nolan r. wallach introduction...testable using the next generation of computers. however, we hope...

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Number 4, Oetober 1993

INVARIANT DIFFERENTIAL OPERATORS ON A REDUCTIVE LIE ALGEBRA

AND WEYL GROUP REPRESENTATIONS

NOLAN R. WALLACH

INTRODUCTION

Let V be a finite-dimensional vector space, and let G be a subgroup of GL( V). Set D( V) equal to the algebra of differential operators on V with polynomial coefficients and D( V) G equal to the G invariants in D( V). If 9 is a reductive Lie algebra over C then ~ egis a Cartan subgroup of g, and if G is the adjoint group of 9 then W is the Weyl group of (g, ~) , Harish-Chandra introduced an algebra homomorphism, J, of D(g)G to D(~)w [H3]. J isgiven by the obvious restriction mapping on the subalgebra of invariant polynomials and on the invariant constant coefficient differential operators, and ker J is the ideal, ..Y, of D(g)G consisting of elements that annihilate all G invariant polynomials. In this paper we prove that if 9 has no factor of type E then J is surjective. We also prove that for general g, the homomorphism is surjective after localizing by the discriminant of g. If go is a real form of 9 and if Go is the adjoint group of go then Harish-Chandra has shown that ..Y is precisely the ideal in D(g)G of operators that annihilate all Go invariant distributions on "completely invariant" open subsets of go [H2]. Our first application of our analysis of J is to give a new proof of this important theorem.

In light of this theorem the space of Go -invariant distributions on a com-pletely invariant open subset of go is a D(g)G-module that "pushes down" to

w w a D(~) -module. To analyze these D(~) -modules we develop a theory anal-ogous to Howe's formalism of dual pairs, proving an equivalence of categories between an appropriate category of D(~)w-modules and the category of all W-modules over C. We show that the D(g)G-module of distributions on go supported in the nilpotent cone of go is (as a D(~)w module) in our category. Thus, to each distribution supported on the nilpotent cone we can associate a (finite dimensional) representation of W. If the distribution is the orbital in-tegral corresponding to a fixed nilpotent element of go then we prove that the representation of W is irreducible and derive a formula for the Fourier trans-form of the orbital integral in terms of W -harmonic polynomials corresponding

Received by the editors April 17, 1991 and, in revised fonn, June 5, 1992. 1991 Mathematics Subject Classification. Primary 22E30, 22E45. Key words and phrases. Research partially supported by an NSF summer grant.

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780 N. R. WALLACH

to this representation of W. In [BV2, BV3, HK] results of this nature were proved in the case when 90 is a Lie algebra over C looked upon as a Lie algebra over R. They prove that the indicated Fourier transform is given in terms of a harmonic polynomial transforming according to the Springer representation associated to the corresponding nilpotent G-orbit in 9 [S]. We use this theorem to prove that our general correspondence between nilpotent Go -orbits of 90 is given by the Springer correspondence for the corresponding G-orbits in 9. In particular, our theory yields a new approach to the Springer correspondence.

We say that 9 is "nice" if d is surjective. (As indicated above one can prove "niceness" for all 9 without ideals of type E.) For nice 9 we show that the surjectivity of d can be used to give a new proof of Harish-Chandra's famous local L 1 theorem for invariant eigendistributions on completely invariant sub-sets of 90 • The question of whether or not 9 is nice is related to a long standing problem concerning Weyl group invariants in two copies of a Cartan subalge-bra. We consider the contragradient action of W on ~... and are interested in the invariants of W in 9'(~ x ~ ... ) (the polynomials on ~ x f), 9'(~ x ~"')w , under the action s/(x, A) = /(S-l X, S-l A). We choose a basis of ~ and the dual basis in ~... and thereby have linear coordinates Xl ' ••• ,xl on ~ and dual linear coordinates {I' ... ,{/ on ~ .... Set P = E{/J/8Xi on 9'(~ x ~ ... ) (P is the usual polarization operator). P stabilizes 9'(~ x f)w , and it has been sug-gested that 9'(~ x ~"')w is the algebra generated by E pk 9'(~)w . For lack of a name let us call this the "polarization hypothesis". If this were true then it is a simple matter to prove that all 9 are good. For 9 of type An the polarization hypothesis can be found in [W]. By a modification of the argument of Weyl it is easy to show that the hypothesis is also true for types Bn and en and G2 • However, the hypothesis is false for Dn for n ~ 4. In the first appendix to this paper we give a counterexample for D4 and introduce what we call the "revised polarization hypothesis". This revision is sufficient to prove "niceness", and it is true for Dn. For F4 even this is false. However, one can prove a result for F4 which is sufficient to prove that it is also nice. The proof for F4 will appear elsewhere. For E6 , E7 , and E8 the question of niceness will most likely be testable using the next generation of computers. However, we hope that there is an elegant theorem on Weyl group invariants (in the spirit of Chevalley's proof that 9'(~)w is a polynomial ring) that will give a uniform argument.

The author began his work on this chain of ideas after a conversation he had with Roger Howe (walking in the Torrey Pines Reserve). In this conversation Howe described his work on the action of the algebra generated by the Casimir polynomial and the Laplacian on 9"'(.5((2, R»Go • An outgrowth of this conver-sation was that it seemed quite likely that D(~)w that is generated by the Weyl group invariant polynomials and the Weyl group invariant differential operators (indeed, Howe sketched a proof of the result for An using a theorem of Weyl alluded to above). We also thank T. Enright, B. Kostant, and D. Vogan for helpful conversations.

1. POLYNOMIAL DIFFERENTIAL OPERATORS INVARIANT UNDER A FINITE GROUP

We begin this section with a simple result that will play an important role in this paper.


DIFFERENTIAL OPERATORS ON A REDUCI1VE LIE ALGEBRA 781

Lemma 1.1. Let,9J' be an algebra over C with unit, and let ~ be a subalgebra of ,9J' containing 1 such that there exists a linear map P of ,9J' onto ~ such that P(I) = 1 and P(ab) = P(a)b for a E,9J', b E ~ . If V is a ~-module then the map V -+ ,9J' ® 9J V given by v 1-+ 1 ® 9J V is injective. Proof. We denote by ,9J' ® V the tensor product over C. Suppose that v E V and 1 ®9Jv = 0 in ,9J' ®9J V. Then there exist ai' ... , am E,9J', VI' ... , vn E V ,and bjj E ~ such that

(1.1) 1 ®V = Eaj(bij ®Vj -1 ®bjjv). jj

If we apply P ® I to both sides of (1.1) then we have (P( 1) = 1)

1 ® v = E P(aj)(bjj ® Vj - 1 ® bjjv) E ~ ® v. ij

Thus 1 ® 9J V = 0 in ~ ® 9J V . Hence v = 0 .

Note. Let ,9J' be an algebra over C with a filtration ,9J'j c ,9J'i+l, Uj.sat i = ,9J' , dim,9J'i < 00. Let G be a compact Lie group acting on .sat byautomor-phisms such that g,9J'j c ,9J'j for all i and g E G. If the corresponding representation of G on ,9J'j is continuous for all i and if ~ = ,9J'G = {a E ,9J'lga = a, g E G} then set

Pa = fo g(a) dg

with d g normalized invariant measure on G. The conclusion of Lemma 1 is therefore true for ~. We will apply Lemma I to this context without further comment.

Let V be a finite-dimensional vector space over C. We will use the notation in Appendix 1. Let G be a subgroup of GL(V). Then G acts on .9(V) by g'f(x)=f(g-I X ) for fE.9(V), XE V, gEG. If DED(V) then we set g. D = gDg- 1 • We note that if g E G then g. Dk(V) C Dk(V). If M is a G-module then we set

MG = {m E Mig· m = m, g E G}.

Then .9(V)G, S(V)G ,and D(V)G are respectively subalgebras of .9(V), S(V) , and D(V). We include the following observation since its proof involves one of the basic ideas in the paper.

Lemma 1.2. If G is afmite group then D(V)G is a simple algebra over C. Proof. If D E D(V) then we use the notation ord(D) for the usual order of D as a differential operator. Let I be a nonzero two sided ideal in D( V) G •

Let DEI be a nonzero element with ord(D) minimal. If f E .9(V)G then ord[f, D) < ord(D). Thus, [f, D) = 0 for all f E .9(V)G. This implies that D(fh) = fDh for all f, g E .9(V)G. Hence D acts on .9(V)G by


782 N. R. WALLACH

multiplication by D· 1 E 9'(V)G. Since 9'(V) is finitely generated as a 9'(V)G-module under multiplication, there exist u I ' •.• ' un E 9'(V)G such that dU I A··· A dUn =F O. This implies that there is an open nonempty subset, U, of V such that u I ' .•. ,un defines a system of holomorphic local coordi-nates on U. Thus if DI E D(V) then

alII DI = L al(ul ' ... ,un) i i

I au l! ••• aUn"

on U. Since qUI' ... , un] is contained in 9'(V)G , we see that if D I9'(V)G = o then DI = O. Hence D is given by multiplication by f = D· 1. Since D =F 0, f =F O. As a right D( V)G -module under multiplication D( V) is finitely generated. Hence there exist D I , ••• ,Dp such that D(V) = EDiD(V)G. Set M = D(V)G / I. Then f· M = O. There exists k such that (adf)k Dj = 0 for all 1 ::; j ::; p. Thus r+ I acts by 0 on D( V) ®D( V)G M. Since D( V) is simple, this implies that D( V) ®D(V)G M = O. So Lemma 1.1 implies that M = O. Thus 1= D(V)G.

We look upon 9'(V) as a D(V)-module under the usual action as differen-tial operators, and we look upon S(V) as a D(V)-module under the obvious identification with D(V)/D(V)9'+(V) (notation as in Appendix 1).

Proposition 1.3. Assume that G isfmite. Let M be a D(V)G-module such that if mE M then dimS(V)Gm < 00 (resp. dim9'(V)G m < 00). If mE M, p E 9'(V)G - {O} (resp. S(V)G - {O}) is such that pm = 0 then m = o. If M is finitely generated as a D( V) G -module then M is of finite length.

Proof. We assume that if m E M then dimS(V)Gm < 00. Let m E M be such that pm = 0 for some p E 9'(V)G - {O}. Set N = D(V)Gm . We show that N = {O}. Set NI = D(V) ®D(V)G N. If q E S(V) (resp. f E 9'(V)) and if D E D(V) then there exists k such that ad(q)k D = 0 (resp. ad(f)k D = 0). Since S(V) is finitely generated as a S(V)G-module under multiplication, this implies that if ml = 1 ® m then dimS(V)m l < 00. Set F = S(V)ml . Then there exists k such that pk F = O. Furthermore, NI = 9'(V)F . We filter NI

by setting g-j NI = Ei~j9'j(V)F . Then

Di(V)g-j NI c g-i+j N I •

Let d be the degree of p. Our assumptions imply that

dimg-j NI ::; dimF L(dimSi(V) - dimSi_dk(V)) ::; C/- I . i~j

Thus Bernstein's theorem (see Appendix 1, Theorem 2) implies that NI = {O}. We now assume that M is finitely generated as a D(V)G-module. Set

MI = D(V) ®D(V)G M.


DIFFERENTIAL OPERATORS ON A REDUCTIVE LIE ALGEBRA 783

If Z}' ••• , Zd are such that D(V) =Ei ziD(V)G and if M = D(V)G F with F a finite-dimensional subspace of M then, if F} = Ei zi ®D(V)G F, F} generates Ml as a D(V)-module. As above, dimS(V)F} < 00. If we filter M} as above then we have a filtration g-k M} of M} so that M} is a filtered module. The argument above implies that

dimg-kM1 ::; Ceo Thus M} is of finite length (see Appendix 1, Theorem 2) as a D(V)-module. We will now show that this implies that M is of finite length as a D( V) G -module. Indeed, let

M ::J MI ::J M2 ::J ..• ::J Mk ::J ..•

be a decreasing family of submodules of M. Set N k equal to the canonical image of D(V) ®D(V)G Mk in MI. Then N k ::J N k+l • Since MI is of finite length, there exists k such that N k = N P for all p 2': k. This implies that if v E Mk and if p 2': k then there exist Di' di E D(V), ei E D(v)G, Vi E M P , Wi EM such that (the following tensor products are over C)

1 ®V = LDi®Vi+ L(diej®wl-di®ejWI)' ijl

Thus, in the notation of Lemma 1.1, we have

1 ® v = L P(D) ® Vi + L(P(di)ej ® WI - P(d) ® ejwl )· ijl

Hence v = EP(Di)Vi E M P • Thus M P ::J Mk. Hence M P = Mk for p 2': k. The proof in this case is now complete.

The proof of the parenthetic statements is the same after we have reversed the roles of 9'(V). and S(V).

Let lfG (resp. lf~) denote the category of all finitely generated D(V)G-modules such that if mE M then there exists k such that if p E Sk(V)G (resp. p E 9'k(V)) then pm = O. If G leaves invariant a symmetric nondegenerate form and if ¢ is as above then the functor M --+ M d~fines an equivalence of categories between lfG and lf~. The following result is a direct consequence of Proposition 1.3.

Corollary 1.4. Assume then G is finite. If M E lfG (resp. C~) andif m E M, p E 9'(V)G - {O} (resp. S(V)G - {O}) is such that pm = 0 then m = O.

We define a D(V) module structure on S(V) as follows. Let Co be the 9'(V)-module, C, with f·l = f(O) . Then as an S(V)-module, D(V) ®9'(V) Co is S(V) ® 1. We look upon S(V) as a G-module in the usual way. Thus with this D(V)-module structure S(V) is a (D(V)G, G)-bimodule (as is 9'(V)). This is the desired structure on S(V). 9'(V) E lfG and S(V) E lf~.


784 N. R. WALLACH

~position 1.5. Assume that G acts completely reducibly on V. Let .7 (resp . .7) denote the set of all isomorphism classes of irreducible finite-dimensional G-modules, M~ such that HomG(M, .9'(V)) :f 0 (resp. HomG(M, S(V)) :f 0) . If A E .7u.7 thenfix J). EA. To each A E.7 (resp. A E.7) there corresponds an irreducible D(V)G-module V;' (resp. V;') such that:

(i) If V;' (resp. V;') is equivalent with Vil (resp. VIl) as a D(V)G-module then A = Jl; and

(ii) As a (D(V)G, G)-bimodule .9'(V) (resp. S(V)) is equivalent with

EB V;' ® J). (resp . EB V;' ® J). ). ;'EsP ;'EJP

Proof. If g E GL(V) , let gt denote the element of GL(V*) given by gt Jl = Jlog- 1 , Jl E V*. Set Gt = {gtlg E G} c GL(V*). Let xl' ... , xn be linear coordinates on V. Let Ij/ be the automorphism of D( V) defined by

G G' Ij/(x) = 8 j and 1j/(8j ) = -xj • Then Ij/(D(V) ) = D(V) , 1j/(.9'(V)) = S(V), and Ij/(gf) = gt Ij/(f) for f E .9'(V) , g E G. Thus if we prove the result for .9'(V) then the result will follow for S(V).

We will be using the following construction throughout the proof of this result. Let A E .7 , and let A * be the class of the contragredient representation of J)., J).* . Then A* E .7. Let e1 , ••• , ed be basis of J)., and let e~ , ... , e; be the dual basis in J).* . If T E HomG(J)., .9'(V)) and S E HomG(J).* , S(V)) then set DT,s = Lj T(e)S(e7). Then DT,s E D(V)G .

If A E.7 then let .9'(V)[A] denote the A-isotypic component of .9'(V). It is clear that as a (D(V)G, G)-bimodule .9'(V) splits into a direct sum of the invariant subspaces .9'(V)[A]. Fix Z;. a nonzero G-invariant irreducible sub-space of .9'(V)[A]. Let Me .9'(V)[A] be a nonzero D(V)G-invariant subspace. Let E = Lj x j 8j • Then E E D(V)G . Thus EM eM. Hence if f E M then every homogeneous component of f is in M.

(1) MnZ;.:f{O}. Indeed, let f EM - {O} be homogeneous of degree r. Then spandgf} is

a direct sum of irreducible G-submodules in the class A. Let T be a nonzero element of HomG(J)., Z;.). Let S E HomG(J).* ,Sr(V)) be such that there exists Jl E J).* with S(Jl)f = 1. Sand Jl exist since f is homogeneous and the pairing between Sr(V) and .9'r(V) given by (plf) = pf E C is perfect. Fix a basis e1 , ••• ,ed of J). such that e~ = Jl and S(e;)f = 0 for i > 1. Then DT ,sf = T(e1)· This implies (1).

Let fEZ;.. Put M = D(V)G f. We use the notation C[G] for the group algebra of G thought of as an abstract group.

(2) M n Z;. = Cf. Indeed, if gEM n Z;. then there exists D E D(V)G such that Df = g.

Thus C[G]g = DC[G]f. So D1Z.l E HomG(Z;., Z;.). Schur's lemma implies that D1Z.l = CI . Thus g E Cf.



(3) M is irreducible as a D(V)G-module. In fact, if MI is a nonzero D(V)G -invariant subspace of V then MI nz;. =I- 0

by (1). Hence f E MI by (2). Thus MI = M. Let f l , ••• , fd be a basis of Z;.' Set Y; = D(V)G 1;. We assert that the

sum V; + ... + JId is direct. Let uik E qG] be such that Uikfi = tJ1k1; (this is possible since qG]lz. = End(Z;.)). Let Vi E Y;, and assume that Li Vi = O. Then Vi = Di1; with Di E D(V)G . Thus

0= uij(v i + ... + vd) = 'LDkuijfk = Di1; = Vi' k

Similarly, uij defines a D(V)G-module isomorphism of ~ onto Y;. Let V;' = D(V)G f for some fEZ;. - {O}. Then since there exists U E qG] with uJ; = f, V; is isomorphic with V;' as a D( V) G -module. This implies that 9"(V)[A] is isomorphic with V;' ~ V;. as a (D(V)G, G)-bimodule. This proves (ii).

If A E!7' then let j(A) be the minimum of j such that 9"/V) n9"(V)[A] =I-

{O}. Suppose that V;' is equivalent with Vil with A, f..t E !7'. We take these modules to be realized as above. Let A implement the equivalence. Let j = j(A). Since AE = EA, A(V;' n 9"/V)) C Vil n 9"j(V), We may assume that Z;. C 9"j(V), Let T E HomG(V;., Z;.) be nonzero, and let !7' E

HomG(V;.* ,Sj(V)) be such that S(V;.*)Z;. =I- O. Set D = DT ,s' Then D(V;' n 9"j(V)) =I- O. Thus

0=1- A(D(V;' n9"j(V))) = DA(V;' n9"/V)) C Vil n9"j(V),

But the fonnula for D implies that D9"/ V) C Z;.' Thus Vil n Z;. =I- O. This implies that A = f..t. This completes the proof of (i) and, hence, of the proposition.

Theorem 1.6. Assume that G is finite. Then !7' = G. If A E G, let V;' E ~G (resp. V;' E ~~) be as in Proposition 1.5.

(i) If M E ~G (resp. ~~) and V is irreducible then M is equivalent with ;. -;. ~

V (resp. V) for some A E G. (ii) If M E ~G (resp. M E ~~) then for each A E G there exists m;. E N

such that M is equivalent with

Eem;.V;' (resp. ~m;.v;,). ;'EG ;'EG

That is, every object in ~G (resp. ~~) splits into irreducible components with finite multiplicities. Proof. As above it is enough to prove the result in the case of ~G' For the sake of completeness, we give the standard proof that !7' = G. Ifg E G and g =I- I then set Ug = {v E Vlgv =I- v}. Then V - Ug = ker(g - I)


786 N. R. WALLACH

which is a proper subspace of V if g i- I. Since G is finite, this implies that V' = ngEG-{I} Ug i- 0. Fix v E V'. Set J = {f E 9'(V)lf(Gv) = O}. Then J is G-invariant and 9'(V)/J is equivalent with qG)* ~ qG] as a G-module.

(1) If M E ~G then D(V) ~D(V)G M E ~ (= ~I})' Indeed, we set U = D(V) ~D(V)G M. We note that as a D(V)G-module

U E ~G' We must show that if p E S+(V) and m E U then there exists k such that pk m = O. Since S(V) is integral over S(V)G , there exist j > 0 and

G ao ' ... , aj _ 1 E S(V) such that j-I

j '"' i P + ~p ai = O. i=O

Clearly, we may assume that ai E S+(V)G. Let m E U. Set Z = S(V)G m. Then ZI = Z + pZ + ... + pj-I Z is p-invariant. Since S( V) is commutative, ZI is S(V)G-invariant. Let Z? = {z E Zllaiz = 0, i = 1, ... , j - I} ZII = {z E Zllarasz = 0, r, s ::::; j - I}, .... Then since ZI is finite dimensional, we have Z? c ZII c .. , c Zi = ZI' Clearly, pZ; c Z;. (*) implies that

. 0 . . I ." I pi ZI = O. We note that aiZ; C Z;- . Thus pi Z; c Z;- (by (*». Hence pj(q+I)Zi = O. Hence pj(q+I)m = O.

Lemma 1.3 now implies that D(V) ~D(V)G M is a finite multiple of 9'(V) as a D(V)-module. Lemma 1.1 implies that M injects in D(V) ~D(V)G M as a D(V)G-submodule. Thus Proposition 1.6(ii) implies that M has a decomposi-tion as in (ii). Since (i) is a consequence of (ii), the theorem follows.

2. DIFFERENTIAL OPERATORS INVARIANT UNDER A WEYL GROUP

Let ~ be a real n-dimensional vector space over R with inner product ( ... , ... ). Let V = (~)c = ~ ~R C. We will denote the Hermitian extension of ( ... , ... ) by the same symbol. Let <I> be a (reduced) root system contained in ~*, and let W c O(~) be the (finite) group generated by the reflections about the hyperplanes a = 0, a E <1>. Let U I "'" un be a set of basic homogeneous invariants which we take to be real valued on ~. Fix <1>+, a system of positive roots for <1>. Let XI' ••• ,xn be a set of linear coordinates on ~ corresponding to an orthonormal basis. Let c;l"'" c;n be the dual coordinates on ~* . If f, g E 9' (~ x ~*) then let {f, g} (the Poisson bracket of f and g) be as in Appendix 1. We say that the pair (W, V) is "good" if the smallest subalgebra of 9'(V x V*) containing 9'(V)w and 9'(V*)w and closed under { ... , ... } is 9'(V x V*)w (here W acts under the diagonal action s(x, A.) = (sx, A.OS-I».

Note that if ~ = ~I EB ... EB V: with W acting irreducibly on V: then W is equal to WI x ... x W d with Wi a finite subgroup of V: generated by reflections. Set Vi equal to the complexification of V: . Then D( V) W is equal



to D(VI)Wl Q9 ••• Q9 D(Vd)Wd and 9'(V)w, S(V)w split into corresponding tensor products. Also, the Poisson bracket "splits" in a manner consistent with the above direct sum decomposition. We therefore see that if (Wi, Vi) is good for i = 1 , .,. , d then (W, V) is good.

Proposition 2.1. If (W, V) is good then D( V) W is the algebra generated by 9'(V)w and S(V)w in D(V).

Proof. We denote by .!B the algebra generated by 9'(V)w and S(V)w in D(V). We set .!Bk =.!B nDk(V) and Dk(V)W = Dk(V) nD(V)w (here we are using the notation of Appendix 1). (J defines an isomorphism of GrD(V) onto 9'(V x V*). W x W acts on ~ x ~* as a finite group generated by reflections under (s, t)(v, A) = (sv, AO t- I), and W acts on ~ x ~* by the diagonal action (as above). We identify W with the subgroup {(s, s)ls E W} c W x W. Under these identifications GrD(V)w = 9'(V x V*)w. We set B = Gr.!B. Then B is a subalgebra of 9'(V x V*)w containing 9'(V)w, 9'(V*)w and closed under { ... , ... } (see the formula for the "top order symbol" of [D I ' D2]

in Appendix 1). Since (W, V) is good, B :J 9'(V x V*)w. Hence B = 9'(V x V*)W. So .!B = D(V)w.

Theorem 2.2. If ~ has no irreducible component isomorphic with the Weyl group of E 6 , E 7 , or E8 then (W, V) is good. In particular, the conclusion of Propo-sition 2.1 holds. Proof. We need only show that if (W, V) is irreducible and not of type E then (W, V) is nice. If (W, V) is not of type F4 this follows from Propositions 2 of Appendix 2 since (in the notation of that appendix) P f = - i- {u, f} with u = Lic;~, Pjf = -{qJi(c;) , f}. In the case of F4 the proof of ;:goodness" is somewhat complicated and will be given elsewhere.

For the rest of this section we will study the module theory of the algebra, .!B, generated by 9'(V)w and S(V)w. Let XI' ... ,xn be linear coordinates on ~ corresponding to an orthonormal basis. We define a conjugate linear anti-automorphism D I-+ D# of D(V) by Xi I-+ 0i' 0i I-+ Xi' If f, g E 9'(V) then we set (f, g) = g# f(O). ( ... , ... ) is an inner product on 9'(V). If DE D(V) then (Df, g) = (j, D#g).

Let n be the product of a system of positive roots for <1>. We observe that n2 E 9'(V)w .

Proposition 2.3. There exists ko such that n 2koD(V)w c.!B .

To prove this result we will introduce some notation and results that will be used throughout this paper. Chevalley's theorem implies that we can choose u l ' ... , un with ui homogeneous of degree di , di :S d i+1 such that det[oiu) = n . Let S + (V) denote the ideal of elements of S( V) , p , such that pi = O. Put S+(V)w = S+(V) n S(V)w. We set JIt? = JIt?(V) = {f E 9'(V)lpf = 0, P E


788 N. R. WALLACH

S+(V)W}. Then the map

9'(V)W ®Jr' -+ 9'(V)

given by f ® h 1-+ fh is a linear bijection. We note that the U j can be chosen such that U j are real valued on ~ and such that if pES + (V) Wand degp < d j then pUj = O.

Let W denote the set of equivalence classes of irreducible representation of ---- W W. If A E W, fix V;. EA. We look upon Homw(V;.' 9'(V» as a 9'(V)

module under the action fT(v)(x) = f(x)Tv(x). Then Homw(V;.' 9'(V» is a free 9'(V)w-module on generators Homw(V;.' Jr'). Furthermore, dim Homw(V;. , Jr') = dim V;..

It is easy to see that it is enough to prove Proposition 2.1 under the assump-tion that the action of W on V is irreducible. We define Tix;) = 0iUj. Then

j

(1) If D E D(V)w is of order (as a differential operator) at most 1 then DE!I.

Indeed, Dl E 9'(V)w c!l. Thus replacing D with D-Dl we may assume that Dl = O. This implies that D is completely determined by its restriction to V·. We therefore see that DW. = E j 'PjTj with 'Pj E 9'(V)w. Thus if we

2 define D j = Ej(ojuj)Oj then D = Ei 'PPi. We set d = EOj • Then

[d, uj ] = 2D j + dU i .

So D j E !I . Hence D E!I as was to be proved. (2) If DE D(V)w then there exists k such that 7C 2k DE!I . The proof of this assertion will take some preparation. If f is a polynomial

in indeterminates Y 1 ' ••• , Y n then

a of o~ af(u1 , ••• , un) = E a(u1 , ••• , Un)a·

Xi j Yj Xi

Set [aij] equal to the inverse to the matrix [Oju). Then 7Ca jj E 9'(V). Set - 2- W • OJ = E j aijoi' Then OJ = 7C OJ E D(V) IS of order 1. Hence OJ E !I. We note that

20f 0jf(u1 , ••• , un) = 7C a(U1 ' ••• , un)' Yj

Let X E ~ be such that 7C(X) i:- O. Let Ube an open connected neighborhood of X in ~R such that U1"'" un define local coordinates on U. If D is a differential operator on U with COO coefficients then we can write



If DE D(V)w then DC[u l , ••• , un] C qUI' '" , Un]' Thus ai ' ... 'i is a 1 n

polynomial. Thus if D E D( V) W has order k then there exist polynomials b. . in n indeterminates such that

't;···,11I

2kD ~ b ( )~il ~in n = L...J i ... i UI "'" un U I ... Un 1 •

on U and hence, on all of ~. This implies (2). If we use the Bernstein filtration on D(V) (see Appendix 1) then the corre-

sponding graded ring is .9 (V x V*). The subring corresponding to the induced grade of D(V)w is .9(V x V*)w relative to the diagonal action of W on V x V* . The Chevalley theorems applied to W x W imply that .9(V x V*)w is free as a .9(V x V*)wxw_module on generators (2(V) ® 2(V*))w. Let E be a filtered subspace of D( V) W such that the corresponding graded sub-space of .9(V x V*)w is (2(V) ® 2(V*))w. Then it is easily seen that as a (.9(V)w, S(V)w)-bimodule (.9(V)w acts by multiplication on the left and S(V)w acts by multiplication on the right), D(V)w is isomorphic under the obvious map (multiplication) with .9(V)w ® E ® S(V)w. Since dimE < 00,

(2) implies that there exists ko such that n2ko E c.£O . Hence n2koD(V)w c.£O . This completes the proof of Proposition 2.3.

If A E W, let .9(V)[A] denote the A-isotypic component of .9(V) with respect to the action of W. In Proposition 1.5 we have seen that for each A E W there exists an irreducible D( V) W -module, VA, such that as a (D( V) W, W)-bimodule,

.9(V) == EB VA ® V;. .. AEW

Furthermore, .9(V)[A] == VA ® V;.. The moduJe VA can be realized as follows. Let Z c .9(V) be a W-invariant irreducible subspace in the class of A. Let h E Z be nonzero. Then D( V) W h is an irreducible D( V) W module isomorphic with VA. Furthermore,

(3) D(V)wh n Z = Ch.

Theorem 2.4. If A E W then .£0 acts irreducibly on VA.

Proof. Fix Z as above and realize VA as D(V)w h = Iv. Let .9(V)[n;2) denote the subalgebra of the algebra of rational functions on V with denominator powers of n2k . That is, .9(V)[n;2] is the algebra of functions generated by XI' ••• , xn ' n-2 . Let N[n;2] be the subspace of .9(V)[n;2] given by qn-2]N.

W W Let D(V)[n;2] (resp. ~n;2)) be the algebra of operators generated by D(V)

(resp . .£0) and n-2 . Then it is easy to see that N[n;2) is a D(V)~2fsubmodule of .9(V)[n;2). Also one can see that if D E D(V)~2) then there exists k such that n2k D E D( V) W • Proposition 2.3 implies that

(i) D(V)~2) = ~n;2] , and


790 N. R. WALLACH

(ii) N[n2) is irreducible as a D(V)~2rmodule. W Indeed, let M c N[n2) be a nonzero D(V)[n2rsubmodule. Let m E M be

nonzero. Then there exists k such that n 2k mEN. Thus D( V) W n 2k m = N. Hence M = N[n2) .

Let M be a nonzero ~-submodule of N. If m/n2k E M[n2) and if D E ~ then D(m/n2k) = n-2rD'm with D' E D(V)w and some r. There exists p such that n 2p D' E ~. Hence D(m/n2k) E Min2). Thus M[n2) is a ~n2) =

D( V)~2rsubmodule of N[n2). So M[n2) = N[n2) . This implies (iii) If MeN is a nonzero ~-submodule and if v E N then there exists

k such that n 2k v EM. Let M be a nonzero ~ -submodule of N. (iv) As a 9'(V)w-module N (resp. M) is free on generators any basis of

N n:Jt' (resp. M n:Jt') . In light of the Chevalley theorems (described above) it is enough to show

that N = 9'(V)w (V n:Jt') (resp. M = 9'(V)w (M n :Jt')). Let B denote either N or M. Let 9'i (V) denote the space of homogeneous polynomials of degree j . Set Bi = B n 9'i (V). Since the Euler operator E x j8j E ~ (see (1) above), B is the direct sum of the Bi. Let jo be minimal such that Bio i- o.

W . . . If p E S+(V) then pBJo c Ei<' B J = O. Thus BJo c :Jt' n B. Assume that

Jo

we have shown that Bi c 9'(V)w (B n:Jt') for jo ::; j < k. If v E Bk and (v,9'(V)W(Bn:Jt')) =0 then for fE9'(V)w, f(O)=O we have

\V,f(LBi))=O. J<k

But, .t E S+(V)w and .tv E Ei<k Bi. Hence,

o = \ v , f ( L Bi) ) = \ f# V , L B i ). J<k J<k

So .tv = O. Since .t is a typical element of S + (V) w, v E :Jt' n B. Hence (v, v) =0. So v =0. Thus Bk c9'(V)w(Bn:Jt').

We are now ready to complete the proof of the theorem. Let hi' ... , hm be a basis of :Jt' n N such that hi' '" , hr is a basis of M n:Jt'. If v E N then v can be written uniquely in the form v = E 1;hj with 1; E 9' (V) W .

(iii) implies that there exists k such that n2kv EM; thus, (iv) implies that n 2kv = Ej<r gjh j with gj E 9'(V)w . Applying (iv) again we find that n 2k 1; = 0 for i > r. Hence v EM. So M = N .

Let AI denote the full subcategory of all ~-submodules, M, such that n2

acts injectively on M. ¥ W

Lemma 2.5. If M E./t then the map M -+ D( V) 0!§ M given by m 1--+ 10.'8 m is injective.



Proof. Suppose that m E M and that 1 ®,q] m = O. Then there exist bi E ~ ,

mj EM, and aij E ncV)W, i = 1, ... , r, j = 1, ... , s, such that

1 ® m = L {aijbi ® mj - aij ® bimj }. ij

Let k be so large that n2k aij E ~ for all i, j. We have 2k ~ 2k 2k n ® m = L.)n aijbi ® mj - n aij ® bimj }.

ij

Hence, n2k ®,q] m = 0 as an element of ~ ®,q] M. This implies that n2k m = 0 . Hence m = O.

Let ~w be the full subcategory of all finitely generated D( V) W modules, M, such that if m E M and PES + (V) W then there exists k such that pk m = 0 (i.e., each p E S+(V)w acts locally nilpotently). Let ~w denote

- W the full subcategory of all MEL such that each p E S+(V) acts locally nilpotently. We note that if M E ~w then D(V)w ®,q] M E ~w. We also note that any ~-submodule of 9'(V) is in ~w.

Theorem 2.6. If A E W then V'~. is an irreducible object in ~w.

(i) If A, /1 E Wand if VA is isomorphic with Vii as ~-modules then ..1.=/1.

(ii) If M E ~w then M is isomorphic with EBAEW mAVA.

Proof. We realize VA as above. Then VA E ~w. Theorem 2.2 implies that VA is irreducible as a ~ -module. Suppose that A, /1 E W, A i= /1 and that T is a ~ -module homomorphism from VA to Vii. Let h E VA, h i= 0, and assume that h is contained in an irreducible W-submodule of 9'(V)[A] and Th = u E Vii. As in the proof of Proposition 1.5 there exists D E D(V)w such that Dh = hand Du = O. Let k be such that n 2k D E ~. Then n 2k u = T(n2kDh) = n 2kDTh = n 2kDu = O. Thus u = O. Since VA is irreducible as a ~-module, this implies that T = O. This implies (i).

We now prove (ii). We have observed that N = D(V)w ®,q] M E ~w. The-orem 1.7 implies that, as a D(V)w-module, N = EBAEwnAVA. Lemma 2.3 implies that the map m t-> 1 ®,q] m defines an injective ~-module homomor-phism of Minto N. This implies (ii).

Let hI' ... ,hn be an orthonormal basis of V, and let h:, ... ,h~ be the dual basis. As in Appendix 1 we have an automorphism, ¢, of D( V) such that ¢(hj) = ih; and ¢(h;) = ihj . Then ¢(D(V)w) = D(V)w and ¢(~) = ~ . Let ~~ be the full subcategory of all finitely generated D( V) W -modules, M, such that if p E 9'+ (V) W then p acts locally nil potently on M. Let ~~ denote the full subcategory of all finitely generated ~ -modules, M, such that if p E 9'+CV)w then p acts locally nilpotently on M and a(n2) acts injectively


792 N. R. WALLACH

- ~ w on M. If M E ~w (resp. M E ~w), we define M to be the D(V) -module (resp. !H-module) with space M and action D· m = a(D)m. We define a functor from ~w to ~~ by M - Xi and if T is in Homw. (M, N) then w T is T as a linear map. Then M - Xi defines an equivalence of categories between ~w and ~~. We therefore have

Theorem 2.7. If.it E W then VA. is an irreducible object in ~~.

(i) If.it, J.l E Wand if VA. is isomorphic with Vll as !H-modules then .it = J.l.

(ii) If M E ~~ then M is isomorphic with EBA.EW mA. VA. .

3. INVARIANT POLYNOMIAL DIFFERENTIAL OPER.ATORS ON A REDUCTIVE LIE ALGEBR.A

Let g be a reductive Lie algebra over C. Let G denote the group of automor-phisms of g generated by those of the form eadx , x E g. We fix a nondegener-ate symmetric bilinear form, B, on g such that B([x, y], z) = -B(y, [x, z]) for x, y, z E g. We assume (as we may) that there exists a real form gu of g such that BI is negative definite. Let ~ be a Cartan subalgebra of g, and

9. let CI> = CI>(g,~) be the root system of g relative to ~. Let W denote the Weyl group of g with respect to ~. We recall Harish-Chandra's construction of a "radial component" map from differential operators on g to differential operators on ~' = {h E ~Ia(h) :f. 0, a E CI>}. For details in the construction the reader could refer to [RRGI, 7 .A.2].

Let V be a finite-dimensional vector space over C. Let U be an open subset of V. Let Dhol (U) denote the algebra of all differential operators on U with holomorphic coefficients. That is, if Xl' ... 'Xn are linear coordinates on U then D E Dhol (U) means that

D = L a/(x)a/ I/I::::;k

with k the order of D and a/ holomorphic on U and some a/ :f. 0 with III = k. As usual, we identify S(V) with the constant coefficient differential operators on V. If D is given as above and if x E U is fixed then we set Dx E S(V) equal to the constant coefficient differential operator with coeffi-cients a/(x).

If X E g then we define r(X) to be the vector field on g given by d -tadX

r(X)f(y) = dtf(e y)t=o.

Then r defines a Lie algebra homomorphism of g into D(g) , and hence r has a canonical extension to the universal enveloping algebra of g, U(g). We define a linear map

r: S(g) ® U(g) - D(g) by r(p®u) = pr(u). We set rx(p®u) = r(p®u)x for x E g. Let symm denote the usual symmetrization map of S(g) onto U(g). Let Z = {x E gIB(x, ~) =


DIFFERENTIAL OPERATORS ON A REDUCTIVE UE ALGEBRA 793

O} = [~, g) = EOE41 go where go is the root space corresponding to a. Let % = symmS(Z). Set

(S(~) ®%/ = L Sj(~) ® symm(Sj(Z)). j+j~k

If h E ~ then r h: (S(~) ®%)k _ Ej~kS/g) = Sk(g). Also,

hl-+r =A hl{s(~)®Z)k h,k

is a polynomial map from ~ to HomC«S(~) ® %)k , Sk(g)). Furthermore, if h E ~' then Ah , k is invertible and there exists a natural number mk such that h 1-+ 7C(h)mk(Ah k)-l is a polynomial map. Here we fix CI>+ , a system of positive roots for CI>; and set 7C = lloE41+ a .

Let Dhol,k(g) denote the space of elements of Dhol(g) of order at most k. Let 8 denote the homomorphism of U(g) to C with kernel U(g)g. Then following Harish-Chandra we define for, D E Dhol,k(g), r(D) E DhOI,k(~') to be the element defined by

-I r(D)h = (I ® 8)Ah,k(Dh).

Notice that there exists m k E N such that 7C mk r(D) E D(~). The definition of r implies that if DE D(g)G then r(D) E DhOI(~')w. We now recall some results of Harish-Chandra related to this construction.

G (1) If D 1 , D2 E Dhol(g) then r(D1D2) = r(DI)r(D2). Since 9 = ~ EEl Z , we have a direct sum decomposition g* = ~* EEl Z* . If p E

S(g) then we look upon p" as an element of 9'(g*). We set Res9/~ (p) = Plh •.

(2) If f E 9'(g)G then r(f) = fj~ . If p E S(g)G then r(p) = 7C-l(Res9/~p)7C. Let J = {D E D(g)G1r(D) = O} . (3) J = {D E D(g)G1D9'(g)G = O}. In the next section we will give a more direct characterization of J . Let .91

be the subalgebra of D(g)G generated by 9'(g)G and S(g)G.

Theorem 3.1. Assume that (W,~) is good (in the sense of§ 2). There exists a homomorphism, t5, from D(g)G to D(~)w with kert5 = J and t5(f) = fj~ for f E 9'.(g)G. t5(p) = Res9/~(p) for p E S(g)G. Furthermore the following sequence of algebra homomorphisms is exact:

O_J_D(g)G~D(~)w -0. Proof. (1) and (2) above imply that if we set t5(D) = 7C 0 r(D) 07C- 1 on .91 then t5 is an algebra homomorphism onto the sub algebra of D(~) generated by 9'(~)w and S(~)w. Since (W,~) is good, Proposition 2.1 implies that t5(.9I) = D(~)w. That t5 extends to a homomorphism of D(g)G to D(f))w is due to Harish-Chandra [H3). Corollary 3.2. If 9 has no simple ideals of type E then the conclusion of Theorem 3.1 is true for g.


194 N. R. WALLACH

This is a direct consequence of Corollary 2.2. We will say that 9 is "nice" if the conclusion of Theorem 3.1 is true for g.

Thus we know that 9 is nice if it contains no simple ideals of type E.

4. A CLOSER EXAMINATION OF THE IDEAL J We retain the notation of the previous section. If X E 9 then we write

det(adX + tI) = L tj dj(X). j

If dim ~ = I then dj = 0 for j < 1 and d/ is a nonzero polynomial function on g. The object of this section is to prove

Lemma 4.1. D E J if and only if there exists kEN such that d; D E (D(g)T(g)) n D(g)G .

We note that if DE D(g)G and there exists kEN such that d; D E D(g)T(g) then d; D.9(g)G = O. Thus D.9(g)G = O. So D E J. Before we begin the preparations for the proof of the converse assertion we first give a conjectural sharpening of the lemma.

Question. If J = (D(g)T(g)) n D(g)G?

We note that the answer to this question would be yes if one could show that the ideal in .9(g x g) generated by the polynomials X, Y 1-+ B(Z , [X, Y)), Z E g, is a prime ideal (it is a well-known result of Richardson that the corre-sponding affine algebraic set is irreducible).

We will now develop some material necessary for our proof of Lemma 4.1. Let qJI' ••• ,qJ/ (l = dim~) be algebraically independent generators of .9(g)G such that each qJj takes real values on gu. Let, for X E g, 'I'i(X) E 9 be defined by B('I'i(X) , Y) = dqJi (Y). Then 'I'i is a polynomial mapping of 9

x to 9 and

(1) [X, 'I'i(X)] = ['I'i(X) , 'I'j(X)] = 0 for all i, j :$ I. This is standard. Indeed, since qJj E .9(g)G ,it is easily seen that [X, 'I'/X)]

= 0 for all X E g. If X E gf = {Y E gld/(Y) =I O} then Cg(X) = {Y E

gl[X, Y] = O} is a Cartan subalgebra. Thus (1) is true for X E gf. Since 'I'j is a polynomial map, (1) is true for all X E 9 .

(2) If X E gf then '1'1 (X), ... , 'I'/(X) is a basis of Cg (X) . Indeed, if X E gf, we may choose ~ = Cg(X). Let XI' ••. ,xl be linear

coordinates on ~. If i is the canonical injection of ~ into 9 then i* (d qJI 1\ ... 1\ d qJ/) = cndxI 1\ ... 1\ dx/

with c =I O. This implies that'l'l (X) , ... , 'I'/(X) are linearly independent. (2) now follows from (1).

One can show that if X E 9 then dim Cg(X) = I if and only if '1'1 (X), ... , 'I'/(X) are linearly independent. This was first observed by Kostant.



(3) If X E g' then Cg(X).L = [X, g]. Indeed, if Cg(X) = fJ then Cg(X).L = Lo:E<f>9o:. Since [fJ, fJ] = 0 and

a(X) =I 0 for all a E fJ, (3) follows.

We regard g' as the affine variety V = {(x, t) E 9 x Cjd/(x)t = 1}. If X E 9 then [X, Xi] E Cg(X).L. Fix Xo E g', and let XI' ... , Xn be a

basis of Cg (Xo).L . Then [Xo ' Xd, ... , [Xo' Xn] is also a basis of Cg (Xo).L . We set

Ux = {X E g'I[X, Xd /\ ... /\ [X, Xn] =I o}. a

Then UXa is Zariski open in V. Put, for X E g, u/X) = [X, X). Then uj

is a polynomial map of 9 to 9 and uj(X) E Cg(X/ . Furthermore, if X E UXo then u l (X), ... , un(X) is a basis if Cg (X).L .

We now take a closer look at the maps Ah k • If X E g' then we define .L k k

AX,k: (8(Cg(X))0symm(8(Cg(X) )) -+8 (g)

to be the restriction of r X' If p E 8(g) , x E U(g) , and g E G then we set g(p 0 x) = gp 0 Ad(g)x.

We have r Ad(g)X(gu) = gr x(u).

If D E DhOI,k(g)G and if g E G then g. DAd(g)X = Dx' If X E UXa then there exists g E G such that Ad(g)Cg(X) = fJ. Thus if DE Dh01,k(g)G and if reD) = 0 then

-I (10 e)(Ax k(Dx)) = 0

] . . J . for all X E g'. We set l{I(X) = l{Il (X)ll ... l{I,(X) 1/ and u(X) = u l (X)}l ... un(X)jn. Let YI , ... , Ym be a basis of 9 (m = n+l), and set y N = yt1 ••• y;m in 8(g). Then if III + IJI ::; k, we have

AX,k(l{I(X/ 0 symm(u(X)J)) = L: aN,],J(X)yN INlg

with aN ] J a polynomial in X. Let ZI' ... ,Zm be a basis of g, and set ZN = Z;l : .. Z:m in U(g). Then these observations imply that if DE Dk(g)G and if reD) = 0 then

(1) A;:k(Dx ) = L: bp ,Q(X)YP 0 zQ IPI+IQI~k

with bp,Q a rational function on X defined for X E UXa and bp,Q = 0 if Q = O. Since Xo E V is arbitrary, the expression (1) is valid for all X E V with bp , Q a regular function on V. Thus

Dx = L: bp,Q(X)YP r(ZQ) 1P1+IQI~k

IQI>O


796 N. R. WALLACH

with bp,Q a regular function on V. The ring of regular functions on V is .9'(g)[d[-I]. Let kEN be such that d; bp ,Q E .9'(g). Then d; DE D(g)r(g). Hence

k G d[ D E D(g)r(g) n D(g) . This completes the proof of the lemma.

5. A THEOREM OF HARISH-CHANDRA

Let go be a reductive Lie algebra over R. Let 9 denote the complexification of go. We assume throughout this section that 9 is nice (see the end of §3). We choose B such that there is a real form gu of 9 such that Big. is negative definite and such that B(go' go) cR. This clearly can be done. We denote by G u the subgroup of G generated by {ead x IX E gu}. Let .9' (go) be the usual Schwartz space of go with the usual Frechet space topology. If f E .9'(go) then we use B to define ~(f) = !T(f) == J E .9'(go) as in Appendix 1. Let 1> be defined as in Appendix 1 for an orthonormal basis of 9 with respect to B. We will also write 1>(D) = D. We note (as in Appendix 1) that !T(Df) = D!T(f) for DE D(g) and f E .9'(go).

Let J be as in the previous sections.

Lemma 5.1. If D E J then D E J . Proof. We first observe that if .9'(gu)G. = {f E .9'(gu)lf(Ad(u)x) = f(x) , u E G u' x E gu} then

(1) If DE D(g)G then DE J if and only if D.9'(gu)Gu = o. Indeed, Lemma 4.1 implies that D.9'(gJG• = 0 if D E J. If D.9'(gu)G. = 0

and if f E .9'(g)G, x E gu let rp E C;'(gu) n .9'(gu)G. be such that rp is identically equal to 1 in a neighborhood of x in gu. Then 0 = (D(rpf))(x) = Df(x). Thus DE J «3) in §3).

G ~ , Now if DE J then !T(Df) = 0 for all f E .9'(gu) u. Thus Df = 0 for

all f E .9'(gu)G • . Since !T is a linear bijection of .9'(gu)G. onto itself, we see ~ G ~

that D.9'(gu) • = o. Thus (1) implies that DE J .

Lemma 5.2. D E J if and only if there exists kEN such that

1>(d;)D E (D(g)r(g)) n D(g)G.

Proof. We note that if X E 9 then (r(X)) = -r(X). Thus G~ G

«D(g)r(g)) n D(g) ) = (D(g)r(g)) n D(g) .

Let D E J , and let DI E J be such that DI = D. Let kEN be such that k G d[ DI E (D(g)r(g)) n D(g) .

Then 'k 'k ~ k ~ G d[ D = d[ DI = (d[ D1) E (D(g)r(g)) n D(g) .

The converse is proved in the same way.



Let GO be the subgroup of G generated by {eadXlx Ego}. If 0 is a subset of go' it will be called invariant if gO c 0 for all g E Go. If 0 is an open invariant subset of go then 0 will be called completely invariant if, for each x EO, Xs EO. Here if X E go then Xs ' XnE go are uniquely determined by the conditions that [Xs ' Xn] = 0, ad Xs is diagonalizable on g, Xn E [go' go]' and ad(Xn) is nilpotent. Set .IY = {X E golX = X n}. Then .IY = {X E golf(X) = f(O) , f E 9'(g)G} (cf. [RRGI,8.A.4.2]).

If Q is an invariant open subset of go and if T E 9' (Q) (distributions on Q) then we define gT(f) = T(g -I f) (gf(x) = f(g -I x)). We set 9' (Q)Go = {T E 9'(Q)lgT = T, g E Go} We now recall a basic theorem of Harish-Chandra ([H2], cf. [RRG1, Theorem 8.3.5].

Theorem 5.3. Let Q be an open completely invariant subset of go' and assume that T E 9' (O)Go is such that dimS(g)G T < 00. If 1Inngl = 0 then T = O.

We will now give a new proof of the following theorem of Harish-Chandra. We note that an affirmative answer to the question in the last section would directly imply the theorem. If 9 is nice our proof is independent of Theorem 5.3 and for such go we show how the next result (combined with our theory) implies Theorem 5.3.

Theorem 5.4. If 0 is an open, nonempty, completely invariant subset of go then

J = {D E D(g)G1D9' (Q)Go = O}.

The first part of our proof follows the same line of the original argument of Harish-Chandra (we will refer to the exposition in Varadarajan [Var]). The proof is by induction on dimg. If dimg < 3 then J = {O}, so the result is clear. Assume the result for 2 :::; dim 9 < r. We look at the case dim 9 = r. We refer to [Var, pp. 149-150] for the reduction of the inductive step to the case when 9 is semisimple and if DE J, suppDT c.IY. Thus Lemma 2 in Appendix 3 implies that DT extends to a tempered distribution on go. Since suppDT c.IY, if f E 9'+(g)G then there exists kEN such that /' DT = O. Applying the Fourier transform, this implies that if p E S+(g)G then there exists

k ~

kEN such that p (DT) = 0 . We now assume that 9 is nice. Let DI E J; then

dimS(g)G DI (DT) < 00.

Since suppD1 (DT) c.IY, we also have

dim9'(g)G DI (DT) < 00.

We consider the operators e = !B( ... , ... ), f = ¢(e) , h = [e, f1 = 'EX/J/oXi + m/2 (here Xi is any system of linear coordinates for go). Then {e, f, h} span a Lie algebra, u, over C isomorphic with 51(2, C). The above implies that dim U(u)DI (DT) < 00. Thus Lemma 8.3.7 in [RRGI] implies that DI (DT) = O. Since Y = J , this implies that J DT = O. Hence D(g)G DT


798 N. R. WALLACH

pushes down (via (5) to a D(~)w-module, N, with the property that if n E N then there exists kEN such that 9'k(~)w n = O. On the other hand, there exists j E N such that 4>(di D E D(g)-r(g) n D(g)G. Thus 4>(d,)j (DT) = O. Hence if no is the element of N corresponding to DT then t5(4)(di)no = O. Corollary 1.4 now implies that no = O. Thus DT = O. This completes the proof if 9 is nice.

We now give the genera\ argument (which uses Theorem 5.3). Lemma 5.2 im-plies thatthere exists kEN such that 4>(d,)kD E D(g)G-r(g). Thus 4>(d,)kDT = O. Hence dtY'(DT) = O. Set S = Y'(DT). Then the observations at the be-ginning of the proof imply that if p E S+(g)G then there exists r such that p'S = O. Since d; S = 0, Slnng' = O. Theorem 5.3 implies that S = O. Hence DT = 0, as was to be proved.

We note that the above result implies that if 0 is an open completely in-variant subset of go then the D(g)G-module .91'(O)Go pushes down, via t5, to a D(~) w -module if 9 is nice. In general it pushes down to a 99 -module.

We now show how one can use Theorem 5.4 to prove Theorem 5.3 in the case when 9 is nice. So assume that 9 is nice. Set g~ = go n g' . Assume that o is open and completely invariant in go and that 1Inng: = O. We show that T=O.

Let Vj be an increasing sequence of open subsets of 0 such that 0 is the union of the Vj and that V j is compact and contained in O. Then 11 uj is of finite order with supp 1Iu C Vj n (go - g~). Hence, for each j there exists

J

kj such that d:j 1luj = 0, kl ~ k2 ~ .... We consider D(g)G T to be the

D(~)w -module, M, via t5 . Let mE M correspond to T. Then we have a de-creasing sequence of D(~)w-submodules, D(~)w 1l2km:J D(~)w 1l2k+2m. Since dimS(~)w m < 00, Proposition 1.3 applies, so M has finite length. Hence there exists k such that

D(~)w 1l2k m = D(~)w 1l2P m

for p ;::: k. Now this implies that if p ;::: k then d;T = DpdfT for an appropriate Dp E D(g)G. We therefore see that if kj ;::: k then

k dk d,1Iu = Dk /1Iu = o. J J J

Since the union of the Vj is 0, we have d; T = O. If we revert to M, we have 1l2k m = O. Hence Proposition 1.3 implies m = O. Thus T = 0 .

Let 9J~(go)Go denote the space of all T E .91' (go)Go such that supp T c ,AI" n go' We now display an implication of the proof of Theorem 5.4.

Proposition S.S. Let T E 9J~(go)Go; then, as a 99-module, .N T E C:V. Proof. In the course of the proof we showed that if f E 9'+(g)G then facts locally nilpotentiy on 9J~(go)Go. It also follows (as in the end of the proof of



Theorem 5.4) that if T E 9~(go)Go and if ¢(d[)T = 0 then T = O. Since J(d[) = 7[2, the result now follows from the definition of ~W'

6. DISTRIBUTIONS SUPPORTED ON THE NILPOTENT CONE AND WEYL GROUP REPRESENTATIONS

We will use the notation and conventions of §5 in this section except that we will take go to be semisimple and B to be the Killing form. Let ,AI be the cone of all nilpotent X E go (i.e., ad X is nilpotent). If X E,AI then there exist Y, H E go such that [X, Y] = H, [H, X] = 2X, [H, Y] = -2Y. Following Kostant we will call {X, Y, H} an s-triple containing X. Thus &x = GoX is a cone with tgX = geIOg(t)adH/2 X for t E R, t > O. If X E go then we set m(X) = dim&xI2.

If S is a closed subset of go' we denote by 9?;(go) the space of all distri-butions on go supported on S. If S is Go-invariant then we set 9;(go)Go =

, ,G 'G - , 9?s(go)n9? (go) o. If T E 9?';y(go) 0 then, as a g-module (via J), .91'T E ~w (see §2 and Proposition 5.5).

Theorem 6.1. As a g·module 9?~(go)Go is equivalent with a direct sum EBAEW mAVA with m A < 00 (notation as in § 1.2).

Proof. If we show that 9~(go)Go is finitely generated as a g-module then the above observation implies that 9?~(go)Go E ~~. The theorem would then follow from Theorem 2.7. We are thus left with the proof of finite generation.

Let XI' ... ,Xm be a basis of g such that B(Xi '. X) = Jij . Let XI' ... , xm be the corresponding linear coordinates on g. Set r: = L:i x;, L\ = L:i X; =

L:i ai2 , and Eo = L:i xiai · Let HI ' ... , H[ be a basis of ~ such that B(Hi' H) = J jj , let Yi be the corresponding linear coordinates on ~,and set r~ = L:i Y; ,

2 2 2 " ~ 2 2 ~ .10 = L:i H j = L:i a laYi ' and E~ = L..Ji yia laYi' Then u(ro) = r~, u(.19 ) = .1~ . Thus

Hence l-m

J(Eo) = E~ + -2-'

If T E 9?~(go)Go then .91' T E t:V. Thus, in particular, Eo acts semisimply on 9~(go)Go. In light of Theorem 2.7, to prove the finite generation of 9~(go)Go it is enough to prove that the dimension of each eigenspace for Eo in 9~(go)Go is finite dimensional. In order to prove this, we set up some notation and a lemma that will be used later in this section. In light of the semisimplicity of Eo on 9~(go)Go the lemma is essentially the same as [BVl, Corollary 3.9]. However, since it is critical to our application, we will include the (not too difficult) proof.


800 N. R. WALLACH

If X E 9 then set gX = ker ad X. If X E go then g; = gX n go is the Lie algebra of G; = {g E GoIgX = X}. Clearly, m(X) = t(dimg - dimgx). Let &; = GoXj , i = 1, ... , r, be the distinct orbits of Go in ./Y. We assume that dimgXj :::; dimgXj+l • Then &,. = {O} and &; c &; U (UdimgXj>dimgXj &j). Set ~ = Uj~p&; (cf. [RRGl, 8.3]).

The first sentence of the following lemma will complete the proof of the theorem.

Lemma 6.2 (compare [BYl, Corollary 3.9]). Eg acts semisimply on 9J~(go)Go p

with finite-dimensional eigenspaces and each eigenvalue of Eg on this space is at most -! (dim 9 + dim gXp). If X E ./Y then if J.l is an eigenvalue of E 9 on 9J ~)go) Go then J.l :::; -! (dim 9 + dim gX) and the eigenspace for -!(dimg + dimgx) has dimension at most 1. Proof. That Eg acts semisimply has already been observed. We prove the first assertion by downward induction on p. If p = r then Xp = {O}. Thus 9J~(go)Go = S(g)G t50 with t50 the Dirac delta function at O. Since Egt50 = -(dl.mg)t5o ' the result follows in this case. Assume the result for p > q; we now prove it for p = q. Let X = Xq . Let {X, Y, H} be an s-triple containing X . Set V = g~ . As in [RRGI, 8.3.6, p. 299] we choose U an open neighborhood of 0 in V so that if cl>(g, Z) = g(X + Z), g E Go' Z E U, then cl> is a submersion of G x U onto an open subset Q of go such that:

(i) Q n ~ = &p , and (ii) (X + U) n ~ = {X} .

Since cl> is a submersion, we may define, for T E 9J~(go)Go, cl>°(T) E p

9J'(U) with suppcl>°(T) = {O} (see [RRGI, p. 301]). Let -J.ll' ... , -J.ld be the eigenvalues of ad H on V counting multiplicities. Then we can choose linear coordinates on V, Y 1 ' ••• , Y d such that

cl>°(EgT) = (L (~J.lj + 1) Yj8~J cl>°(T).

We note that 'E(J.lj + 1) = dimg and that dim V = dimgX by TDS theory. We also note that cl>°(T) E S(Vdt5v,o = 9J (t5v ,o the Dirac delta for V sup-ported at 0). The eigenvalues of 'E(!J.lj + l)y j8/8Yj on 9J are of the form - 'E(!J.lj + l)aj with aj > 0, each has finite multiplicity and the eigenspace for - 'E(!J.lj + 1) is Ct5v ,o' Now the observations using TDS theory imply that

L(!J.lj+ 1) = !(dimg+dimgx ).

If cl>°(T) = 0 then T E 9J:/ (g )Go. The first assertion of the lemma now o/'p+1 °

follows. As for the second, the proof of the first part implies that if T E 9J '.,. (g )Go is an eigenvector for E with eigenvalue - -21 (dim 9 + dim gX) then

~x ° 9



~o (T) E Cc5v . 0 and if ~o (T) = 0 then T = O. This proves the second assertion of the lemma.

If X E go then we recall that there is a canonical Go invariant measure (the Kostant-Kirillov measure) on &x given as follows. If y E &x and if U, v E T(&x)y with U = [y, U], v = [y, V], U, V E go' then set wy(u, v) = B(y, [U, V]). Then w defines a symplectic structure on &x and IIx = wm(X) defines a volume form on &x. In [RR] it has been shown that if X E,AI" and f E Cc(go) then

r fllX = Tx(f) l~x

is defined by an absolutely convergent integral and that T x defines a Go-

invariant Radon measure on go. Thus Tx E 9~(go)Go. Clearly, supp Tx c &x. Theorem 6.1 implies that as a D(~)w-module we have

D(g)G Tx = EB mx(A)V).. ).EW

We have therefore assigned to each X E,AI" a function m x: W -+ N. If A E W then set j(A) = min{jl Homw(J';', 2j) =I O}. Put j(X) = min{j(A)lmx(A) =I OJ. Theorem 6.3. If X E ,AI" then j (X) = ! (dim gX - l). Furtherm~re, {A. E

Wlmx(A) > O} consists of one element, AX with mx(Ax) = 1, and

dim HomwU;'x ,2j(X)) = 1.

This result will take some preparation. We note that there are choices of invariant measures on Go and G: = {g E GlgX = X} such that if f E ~(go) then

Tx(f) = 1 f(gX)d(gG:). Go/G;

If hE Go and if hX = cX for some c E R then

r f(ghX)d(gG:)=det(h l x)-I r f(gX)d(gG:) lG /Gx go lG /Gx

o 0 0 0

for f E Cc(go) . As above, we will look upon the N-module NTx as a g-module via c5.

Lemma 6.4. Let XE,AI". Then E"Tx=-!(dimg: +1)Tx.

Proof. We note that EgTx(f) = Tx(e~f) = -mTx(f) - Tx(Egf). If X = 0 then To is the Dirac delta function at O. Thus To(Egf) = O. Thus in this case EgTO = -mTo . Otherwise, let {X, Y, H} be an s-triple containing X. Then

f(ge(logt)ad H/2 X) = f(tgX)


802 N. R. WALLACH

for g E Go' t > O. Thus

E f(gX) =!!:.- f(ge(IOgt)ad H/2 X ). g dtlt=l

Hence, if we apply the observation above, we have

EgTX = (~tr ad H1gx - m) Tx-

Let gX = EB g; with g; the A eigenspace for ad H in gX. Then standard TDS theory implies that

L dim(g;)(A + I) = dim g. J.

Hence tr ad H1gx = dimg - dimgX

We have thus shown that

(*) EgTX = -~(dimg+dimgX)Tx-This formula is valid if X = 0 .

We note as above that f5(Eg + !f) = E~ + i. Since m = dimg, the lemma follows.

We will now give another interpretation of the D(~)-module, S(~). Let f50 E g;(~)* be defined by f50(f) = f(O). If D E D(~) and if A E g;(~)*

then set DA = A 0 DT. Then our module structure on S(g) is just the module D(~)f5o .

Lemma 6.5. If A E W then the highest eigenvalue of E~ on VJ. is -I - j(A) and it has multiplicity dim Homw(J-i '~(J.))'

Proof. We realize VJ. as tBpf50 with p E ¢(~(J.)[A]). We note that

E~pf5o = (-I - j(A))pf5o'

The lemma now follows from Theorems 1.6 and 2.7.

Lemma 6.6. If X E ,AI' then the highest eigenvalue of Eg on Sf'Tx is - t (dim 9 + dim gX) and it occurs with multiplicity I. Proof. In light of (*) in the proof of Lemma 6.4, the eigenspace for Eg with eigenvalue - t (dim 9 + dim gX) has positive dimension. The result now follows from the second assertion of Lemma 6.2.

Proof of Theorem 6.3. As a tB-module, M = Sf'Tx is isomorphic with EBJ.EWmX(A)VJ.. Thus the highest eigenvalue of E~ on M is -1- j(X) with multiplicity equal to

a = L dim Homw(J-i '~(x))mx(A). J.EW j(J.)=j(X)


DIFFERENTIAL OPERATORS ON A REDUCTIVE LIE AWEBRA 803

Lemma 6.3 in combination with Lemma 6.4 implies that -/ - j(X) = _!(dimgX + /) and that a = 1. Thus j(X) = !(dimgX - /) and, since a = 1, {A. E Wlmx(A.) > 0, j(A.) = j(X)} consists of one element AX with mx(Ax ) = 1 and dim Homw(J'J.x ,~(X)) = 1. On the other hand, M is gen-erated by one element in the -/ - j(X) eigenspace of M. Since M splits into a direct sum of modules isomorphic of the form VA and mx(A) = 0 for j(A.) < j(X) , mx(A) = 0 if j(A) = j(X) and A =f. AX' it follows that M ~ VAx. This completes the proof of the theorem.

Since &x is a cone for X E ,AI, Tx is a homogeneous distribution on go' Hence Tx is tempered. We will now analyze Tx. Choose £) such that

G~ ~

£)0 = £) n go is a real form of £). We note that 8+(g) Tx = O. Thus Tx I is Isons

a real analytic function which completely determines Tx (cf. Theorem 5.3).

Theorem 6.7. Let X E ,AI . If C is a connected component of £)0 n £)' then there exists he E .:fj(X) [Axl such that Tx,c = he/n.

Proof. If f E .9'(£)0) then we define

c;z- A 1 ( -iB(y x) J f(x) = f(x) = (2n)I/2 1f)0 f(y)e ' dy.

Here dy is the Lebesgue measure on £)0 corresponding to a pseudo-orthonormal basis of £)0' If T E .9"(£)0) and if f E ~/£)) (resp. p E 8j (£))) then (fT) =

¢>(f)i (resp. (pt) = ¢>(p)i, where ¢> is defined as in Appendix 1 corresponding to an orthonormal basis of £). Thus we have

(i) o(D) = ¢>(o(D)) . We also note that if M is the ~-module VA with action given by D· m =

Dm then M ~ VA . With these observations in hand we can prove the theorem. Set T = Tx' As a ~-module NT is isomorphic with vAx. Let C be as

in the statement. If DEN then r(D)~e = n-IO(D)n~e' Since ~+(g)GT = G~

0, 8+ (g) T = O. Thus w ~ w ~

nr(8+(g) )1Ie = 8+(£)) n1le = O.

This implies that n~e = he with he E Jt'. Since Ef) = -Ef) + / , we see that Ef)He = j(X)he' The above observations imply that D(£))w he ~ VAx as a D(£))w-module. Hence he E .:fj(x)[A.xl. This completes the proof.

We now look at a special case of these results. Let gl be a semisimple Lie algebra over C. Let go denote gl as a Lie algebra over R. Let ul be a compact form of gl . Let X denote complex conjugation of X E gl with respect to ul Then we identify go with the subalgebra {(X, X) E gl x gliX E gl} of gl x gl . If £)1 is a Cartan subalgebra of gl then £)0 = {(X, X)IX E £)I} is a Cartan subalgebra of go' With these identifications 9 = gl X gl and £) = £)1 X £)1 .


804 N. R. WALLACH

W(9, ~) = W(91' ~I) X W(91' ~I)' We set W = W(9, ~), »'t = W(91' ~I)· Then {(s, s)ls E WI} = {s E WI there exists g E Go such that gl~ = s}. If A, J.l E ~ then we denote by A®J.l their exterior tensor product as an element of W. We fix a system of positive roots, PI' for the roots of 91 with respect to ~I • Then P = {(o, 0)10 E PI}U {(O, 0)10 E PI} is a system of positive roots for 9 with respect to ~. This implies that if 1t1 = TIaEplo and 1t = IIaEpo

then 1t«X, X» = 1t1 (X)1t1 (X).

We note that .w'(9) = .w'(91) ®.w'(91) and g(9, h) = g(91' ~I) ®g(91' ~I)· Thus the irreducible g (9, ~ )-module corresponding to A ® J.l in the category C' -;'-p. wisV®V.

Theorem 6.S. If X E 91 is nilpotent then there exists axE WI such that A(X,X) = ax®ax (here o®P(SI' S2) = o(SI) ® P(S2» and j(ax ) = !(dim 9; - dim ~I). Furthermore, if XI' X2 are nilpotent elements of 91 and if ax = ax then GIXI = GIX2·

I 2

Note. The above result implies that for a semisimple Lie algebra over C, we have constructed an injective map from the set of nilpotent orbits of the adjoint group into the set of equivalence classes of irreducible representations of its Weyl group. At the end of this section we will use the results of [BV2, BV3, HK] to show that this correspondence is the Springer correspondence [S].

Proof· Since ~o - (~o n ~') is of (real) codimension 2 in ~o' we see that ~o n ~' is connected. Set T = T(X,X) and h~on~' = hx . If we had chosen a different Cartan subalgebra then it would be of the form g~o with g E Go. Since T (hence T) is Go-invariant, the corresponding" h" would be given by h(gH) = hx(H) for H E ~o. Also by the Go-invariance of T and the ~-invariance of 1t , we see that w h x = h x for w E ~. This implies that (II). ) wo =f. O. Schur's

x Lemma, the fact that every irreducible representation of »'t is defined over R, and Theorem 6.3 imply that:

(i) AX = ax ® ax' and (ii) dim(II).)w" = dim(~(x,X)[Ax])wo = 1.

This implies that if XI' X2 are nilpotent elements of 91 and if A(XI ,XI) =

A(X2 ,X2) then there exists c E C such that 1(XI'XI) = C1(X2 ,X2) on 9~ and, hence, on 90 by Theorem 5.5. This implies that T(XI'XI) = c1(X2'X2). Since 19(XI ,XI ) is the unique open orbit in its closure, this completes the proof.

We now return to the general case. The main result in this section is

Theorem 6.9. Let X E./Y. Ifwe look upon X as a nilpotent element of 9 then Ax=ax · Proof. We note that GoX is open in GXn90. We look upon GX as a complex submanifold of 9. By its very definition, lIx extends to a holomorphic 2m(X) form on GX. We will use the same symbol for this extension. Then up to



a scalar multiple, lI(X,X) = lIx 1\ vx · Let Yx = {f E 9'(g)lf(&x) O}. If x E &x then there exists an open neighborhood, no of x in go' and It ' ... , f m- 2m(x) E Yx that are real valued on no and such that no n &x = {y E nolf;(y) = O}. We may also assume that there is an open neighborhood, n, of GX in 9 such that n n go = no and n n GX = {y E'nlf;(y) = O}. If we shrink 0, we may assume that 0 n GX is connected and there are local holomorphic coordinates Xl' ••• ' Xm on n such that Xj(x) = 0 and x j+2m(X) = f; for i ~ 1 and that Xl' ••• ' Xm restricted to no are (real) local coordinates on no. Furthermore, may assume that these coordinates on no satisfy the condition of Lemma 3 in Appendix 3 for M = &x and that after reordering {RexI' ... , Rexm , ImxI ' ... ,Imxm} satisfy the conditions of Lemma 3 in Appendix 3 for M = GX. In the notation of Lemma 3 in Appendix 3, if rto is the" rt" for (J) = lIx then rto extends to a holomorphic function on GX n n and the" rt" for lI(X ,X) is rtotlo .

If D E D(g)G then we can think of D as a holomorphic differential operator on 9 and then we denote it by D® 1 . We can think of D as an antiholomorphic differential operator on 9 and then we denote it by 1 ® D. Then D(g x g)GXG is just D(g)G ® (D(g)G)- . Also the corresponding".9f " is .9f ®.9f . Set .fx = {D E D(g)G1DTx = O}; then Theorem 5.4 implies that .fx ::> J. The local criterion in Lemma 3 in Appendix 3 implies that if D E .fx and if (in the notation of Appendix 3)

T ~ I J D = L..tal,iJ a

I,J then for each J we have (1) ~ III I 0 L..t(-I) a (rtoal,J) =

I

on 0 0 n & X. Since all the terms extend to holomorphic functions on 0 n G X , (1) is true on 0 n GX. Now the local condition that (D ® J)T = 0 is that for each J

L(-I)l/lal(rtol1oal,J) = 0 I .

on n n GX with al looked upon as partial derivatives from the holomorphic tangent space. Thus

L(-I)l/lal(rtotloal,J) = tloL(-1)l/lal(rtoal,J) = O. I I

Lemma 3 in Appendix 3 implies that (D ® 1) 1(x ,X) = O. Similarly (1 ® D) T(X ,X) = O. Thus the cyclic .9f ®.9f -module generated by 1(x, X) is a quotient of (~/.fx) ® (~ /.fx). But as a g-module, .9f /.fx is isomor-phic with VAX (Theorem 3). This implies that Vax ® Vax is isomorphic as a

-- ~A ~A g®g-module with a quotient of V x ® V x . Thus Ax®Ax = (Jx®(Jx. Hence Ax = (Jx·

We now close the circle of ideas that we have developed in this section by quoting an important theorem of [BV2; BV3; HK, Theorem 8.2, p. 357].


806 N. R. WALLACH

Theorem 6.10. If 9 is a semisimple Lie algebra over C then the correspondence &'X f-t a x is the Springer correspondence.

Proof. In [HK] it is shown that (1(x ,X))I~I = h/(1C 0 it) and that h is a (W x W)-harmonic polynomial whose cyclic space under W x W is in the class of A 0 A with A the Springer representation associated with X. Theorem 6.7 now implies the result.

We conclude this section with a complete description of g-.:r(g)Ga as a s(-module in the special case when go is a semisimple Lie algebra over C looked upon as a Lie algebra over R. We use the notation and conventions established above for this special case. Theorem 6.11. As a !B(g, ~)-module g-.:r(go)Go is isomorphic with

E9 V"0 V". "EW,

Proof. Theorem 6.1 implies that as a !B -module , G ffi -" ~11 g-.#'(go) 0 ~ W m",11 V 0 V .

",I1EW,

Let M be an irreducible nonzero s( -submodule of g-.:r(go)Ga • Then since 9'+(g)G acts locally nilpotently on M, there exists T =1= 0, T E M such that 9'+(g)w T = O. Thus S+(g)Gr = O. This implies that there exists h E ,;r'(~I) 0,;r'(~1) such that

r(X, X) = heX , X)/1C I (X)1C I (X) for X E ~o n g'. Furthermore, !B h is irreducible and in ~w. Thus, !B h ~ " ~ V 0 VI1 for some A, f.l E WI' Hence, h E 9'(~)[A 0 f.l]. But, if s E WI then r(sX, sX) = r(X, X) and 1C I (SX)1C I (sX) = 1C(X)1C(X) for all x E ~I' Thus relative to the diagonal action of ~ on TJ. 0 VI1 there is a nonzero fixed vector. This implies that A = f.l. Let Wo be the diagonal subgroup of WI x WI . Then we have shown that 1C.M;~angl is an irreducible !B-submodule of 9'(~)[A0A]wa . The results of § 1 easily imply that 9'(~)[A 0A]wa is irreducible as a !B-module. We therefore see that m",11 :::; 1 and m",11 = 0 if A =1= f.l.

To prove that m" " = 1, we will make use of Harish-Chandra's theory of orbital integrals. Let 'n l denote the sum of the root spaces in gl corresponding

- h to a E PI' Set n = {(X, X)IX E n l }. Let, for f E 5"'(go) , <1>; be as in [RRG1, 7.3.6]. There is a choice of Lebesgue measure on n such that (see [RRG1, 7.3.8(4)]) if H E ~o n g' then

<1>ja(H) = 1 ;(H +X)dX

where if K is the maximal compact subgroup of Go corresponding to u and if dk is normalized invariant measure on K then

leX) = tf(kX)dk.



This implies that if 1 E .9'(go) then <l>~ E .9'(~o) . If h E K(~)w" n9'(~)[A.®A.] then we set

Sh (I) = 1 h(X, X)<l>~ (X) dX. ~o

Here we choose the Lebesgue measure on ~o corresponding to a pseudo-ortho-normal basis relative to the form B. If p E .9'(g)G then

<l>!j = c5(p)<l>~.

Thus if DT is the formal adjoint of D E D(~o) then Sh(Pf) = SJ(p)Th(/) for P E S(g)G. This implies that if we set Th(f) = Sh(j) then 9'+(g)G Th = o. Hence supp Th c./Y n go .

Since h(sX, sX) = h(X, X) for X E ~I ' we can define a COO function 'Ph on gong' by 'Ph(gH) = h(H) for H E ~ong'. Then noting that if II = dim~1 then 1=211 and d/(X) ~ 0 for X E go' we have

1 'Ph (X) A

Th(/) = 1/2 / (X) dX. gong' d2/(X)

Thus Th E ~;"'(go)Go. Now Th(f) = Th(.9T- I I) = Sh(.9T(.9T- I f)) = Sh(f). So Th = Sh and

S (X) _ h(X) h - n(X)n(X)

for X E ~o n g'. In light of the material at the b~ginning of the proof of this theorem, it follows that .SiI Th is isomorphic with VA. ® VA. when looked upon as a B-module.

ApPENDIX 1. POLYNOMIAL DIFFERENTIAL OPERATORS

The purpose of this appendix is to compile basic theory of the Weyl algebra that will be used in the body of the paper. Let V be an n-dimensional vector space over C, and let V* denote the dual space. Let 9'(V) denote the algebra of all polynomials on V, and let S(V) the symmetric,algebra on V. 9'k(V) (resp. Sk(V)) will denote the space of elements in 9'(V) (resp. S(V)) homo-

k k geneous of degree k. We set 9'+(V) = Ek>o9' (V), S+(V) = Ek>oS (V). We also use the notation D( V) for the algebra of differential operators on V with polynomial coefficients. If v E V then we look upon v as a differen-tial operator on V using the operation vl(x) = 8vl(x) = ftl(x + tV)lt=o for 1 E 9'(V). The corresponding map v 1--+ 8v induces an injective algebra homo-morphism of S(V) into D(V) with image the algebra of constant coefficient differential operators on V. Thus if U E S(V), we will look upon U as a con-stant coefficient differential operator on V. If 1 E 9'(V) then we look upon 1 as a differential operator on V relative to polynomial multiplication. With these identifications the map 9'(V) ® S(V) to D(V) given by 1 ® U 1--+ Iu


808 N. R. WALLACH

defines a linear bijection. If VI' ••• 'Vn is a basis of V and if Xl' ... Xn is the corresponding dual basis then we have identified Vj with a~ .

We use standard multi-index notation. If I = (iI' ... ,'in), i j E N (the nonnegative integers), then we set III = i l + ... + in' We write Xl = X;I ... x!n and

I alII a =. . . axil ... ax'n

1 n

We will also write aj = a~;' If D E D(V) then D = EIII+IJI9 aI ,JXI a J for some kEN and aI, J E C. The minimum of such k is called the Bernstein degree of D. We set Dk (V) equal to space of all D with Bernstein degree less than or equal to k. With this filtration D( V) is a filtered algebra with corresponding graded algebra, Gr D( V) , isomorphic with .9 (V x V*) = .9 (V) ® .9(V*). If el , ... ,en are the dual linear coordinates on V* to Xl"" , Xn on V (i.e., ej(A.) = A.(v)) and if we set (Jk(D) = EIII+IJI=kaI,JxleJ then the map

(Jk: Dk(V)/Dk-\V) -+ .9k(V x V*)

gives the isomorphism. We also note that if DiE Dk; (V), i = 1, 2, then [D l , D2] E Dk l+k2-2(V). Furthermore,

(Jk +L- _2([Dl , D2]) = {(Jk (Dl ), (Jk (D2)) I "'2 I 2

with {... , ... } the usual Poisson bracket given by

",(afag afag ) {f, g} = ~ ax. ae . ..,.. ae· ax.

j I I I I

for f, g E .9(V x V*). We now record some (standard) results.

Proposition 1. D( V) is a simple algebra over C.

For a proof see, for example, [Eh, Proposition 1.1]. If M is a D( V)-module with a filtration, Mo c Ml C M2 C ... such that

Di(V)Mj c Mj+j' then we say that M is a filtered D(V)-module. We note with the obvious action of D(V) on .9(V) and the degree filtration that .9(V) is an irreducible D( V)-module.

Theorem 2. Suppose that M is a filtered D( V)-module with dim Mq < 00 and that dimMq :::; cqP /p! for some 0 < c < 00 and all q sufficiently large. If p < n then M = 0 and if p = n then M has length at most c as a D( V)-module.

A proof of this result of J. Bernstein can be found in [Eh, Propositions 1.12 and 1.15].

The next theorem (an algebraic version of the Stone·-Von Neumann theorem) is a special case of the Kashiwara Lemma of 9 -module theory. Since it plays a critical role in this paper, we will record a relatively simple proof.



Lemma 3. Let M be a finitely generated D( V)-module such that if m E M and PES + (V) then there exists k such that pk m = O. Then M is D( V)-isomorphic with a finite multiple of .9(V). In particular, M splits into a finite direct sum of irreducible D( V)-modules. Proof. Let !?f be the category of all D( V)-modules M satisfying the S + (V)-nilpotence assumption of the lemma. We look upon V as an abelian Lie algebra and M as a V-module under the action V· m = 8v • m. We show that if M E !?f then HI(V, M) = O. Let Mk = {m E MISk(V)m = O}. Then ~ = {OJ and Uk>o Mk = M. Let Xl' ... ,xn be linear coordinates on V. Put E = EXj 8 j • Then EMk c Mk for all k. Let WE ZI(V, M); then there exists k such that w(V) c Mk. Since W E ZI(V, M), 8j w(8) = 8jw(8J. Define T(w) = Exj w(8J E Mk+l . Then

dT(w)(8) = L8jx jw(8j) = w(8) + LXj8jw(8j) = (E + 1)w(8). j j

If mE Mk then we assert that (E -k+ l)m E M k- l . Indeed, if mE Mk and v E Sk_l (V) then vEm = [v, E]m + Evm. The formula for E implies that Evm = O. Now [v, E] = (k - l)v. Thus veE - k + l)m = O. We therefore see that

(dT(w) - kw)(V) c M k- l . Since ~ = {O}, this implies our vanishing assertion.

We note that Ml = ~(V, M). If mE Ml - {OJ then D(V)m = .9(V)m. Thus D(V)m is a nonzero quotient of the irreducible D(V)-module, .9(V). So it is equivalent with .9(V). The above vanishing of first cohomology easily implies that, as a D(V)-module, M is isomorphic with .9(V)®MI with D(V) acting on the first factor. This completes the proof.

We define an isomorphism, ¢>, of D(V) by ¢>(x) = iaj8xj and ¢>(8j8xj) = iXj • We note that ¢> depends on the choice of basis {xJ of V*. If ~ is a sub algebra of D(V) such that ¢>(~) = ~ and if M is a ~-module then we denote by Xi the D( V)-module with total space M and with action D . m = ¢>(D)m .

Let ~ be a real form of V, and let B be a nondegenerate, symmetric, bilinear form on ~. We will also denote by B the complex bilinear extension of B to V. Let dx denote the Lebesgue measure on ~ corresponding to a basis {Vj} of ~ such that B(vp v) = ±c5jj . Let 9(~) denote the Schwartz space of ~. We use B to define a Fourier transform !Fn(f) = J by

fA(x) = 1 ( f( )e-jB(y,x) d . (2nt/21v" y y

If U is an open subset of ~ then we look upon D( V) as a subalgebra of the algebra of differential operators on U with COO coefficients. If {xJ is defined as above with {v J an orthonormal basis of V then

!Fn(Df) = ¢>(D)!Fn(f).


810 N. R. WALLACH

ApPENDIX 2. SOME INVARIANT THEORY

Let ~ be a vector space over R with inner product (... , ... ). Let <I> be a root system in ~ (cf. [Bo]), and let W denote the corresponding Weyl group. Let V denote the complexification of ~. We extend ( ... , ... ) to a symmetric nondegenerate C-bilinear form on V. We choose u l ' ••• , un to be basic (homogeneous) W-invariant polynomials in 9'(V) such that u j is real valued on ~. Fix <1>+ a system of positive roots in <1>. Let e l , ••• ,en be an orthonormal basis of ~, and let Xl' '" ,xn be the corresponding linear coordinates on ~ and V. Let el , ... ,en be the linear coordinates on ~* corresponding to the dual basis to e l , .•. , en' We look upon V* as a W-module under the contragredient action and write sA. = A. 0 S -1 for A. E V*, S E W. In this section we will be studying the action of W on V x V* given by s(v, A.) = (sv , sA.) .

We set P = Eje/J/8xj. If f E 9'(V) then pkf is called a polarization of f. We identify 9'(V) (resp. 9'(V*)) with the polynomials on 9'(V x V*) that depend only on Xl' ••. 'Xn (resp. el , •.• , en)' The basic question of this section is to find a method of generating 9' (V x v*) W using 9' (V) W • Based on a result in [W] for the symmetric group (and its easy extension to Weyl groups of type Bn = en) it has been suggested (sometimes conjectured) that 9'(V x V*)w is generated as an algebra by the polarizations of the elements of 9'(V)w. We will call this suggestion the polarization hypothesis. Unfortunately, this hypothesis is false for Weyl groups of type Dn for n ~ 4. Before we go on to positive results we give an example for D 4 of an invariant that is not in the algebra generated by polarizations of elements of 9'(V)w .

We will use the notation of [Bo]. Let W be the Weyl group of type D4 ,

and use the coordinates Xj = ej . We take uj = E j xJj for i = 1, 2, 3 and u4 = X l x2x3X 4 • We note that, if 1 ~ i ~ 3, u j is invariant under the bigger Weyl group, W', of B4 • Let (() E Wi be the element such that wei = ei , i = 1 , 2, 3, and we4 = -e4 • We set

333 3 U = u(x, e) = el X2X3X4 + Xl e2X3X4 + Xl X2e3X4 + Xl x2x3e4·

Then U E 9'(V x V*)w. We assert that U is not contained in the algebra generated by the polarizations of the elements of 9' (V) W • Since P is a vector field it is enough to show that U is not in the algebra generated by 1 and pk U j for k = 1, 2, 3, 4, 5, 6 and j = 1, 2, 3, 4. So assume that U is in this algebra. Since wu = -U, this implies (assuming the polarization hypothesis for B4 , which is correct) that

2 3 4 U = fau4 + ftPu4 + J;p U4 + J;p U4 + ~P u4

with 1; in the algebra generated by the polarizations of u l ' u2 ' u3 and v = Ej xJ . We may assume that the 1; are homogeneous in x, e. Bidegree consid-erations imply that fa and ~ are O. Also, we may assume that 1; has bidegree (i - 1, 3 - i) (the first coordinate is the degree in x, the second is the degree


DIFFERENTIAL OPERATORS ON A REDUCTIVE LIE ALGEBRA gil

in ~). Degree considerations imply that J; = CiP3- iU I • If we expand u l p 3u4

into monomials, we see that the term xi ~2~3~4 occurs with a positive coefficient but cannot occur in the expansion of u or p3-iUIpiU4 for i = 1,2. Thus h = o. If we expand PU IP2U4 then the monomial X~X2~1~3~4 has a positive coefficient but cannot occur in U or P2UI Pu4 . Hence fz = o. But now we have the contradiction that U = ci P2UI Pu4 .

Thus the polarization hypothesis is false in general. We now introduce a new hypothesis that we will prove all irreducible Weyl groups but F4 , E6 , E7 , E g • We return to the general notation of this appendix. Revised polarization hypothesis. Set

~ au. a Pi = ~ 0:/ (~) ax ;

j'oJ I

then g (V x V*) W is generated as an algebra by 1 and the elements p. . .. P. U . , II I, J

r = 0, 1, ... , ik = 1, ... , n, j = 1 , ... , n. We state a simple lemma whose proof will be left to the reader.

Lemma 1. If the revised polarization hypothesis (resp. the polarization hypoth-esis) is true for every irreducible factor of W then it is true for W. If after adjoining a variable xn+1 and dual variable ~n+1 with sXn+1 = xn+I' SEW, the revised polarization hypothesis (resp. the polarization hypothesis) is true then it is true for W on ~.

Proposition 2. The polarization hypothesis is true for Weyl groups of type An' En = Cn' G2 . The revised polarization hypothesis is true for Dn , n ~ 4. Proof. We first consider the case of G2 . We note that the argument that we use would apply to any dihedral group. Let ai' a 2 be simple roots for <1>+ . Set XI = al/llaill , and take e2 such that a l (e2) = o. This defines XI' x2 . Set Sl = sal· We take as basis generators for the invariants of Sl' u = X~ + xi and x2 • The polarization hypothesis is trivial for the group generated by Sl • Thus if f E g(V x V*)w then f is a polynomial in u, Pu, x2 ' and ~2. We write f = L.p,q'Ppq(u, Pu)~~i. Let Av(g) = I~I L.sEwsg. Then since su = u, SPu = Pu for SEW,

f = Av(f) = L 'Pp,q(U, PU) Av(xf~i)· p,q

If q = 0 then since Av(xf) is W -invariant it is a polynomial in uI ' u2 . If q > 0 then g = Av(~+q) is invariant, so it is a polynomial in u I ' u2 and

f= CL'Pp,q(u, Pu)Pq(g) p,q

with C-I=(p+q)(p+q-l) ... (p+l). We note that this argument is a variant of Weyl's original argument of An'

We now consider the cases An_I' En' Dn. For all of these cases we take Xi = ei


812 N. R. WALLACH

as in [Bo]. We will use the symbol

( TI.l'SI Tk .l'Sk) _ 1 '"' TI .l'SI Tk .l'Sk XI ~I ... xk ~k -, L..J xwl~wl ... xwk~wk' n. wESn

1 $ k $ n.

Then in the An_I case every j E .9'(V x V*)w is a linear combination of terms as above. For Bn the same is true as long as we restrict our symbols to rj + Sj

even. For Dn this is true if for k < n the rj + Sj are even and for k = n the rj + Sj are either all even or all odd.

Let A be the algebra generated by 1 and the polarizations of the basic in-variants. We show that the symbols above for k < n are in A by induction on k. If k = 1 then

(rl +sl)···(rl + l)(x;I<!:I) = pSI (X;I+SI).

Assume the assertion for 1 $ k - 1 < n. If k < n then T S T S T S n-k+1 T S T S . (XII<!II)(X22<!22 .. ·xkk<!n = (XII<!II ... Xkk<!kk) + symbols wIth smaller k. n

For An_I and Bn this argument also works for k = n. This completes the proof for An_I and Bn. If k = n in the case of Dn and if all of the rj + Sj

are even then the same argument applies. We are thus left with the case when rj + Sj is odd for all i = 1, ... , n, and we may assume the result for Bn.

We set L2j_I=Lj<!;k-18/8xj. We note that LCL2k_ICLj9I.9'(V*)wPj.

(Here we take uj = Lj x;j for j = 1, ... , n -1 and un = XIX2··· x n .) Indeed if vk(<!) = Lj <!;k then

Since vk is a polynomial in uI ' ... , un' the assertion follows from the chain rule. We now prove that if A I is the algebra generated by the polynomials LjILj2···Lj/ with jj odd and jE.9'(V)w then

U = (X;I<!:I .. . x~n<!~n) E Al

by induction on the number, q, of sJ =1= o. If q = 0, then U E .9'(V)w CAl. Assume 0 $ q $ k - 1 . We now consider the case when q = k. Set p = n - k. Then we may (after relabeling) assume that

_ ... (T1-I ... Tp-I Tp+I-I.l'Sp+I ... Tn-I.l'Sn) U - XI xn XI xp Xp+I ~p+1 xn ~n·

The result for Bn now implies that U EA. Thus we may assume that rp+1 = o. Consider

L (TI Tp Tp+2.l'Sp+2 Tn .l'Sn) S XI· .. Xp Xp+I Xp+2~P+2 ... Xn ~n p+1

p T T.-I S T T S ='"'r.(x l ···X 1 <!p+I··· XPX ···Xn<!n)+U+V L..J ] I ] ] P p+1 n n

j=1



where v is a sum of symbols with at most q - 1 nonzero exponents for the ~j •

We note that all of the terms in the first sum have at least one pair (rj' s) = (1 , 0) .

Thus we may assume that

If we now repeat the above argument then

L ( 'p 'p+2):Sp+2 .rn):sn) S XI·· ·xp xp+ I Xp+2"'P+2·· ·xn "'n p+l

p ,.-1 S , , S = Lrj(xl •· ·x/ ~rl .. ·X;Xp+I •. ·xnn~nn) + 2u + V j=2

with v in Al (smaller number of nonzero ~j exponents) and each term having two exponents (1, 0). If we continue in this way, we may assume that

U - (x x ... X ):Sp+l ••• x'n):sn) - I 2 p"'p+1 n "'n .

Now Ls (X I X 2 •• ·XpXp+I •• • x~n~~n) = (p + l)u + V

p+l

with v E A I . This completes the proof.

ApPENDIX 3. SOME OBSERVATIONS ABOUT DISTRIBUTIONS

In this appendix we first give a slight generalization of the well-known re-sult that a homogeneous distribution on Rn is tempered. We use it to show that certain distributions, locally defined, on a real semisimple Lie algebra ex-tend to tempered distributions. If n is a compact subset of Rn then we set C; (Rn) equal to the space of all smooth functions on Rn such that supp fen with the topology given by the seminorms ql(f) = sUPxEn 181 f(x)1 for I = (iI' ... , in)' ij E N (= {O, 1, 2, ... }). Here we use standard multi-index notation as in Appendix 1.

As usual, a distribution on Rn is a linear function, T, on C;"(Rn) such that T is continuous on each space C;(Rn ). As a customary, we set 9'(Rn )

equal to the space of all distributions on Rn . If f E C,oo(Rn) then we set

Pk I(f) = sup IIxll k l81 f(x)l· , xERn

Here IIxII 2 = x~ + ... + x; (as usual) and k ~ O. As is usual, we denote by .9"(Rn) the space of all f E Coo(Rn) such that Pk ,/(f) < 00 for all k ~ 0, I endowed with the topology induced by the seminorms Pk, I. If T E

9' (Rn) then (as usual) we say that T is tempered if T extends to a continuous functional on .9"(Rn).

We set E = Ei x j 8 /8xj • The following is a mild extension of a well-known result and is no doubt also well known. The proof involves standard methods of distribution theory.


814 N. R. WALLACH

Lemma 1. If T E 9' (Rn) is such that dim C[E]T < 00 then T is tempered.

We will now apply this lemma to the case of interest in this paper. Let G be a semisimple group of inner type (cf. [RRGI, 2.2.8]). Let g be the Lie algebra of G, and let 0 be a Cartan involution of G. As usual, B will denote the Killing fomi of g. We note that (X, Y) = -B(OX, Y) defines an inner product on g. Using an orthonormal basis of 9 with respect to ( ... , ... ) we may identify g with Rn with n = dim g. If X E 9 then X is said to be nilpotent if Ad X is nilpotent. Let ,Af denote the variety of nilpotent elements of g. If f E C<Xl(O) and if g E G then we set r(g)f(X) = f(Ad(g)-IX). We set 9'(0)G = {T E 9'(n)IT(r(g)f) = T(f) for all g E G, f E C;'(On. The following result is due to Harish-Chandra. However, his proof is quite complicated (see [Var, pp. 128-137]).

Lemma 2. Let 0 be open in 9 and completely invariant (see § 5). If On,Af =1= 0 then ,Af cO. If T E 9' (O)G and if supp T c,Af then T extends to a tempered distribution on g. That is, there exists a tempered distribution S on 9 such that T(f) = S(f) for f E c;,(n). Proof. If 0 n,Af =1= 0 then, if X EOn,Af, Xs = 0 EO. If X E ,Af

and if {X, Y, H} is an s-triple in 9 then e-tadH X = e-2tX. Thus if t is sufficiently large e-tadH X En. Hence X En. According to [Var, The-orem 28, p. 19] there exists a E COO(n)G such that suppa cO, and a is identically equal to 1 in a neighborhood of O. Thus if X E ,Af then a(X) = a(e-tadH X) = lim t--++oo a(e-tadH X) = a(O) = 1. Thus a is identi-cally equal to 1 in a neighborhood of ,Af. We set S = aT. Then S E 9' (g)G and S(f) = T(f) for f E C;'(O). Thus to prove the lemma it is enough to show that S is tempered. We note that the support of S is contained in ,Af . Hence Lemma 8.3.7 in [RRGI] implies that dimC[E]S < 00. Lemma 1 implies that S is tempered.

We will now prove another technical result that will be used in §7. Let M be an m-dimensional submanifold of Rn • Assume that for each u E M there exists a neighborhood, 0, of u in Rn and local coordinates {Xl' ... ,Xn} on n such that Xj(u) =0, l::;i::;n,and MnO={yEnlxj(y)=O, i>m}. Let w be a smooth d-form on M that defines a volume form on M. We define a distribution on Rn by T(f) = fM fw, f E C;'(Rn ) (note that our local condition implies that T is in fact a Radon measure on Rn). If D is a differential operator on Rn then let DT denote its formal adjoint with respect to Lebesgue measure.

Lemma 3. Let D be a differential operator on Rn such that DT = O. Let u EM, and let 0 and Xl' ... ,xn be as above. Define,., E COO(O n M) by

w1nnM = ,.,dxl 1\ ... 1\ dxm.

Write DT = E/ J a/ iJ/ aJ with the"]" multi-indices of the form (iI' ... , im, 0) and the" J" 'mulii-indices of the form (0, jm+l ' ... , jn)' Then for each J


DIFFERENTIAL OPERATORS ON A REDUcrIVE LIE ALGEBRA 815

we have ~)-I)III(l(l1al,J)(u) = O.

I

Furthermore, if D is a differential operator on Rn and D satisfies all of these local conditions then DT = o. Proof. Assume that D is of order k. Let rp E C;'(Q) be such that the support of rp is contained in the set R = {y E QI IXj(Y)1 < r, 1 ::=; i ::=; n}. Let V={YEQlxj(y)=O, i>m, IXj(Y)I<r, l::=;i::=;m}.Then

DT(rp) = Ll11al,J(al{/rp)dXII\···l\dXm I,J v

= L(-I),1'l al(l1al,J)aJrpdXII\···l\dxm· I,J v

By a simple iteration of [Ho, Theorem 1.2.6], if 'I' E C;'(V) then then for each J = (0, jm+I' ... ,jn) with IJI ::=; k there exists rp E C;'(R) such that aKrpw = 0J,K'I' for all K = (0, km+l , ••. ,kn ) with IKI ::=; k. This clearly implies the first assertion. The converse also follows from the above formula.

NOTE ADDED IN PROOF

Lance Small has observed that Lemma 1.2 can be deduced from the results in S. Montgomery, Fixed rings offinite automorphism groups of associative rings, Lecture Notes in Math., vol. 818, Springer, Berlin, 1980.

REFERENCES

[BVl] D. Barbasch and D. Vogan, The local structure o/characters, J. Funct. Anal. 37 (1980), 27-55.

[BV2] __ , Primitive ideals and orbital integrals in complex classical groups, Math. Ann. 259 (1982),153-199.

[BV3] __ , Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra 80 (1983),350-382.

[B) J. N. Bernstein, The analytic continuation 0/ generalized junctions with respect to a param-eter, Funct. Anal. Appl. 6 (1972), 26-40.

[Bo] N. Bourbaki, Groupes et Algebras de Lie, Chapitres IV, V, et VI, Hermann, Paris, 1968. [Eh] F. Ehlers, The Weyl algebra, Algebraic D-modules, Chapter V, Perspect. Math., vol. 2,

Academic Press, Boston, MA, 1987. [HI] Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer. J. Math. 79

(1957),87-120. [H2] __ , Invariant eigendistributions on a semisimple Lie algebra, Inst. Hautes Etudes Sci.

Publ. Math. 27 (1965), 5-54. [H3] __ , Invariant differential operators and distributions on a semisimple Lie algebra, Amer.

J. Math. 86 (1964), 534-564. [Ho] L. Hormander, The analysis 0/ linear partial differential operators. I, Springer-Verlag,

Berlin, 1990. [HK] R. Hotta and M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra,

Invent. Math. 75 (1984), 327-358.


816 N. R. WALLACH

[K] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 8S (1963), 327- 404.

[RR] R. Ranga Rao, Orbital integrals in reductive groups, Ann. of Math. (2) 96 (1972), 505-510. [S] T. A. Springer, Trigonometric sums, Green functions offmite groups and representations of

Weyl groups, Invent. Math. 36 (1976), 209-224. [Var] V. S. Varadarajan, Harmonic analysis on real reductive groups, Lecture Notes in Math.,

vol. 576, Springer-Verlag, Berlin, 1977. [RROI] N. R. Wallach, Real reductive groups. I, Academic Press, Boston, MA, 1988. [WI H. Weyl, The classical groups, their invariants and representations, 2nd. ed., Princeton

University Press, Princeton, NJ, 1949.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, SAN DIEGO, LA JOLLA, CALIFORNIA 92093-0001

E-mail address: nwallach@ math.ucsd.edu


nolan r. wallach introduction...testable using the next generation of computers. however, we hope...

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