

(3.2)

where η := η(f)(3r). But

〈A(u∇ζk), u∇ζk〉= 〈A∇ζk,∇ζk〉u2 = 〈A∇ζk,∇(u2ζk)〉 − 2〈A(u∇ζk), ζk∇u〉

≤ 〈A∇ζk,∇(u2ζk)〉+12〈A(u∇ζk), u∇ζk〉+ 2〈A(ζk∇u), ζk∇u〉.


That is,

〈A(u∇ζk), u∇ζk〉 ≤ 2〈A∇ζk,∇(u2ζk)〉+ 4〈A∇u,∇u〉ζ2k .(3.3)

Using (3.3) in (3.2), we obtain∫2B∗〈A∇ζk,∇u2〉

≤ 4η∫

2B∗〈A∇u,∇u〉+ 1

2η

∫2B∗〈A∇ζk,∇(u2ζk)〉

+1η

∫2B∗〈A∇u,∇u〉ζ2

k

= 4η∫

2B∗〈A∇u,∇u〉+ 1

2η

∫2B∗|f |u2ζk +

1η

∫2B∗〈A∇u,∇u〉ζ2

k +12ηγk,

where

γk :=∫

2B∗〈A∇ζk,∇(u2ζk)〉 −

∫2B∗|f |u2ζk.

Using this in (3.1), and recalling that supp(u) ⊂ B, and 0 ≤ ζk . η a.e. onB, we obtain

δk +∫B|f |u2 . 5η

∫B〈A∇u,∇u〉+ 1

2

∫B|f |u2 +

12ηγk.(3.4)

By (2.2), ϕk := u2ζk is easily seen to be bounded in H0(2B∗). There-fore there is a weakly convergent subsequence which we continue to denoteby ϕk. Using this subsequence, and recalling that ζ ∈ H0(2B∗) is theLax-Milgram solution of L0ζ = |f |χB in 2B∗ and ζ = 0 on ∂(2B∗) , we seeby Remark 2.2 that δk → 0, and γk → 0 as k → ∞. Therefore, taking thelimit as k →∞ in the inequality (3.4), we conclude∫

B|f |u2 . 5η

∫B〈A∇u,∇u〉+ 1

2

∫B|f |u2,

from which follows the desired result when u ∈ Lip0(B).To prove the Lemma for u ∈ H0(B), suppose u = uk, uk ∈ Lip0(B).

For each k, we have ∫B|f |u2

k . 10η∫B〈A∇uk,∇uk〉.

Take a subsequence of uk that converges pointwise a.e. to u on B. Byappealing to (2.3), and Fatou’s Lemma we get the desired result after takingthe limit as k →∞.

Remark 3.1. Let f ∈ K, and B ⊂⊂ B0 be a ball. Using Holder inequality,followed by an application of Lemma 3.1, the map

ϕ 7→∫Bfϕ

476 AHMED MOHAMMED

is seen to be continuous on H0(B). Therefore, by the Lax-Milgram theoremthere is a unique ζ ∈ H0(B) such that

a0(ζ, ϕ) =∫Bfϕ , for ϕ ∈ H0(B).

We will also refer to ζ as the Lax-Milgram solution of L0ζ = f on B, ζ =0 on ∂B.

Lemma 3.2. Let f ∈ K, and B ⊂ B0 be a ball of radius r.(1) If Gρ is the approximate Green function of L0 on B, then∫

B|f |Gρ . η(f)(2r).

(2) If ξ ∈ H0(B) is the Lax-Milgram solution of L0ξ = |f | in B, then

ξ(x) . η(f)(2r), for a.e x ∈ B.

Proof. First we show that if ζ ∈ H0(2B) is the Lax-Milgram solution ofL0ζ = |f |χB in 2B, then ζ(x) . η(f)(2r) for a.e. x ∈ 2B. To this end, letus write |f |(k) := |f |∧(kυ), k = 1, 2, 3 · · · , and |f |(0) := |f |. As in the proofof Lemma 3.1, we can argue that |f |(k) → |f | pointwise a.e. on B. By (2.1)and Lemma 3.1 the map

ϕ ∈ H0(2B) 7→∫B∗|f |(k)ϕ,

is seen to be continuous on H0(2B) for all k = 0, 1, 2, · · · . Let ζ(k) ∈ H0(2B)be the Lax-Milgram solution of L0ζ = |f |(k)χB∗ on 2B. Here B∗ is a ballconcentric to B but with radius (1− ε)r for small ε > 0. For k = 1, 2, 3, · · · ,note that |f |(k)υ−1 ∈ Lt′υ (B), t < σ. Then, by the representation formula ofLemma 2.2, we have for a.e. x ∈ B, and k = 1, 2, · · · ,

ζ(k)(x) =∫B∗Gr(x, y)|f |(k)(y) dy ≤

∫B∗G(x, y)|f |(k)(y) dy

≤∫

2Br(x)∩2B0

G(x, y)|f(y)| dy ≤ η(f)(2r),

where Gr and G denote the Green functions of L0 on 2B and 2B0 respec-tively. We have used the fact that Gr ≤ G on B, which is valid by theweak maximum principle, Lemma 2.3. By Lemma 2.4, there is a sequenceζ(k)m in Lip0(2B) such that ζ(k)

m → ζ(k) in H0(2B), and ζ(k)m . η(f)(2r) a.e.

on a ball concentric with B, and of radius strictly between that of B∗, andB. Now let us observe that ζ(k), for k = 1, 2, 3, · · · , is the weak solutionof L0ζ = 0 on 2B r B∗ such that Cη(f)(2r) − ζ(k)

m ≥ 0 on a neighborhoodof ∂(2B r B∗). Therefore by the weak maximum principle, Lemma 2.3,we conclude that ζ(k) . η(f)(2r) on 2B r B∗, and hence on 2B. To show


that the same bound holds for ζ := ζ(0), let Gρ∗ be the approximate Greenfunction of L0 on 2B, and let us observe that

−∫Bρ

(ζ− ζ(k))υ = a0(Gρ∗, ζ− ζ(k)) = a0(ζ− ζ(k), Gρ∗) =∫B

(|f |− |f |(k)χB∗)Gρ∗,

for all k. Since |f |Gρ∗ ∈ L1(B), we invoke the Lebesgue dominated conver-gence theorem to conclude that

−∫Bρ

ζυ = limk→∞

−∫Bρ

ζ(k)υ +∫B|f |(1− χB∗)Gρ∗

≤ Cη(f)(2r) +∫BrB∗

|f |Gρ∗.

After letting ε→ 0, we obtain

−∫Bρ

ζυ . η(f)(2r).

This leads to the claimed estimate after taking the limit as ρ → 0, namelywe get,

ζ(x) . η(f)(2r), for a.e. x ∈ 2B.(3.5)

We now use this result to prove the statement in (1). To see this, let ustake the Lax-Milgram solution ζ of L0ζ = |f |χB in H0(2B). Let Gρ, and Gρ∗be the approximate Green functions of L0 on B, and 2B respectively. SinceGρ∗ −Gρ is a solution of L0 in B, and Gρ∗ − Gρ ≥ 0 near ∂B, by Lemma 2.3we note that Gρ ≤ Gρ∗ on B. Therefore, by (3.5) above,∫

B|f |Gρ ≤

∫B|f |Gρ∗ = a0(ζ,Gρ∗) = a0(Gρ∗, ζ) = −

∫Bρ

ζυ . η(f)(2r).

The statement in (2) is now an easy consequence of (1). To see this, letξ ∈ H0(B) be the Lax-Milgram solution of L0ξ = |f | in B. Let Gρ be theapproximate Green function of L0 on B. Then

−∫Bρ

ξυ = a0(Gρ, ξ) = a0(ξ,Gρ) =∫B|f |Gρ . η(2r).

Taking the limit as ρ→ 0, we obtain the desired result.

The next Lemma is a slight extension of (2.3), and we will use it repeat-edly.

Lemma 3.3. Let u = uk, ϕ = ϕk be in H(B). If ζk is a boundedsequence in L∞(B) that converges pointwise a.e. to ζ ∈ L∞(B), then∫

B〈A∇uk,∇ϕk〉ζk →

∫B〈A∇u,∇ϕ〉ζ, as k →∞.

478 AHMED MOHAMMED

Proof. Since ∫B|〈A∇uk,∇ϕk〉ζk − 〈A∇u,∇ϕ〉ζ|

is not bigger than

‖ζk‖∞∫|〈A∇uk,∇ϕk〉 − 〈A∇u,∇ϕ〉|+

∫|〈A∇u,∇ϕ〉ζ||ζk − ζ|,

and 〈A∇u,∇ϕ〉ζ ∈ L1(B) the Lemma follows from (2.3), and the Lebesguedominated convergence theorem.

Let us now consider the general elliptic operator:

Mu := −div(A(x)∇u+ c(x)u) + b(x) · ∇u+ V (x)u,

where, in addition to (1.5) we also assume that |c|2ω−1 ∈ K. With M , andits adjoint operator

M∗u := −div(A(x)∇u+ b(x)u) + c(x) · ∇u+ V (x)u,

we associate the bilinear forms D(., .) and D∗(., .) as follows. Fix a ballB ⊂⊂ B0 of radius r, and let

D(u, ϕ) :=∫B〈A∇u,∇ϕ〉+ c(x) · (∇ϕ)u+ b(x) · ∇uϕ+ V uϕ,

and D∗(u, ϕ) := D(ϕ, u), for all u, ϕ ∈ Lip0(B). Observe that by Holderinequality and Lemma 3.1, it follows

|D(u, ϕ)− a0(u, ϕ)| . ϑ(r)‖u‖0‖ϕ‖0, u, ϕ ∈ Lip0(B),(3.6)

where ϑ(r) := (η(|c|2ω−1)(3r))1/2 + (η(|b|2ω−1)(3r))1/2 + η(V )(3r). There-fore, we get

|D(u, ϕ)| . (1 + ϑ(r))‖u‖0‖ϕ‖0, u, ϕ ∈ Lip0(B).(3.7)

Thus if u = uk, ϕ = ϕk, uk, ϕk ∈ Lip0(B) are elements of H0(B) thenthe above inequality shows that D(uk, ϕk) is a Cauchy sequence and hencelimkD(uk, ϕk) exists. Therefore we define

D(u, ϕ) := limkD(uk, ϕk).

Having defined D(u, ϕ) for u, ϕ ∈ H0(B), the inequality (3.7) still holdsfor any u, ϕ ∈ H0(B). As a result of this inequality we see that for a fixedu ∈ H0(B), the map ϕ 7→ D(u, ϕ) is a continuous linear functional onH0(B).

Using (3.6) one also obtains a0(u, u)(1 − Cϑ(r)) . D(u, u), for u ∈Lip0(B). Therefore for sufficiently small r0, and all 0 < r ≤ r0, we have

‖u‖20 . D(u, u), u ∈ H0(B),

so that D(., .) is a coercive bilinear form on H0(B).


Given f ∈ K, we shall say that u ∈ H0(B) is a weak solution of Mu = fin B if

D(u, ϕ) =∫Bfϕ,

for all ϕ ∈ H0(B). Similar statements and definitions hold for the adjointoperator M∗ and the associated bilinear form D∗(., .).

The following remark will be useful at several stages in our subsequentproofs.

Remark 3.2. Let f ∈ K, and u = uk ∈ H0(B) be a weak solution ofMu = f in B. If vk is a bounded, weakly convergent sequence in H0(B),then

limk→∞

[D(uk, vk)−

∫Bfvk

]= 0.

This can be verified along the lines of argument given in Remark 2.2, sincethe linear functionals

ϕ 7→ D(u, ϕ), and ϕ 7→∫Bfϕ

are continuous on H0(B).

Lemma 3.4. Suppose fυ−1 ∈ Lpυ(B) for some p > σσ−1 , and some B ⊂ B0,

where B is a ball of raduis r. Then there is a unique solution u of L0u = fin H0(B), and the following estimate holds.

‖u‖L∞(B) ≤ Cr2υ(B)

1p′

ω(B)‖fυ−1‖Lpυ(B).

Proof. The existence and uniqueness follows by the Lax-Milgram Theoremas pointed out in Remark 2.1.

Let Gρ be the approximate Green function of L0 on B with pole y ∈ B.Then

−∫Bρ

uυ = a0(Gρ, u) = a0(u,Gρ) =∫BfGρ,

so that by Holder’s inequality,∣∣∣∣∣−∫Bρ

uυ

∣∣∣∣∣ =∣∣∣∣∫BfGρ

∣∣∣∣ ≤ ‖fυ−1‖Lpυ(B)

(∫B

(Gρ)p′υ

) 1p′

≤ Cr2υ(B)1p′


In the last inequality, we used, (see [3]) the fact that, when 1 < p′ < σ,(∫B

(Gρ)p′υ

) 1p′

≤ Cr2υ(B)1p′

ω(B),(3.8)

480 AHMED MOHAMMED

where C is independent of ρ and the pole of Gρ. If we now let ρ → 0, weconclude

|u(y)| ≤ Cr2υ(B)1p′


This and the arbitrariness of y ∈ B establishes the Lemma.

Remark 3.3. Let Gρ be the approximate Green function of L0 on B. ThusGρ is the Lax-Milgram solution of L0G

ρ = υχBρ [υ(Bρ)]−1. Therefore, byLemma 3.4, ‖Gρ‖L∞(B) ≤ C for some constant C depending on ρ, the poleof Gρ, and υ.

For the next Lemma, given B ⊂ B0 we take f ∈ L1(B) such that the map

ϕ 7→∫Bfϕ, ϕ ∈ H0(B)

is continuous on H0(B). Furthermore, we require that∫B|f |Gρ = O(1), as ρ→ 0+,

where Gρ is the approximate Green function of L0 on B.

Lemma 3.5. If u ∈ H0(B) is the unique solution of L0u = f on B, thenu has a representative u = uk, uk ∈ Lip0(B) such that ‖uk‖∞ ≤ Muniformly in k for some constant M .

Proof. Let u(+) ∈ H0(B), and u(−) ∈ H0(B) be the solutions of L0u(+) =

f+, and L0u(−) = f− respectively. Here f+ := max0, f, and f− :=

max0,−f. If u(+), and u(−) are the associated functions, then

−∫Bρ

u(±)υ = a0(Gρ, u(±)) = a0(u(±), Gρ) =∫Bf±Gρ ≤ C.

Taking the limit as ρ → 0, we conclude that u(±) ≤ C a.e. on B. ByLemma 2.4 (see Remark 2.4), the solutions u(+), and u(−) have represen-tatives u(+) = u(+)

k , u(+)k ∈ Lip0(B), and u(−) = u(−)

k , u(−)k ∈ Lip0(B)

such thatu

(+)k . C, and u

(−)k . C

a.e. on B. By Lemma 2.1, we can in fact choose such representatives tosatisfy

0 ≤ u(+)k . C, and 0 ≤ u(−)

k . C

a.e. on B. Now let u∗ = u(+)k − u(−)

k . It is easy to verify that u∗ ∈ H0(B)is a solution of L0u = f on B. By uniqueness, we must then have u =u(+)

k − u(−)k , and this representation satisfies the condition stated in the

Lemma.


From now on, we will assume that c ≡ 0 in the bilinear forms D(., .), andD∗(., .).

Theorem 3.1. Let B ⊂⊂ B0 be a ball of radius r, and let fυ−1 ∈ Lpυ(B)for some p > σ

σ−1 . Then there is r0 such that if 0 < r ≤ r0, there is a uniquesolution u ∈ H0(B) of Lu = f in B and it satisfies the estimate

‖u‖L∞(B) ≤ Cr2υ(B)

1p′

ω(B)‖fυ−1‖Lpυ(B),

for some constant C.

Proof. Choose r0 such that the bounded bilinear form D(., .) is coercive onH0(B), whenever B is a ball of radius r, with 0 < r ≤ r0. Since ϕ→

∫B fϕ

is a continuous linear functional on H0(B), by the Lax-Milgram theoremthere is a unique u ∈ H0(B) such that

D(u, ϕ) =∫Bfϕ, ϕ ∈ H0(B);

that is Lu = f in B. Moreover, by Holder’s and Sobolev inequality,

‖u‖0 ≤ C‖fυ−1‖Lpυ(B).

We want to show that for some constant C,

‖u‖L∞(B) ≤ Cr2υ(B)

1p′


Let u−1 ≡ 0, and we inductively define uj ∈ H0(B), j = 0, 1, 2, · · · , asthe unique element for which

a0(uj , ϕ) =∫B

(f − b · ∇uj−1 − V uj−1) ϕ, for all ϕ ∈ H0(B),

so that uj is the solution of L0u+ b · ∇uj−1 + V uj−1 = f in H0(B). This ispossible, since for a given uj−1 ∈ H0(B), the map

ϕ 7→∫B

(f − b · ∇uj−1 − V uj−1) ϕ,

is a continuous linear functional on H0(B). Suppose that Gρ is the approxi-mate Green function of L0 on B. We now claim that for each j = 0, 1, 2, · · · ,we can choose a representative uj = u(k)

j , u(k)j ∈ Lip0(B), such that

‖u(k)j ‖∞ ≤Mj , and

∫B|∇uj |2Gρω = O(1), as ρ→ 0+,(3.9)

for some positive constant Mj independent of k. We show this by inductionon j. Since Gρ is essentially bounded, and since Gρ ≥ 0, by Lemma 2.4 (orsee Remark 2.4) we can take a representative Gρ = Gρk, G

ρk ∈ Lip0(B)

such that 0 ≤ Gρk ≤ C a.e. on B for some constant C independent of k.

482 AHMED MOHAMMED

Since u0 ∈ H0(B) is the solution of L0u = f , by Lemma 3.5 we can choosea representative u0 = u(k)

0 , u(k)0 ∈ Lip0(B) such that u(k)

0 is uniformlybounded on B. Consequently, one can use (2.2) to show that u(k)

0 Gρk,and (u(k)

0 )2 are bounded in H0(B). Then for some subsequences, ϕ(k)0 :=

u(k)0 Gρk, and ψ(k)

0 := (u(k)0 )2 are weakly convergent in H0(B). Using a further

subsequence if necessary, we can assume that ϕ(k)0 → u0G

ρ a.e. on B. Letus now observe that∫

B

⟨A∇u(k)

0 ,∇u(k)0

⟩Gρk

=∫B

⟨A∇u(k)

0 ,∇(u

(k)0 Gρk

)⟩− 1

2

∫B

⟨A∇Gρk,∇

(u

(k)0

)2⟩

= δk +∫Bfu

(k)0 Gρk −

12−∫Bρ

(u

(k)0

)2υ

≤ δk +∫B|f |u(k)

0 Gρk,

where

δk :=∫ ⟨

A∇u(k)0 ,∇

(u

(k)0 Gρk

)⟩− 1

2

∫ ⟨A∇Gρk,∇

(u

(k)0

)2⟩

−∫fu

(k)0 Gρk +

12−∫Bρ

(u

(k)0

)2υ,

and the first three integrals are over B. We now take the limit as k → ∞.By Remark 2.2, we observe that δk → 0. Then by Lemma 3.3, Lebesguedominated convergence theorem, and the fact that δk → 0, we obtain∫

B〈A∇u0,∇u0〉Gρ ≤

∫B|f |u0G

ρ.

Using (1-1), this leads to the estimate∫B|∇u0|2Gρω ≤ ‖u0‖∞

∫B|f |Gρ ≤ C‖u0‖∞r2

υ(B)1p′

ω(B)‖fυ−1‖Lpυ(B)

= A0‖u0‖∞‖fυ−1‖Lpυ(B),

where we have also used (3.8) in the penultimate inequality and A0 stands

for the expression Cr2υ(B)1p′ /ω(B). This completes the first induction step.

Let us now suppose that uj has a representative uj = u(k)j , u

(k)j ∈

Lip0(B) and that (3.9) holds for the index j and some constant Mj . Thenby Lemma 3.2, Holder’s inequality, and assumption (3.9), we see that∫

B|f − b · ∇uj − V uj |Gρ ≤ C,


for some constant C independent of ρ. Thus by Lemma 3.5, we can find arepresentative uj+1 = u(k)

j+1, u(k)j+1 ∈ Lip0(B) such that ‖u(k)

j+1‖∞ ≤ Mj+1

on B for some positive constant Mj+1 independent of k. The rest of theargument proceeds in exactly the same way as for the j = 0 case. Thiscompletes the induction, thereby proving the claim that (3.9) holds for allj.

Now let ξj := uj − uj−1, for j = 0, 1, 2, · · · , where we take the represen-tation ξj := ξ(k)j with ξ(k)j := u

(k)j − u

(k)j−1. Then the ξj satisfy

a0(ξj , ϕ) = −∫B

(b · ∇ξj−1 + V ξj−1

)ϕ,

for all ϕ ∈ H0(B), and j = 1, 2, . . . .

As a result of (3.9), we have

‖ξj‖∞ <∞, and∫B|∇ξj |2Gρω = O(1), as ρ→ 0+.

For notational convenience, let us introduce the following. For some suffi-ciently small ρ0, and for j = 0, 1, 2, . . . , let

ϑ :=√η(|b|2ω−1)(3r) + η(V )(3r), and τj := sup

0<ρ≤ρ0

(∫B|∇ξj |2Gρω

) 12

.

Using these notations, and using Lemma 3.2, we find that

−∫Bρ

ξjυ = a0(Gρ, ξj) = a0(ξj , Gρ) = −∫B

(b · ∇ξj−1 + V ξj−1

)Gρ

≤(∫

B|b|2ω−1Gρ

) 12

·(∫

B|∇ξj−1|2Gρω

) 12

+ ‖ξj−1‖∞∫B|V |Gρ

≤ ϑ(τj−1 + ‖ξj−1‖∞

).

After letting ρ→ 0, we obtain

‖ξj‖∞ ≤ ϑ(τj−1 + ‖ξj−1‖∞

).(3.10)

As a consequence of (2.2), and (3.9) one can see that the sequence ξ(k)j Gρk is

bounded inH0(B). Then for an appropriate subsequence, ϕ(k)j := ξ

(k)j Gρk and

ψ(k)j := (ξ(k)j )2 are weakly convergent in H0(B). Without loss of generality,

484 AHMED MOHAMMED

we can assume that ϕ(k)j → ξjG

ρ pointwise a.e. on B. We now observe that∫B〈A∇ξ(k)j ,∇ξ(k)j 〉G

ρk =

∫B〈A∇ξ(k)j ,∇(ξ(k)j Gρk)〉 −

12

∫B〈A∇Gρk,∇(ξ(k)j )2〉

= δ(k)j +

∫B

(b · ∇ξj−1 + V ξj−1

)ξ(k)j Gρk −−

∫Bρ

(ξ(k)j )2υ

≤ δ(k)j +∫B

(b · ∇ξj−1 + V ξj−1

)ξ(k)j Gρk,

where δ(k)j is given by∫B〈A∇ξ(k)j ,∇(ξ(k)j Gρk)〉 −

12

∫B〈A∇Gρk,∇(ξ(k)j )2〉

−∫B

(b · ∇ξj−1 + V ξj−1

)ξ(k)j Gρk +−

∫Bρ

(ξ(k)j )2υ.

Notice that by Remark 2.2, δ(k)j → 0 as k →∞. Therefore taking the limit inthe last inequality, as k →∞, applying Lemma 3.3, and the Lebesgue dom-inated convergence theorem, followed by an application of Holder inequalityand Lemma 3.1, we obtain∫

B|∇ξj |2Gρω ≤ ϑ‖ξj‖∞

(τj−1 + ‖ξj−1‖∞

).

Using (3.10) to estimate ‖ξj‖∞ in the above inequality, we get

τj ≤ ϑ(τj−1 + ‖ξj−1‖∞

), j = 1, 2, . . . .

The sum ‖ξj‖∞ + τj can thus be estimated as

‖ξj‖∞ + τj ≤ 2ϑ(τj−1 + ‖ξj−1‖∞

), j = 1, 2, . . . .(3.11)

Observe that τ0 + ‖ξ0‖∞ ≤ (A0‖fυ−1‖Lpυ(B)‖u0‖∞)1/2 + ‖u0‖∞. But byLemma 3.4, we recall ‖u0‖∞ ≤ A0‖fυ−1‖Lpυ(B). Therefore from (3.10), and(3.11) one obtains by induction

‖ξj‖∞ ≤ 2j−1ϑj(τ0 + ‖ξ0‖∞

)≤ (2ϑ)jA0‖fυ−1‖Lpυ(B), j = 1, 2, . . . .

(3.12)


An application of Cauchy-Schwarz inequality, and Lemma 3.1 leads usalso, on using (1.1), to observe that∫

B〈A∇ξj ,∇ξj〉 = a0(ξj , ξj) = −

∫B

(b · ∇ξj−1 + V ξj−1

)ξj

≤(∫

B|b|2ω−1ξ2j

) 12

·(∫

B|∇ξj−1|2ω

) 12

+(∫

B|V |ξ2j

) 12

·(∫

B|V |ξ2j−1

) 12

≤ 2ϑ∫B〈A∇ξj ,∇ξj〉+

12

∫B|∇ξj−1|2ω.

Therefore

βj ≤1√

2(1− 2ϑ)βj−1, where βj :=

(∫B〈A∇ξj ,∇ξj〉

) 12

, j = 1, 2, . . . .

Thus

‖ξj‖0 = βj ≤

(1√

2(1− 2ϑ)

)jβ0 =

(1√

2(1− 2ϑ)

)j‖ξ0‖0.(3.13)

Now, from (3.12) we observe that

‖um − uk‖∞ ≤m∑

j=k+1

‖ξj‖∞ ≤ A0‖fυ−1‖Lpυ(B)

m∑j=k+1

(2ϑ)j−1, for m > k.

Also, from (3.13) we obtain

‖um − uk‖0 ≤m∑

j=k+1

‖ξj‖0 ≤ ‖ξ‖0m∑

j=k+1

(1√

2(1− 2ϑ)

)j, for m > k.

Thus, if we further choose r0 such 4ϑ(r) < 1 for 0 < r ≤ r0, then weconclude that um, and uk are Cauchy sequences in L∞(B), and H0(B)respectively. So let us take u∗ ∈ H0(B) such that um → u∗ in H0(B). Nowlet ϕ ∈ Lip0(B) be arbitrary. Then, we have

D(u∗ − u, ϕ) = D(u∗ − um, ϕ) +D(um, ϕ)−D(u, ϕ).

But

D(um, ϕ)−D(u, ϕ) = a0(um, ϕ) +∫B

(b · (∇um)ϕ+ V umϕ)−∫Bfϕ

= a0(um, ϕ)− a0(um+1, ϕ) = a0(um − um+1, ϕ).

486 AHMED MOHAMMED

Therefore, for m ≥ 1

|D(u∗ − u, ϕ)| . ‖u∗ − um‖0‖ϕ‖0 + |D(um, ϕ)−D(u, ϕ)|. ‖u∗ − um‖0‖ϕ‖0 + |a0(um − um+1, ϕ)|. (‖u∗ − um‖0 + ‖um+1 − um‖0) ‖ϕ‖0.

Taking the limit as m → ∞, we obtain D(u∗ − u, ϕ) = 0 for ϕ ∈ Lip0(B).Since Lip0(B) is dense in H0(B), and the bilinear form D(., .) is coerciveon H0(B) we conclude that u = u∗. Since um is a Cauchy sequence inL∞(B), by uniqueness of limits we know that um → u in L∞(B). But,

‖um‖∞ ≤m∑k=1

‖ξk‖∞ ≤ CA0‖fυ−1‖Lpυ(B)

m∑k=1

(2ϑ)k−1.

Therefore, since

‖u‖∞ ≤ ‖um − u‖∞ + C(r)A0‖fυ−1‖Lpυ(B),

letting m→∞, and recalling the value of A0, gives the desired estimation.

Remark 3.4. Let fυ−1 ∈ Lpυ(B) for some p > σσ−1 . If L∗u = f for u ∈

H0(B), then Lu = f , where L := −div(A(x)∇) − b(x) · ∇ + (V − divb).Therefore, if |b|2ω−1, divb, V ∈ K, then by the above theorem, we alsohave the estimate

‖u‖L∞(B) ≤ Cr2υ(B)

1p′

ω(B)‖fυ−1‖Lpυ(B),

for some constant C.

For the rest of the paper we will require an additional condition on thecoefficient b of the operator L. Thus, in addition to the condition (1.5) onthe coefficients b, and V of L, we impose the following:

divb ∈ K.(3.14)

Let B ⊂⊂ B0 be a ball of radius sufficiently small that the boundedbilinear form (with c ≡ 0) D∗(u, ϕ) is coercive on H0(B). Let y ∈ B, andρ > 0 such that Bρ := Bρ(y) ⊂ B. Since the map

ϕ 7→ −∫Bρ

ϕυ

is a continuous linear functional on H0(B), by Lax-Milgram theorem thereis a unique Gρ ∈ H0(B) such that

D∗(Gρ, ϕ) = −∫Bρ

ϕυ, ϕ ∈ H0(B).


Following [3], we call Gρ the approximate Green function of L on B withpole y. Note that by Theorem 3.1 (see Remark 3.4), ‖Gρ‖L∞(B) ≤ C forsome constant C depending on ρ, the pole of Gρ, and υ.

In the following Lemma, B ⊂ B0 is a ball of radius so small that thebilinear form D(., .) is coercive.

Lemma 3.6. Suppose u ∈ H0(B), and D(u, ϕ) ≥ 0 whenever 0 ≤ ϕ ∈Lip0(B). Then u ≥ 0.

Proof. Let u=uk, uk ∈ Lip0(B). Since ∇|uk| = (sgnuk)∇uk where uk 6=0, and (sgnuk)|uk| = uk, it follows that for each k

D(uk, |uk|) = D(|uk|, uk), and D(|uk|, |uk|) = D(uk, uk).

Since a0(uk, |uk|) = a0(|uk|, uk) also, the sequence |uk| is bounded inH0(B), and thus a subsequence which we continue to write as |uk| con-verges weakly to some v ∈ H0(B). Since ϕ 7→ D(u, ϕ) is continuous onH0(B) and D(u, |uk| − uk) ≥ 0 by hypothesis, it follows that

0 ≤ limk→∞

D(u, |uk| − uk) = D(u, v − u),

so that D(u, u) ≤ D(u, v). Then D(u, u) = αD(u, v) for some 0 < α ≤ 1.Let us now observe that

0 ≤ ‖u− α|uk|‖20 . D(u− α|uk|, u− α|uk|)= D(u, u)− 2αD(u, |uk|) + α2D(|uk|, |uk|)= D(u, u)− 2αD(u, |uk|) + α2D(uk, uk).

Taking the limit as k →∞, the last inequality reduces to

0 ≤ limk→∞

‖u− α|uk|‖20 . D(u, u)− 2αD(u, v) + α2D(u, u)

≤ D(u, u)− 2D(u, u) + α2D(u, u) = (α2 − 1)D(u, u).

Hence0 ≤ lim

k→∞‖u− α|uk|‖20 . (α2 − 1)D(u, u) ≤ 0.

That is, α|uk| → u in H0(B) as k → ∞, showing that u is the limit inH0(B) of α|uk| ≥ 0. From this, it also follows that α|uk| → u in L2

υ. Butalso uk → u in L2

υ(B). Therefore we must have α|u| = u a.e. on B. Thusα = 1, and hence u = |uk|.

The following lemma will be useful.

Lemma 3.7. Let B ⊂⊂ B0 be a ball of radius r, and Gρ be the approximateGreen function of L on B. There is r0 > 0 such that if 0 < r ≤ r0, then∫

B

(|b|2ω−1 + |V |

)Gρ ≤ Cη(|b|2ω−1 + |V |)(2r),

where C is a constant independent of ρ and the pole of Gρ.

488 AHMED MOHAMMED

Proof. Since

ϕ 7→∫B

(|b|2ω−1 + |V |

)ϕ,

is a continuous linear functional on H0(B), by Lax-Milgram theorem letζ ∈ H0(B) be the unique solution of

a0(ζ, ϕ) =∫B

(|b|2ω−1 + |V |

)ϕ, ϕ ∈ H0(B).

By Lemma 3.2, ζ(x) . η := η(|b|2ω−1 + V )(2r) for a.e. x ∈ B. ByLemma 2.1, and Lemma 2.4 (or Remark 2.4), let us pick a sequence ζk ∈Lip0(B) such that 0 ≤ ζk . η a.e. on B. Let Gρ be the approximateGreen function of L on B. By Remark 3.4, Gρ is essentially bounded, andLemma 3.6 shows Gρ ≥ 0. Thus by Lemma 2.4 (see Remark 2.4), we canpick a representative Gρ = Gρk, G

ρk ∈ Lip0(B) such that for some constant

C independent of k, we have 0 ≤ Gρk ≤ C a.e. on B. Let us now observethat ∫

B

(|b|2ω−1 + |V |

)Gρk

= δk +∫B〈A∇ζk,∇Gρk〉

= δk + γk −∫B

(b · (∇ζk)Gρk + V ζkG

ρk

)+−∫Bρ

ζkυ

≤ δk + γk +√η

∫B|b|2ω−1Gρk +

1√η

∫B〈A∇ζk,∇ζk〉Gρk

+ η

∫B|V |Gρk + η.

Here δk and γk are given by

δk :=∫B

(|b|2ω−1 + |V |

)Gρk −

∫B〈A∇ζk,∇Gρk〉,

and γk := D∗(Gρk, ζk)−−

∫Bρ

ζkυ.

By Remark 2.2, and Remark 3.2 respectively, we notice that δk → 0, andγk → 0 as k → ∞. We thus take the limit as k → ∞. By Lemma 3.3, andthe Lebesgue dominated convergence theorem, we obtain∫

B

(|b|2ω−1 + |V |

)Gρ(3.15)

≤ √η∫B|b|2ω−1Gρ +

1√η

∫B〈A∇ζ,∇ζ〉Gρ + η

∫B|V |Gρ + η.


As a result of (2.2), we see that ζkGρk is a bounded sequence in H0(B).Therefore, we pick a subsequence still denoted by ζkGρk that convergesweakly in H0(B), and such that ζkG

ρk → ζGρ pointwise a.e. on B. Using

this subsequence we have∫B〈A∇ζk,∇ζk〉Gρk =

∫B〈A∇ζk,∇(ζkG

ρk)〉 −

12

∫B〈A∇Gρk,∇ζ

2k〉

= δk +∫B

(|b|2ω−1 + |V |

)Gρkζk

+12

(∫B

(b · (∇ζ2

k)Gρk + V Gρkζ

2k

)+ γk −−

∫Bρ

ζ2kυ

)

≤ δk +12γk + η

∫B

(|b|2ω−1 + |V |

)Gρk

+12

(η2

∫B

(|b|2ω−1 + |V |

)Gρk +

∫B〈A∇ζk,∇ζk〉Gρk

),

where

δk := a0(ζk, ζkGρk)−

∫B

(|b|2ω−1 + |V |

)Gρkζk,

and γk := −D∗(Gρk, ζ2k) +−

∫Bρ

ζ2kυ.

Again by Remark 2.2, and Remark 3.2 respectively, we see that δk → 0, andγk → 0 as k → ∞. We thus take the limit as k → ∞. By Lemma 3.3, andthe Lebesgue dominated convergence theorem, we obtain∫

B〈A∇ζ,∇ζ〉Gρ ≤ η

∫B

(|b|2ω−1 + |V |

)Gρ

+12

(η2

∫B

(|b|2ω−1 + |V |

)Gρ +

∫B〈A∇ζ,∇ζ〉Gρ

).

Putting this last inequality back into (3.15), we see that there is r0 suchthat for all 0 < r ≤ r0,∫

B

(|b|2ω−1 + |V |

)Gρ ≤ Cη(|b|2ω−1 + |V |)(2r),

as required.

We now have all the needed ingredients to demonstrate the uniform inte-grability of the approximate Green functions of L. We use the methods in[3] (see also [5], [8]) to prove the integrability theorem.

Theorem 3.2. Let B ⊂⊂ B0 be a ball of radius r. Suppose Gρ is theapproximate Green function of L on B, where we assume that the coefficients

490 AHMED MOHAMMED

of L satisfy the conditions (1.5) and (3.14). Then for 0 < p < σ there is apositive constant, independent of ρ and the pole, such that(

−∫B

(Gρ)pυ

) 1p

≤ C r2

ω(B),

when r is sufficiently small.

Proof. Let Gρ be the approximate Green function of L on B. By Lemma 3.6,Gρ ≥ 0 on B. Therefore we pick a representative Gρ = Gρk, G

ρk ∈ Lip0(B)

such that Gρk ≥ 0.Now, for t > 0, let us define

ϕk :=[1t− 1Gρk

]+

=[1t− 1Gρk

]χGρk>t

.

Then ϕk ∈ Lip0(B), and

∇ϕk =∇Gρk(Gρk)

2χGρk>t

.

Since ‖ϕk‖20 ≤ ‖Gρk‖

20/t

4, we can pick a subsequence, still denoted by ϕkthat converges weakly in H0(B). With such subsequence, we observe that∫

B〈A∇Gρk,∇ϕk〉 = δk −

∫B

(b · ∇ϕk + V ϕk)Gρk +−

∫Bρ

ϕkυ,(3.16)

where

δk := D∗(Gρk, ϕk)−−

∫Bρ

ϕkυ.

Using the Cauchy-Schwarz inequality, and noting that ϕk ≤ 1/t on B, weestimate∫B

(b · ∇ϕk + V ϕk)Gρk ≤

12t

∫B|b|2ω−1Gρk +

t

2

∫B|∇ϕk|2Gρkω +

1t

∫|V |Gρk

≤ 1t

∫B

(|b|2ω−1 + |V |)Gρk +t

2

∫Gρk>t

|∇Gρk|2

(Gρk)4Gρkω

≤ 1t

∫B

(|b|2ω−1 + |V |)Gρk +12

∫Gρk>t

|∇Gρk|2

(Gρk)2ω

≤ 1t

∫B

(|b|2ω−1 + |V |)Gρk +12

∫B〈A∇Gρk,∇ϕk〉.

Taking this last estimation, and using again the fact that ϕk ≤ 1/t on Bwe get from (3.16) that∫

B〈A∇Gρk,∇ϕk〉 ≤ 2|δk|+

2t

∫B

(|b|2ω−1 + |V |)Gρk +2t.


We now take the limit as k → ∞. By Remark 3.2 we know that δk → 0 ask →∞. Using this fact and applying Lemma 3.7, we get

limk→∞

∫B〈A∇Gρk,∇ϕk〉 ≤

2t

∫B

(|b|2ω−1 + |V |)Gρ +2t

≤ 2t(1 + η(|b|2ω−1 + V )(3r)).

Hence by the degeneracy condition, we obtain the inequality

lim supk→0

∫Gρk>t

|∇Gρk|2

(Gρk)2ω ≤ 2

t(1 + η(|b|2ω−1 + V )(3r)).

Now letψk :=

[logGρk − log t

]+ =[logGρk − log t

]χGρ

k>t

.

Then ψk ∈ H0(B), and ∇ψk =(∇GρkGρk

)χGρk>t

. Therefore, for sufficientlysmall r the last inequality now reads

lim supk→∞

∫B|∇ψk|2ω ≤

C

t,

and hence by Sobolev inequality (2.1), we get

lim supk→∞

(1

υ(B)

∫B∩Gρk>t

log(Gρkt

)qυ

) 2q

≤ C r2

ω(B)C

t.

Restricting the integration to Gρk > 2t, we get

(log 2)2 lim supk→∞

(υ(Gρk > 2t)

υ(B)

) 2q

≤ C r2

ω(B)1t.

The above inequality remains valid if we replace 2t by t. Also by usingfurther subsequence if necessary, we may assume that Gρk → Gρ pointwisea.e. on B. Thus χGρ>t ≤ lim inf χGρk>t a.e., and by Fatou’s lemma,

υ(Gρ > t) ≤ lim infk→∞

υ(Gρk > t).

Therefore we obtain,

υ(Gρ > t) ≤ C(

r2

ω(B)

)σ 1tσυ(B).

The theorem then follows from this estimate, and the formula∫B

(Gρ)p = p

∫ ∞0

tp−1υ(Gρ > t)

≤ pυ(B)∫ A

0tp−1 + p

∫ ∞A

tp−1υ(Gρ > t),

where A = r2/ω(B).

492 AHMED MOHAMMED

Acknowledgments. I would like to record my indebtness to Professor C.E.Gutierrez for his constant encouragement and his invaluable advice through-out the preparation of this paper. I would also like to thank the referee forthe timely and careful reading of the manuscript.

References

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[3] , Existence and estimates of Green’s function for degenerate elliptic equations,Scuola Norm. Sup., Pisa, XV. Fasc. II (1989), 309-340, MR 91b:35045.

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[9] J.O. Stromberg and A. Torchinsky, Weighted Hardy spaces, in ‘Weighted HardySpaces’ (Lecture notes in Mathematics), 1381, Springer Verlag, New York, (1989),83-128, MR 90j:42053, Zbl 676.42021.

Received September 10, 1999 and revised February 15, 2000. This research was partiallysupported by Kuwait University grant SM-137.

Temple UniversityPhiladelphia, PA 19122E-mail address: [email protected]




















PERIODIC SUBWORDS IN 2-PIECE WORDS

C.M. Weinbaum

We find families of words W where W is a product of kpieces for k=2. For k=3,4,6, W arises in a small cancella-tion group with single defining relation W=1. We assume Winvolves generators but not their inverses and does not havea periodic cyclic permutation (like XY...XYX for nonemptybase word XY). We prove W or W written backwards equalsABCD where ABC, CDA are periodic words with base wordsof different lengths. One family includes words of the formEFGG for periodic words G, E, F with the same base wordand increasing lengths. Other W are found using Mathematica.

1. Introduction.

A small cancellation condition on a group’s defining relations yields, forexample, a solution to the conjugacy problem. See [1, 4]. There are 3types of such conditions. Each includes a condition C(k) for k = 3, 4 or 6,depending on the type. For a group G with one defining relation R = 1,C(k) involves the set [R] of cyclic permutations of R and of R−1. A pieceis a nonempty, initial subword of 2 distinct members of [R]. C(k) requiresthat no word in [R] is a product of fewer than k pieces.

To study a “large cancellation” group G and avoid all small cancellationtypes, we can use the condition that R is a product of 2 pieces. What doessuch a word R look like? For simplicity, in this paper we consider wordsinvolving generators but not their inverses. In particular, we study 2-piecewords R, meaning

R involves generators but not their inverses and R is a product of 2 pieces.(1.1)

An attempt to classify these words led to the results in this paper. Theseresults also lie in the field of combinatorics on words which is surveyed in[3].

2. Summary of results.

For convenient exposition, from now on, a word W is a finite sequence ofletters taken from some alphabet; |W | = length of W ; the empty word=1;|1| = 0. Write W ∼ V if W, V are cyclic permutations. W is 2-piece if

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494 C.M. WEINBAUM

(∃U, V, Y, Z 6= 1) W = UV ∼ UY ∼ V Z, Z 6= U, Y 6= V. (U, V ) is a 2-piece pair. W is periodic if (∃Q 6= 1, P, k ≥ 2) W = P (QP )k. X < Wmeans W = XY, Y 6= 1. W is biperiodic if (∃P,Q,R, S,m, n) W = UV,U = P (QP )m, V = R(SR)n, SR < U, QP < V, 1 6= QP 6= SR 6= 1,URS 6= SRU and V PQ 6= QPV ; m,n ≥ 1.

The main results are: If a 2-piece word W has no periodic cyclic per-mutation then W or W written backwards is biperiodic. Each biperiodicword is 2-piece. W is biperiodic and |S| < |RS| < |PR| < PQ| < |RSR|is equivalent to (∃A,B, a, b, c,m, n) W = A(BA)b(A(BA)c)m(A(BA)a)n to-gether with AB 6= BA, 1 < a < b < c ≤ 2a, m, n ≥ 1. Such a word W is notperiodic for n ≥ 2. Two other similar equivalences are proved. Other as yetunclassified biperiodic words are found using Mathematica.

The title of the paper refers to the periodic subwords P (QP )m+1,R(SR)n+1 which begin word W = P (QP )mR(SR)n and its cyclic permu-tation R(SR)nP (QP )m, respectively, whenever W is biperiodic and hence2-piece.

3. Terminology.

Terminology in the previous section is augmented as follows. Let A,B bewords over some alphabet. The concatenation of words A,B is written as aproduct AB. The product of k copies of A, written Ak, is a power of A ifk ≥ 0, with A0 = 1, and a proper power if k ≥ 2. Call W simple if W is nota proper power. Note the empty word E = 1 is not simple since E = E2.If W = XY Z then X,Y, Z are factors of W ; write X,Y, Z ⊆ W . X,Z areleft and right factors; write X ≤ W , W ≥ Z. If U is a factor of W then Uis major if 2|U | ≥ |W | and proper if U 6= W. Proper left and right factorsof W are indicated by X < W, W > Z. Denote W written backwards byW ∗, the reverse of W . As in [4, p. 153], the period π(W ) of a word W isthe minimum length of the words admitting W as a factor of some of theirpowers. Equivalent definitions of periodic are in Theorem 4.13. Call Wplain if W has no periodic cyclic permutation.

We restate a definition to enable later reference to its parts:

Definition 3.1. Word W is biperiodic using U, V, P,Q,R, S,m, n if W =UV for words U, V such that:

U = P (QP )m for some words Q 6= 1, P and some integer m ≥ 1.(3.1a)

V = R(SR)n for some words S 6= 1, R and some integer n ≥ 1.(3.1b)

SR < U, QP < V .(3.1c)

1 6= QP 6= SR 6= 1.(3.1d)

URS 6= SRU , V PQ 6= QPV .(3.1e)

PERIODIC SUBWORDS IN 2-PIECE WORDS 495

4. Preliminaries.

Let A,B, . . . denote words and a, b, . . . denote integers in the following Lem-mas. Lemmas 4.1–4.3 are found in Propositions 1.3.4, 8.1.1 and Theorem8.1.2 in [3]. Lemmas 4.4, 4.5, 4.6, 4.8, 4.9, 4.10, 4.12 are in the followingPropositions in [2]: 1.2, 1.3, 1.4, 1.4′′, 1.8, 1.16, 1.23 . Prove Lemma 4.7from Lemma 4.6 and Lemma 4.11 from Lemma 4.10 using reverse words.Use S simple if and only if S∗ simple.

Lemma 4.1. Y, Z 6= 1, XZ = Y X imply (∃n ≥ 0, U, V ) Y = UV, Z = V U,X = U(V U)n.

Lemma 4.2. π(W ) = Min|W | − |V | where V < W > V.

Lemma 4.3. p = π(XY ), q = π(Y Z), d = gcd(|X|, |Y |), |Y | ≥ p + q − dimply p = q = π(XY Z).

Lemma 4.4. If XY = Y X then (∃S simple, a, b ≥ 0) X =a, Y =b.

Lemma 4.5. If S, T simple, Sa = T b, a, b ≥ 1 then S = T.

Lemma 4.6. If S is a simple word and PS ≤ Sn, n ≥ 1 then P is a powerof S.

Lemma 4.7. If S is a simple word and Sn ≥ SP , n ≥ 1 then P is a powerof S.

Lemma 4.8. If PBA ≤ A(BA)r, r ≥ 1, P 6= 1 with BA simple then(∃e ≥ 0) P = A(BA)e.

Lemma 4.9. A cyclic permutation of a simple word is a simple word.

Lemma 4.10. X ≤ Y eX, e > 0, Y 6= 1 imply (∃t ≥ 0, E) t, E unique,X = Y tE, E < Y.

Lemma 4.11. XY e ≥ X, e > 0, Y 6= 1 imply (∃t ≥ 0, E) t, E unique,X = EY t, Y > E.

Lemma 4.12. If XY Z = ZY X, X 6= 1, Z 6= 1 then (∃a, b, c ≥ 0, U, V )X = U(V U)a, Y = V (UV )b, Z = U(V U)c and the word UV is simple.

Lemma 4.13. Equivalent conditions on a word W are:

W has a proper major left factor which is also a right factor.(4.13a)

W = Y X = XZ, |X| ≥ |Y | > 0.(4.13b)

(∃k ≥ 2, U 6= 1) W ≤ Uk, |W | ≥ 2|U |.(4.13c)

W is periodic, that is, (∃B 6= 1, A,m ≥ 2) W = A(BA)m.(4.13d)

|W | ≥ 2π(W ) and W 6= 1.(4.13e)

Proof. (a) if and only if (b): Use definitions.

496 C.M. WEINBAUM

(b) implies (c): Deduce |W | = |Y X| ≥ |Y Y | = 2|Y | and W < YW. ByLemma 4.10, W = Y tE < Y t+1 for some word E with t ≥ 1 because|E| < |Y | < |W |.

(c) implies (d): From (c), (∃t ≥ 2) t|U | ≤ |W | < (1 + t)|U |. So (∃B 6=1, A) U = AB, W = U tA, proving (d).

(d) implies (b): Use X = A(BA)m−1, Y = AB, Z = BA.(d) implies (e): From (d), W < (AB)m+1 and so π(W ) ≤ |AB|. It

follows that |W | = |A(BA)m| ≥ m|BA| ≥ 2|AB| ≥ 2π(W ), yielding(e).

(e) implies (c): In general, π(W ) ≤ |W |. From (e), (∃V 6= 1, k ≥ 1)|V | = π(W ), W ⊆ V k. k ≥ 2 since 2|V | = 2π(W ) ≤ |W | ≤ k|V |.Then (∃U ∼ V ) W ≤ Uk.

Lemma 4.14. If W = XY eZ, Z ≤ Y ≥ X, Y 6= 1, e ≥ 1 then (∃B 6=1, A, p) 0 ≤ p ≤ 2, W = A(BA)e+p, |AB| = |Y |, XZ = A(BA)p.

Proof. (∃C,D) Y = ZC = DX. Let Y1 = XD, X1 = XZ. ThenX1 ≤W =(Y1)eX1. Apply Lemma 4.10 to X1 ≤ (Y1)eX1. (∃p ≥ 0, A) X1 = (Y1)pA,A < Y1. So (∃B 6= 1) Y1 = AB. Hence W = (AB)e(AB)pA = A(BA)e+p;XZ = X1 = (AB)pA = A(BA)p. Also p ≤ 2 since |Y pA| = |(Y1)pA| =|X1| = |XZ| ≤ |Y 2|.

Lemma 4.15. If AB 6= BA then (∃C,D 6= 1, a, b ≥ 0) CD simple, A =C(DC)a, B = D(CD)b. For t ≥ 0, A(BA)t = C(DC)p(t), (AB)t =(CD)q(t), p(t) = (a+ b+ 1)t+ a, q(t) = (a+ b+ 1)t.

Proof. (∃ simple S, e ≥ 1) AB = Se. (∃a, b ≥ 0, C,D) S = CD, A =C(DC)a, B = D(CD)b. C,D 6= 1 else A,B are powers of the same wordand AB = BA, a contradiction.

Lemma 4.16. Y, Z 6= 1, XZ = Y X imply (∃r ≥ 0, s ≥ 1, C,D) CDsimple, Y = (CD)s, Z = (DC)s, X = C(DC)r.

Proof. By Lemma 4.1, (∃n ≥ 0, U, V ) Y = UV, Z = V U, X = U(V U)n.(∃ simple S) UV is a power of S. (∃i, j ≥ 0, C,D) S = CD, U = C(DC)i,V = D(CD)j . Use r = i+ n(i+ j + 1), s = i+ j + 1.

5. General results.

Each 2-piece word is simple (Theorem 5.5). If W = UV is 2-piece then U, Vappear again as factors of cyclic permutations of W. If U, V appear at leasttwice in W then W is periodic (Theorem 5.6). If a 2-piece word W is plainthen W or W ∗ is biperiodic (Theorem 5.8). Each biperiodic word is 2-piece(Theorem 5.9). We start with some easily verifiable remarks.


Remark 5.1. If PQ 6= QP and m ≥ 2 then the periodic word W =P (QP )m is 2-piece, by definition, using U = P (QP )m−1, V = QP, Y = PQ,Z = QPP (QP )m−2.

Remark 5.2. A word is simple, periodic, 2-piece or biperiodic if and onlyif its reverse has the same property.

Remark 5.3. If (U, V ) is a 2-piece pair then so are (V,U) and (V ∗, U∗).

Given a 2-piece word W,W ∗ inherits properties as follows:

Remark 5.4. If a 5-tuple (W,U, V, Y, Z) of words satisfies W = UV ∼Y U ∼ ZV, 16= U 6= Z, 1 6= V 6= Y then so does (W ∗, V ∗, U∗, Z∗, Y ∗).

Theorem 5.5. Each 2-piece word is a simple word.

Proof. Let W be 2-piece word, W = UV ∼ UY ∼ V Z, 1 6= U 6= Z,1 6= V 6= Y. (∃A,B) W = AB, UY = BA. Suppose W is not simple. Then(∃ simple X) W = Xm, m ≥ 2.

If |X| ≤ |U | then (∃C) U = XC. So AX ≤ AUY ≤ ABAB = X2m. ByLemma 4.6, A is a power of X, so is W and hence so is B. Thus AB = BA.Then UV = AB = BA = UY implies V = Y, a contradiction.

If |X| > |U | then |X| ≤ |V |. By Remark 5.3, W ∗ = V ∗U∗ is 2-pieceand W ∗ = (X∗)m. Get a contradiction for W ∗, V ∗, U∗ as previously withW,U, V.

Theorem 5.6. If U, V 6= 1 each appear at least twice in W = UV then Wis periodic. In other words, W = UV = IUJ = KV L and U, V, I, L 6= 1imply W is periodic.

Proof. Assume W = UV = XUT = RV Y and U, V,X, Y 6= 1. The changein letters allows a more pleasing factorization W = RST in Case 1.

Case 1: |X| < |U |; |Y | < |V |. Then (∃F,G) UGT = XUT = W =RFV = RV Y, |U | > |X| = |G| > 0, |V | > |F | = |Y | > 0. So U =RF, V = GT, W = RFGT . Let S = FG. Then W = RST. HenceUGT = W = RST = W = RFV imply UG = RS, ST = FG. ApplyLemma 4.2 to W1 = RS, W2 = ST.

So π(RS) ≤ |RS| − |U |, π(ST ) ≤ |ST | − |V | since RS = UG = XU,ST = FV = V Y. Then |S| = |F | + |G| = |RS| − |U | + |ST | − |V | ≥π(RS) + π(ST ). By Lemma 4.3, π(RST ) = π(RS) = π(ST ). Since|W | ≥ |S| ≥ π(RS) + π(ST ) = 2π(W ), W is periodic by (4.13e) inLemma 4.13.

Case 2: |U | ≤ |X|; |V | ≤ |Y |. Then |U | ≤ |UT | ≤ |V |, |V | ≤ |RV | ≤ |U |.So |U | = |V |, U = V and hence W = UU is periodic.

Case 3: |X| < |U |; |V | ≤ |Y |. Then (∃F ) UFT = XUT = W with|U | > |X| = |F | > 0. So UF = XU. By Lemma 4.16, (∃C,D, r ≥0, s ≥ 1) CD simple, U = C(DC)r, F = (DC)s, X = (CD)s. r ≥ 1

498 C.M. WEINBAUM

since |U | > |X| ≥ |CD|. DC ≤ V since V = FT . Thus |W | ≥ |UF | ≥2|CD|. Also (∃P,Q) U = PV Q.

Then PDC ≤ PV ≤ U = C(DC)r. Also DC ∼ CD implies DCsimple by Lemma 4.9. By Lemma 4.8, P = C(DC)e, e ≥ 0 implyingV ≤ (DC)r−e. So W ≤ C(DC)2r−e; hence π(W ) ≤ |CD|. Then |W | ≥2|CD| ≥ 2π(W ) implies W is periodic by (4.13e) in Lemma 4.13.

Case 4: |U | ≤ |X|; |Y | < |V |. Then V ∗, U∗ each appear at least twice inW ∗ = V ∗U∗. Apply Case 3 to W ∗; get W ∗ periodic. By Remark 5.2,W is periodic.

Lemma 5.7. Let W = UV be a plain word. Assume cyclic W has 2ndoccurrences U ′′, V ′′ of the words U, V, respectively. Then (i) U,U ′′ overlapand V, V ′′ overlap and (ii) U ′′ is a factor of one of the words UV, V U andV ′′ is a factor of the other.

Proof. First prove results (1)-(7).(1) Conclusions for U, V, U ′′, V ′′ apply to V ∗, U∗, (V ′′)∗, (U ′′)∗. By Re-marks 5.2, 5.3, 5.4, assumptions on W,U, V, U ′′, V ′′ apply to W ∗, V ∗, U∗,(V ′′)∗, (U ′′)∗, respectively.(2) Neither UV nor V U has both factors U ′′, V ′′. Use Theorem 5.6.(3) U,U ′′ overlap or V, V ′′ overlap. If not then U has factor V ′′, V hasfactor U ′′. Therefore U = V, W = UU, W is periodic, contradicting W isplain.(4) U,U ′′ overlap implies (U ′′ ⊆ UV or U ′′ ⊆ V U). Suppose U,U ′′

overlap and U ′′ is not a factor of UV or V U . Then (∃A,B,C 6= 1) W =ABCV, U = ABC, U ′′ = CV A. So CV < ABCV > CV, AB < CV AB >AB. Then ABCV or CV AB has a major left and right factor, namely, CVor AB, respectively. Thus ABCV or CV AB is periodic by Lemma 4.13,contradicting W is plain. Thus (4) is true.(5) V, V ′′ overlap implies (V ′′ ⊆ V U or V ′′ ⊆ UV ). Use (1), (4).Get V ∗, (V ′′)∗ overlap implies ((V ′′)∗ ⊆ V ∗U∗ or (V ′′)∗ ⊆ U∗V ∗). Thisimplies (5).(6) U,U ′′ overlap implies V, V ′′ overlap. If not then U,U ′′ overlap butV, V ′′ do not. So V ′′ ⊆ U . Also U ′′ ⊆ UV or U ′′ ⊆ V U. Then U,U ′′, V, V ′′ ⊆UV (or V U), contradicting (2).(7) V, V ′′ overlap implies U,U ′′ overlap. Use (1), (6). Therefore V ∗, (V ′′)∗

overlap implies U∗, (U ′′)∗ overlap. This implies (7).Now (i) follows from (3), (6), (7) and (ii) follows from (i), (2), (4), (5).

Theorem 5.8. Each plain, 2-piece word W is biperiodic or its reverse isbiperiodic.


Proof. By Remarks 5.2 and 5.3, W ∗ is plain, 2-piece. Since W = UVsatisfies the 2-piece condition, cyclic W has 2nd occurrences U ′′, V ′′ of thewords U, V, respectively. By Lemma 5.7, there are 2 cases:

Case 1: U ′′ ⊆ UV, V ′′ ⊆ V U . Using the 2-piece property and Lemma 5.7Part (i), it follows that UV = W = UDB = CU ′′B, V U = V FG =EV ′′G, U = FG, V = DB, Y = BC, Z = GE, |U | > |C|, |V | > |E|for some words B,C,D,E, F,G 6= 1. Since |U | > |C|, |V | > |E|, wecan apply Lemma 4.1 to UD = CU and V F = EV.

(∃Q 6= 1, P,m ≥ 1) U = P (QP )m, C = PQ, D = QP and (∃S 6=1, R, n ≥ 1) V = R(SR)n, E = RS, F = SR. Thus UV = UDB =UQPB, QP < V = R(SR)n and V U = V FG = V SRG, SR < U =P (QP )m, implying Conditions (3.1a), (3.1b) and (3.1c).

If |PQ| = |RS| then QP < V = R(SR)n implies QP = RS. SoW = PXm+nR for X = QP and W is periodic by Lemma 4.14, acontradiction. So (3.1d) is true.

If URS = SRU then FU = SRU = URS = UE = FGE = FZand U = Z, not true. If QPV = V PQ then DV = QPV = V PQ =DBC = DY and V = Y, not true. So URS 6= SRU, QPV 6= V PQ,(3.1e) is true, making W biperiodic.

Case 2: V ′′ ⊆ UV , U ′′ ⊆ V U. Then (V ′′)∗ ⊆ V ∗U∗, (U ′′)∗ ⊆ U∗V ∗. ByRemark 5.4, Case 1 applies to W ∗ so W ∗ is biperiodic.

Theorem 5.9. Each biperiodic word is a 2-piece word.

Proof. Let W = UV be biperiodic using P,Q,R, S, U, V,m, n. V U is biperi-odic using R,S, P,Q, V, U, n,m. By symmetry and Remark 5.3, we may as-sume |SR| < |PQ|. By Conditions (3.1a), (3.1b) and (3.1c), SR < PQand QP < R(SR)n. Define F, J by SRF = PQ, QPJ = R(SR)n. We nowcheck that W satisfies the definition of being 2-piece by using Y = JPQ,Z = FP (QP )m−1RS.

UY = P (QP )mJPQ ∼ P (QP )m+1J = UQPJ = UV

V Z = R(SR)nFP (QP )m−1RS ∼ SRFP (QP )m−1R(SR)n

= PQP (QP )m−1V = UV.

If Y = V then V PQ = QPJPQ = QPY = QPV, a contradiction. If Z = Uthen we get a contradiction from SRU = SRZ = PQP (QP )m−1RS = URS.Thus W is 2-piece.

6. Factoring some 2-piece words.

The 2-piece words to be factored are two types of biperiodic words. They arecalled biperiodic-1 and biperiodic-2 words. They are 2-piece by Theorem 5.9.

500 C.M. WEINBAUM

Their factorizations are called binary-1 and binary-2 words and have factorsA,B,AB 6= BA. See Theorems 6.10 and 6.11. A 3rd type of biperiodic word,called a biperiodic-3 word, has a cyclic permutation possessing a 3rd typeof factorization, a binary-3 word (Theorem 6.12). Each binary-3 word has abinary-2 cyclic permutation (Remark 6.4). Likewise, each biperiodic-3 wordhas a biperiodic-2 cyclic permutation, using Theorem 6.12, Remark 6.4 andTheorem 6.11.

Results in this section, together with Remark 5.3, will show that 2-piecewords can be found using the above factorizations and their reverses. Moreprecisely, we have:

Theorem 6.1. Binary-1 and binary-2 words and their reverses are 2-piecewords. Each binary-3 word has a 2-piece cyclic permutation and so does itsreverse.

The types of biperiodic words and factorizations are defined as follows:

Definition 6.2. A word W is biperiodic-1, biperiodic-2 or biperiodic-3 if Wsatisfies Definition 3.1 together with (6.2a), (6.2b) or (6.2c), respectively.

|S| < |RS| < |PR| < |PQ| < |RSR|.(6.2a)

P = 1, |SR| < |Q|.(6.2b)

|RS| ≤ |P | < |PQ|.(6.2c)

Definition 6.3. A wordW is binary-1, binary-2 or binary-3 if (6.3a), (6.3b)or (6.3c), respectively, with AB 6= BA. Call such W binary. Terminologyfor later use: W is binary-1 using A,B : AB 6= BA and W is binary-1 forA,B, h, i, j. Similar terminology applies to binary-2 and binary-3.

W = A(BA)i(A(BA)j)m(A(BA)h)n, 1 < h < i < j ≤ 2h, m, n ≥ 1.(6.3a)

W = (A(BA)i)m(AB)j , 1 ≤ i, i+ 1 ≤ j, m, n ≥ 1.(6.3b)

W = (A(BA)i)mA(BA)j , 1 ≤ i, i+ 2 ≤ j, m, n ≥ 1.(6.3c)

Remark 6.4. Each binary-3 word W has a cyclic permutation V which isbinary-2. In particular, if W satisfies (6.3c), use V = (A(BA)i)m+1(AB)j−i.

The proof that the biperiodic-1 and binary-1 conditions are equivalent fora word W requires 3 lemmas involving closely related conditions defined asfollows:

Definition 6.5. A word W is biperiodic-1∗ if W satisfies (3.1a)-(3.1d) withm = n = 1. Notice the omission of (3.1e). In other words, W satisfies:

(∃P,Q,R, S) W = PQPRSR, SR < PQ, QP < RSR,(6.5a)

|S| < |RS| < |PR| < |PQ|.

Definition 6.6. W is binary-1∗ if (6.3a) with m = n = 1. (AB 6= BA notrequired.)


Lemma 6.7. Word W is biperiodic-1∗ if and only if

(∃F, I, J, P,Q,R, S, T, U) all words 6= 1 except possibly S and(6.7a)

W = PQPRSR, SRF = PQ, QP = RSI, R = IJ , P = ST , Q = RU .

Proof. Assume (6.5a). Then SR < PQ, QP < RSR imply PQ = SRF,QP = RSI, R = IJ for some words F, I, J 6= 1. |RS| < |PR| < |PQ| imply|S| < |P |, |R| < |Q|. Using these inequalities and SR < PQ, QP < RSRwe get P = ST, Q = RU for some words T,U 6= 1. Thus (6.7a) is true.(6.5a) follows easily from (6.7a).

Lemma 6.8. If W is biperiodic-1∗ then W is binary-1∗ using A,B : B 6= 1.

Proof. By Lemma 6.7, we can assume W satisfies (6.7a) from which wededuce:

(1) (∃V 6= 1) F = V U since STRU = PRU = PQ = SRFand so F > U .

(2) UP = SI since RUPJ = QPJ = RSIJ .(3) |T | < |I| since |UST | = |UP | = |SI|,

|UT | = |I|.(4) RV = TR since (1) and

SRV U = SRF = PQ = STRU .(5) (∃K 6= 1) I = TK since (3), (4) and R = IJ .(6) Q = KJF since PQ = SRF = SIJF

= STKJF = PKJF .(7) KJFST = TKJSTK by (5), (6) and KJFST = QP

= IJSI = TKJSTK.(8) TK = KT since, from (7), K ≤ TK ≥ T.(9) (∃N 6= 1, r, s ≥ 1) K = N r,

T = N s using (8) and Lemma 4.4.(10) SN r = SK = US, since (8), (5), (2) and P = ST imply

hence SN r > S SKT = STK = SI = UP = UST .(11) JFS = TJSK since (7) and (8).(12) JV US = TJSK since (11) and (1).(13) JV = TJ = N sJ,

hence J < N sJ using (12), (10) and (9).(14) (∃t ≥ 0, D) S = DN t, by applying Lemma 4.11

N > D to (10) SN r > S.(15) (∃u ≥ 0, L) J = NuL, by applying Lemma 4.10

L < N to (13) J < N sJ .(16) PR = DN bL, since PR = STIJ

b = r + 2s+ t+ u = DN tN sN r+sNuL.

502 C.M. WEINBAUM

(17) UPR = DN cL, since UPR = USTIJc = 2r + 2s+ t+ u = SKTIJ = SKTKTJ .

(18) SR = DNaL, since SR = SIJ = STKJa = r + s+ t+ u = DN tN r+sNuL.

(19) (∃C,M) N = LM = CD using (14) and (15).

Using (18) and (19) with Lemma 4.14 and (16) and (19) with Lemma 4.14,we have:

(∃B 6= 1, A, p ≥ 0) SR = A(BA)a+p, |AB| = |N |,DL = A(BA)p, |A| < |N | and hence AB = DC.

(∃G 6= 1, H, q ≥ 0) PR = G(HG)b+q, |GH| = |N |,DL = G(HG)q, |G| < |N | and hence GH = DC.

So p|N |+ |A| = |DL| = q|N |+ |G|, implying p = q, |A| = |G|. Then A = G,B = H since AB = DC = GH. So PR = A(BA)b+p. Similarly from (17)and (19) with Lemma 4.14, UPR = A(BA)c+p. From (16), (17) and (18),1 < a < b < c ≤ 2a. So 1 < a+p < b+p < c+p ≤ 2a+p ≤ 2a+2p = 2(a+p).Let h = a+ p, i = b+ p, j = c+ p.

Then W = PQPRSR = (PR)(UPR)(SR) = A(BA)iA(BA)jA(BA)h

satisfies (6.3a) for m = n = 1. So W is binary-1∗.

Lemma 6.9. If W is binary-1∗ using A,B : A 6= 1 then W is biperiodic-1∗.

Proof. Assume word W satisfies (6.3a) with A 6= 1, m = n = 1. Sincei − h ≤ i + h − j, there exists an integer r with 0 < i − h ≤ r ≤ i + h − j.Therefore 0 ≤ h− i+ r, j + r ≤ i+ h, 0 < j − h ≤ i− r, j − i+ r ≤ h. Also

(AB)j−i+r < A(BA)h(∗)

since A 6= 1.Define P = (AB)r, Q = A(BA)i−r(AB)j−i, R = A(BA)i−r, S =

(AB)h−i+r. Note that P,Q,R > 1, S ≥ 1, W = PQPRSR.W is biperiodic-1∗ with (6.5a) true because:

SR < PQ since SR = A(BA)h < A(BA)i(AB)j−i = PQ, h < i.

QP < RSR since QP = A(BA)i−r(AB)j−i+r

< A(BA)i−rA(BA)h = RSR using (∗).

|RS| < |PR| since |S| = |(AB)h−i+r| < |(AB)r| = |P |, AB 6= 1, h < i.

|PR| < |PQ| since |R| = |A(BA)i−r| < |A(BA)i−r(AB)j−i| = |Q|.

Theorem 6.10. A word is biperiodic-1 if and only if it is binary-1.


Proof. Assume that W is biperiodic-1 with P,Q,R, S as in (3.1a)-(3.1e) and(6.2a). Word W1 = PQPRSR is biperiodic-1∗. By Lemma 6.7, (∃F, I, J, T,U) satisfying Condition (6.7a). By Lemma 6.8, W1 is binary-1∗ and satisfies(6.3a) for m = n = 1 and some B 6= 1. As in the proof of Lemma 6.8:

W1 = PQPRSR = PRUPRSR,

PR = A(BA)i, UPR = A(BA)j , SR = A(BA)h.

So W = P (QP )mR(SR)n = P (RUP )mR(SR)n = PR(UPR)m(SR)n im-plying (6.3a).

If AB = BA then by Lemma 4.4, A,B (and hence W ) are powers of thesame word. So W is a proper power, not simple. By Theorems 5.9 and5.5, W is 2-piece and simple, a contradiction. Thus AB 6= BA and W is abinary-1.

Now assume W is binary-1. Define P,Q,R, S as in proof of Lemma 6.9so that:

1 < SR < PQ, QP < RSR.(6.10a)

(PQ)m = A(BA)i(A(BA)j)m−1(AB)j−i.(6.10b)

(PQ)mPR = A(BA)i(A(BA)j)m−1A(BA)j = A(BA)i(A(BA)j)m.(6.10c)

(PQ)mPR(SR)n = A(BA)i(A(BA)j)m(A(BA)h)n = W.(6.10d)

W = UV for U = (PQ)mP, V = R(SR)n. Then (6.10a) implies (3.1a)-(3.1d)are true.

If URS = SRU then A(BA)h+1 ≤ A(BA)i <PQ <URS and A(BA)hAB≤ SRP < SRU imply A(BA)h+1 = A(BA)hAB. So BA = AB, a contra-diction. Thus URS 6= SRU. If V PQ = QPV then V PQ > Q > (AB)j−i >AB, QPV > V > SR > BA imply AB = BA, a contradiction. SoV PQ 6= QPV, (3.1e) is true and W is biperiodic-1.

Theorem 6.11. A word is biperiodic-2 if and only if it is binary-2.

Proof. Assume W is biperiodic-2. Then U = Qm, 1 < SR < Q < R(SR)n =V, implying RS = SR. By Lemma 4.4, R = Xr, S = Xs, X 6= 1, r, s ≥ 1.Since SR < Q < R(SR)n, (∃A 6= 1, B, t ≥ 0) Q = A(BA)r+s+t, X = AB.Then W = (A(BA)i)m(AB)j using i = r + s + t, j = r + n(r + s). AlsoXi < XiA = Q < R(SR)n = Xj implies i < j, i+1 ≤ j. Thus (6.3b) is true.If AB = BA then by Lemma 4.4 W is a proper power, not simple. But Wis 2-piece, simple by Theorems 5.9 and 5.5, a contradiction. So AB 6= BAand W is binary-2.

Now assume W is binary-2. Use Q = A(BA)i, P = R = 1, S = AB,n = j, U = Qm, V = R(SR)n. Then (3.1a)-(3.1d) and (6.2b) P = 1,|SR| < |Q| are true. Suppose URS = SRU. Then

(A(BA)i)mAB = URS = SRU = AB(A(BA)i)m.

504 C.M. WEINBAUM

Hence, AB = BA, a contradiction. Suppose V PQ = QPV. Then

(AB)nA(BA)i = V PQ = QPV = A(BA)i(AB)n

hence, AB = BA, a contradiction. Thus (3.1e) is true and W is biperiodic-2.

Theorem 6.12. Each biperiodic-3 word has a binary-3 cyclic permutation.Each binary-3 word has a biperiodic-3 cyclic permutation.

Proof. Assume W is binary-3. Then W ∼ W1 = BA(A(BA)i)mA(BA)j−1.Use P = BA,Q = A(BA)i−1, R = A, S = B, n = j. Thus (3.1a)-(3.1d)and (6.2c) are true. If URS = SRU then URS > AB, SRU > P = BAimply AB = BA, a contradiction. If V PQ = QPV then A(BA)iAB =QPRS < QPV and A(BA)i+1 < A(BA)j = V < V PQ imply AB = BA, acontradiction. So (3.1e) is true and W has a biperiodic-3 cyclic permutationW1.

Now assume W is biperiodic-3. Then (3.1a)-(3.1c), |RS| ≤ |P | < |PQ|imply SR ≤ P, RS < QP, QP < R(SR)n. Thus (∃I, J 6= 1, T ) QP = RSI,QPJ = R(SR)n, P = SRT.

(1) QSRT = QP = RSI,IJ = R(SR)n−1 using R(SR)n = QPJ = RSIJ .

(2) I = V T for some V,|V | = |Q| from (1).

(3) QSRTJ = V TJSR since QSRTJ = QPJ = RSIJ= R(SR)n = IJSR = V TJSR.

(4) Q = V from (3).(5) SRTJ = TJSR from (3).(6) QSRTJ = RSIJ

= RSV TJ = RSQTJ using (1), (2) and (4).(7) RSQ = QSR from (6).(8) (∃a, b, c ≥ 0, C,D) from (7) and Lemma 4.11.

R = C(DC)a, S = D(CD)b,Q = C(DC)c, CD simple

(9) DC is simple; SR = (DC)a+b+1 by Lemma 4.9 and (8).(10) (∃t ≥ 1) TJ = (DC)t from (9), (5) and Lemmas 4.4

and 4.5.(11) (∃d ≥ 0, F 6= 1, G) T = (FG)dF, from (10).

DC = FG(12) R(SR)n = C(DC)a((DC)a+b+1)n from (8).

= C(DC)r withr = a+ n(a+ b+ 1) ≥ n ≥ 2

(13) |C(DC)c+1| ≤ |C(DC)a+b+c+1| from (8) and |RS| ≤ |P |.= |QSR| ≤ |QP |


(14) |QP | < |R(SR)n| from QP < R(SR)n and (12).= |C(DC)n(a+b+1)+a|

(15) c+ 1 < n(a+ b+ 1) + a = r from (13) and (14).(16) QP = C(DC)eF, by (1) QP = QSRT, (8) and (11).

e = a+ b+ c+ d+ 2(17) (∃B 6= 1, A, p ≥ 0) by (16), QP = X1(Y1)eZ1,

QP = A(BA)e+p, AB = CD. X1 = C, Y1 = DC = FG, Z1 = F ;Let i = e+ p so that i ≥ 2. apply Lemma 4.14 to

W1 = X1(Y1)eZ1.(18) R(SR)nP = C(DC)fF, by P = SRT, (8) and (11).

f = r + a+ b+ d+ 1(19) (∃M 6= 1, L, q ≥ 0) by (17), R(SR)nP = X2(Y2)eZ2,

R(SR)nP = L(ML)f+q, X2 = C, Y2 = DC = FG, Z2 = F ;LM = CD. Let j = f + q. apply Lemma 4.14 to

W2 = X2(Y2)fZ2.(20) |A| ≡ |CF | Modulo |DC|; from (16), (17) and |BA| = |DC|.

|A| ≤ |DC|(21) |L| ≡ |CF | Modulo |DC|; from (18), (19) and |ML| = |DC|.

|L| ≤ |DC|(22) |A| = |L|, A = L, B = M, by (20) and (21).

L(ML)j = A(BA)j

(23) (j − i)|DC| = (j − i)|AB| since (17) AB = CD.= |A(BA)j | − |A(BA)i| from (19), (22) and (17).= |R(SR)nP | − |QP | from (12) and (8).= |R(SR)n| − |Q|= |C(DC)r| − |C(DC)c|= (r − c)|DC|

(24) j − i = r − c > 1 from (23) and (15).and hence i+ 2 ≤ j

(25) W has cyclic permutation from (17), (19) and (22).W3 = (QP )mR(SR)nP= (A(BA)i)mA(BA)j

(26) W3 satisfies (6.3c) from (17), (24) and (25).

If AB = BA, it follows that W3, W are proper powers, not simple by (25)and Lemma 4.4. However, W is 2-piece, simple by Theorems 5.9 and 5.5, acontradiction. So AB 6= BA and W3 is binary-3.

7. Some nonperiodic binary words.

We prove that binary words, with some restrictions on their exponents, arenot periodic. Details are in Theorems 7.5, 7.6 and 7.7. In the proofs, ABsimple, A,B 6= 1 can be assumed in (6.3a)-(6.3c) instead of AB 6= BAbecause of the following lemma.

506 C.M. WEINBAUM

Lemma 7.1. For k = 1, 2 or 3, equivalent properties for a word W are:

(i) binary-k using A,B : AB 6= BA;

(ii) binary-k using C,D : C,D 6= 1, CD simple.

Proof. Assume (ii) for W. Then CD 6= DC by Lemma 4.4. Use A = C, B =D to get (i). Now assume (i) for W . By Lemma 4.14, (∃C,D 6= 1, a, b ≥ 0)CD simple, A = C(DC)a, B = D(CD)b. Define p(t) = (1 + a + b)t + a,q(t) = (1 + a+ b)t for t ≥ 0.

Assume k = 1. Then p(h) < p(i) < p(j) since p(t) is strictly increasing.Since j ≤ 2h, p(j) ≤ p(2h) = (1+ a+ b)2h+ a ≤ (1+ a+ b)2h+2a = 2p(h).Also 1 < p(h) since 1 < h ≤ p(h). So W is binary-1 for C,D, p(h), p(i), p(j).

Assume k = 2. Then p(i)+1 = (1+a+b)i+a+1 ≤ (1+a+b)(j−1)+a+1 =q(j)− b ≤ q(j) since i ≤ j − 1. 1 ≤ p(i) since 1 ≤ i ≤ p(i). So W is binary-2for C,D, p(i), q(j).

Assume k = 3. Then 0 < p(i) since 0 < i ≤ p(i). Since i ≤ j − 2,

p(i) + 2 = (1 + a+ b)i+ a+ 2 ≤ (1 + a+ b)(j − 2) + a+ 2

= q(j)− a− 2b ≤ q(j).So W is binary-3 for C,D, p(i), p(j). Thus (ii) is true for W for k = 1, 2, 3.

By Lemma 7.1, binary-1 and binary-3 words are products of words Xk

defined below. Results about such products appear in the next two lemmas.

Definition 7.2. For fixed wordsA,B 6= 1, AB simple, defineXk = A(BA)k,k ≥ 0.

Lemma 7.3. Let G = Xa1 . . .mXam , H = Xb1 . . .mXbn , 1 ≤ ai, 1 ≤ bj ,1 ≤ i ≤ m, 1 ≤ j ≤ n with 2 ≤ m,n. Assume G = H. Then m = n, ai = bifor 1 ≤ i ≤ m.

Proof. If not, pick least integer k ≥ 1 with ak 6= bk. Assume ak < bk sothat Xak

BA ≤ Xbk. Therefore k < m, TXak

AB < TXakXak+1

≤ G,TXak

BA < TXbk≤ H for T = Xa1 . . .mXak−1

. So AB = BA; hence ABnot simple by Lemma 4.4, a contradiction.

Lemma 7.4. Let W = Xa1 . . .mXam , 1 ≤ ai, 1 ≤ i ≤ m, m ≥ 2. Assume(∃F ) F ≤W ≥ F.

If |Xa1 | < |F | then (∃s, b) F = Xa1 . . .mXasXb, 1 ≤ s < m, 1 ≤ b ≤ as+1.(7.4a)

If |Xam | < |F | then (∃t, c) F = XcXat . . .mXam , 1 < t ≤ m, 1 ≤ c ≤ at−1.(7.4b)

Proof. Assume |Xa1 | < |F |. Using F < W, W = Xa1 . . .mXam induces afactorization F = Y1 . . .mYrZ where Yk = A or Yk = B for 1 ≤ k ≤ r and


Z equals P or Q for some words P,Q satisfying 1 ≤ P < A, 1 ≤ Q < B.Also r ≥ 3 since |F | > |Xa1 |, a1 ≥ 1. Since AAA, BB do not appear inW = Xa1 . . .mXam , Y = Yr−2Yr−1Yr equals BAA, AAB, BAB or ABA.To prove (7.4a) it suffices to prove that Y = ABA and Z = 1.

The five cases for Y Z are BAAQ, AABP, BABP, ABAP, ABAQ. Asshown below, only Case 4 with P = 1 and Case 5 with Q = 1 can occur. Soindeed Y = ABA, Z = 1.

Case 1 BAAQ: W > F, W > BABA imply BABA > BAAQ. ByLemma 4.7, AQ is a power of BA but |AQ| < |BA|. So AQ = 1,contradicting A 6= 1.

Case 2 AABP : W > F, W > ABA imply ABA > ABP. ABA =RABP for some R 6= 1 with |RP | = |A|. So ABAB = RABPB,RAB < ABAB, AB simple. By Lemma 4.6, R is a power of AB but|R| ≤ |A| < |AB|. So R = 1, a contradiction.

Case 3 BABP : W > F, W > ABA imply ABA > ABP as in Case 2.Case 4 ABAP : W > F, W > BABA imply BABA > BAP . By

Lemma 4.7, P is a power of BA but |P | < |BA|. So P = 1.Case 5 ABAQ: W > F, W > BABA imply BABA > BAQ. By

Lemma 4.7, Q is a power of BA but |Q| < |BA|. So Q = 1.Now assume |Xam | < |F |. So W ∗ = (Xam)∗ . . .m(Xa1)

∗ and (Xat)∗ =A∗(B∗A∗)t, t ≥ 0. AB simple implies BA simple by Lemma 4.9. (BA)∗ issimple by Remark 5.2. A∗B∗ is simple since A∗B∗ = (BA)∗. Note that F ∗ ≤W ∗ ≥ F ∗. Apply (7.4a) to W ∗, F ∗. So (∃t, c) F ∗ = (Xam)∗ . . .m(Xat)(Xc)∗,1 < t ≤ m, 1 ≤ c ≤ at−1. Take reverses to get (7.4b).

Theorem 7.5. Each binary-1 word W with n ≥ 2 is not periodic.

Proof. By Lemma 7.1, Definition 7.2, W = Xi(Xj)m(Xh)n, 1 < h < i <j ≤ 2h for some A,B 6= 1, AB simple. By (4.13a), it suffices to provethat each major left factor of W which is also a right factor is equal toW . Suppose F ≤ W ≥ F, 2|F | ≥ |W |. Then |F | > |Xi|, |F | > |Xh| since|Xi| < |Xj | > |Xh|. By Lemma 7.4, F = XiGHXh where G is a product ofone or more Xj and H is a product of one or more Xh. It also follows fromLemma 7.4 that:

XiGH and the start of W have the same Xk factors.(7.5a)

GHXh and the end of W have the same Xk factors.(7.5b)

|F | ≤ |W | implies |GH| ≤ |(Xj)m(Xh)n−1|. It follows that:

GH and the start of (Xj)m(Xh)n−1 have the same Xk factors.(7.5c)

GH and the end of (Xj)m(Xh)n−1 have the same Xk factors.(7.5d)

(7.5c) implies G = (Xj)m. (7.5d) implies H = (Xh)n−1. Thus F = W asrequired.

508 C.M. WEINBAUM

Theorem 7.6. Each binary-2 word W with m(i + 1) ≤ j, 3 ≤ j is notperiodic.

Proof. By Lemma 7.1 and Definition 7.2, W = (Xi)m(AB)j , 1 ≤ i, i+1 ≤ j,m ≥ 1 for some A,B 6= 1, AB simple. By (4.13a) it suffices to show thatW has no proper major left factor which is also a right factor. Suppose Fis such a factor, 2|F | ≥ |W |, F < W > F. We show this implies AB = BA,a contradiction.

Since m(i + 1) ≤ j, |(Xi)m| = |Am(BA)mi| < |(AB)m(i+1)| ≤ |(AB)j |.Then 2|F | ≥ |W | and |(Xi)m| < |(AB)j | imply |(Xi)m| < |F |. Using F < W,F has one of the forms:

(Xi)m(AB)rP, (Xi)m(AB)sAQ, (Xi)mR

where 1 ≤ r < j, 0 ≤ s < j, 1 ≤ P < A, 1 ≤ Q < B, 1 < R < A.

Case 1. F = (Xi)m(AB)rP : W = (Xi)m(AB)j > F, r < j imply (AB)j

> (AB)rP. By Lemma 4.7, P = 1. Then F > BA, W > AB implyAB = BA.

Case 2. F = (Xi)m(AB)sAQ: W > F implies (AB)j > (AB)sAQ and(AB)j > AQ. By Lemma 4.7, AQ is a power of AB. Since |AQ| <|AB|, we have AQ = 1, A = 1, contradicting AB 6= BA. So this casecannot occur.

Case 3. F = (Xi)mR: W > F, 3 ≤ j, 1 ≤ i imply (AB)3 > ABAR.By Lemma 4.7, AR = (AB)t for some t ≥ 0. Here t ≤ 1 since 0 <|R| < |A|. If t = 0 then A = 1, a contradiction. If t = 1 then R = B,F > BA, W > AB so that AB = BA.

Theorem 7.7. Each binary-3 word W is not periodic.

Proof. By Lemma 7.1 and Definition 7.2, W = (Xi)mXj , 1 ≤ i, i + 2 ≤ j,m ≥ 1 for some A,B 6= 1, AB simple. By (4.13a) it suffices to show thatW has no proper major left factor which is also a right factor. Suppose Fis such a factor, 2|F | ≥ |W |, F < W > F. We show this implies AB = BA,a contradiction.

Since |W | ≥ |XiXj | > |XiXi| = 2|Xi| we have 2|F | ≥ |W | > 2|Xi|. By(7.4a), F has one of the forms: (Xi)r, (Xi)sXb, (Xi)mXc where 1 ≤ r ≤ m,1 ≤ s < m, 1 ≤ b < i, 1 ≤ c < j. Thus F has a right factor XiXa forsome a, 1 ≤ a < j and hence F > BAXa. Also W > ABXa. ThereforeAB = BA.

8. Computing possibly biperiodic words.

Let W = UV, SR < P (QP )m = U, QP < R(SR)n = V, 0 < |RS| < |PQ| <|U |, m, n ≥ 1. W may be biperiodic. p = |PQ|, q = |SR|, d = |U | − p,e = |V | − q satisfy 0 < d, 0 < e, q < p < q + e. Function g (see below) with


inputs p, q, d, e, generates such a W as a list of integers ≥ 1. Format for gcomes from Mathematica software, version 3.0.

g[p , q , d , e ] := Module[i, k, n = p+ q + d+ e, w,If [!((0 < d)&&(0 < e)&&(q < p)&&(p < q + e)),

Return[“Invalid Input”]];w = Join [ Range[q], Range[n− 2q], Range[q]];For[k = n− q, k ≥ p+ d+ 1, k −−, w[[k]] = w[[k + q]]];For[k = p+ d, k ≥ q + 1, k −−, w[[k]] = w[[k + p]]];For[k = q, k ≥ 1, k −−, i = w[[k + p]];w = w/.(k− > i)];w].

The observed output from g is (unpredictably) either biperiodic or a properpower.

In Mathematica, Range[q] is the list of positive integers from 1 to q.Join[a, b, c] concatenates lists a, b, c. The code k- - indicates integer k isdecreased by 1 after each stage of a loop. w[[i]] = i-th element of the list w.The code w = w /. (k − > i) rewrites list w by replacing each instance ofthe current value of k in list w by the current value of i.

9. Examples.

We give 2 sets of examples of biperiodic words W = PQPRSR over thealphabet a, b. In Example (9.1), |PR| < |RS|, P 6= 1. In Example (9.2),|PQ| < |PR|, P 6= 1. Therefore (6.2a)-(6.2c) are not true. These examplesinclude g[24, 18, 3, 10] and g[24, 18, 15, 14] for the function g defined in theprevious section.

Example 9.1. W = (CCDDCD)2D(ab)j−i, C = a(ba)i, D = a(ba)j , 0 <i < j ≤ 2i.

Let P = C, Q = CDDCD, U = PQP = CCDDCDC, R = CD(ab)j−i,S = CC(ab)j−i, V = RSR = CDDCDD(ab)j−i. W is biperiodic because:

SR < PQ since SR = CCDD(ab)j−i,PQ = CCDDCD = CCDD(ab)iaD, j ≤ 2i.

QP < RSR since C < D, QP = CDDCDC,RSR = CDDCDD(ab)j−i.

|PR| < |RS| since |PR| = |CCD(ab)j−i| < |CCDDC(ab)j−i| = |RS|.URS 6= SRU else URCCD = URSC = SRUC = SRCCDDCDCC,

CCD = DCC, not true.QPV 6= V PQ else QPCDDCDDD = QPV C = V PQC

= V PCDDCDC, DD > DC, not true.

Using i = 1, j = 2 and shorthand 2 = ab, 3 = aba, 5 = ababa,W = (335535)(335535)52. Rewrite W by replacing the letters a, b with thesymbols 1, 2, respectively. The resulting word, written as a list, is equal tog[24, 18, 3, 10].

510 C.M. WEINBAUM

Example 9.2. W = X(XZY )2XXY, X = (abb)hab, Y = (abb)iab, Z =(abb)2h+1ab, 0 < h < i.

Let P = XXX, Q = bX(abb)i−h, PQ = XXZ(abb)i−h, QP = bXY XX,U = PQP = XXZYXX, R = bXY, S = (abb)ha, V = RSR = bXY XXY.W is biperiodic because:

SR<PQ, QP <RSR since SR = XXY .|PQ| < |PR| since i− h < i implies Q < R.URS 6= SRU since URS > S > ba, SRU > X > ab and ab 6= ba.V PQ 6= QPV since V PQ > Q > bb, QPV >Y >ab and bb 6= ab.

Using h = 1, i = 2 and shorthand 5. = X, 8. = Y, 11. = Z, W =5.5.11.8.5.11.8.5.5.8. Rewrite W by replacing the letters a, b with the sym-bols 1, 2, respectively. The resulting word, written as a list, is equal tog[24, 18, 15, 14].

References

[1] E. Ghys and P. de la Harpe, Sur les Groupes Hyperboliques d’apres Mikael Gro-mov, Birkhauser, Boston, MA, 1990 (see Appendix, in English, On small cancellationgroups, by Ralph Strebel), MR 92f:53050, Zbl 731.20025.

[2] Ju.I. Hmelevskii, Equations in free semigroups, Proc. Steklov and Inst. Mat., Am.Math. Soc. Transl., 107 (1976), MR 52 #14094, Zbl 326.02032.

[3] M. Lothaire, Combinatorics on Words, Cambridge University Press, 1997,MR 98g:68134, Zbl 874.20040.

[4] R.C. Lyndon and P.E. Schupp, Combinatorial Group Theory, Berlin and New York,Springer-Verlag, 1977, MR 58 #28182, Zbl 368.20023.

Received May 17, 1999 and revised April 10, 2000.

Department of Mathematics1020 Souza DriveEl Dorado Hills, CA 95762E-mail address: [email protected]










CONTENTS

Volume 199, no. 1 and no. 2

Christine Bessenrodt: On mixed products of complex characters of the doublecovers of the symmetric groups 257

Etienne F. Blanchard and Kenneth J. Dykema: Embeddings of reduced free productsof operator algebras 1

Vera Carrara, J. Scott Carter and Masahico Saito: Singularities of the projections ofsurfaces in 4-space 21

J. Scott Carter with Vera Carrara and Masahico Saito 21

Ildefonso Castro, Cristina R. Montealegre and Francisco Urbano: Closed conformalvector fields and Lagrangian submanifolds in complex space forms 269

Wenxiong Chen and Congming Li: Prescribing scalar curvature on Sn 61

Wenxiong Chen and Congming Li: Prescribing scalar curvature on Sn 61

Anne Cumenge: Zero sets of functions in the Nevanlinna or theNevanlinna–Djrbachian classes 79

S.B. Damelin, P.D. Dragnev and A.B.J. Kuijlaars: The support of the equilibriummeasure for a class of external fields on a finite interval 303

P.D. Dragnev with S.B. Damelin and A.B.J. Kuijlaars 303

Kenneth J. Dykema with Etienne F. Blanchard 1

Minking Eie with Kwang-Wu Chen 41

Tobias Ekholm: Invariants of generic immersions 321

Peter Friis and Mikael Rørdam: Approximation with normal operators with finitespectrum, and an elementary proof of a Brown–Douglas–Fillmore theorem 347

Chris Jantzen and Henry H. Kim: Parametrization of the image of normalizedintertwining operators 367

Henry H. Kim: Residual spectrum of split classical groups; contribution from Borelsubgroups 417

Henry H. Kim: Residual spectrum of split classical groups; contribution from Borelsubgroups 417

A.B.J. Kuijlaars with S.B. Damelin and P.D. Dragnev 303

Soo Teck Lee and Chen-bo Zhu: On the (g, K )-cohomology of certain theta lifts 93

400

Congming Li with Wenxiong Chen 61

Wim Malfait: Homotopically periodic maps of model aspherical manifolds 111

David Manderscheid: Quadratic base change for p-adic SL(2) as a thetacorrespondence II: Jacquet modules 447

Ahmed Mohammed: Uniform integrability of approximate Green functions of somedegenerate elliptic operators 467

Cristina R. Montealegre with Ildefonso Castro and Francisco Urbano 269

Mara D. Neusel: The transfer in the invariant theory of modular permutationrepresentations 121

R.C. Orellana: The Hecke algebras of type B and D and subfactors 137

Shu-Yen Pan: Splittings of the metaplectic covers of some reductive dual pairs 163

Mikael Rørdam with Peter Friis 347

Masahico Saito with Vera Carrara and J. Scott Carter 21

Sergio L. Silva: On isometric and conformal rigidity of submanifolds 227

Wojciech Szymanski: Simplicity of Cuntz–Krieger algebras of infinite matrices 249

Francisco Urbano with Ildefonso Castro and Cristina R. Montealegre 269

C.M. Weinbaum: Periodic subwords in 2-piece words 493

Chen-bo Zhu with Soo Teck Lee 93

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http://pjm.math.berkeley.edu/about/journal/submissions.html





On mixed products of complex characters of the double covers of thesymmetric groups 257

CHRISTINE BESSENRODT

Closed conformal vector fields and Lagrangian submanifolds in complexspace forms 269

ILDEFONSO CASTRO, CRISTINA R. MONTEALEGRE AND FRANCISCOURBANO

The support of the equilibrium measure for a class of external fields on afinite interval 303

S.B. DAMELIN, P.D. DRAGNEV AND A.B.J. KUIJLAARS

Invariants of generic immersions 321TOBIAS EKHOLM

Approximation with normal operators with finite spectrum, and anelementary proof of a Brown–Douglas–Fillmore theorem 347

PETER FRIIS AND MIKAEL RØRDAM

Parametrization of the image of normalized intertwining operators 367CHRIS JANTZEN AND HENRY H. KIM

Residual spectrum of split classical groups; contribution from Borelsubgroups 417

HENRY H. KIM

Quadratic base change for p-adic SL(2) as a theta correspondence II:Jacquet modules 447

DAVID MANDERSCHEID

Uniform integrability of approximate Green functions of some degenerateelliptic operators 467

AHMED MOHAMMED

Periodic subwords in 2-piece words 493C.M. WEINBAUM

PacificJournalofM

athematics

2001Vol.199,N

o.2

PacificJournal ofMathematics


volume 199 no. 2 june 2001 journal of mathematics - msp · pacific journal of mathematics volume...

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