on the centre of two-parameter quantum groupskhlee/papers/tqm.pdfthe centre of two-parameter quantum...

Proceedings of the Royal Society of Edinburgh 136A 445ndash472 2006

On the centre of two-parameter quantum groups

Georgia BenkartDepartment of Mathematics University of Wisconsin MadisonWI 53706 USA (benkartmathwiscedu)

Seok-Jin KangDepartment of Mathematical Sciences Seoul National UniversitySan 56-1 Shinrim-dong Kwanak-ku Seoul 151-747 Korea(sjkangmathsnuackr)

Kyu-Hwan LeeDepartment of Mathematics University of Connecticut StorrsCT 06269 USA (khleemathuconnedu)

(MS received 8 April 2004 accepted 28 July 2005)

We describe PoincarendashBirkhoffndashWitt bases for the two-parameter quantum groupsU = Urs(sln) following Kharchenko and show that the positive part of U has thestructure of an iterated skew polynomial ring We define an ad-invariant bilinearform on U which plays an important role in the construction of central elements Weintroduce an analogue of the Harish-Chandra homomorphism and use it to determinethe centre of U

1 Introduction

In this paper we determine the centre of the two-parameter quantum groups U =Urs(sln) which are the same algebras as those introduced by Takeuchi in [3536] but with the opposite co-product As shown in [4 5] these quantum groupsare Drinfelrsquod doubles and have an R-matrix They are related to the downndashupalgebras in [23] and to the multi-parameter quantum groups of Chin and Musson [8]and Dobrev and Parashar [10] In the analogous quantum function algebra settingallowing two parameters unifies the DrinfelrsquodndashJimbo quantum groups (r = q s =qminus1) in [11] with the DipperndashDonkin quantum groups (r = 1 s = qminus1) in [9]

For the one-parameter quantum groups Uq(g) corresponding to finite-dimensionalsimple Lie algebras g there is a sizeable literature [7 15 21ndash28 30ndash32 37ndash39] deal-ing with PoincarendashBirkhoffndashWitt (PBW) bases For the multi-parameter quantumgroups associated with g of classical type Kharchenko [21] constructed PBW basesby first determining GrobnerndashShirshov bases for them We show in this paper thatKharchenkorsquos results when applied to the algebra U = Urs(sln) yield useful com-mutation relations which enable us to prove that the positive part U+ of U hasthe structure of an iterated skew polynomial ring As a consequence of that resultU+ modulo any prime ideal is a domain The commutation relations also playan essential role in [6] where finite-dimensional restricted two-parameter quantum

445ccopy 2006 The Royal Society of Edinburgh

446 G Benkart S-J Kang and K-H Lee

groups urs(gln) and urs(sln) are constructed when r and s are roots of unity Theserestricted quantum groups are Drinfelrsquod doubles and are ribbon Hopf algebras undersuitable restrictions on r and s

Much work has been done on the centre of quantum groups for finite-dimensionalsimple Lie algebras [1 12 19 28 29 34 37] and also for (generalized) KacndashMoody(super)algebras [13 16 20] The approach taken in many of these papers (andadopted here as well) is to define a bilinear form on the quantum group whichis invariant under the adjoint action This quantum version of the Killing form isoften referred to in the one-parameter setting as the Rosso form (see [34]) The nextstep involves constructing an analogue ξ of the Harish-Chandra map It is straight-forward to show that the map ξ is an injective algebra homomorphism The maindifficulty lies in determining the image of ξ and in finding enough central elementsto prove that the map ξ is surjective In the two-parameter case a new phenomenonarises the n odd and n even cases behave differently Additional central elementsarise when n is even which complicates the description in that case

Our paper is organized as follows In sect 2 we briefly recall the definition and basicproperties of the two-parameter quantum group U = Urs(sln) In sect 3 we describethe commutation relations which determine a GrobnerndashShirshov basis and allowa PBW basis to be constructed and we prove that the positive part of U has aniterated skew polynomial ring structure The next section is devoted to the con-struction of a bilinear form and the proof of its invariance under the adjoint actionIn the final section we define a Harish-Chandra homomorphism ξ and determinethe centre of U by specifying the image of ξ and constructing central elementsexplicitly

2 Two-parameter quantum groups

Let K be an algebraically closed field of characteristic 0 Assume that Φ is a finiteroot system of type Anminus1 with Π a base of simple roots We regard Φ as a subsetof a Euclidean space R

n with an inner product 〈middot middot〉 We let ε1 εn denote anorthonormal basis of R

n and suppose that Π = αj = εj minus εj+1 | j = 1 nminus 1and that Φ = εi minus εj | 1 i = j n

Fix non-zero elements r s in the field K Here we assume r = s Let U = Urs(gln)be the unital associative algebra over K generated by elements ej fj (1 j lt n)and aplusmn1

i bplusmn1i (1 i n) which satisfy the following relations

(R1) the aplusmn1i bplusmn1

j all commute with one another and aiaminus1i = bjb

minus1j = 1

(R2) aiej = r〈εiαj〉ejai and aifj = rminus〈εiαj〉fjai

(R3) biej = s〈εiαj〉ejbi and bifj = sminus〈εiαj〉fjbi

(R4) [ei fj ] =δijr minus s (aibi+1 minus ai+1bi)

(R5) [ei ej ] = [fi fj ] = 0 if |iminus j| gt 1

(R6) e2i ei+1 minus (r + s)eiei+1ei + rsei+1e2i = 0

eie2i+1 minus (r + s)ei+1eiei+1 + rse2i+1ei = 0

The centre of two-parameter quantum groups 447

(R7) f2i fi+1 minus (rminus1 + sminus1)fifi+1fi + rminus1sminus1fi+1f

2i = 0

fif2i+1 minus (rminus1 + sminus1)fi+1fifi+1 + rminus1sminus1f2

i+1fi = 0

Let U = Urs(sln) be the subalgebra of U = Urs(gln) generated by the elementsej fj ωplusmn1

j and (ωprimej)

plusmn1 (1 j lt n) where

ωj = ajbj+1 and ωprimej = aj+1bj

These elements satisfy (R5)ndash(R7) along with the following relations

(R1prime) the ωplusmn1i (ωprime

j)plusmn1 all commute with one another and ωiω

minus1i = ωprime

j(ωprimej)

minus1 = 1

(R2prime) ωiej = r〈εiαj〉s〈εi+1αj〉ejωi and ωifj = rminus〈εiαj〉sminus〈εi+1αj〉fjωi

(R3prime) ωprimeiej = r〈εi+1αj〉s〈εiαj〉ejω

primei and ωprime

ifj = rminus〈εi+1αj〉sminus〈εiαj〉fjωprimei

(R4prime) [ei fj ] =δijr minus s (ωi minus ωprime

i)

Let U+ and Uminus be the subalgebras generated by the elements ei and fi respec-tively and let U0 and U0 be the subalgebras generated by the elements aplusmn1

i bplusmn1i

1 i n and ωplusmn1i (ωprime

i)plusmn1 1 i lt n respectively It now follows from the defining

relations that U has a triangular decomposition U = UminusU0U+ Similarly we haveU = UminusU0U+

The algebras U and U are Hopf algebras where the aplusmni bplusmni are group-like ele-

ments and the remaining co-products are determined by

∆(ei) = ei otimes 1 + ωi otimes ei ∆(fi) = 1 otimes fi + fi otimes ωprimei

This forces the co-unit and antipode maps to be

ε(ai) = ε(bi) = 1 S(ai) = aminus1i S(bi) = bminus1

i

ε(ei) = ε(fi) = 0 S(ei) = minusωminus1i ei S(fi) = minusfi(ωprime

i)minus1

Let Q = ZΦ denote the root lattice and set Q+ =oplusnminus1

i=1 Z0αi Then for anyζ =

sumnminus1i=1 ζiαi isin Q we adopt the shorthand

ωζ = ωζ11 middot middot middotωζnminus1

nminus1 ωprimeζ = (ωprime

1)ζ1 middot middot middot (ωprime

nminus1)ζnminus1 (21)

Lemma 21 (Benkart and Witherspoon [4 lemma 13]) Suppose that

ζ =nminus1sum

i=1

ζiαi isin Q

Then

ωζei = rminus〈εi+1ζ〉sminus〈εiζ〉eiωζ ωζfi = r〈εi+1ζ〉s〈εiζ〉fiωζ

ωprimeζei = rminus〈εiζ〉sminus〈εi+1ζ〉eiω

primeζ ωprime

ζfi = r〈εiζ〉s〈εi+1ζ〉fiωprimeζ


There is a grading on U with the degrees of the generators given by

deg ei = αi deg fi = minusαi degωi = degωprimei = 0

Then since the defining relations are homogeneous under this grading the algebraU has a Q-grading

U =oplus

ζisinQ

Uζ

We also haveU+ =

oplus

ζisinQ+

U+ζ and Uminus =

oplus

ζisinQ+

Uminusminusζ

where U+ζ = U+ cap Uζ and Uminus

minusζ = Uminus cap Uminusζ Let Λ =

oplusni=1 Zεi be the weight lattice of gln Corresponding to any λ isin Λ is an

algebra homomorphism λ U0 rarr K given by

λ(ai) = r〈εiλ〉 and λ(bi) = s〈εiλ〉 (22)

For any λ =sumn

i=1 λiεi isin Λ we write

aλ = aλ11 middot middot middot aλn

n and bλ = bλ11 middot middot middot bλn

n (23)

Let Λsl =oplusnminus1

i=1 Z13i be the weight lattice of sln where 13i is the fundamentalweight

13i = ε1 + middot middot middot + εi minus i

n

nsum

j=1

εj

and let

Λ+sl

= λ isin Λsl | 〈αi λ〉 0 for 1 i lt n =nminus1sum

i=1

li13i

∣∣∣∣ li isin Z0

denote the set of dominant weights for sln We fix the nth roots r1n and s1n

of r and s respectively and define for any λ isin Λsl an algebra homomorphismλ U0 rarr K by

λ(ωj) = r〈εj λ〉s〈εj+1λ〉 and λ(ωprimej) = r〈εj+1λ〉s〈εj λ〉 (24)

In particular if λ belongs to Λ then the definition of λ(ωj) and λ(ωprimej) coming

from (22) coincides with (24)Associated with any algebra homomorphism ψ U0 rarr K is the Verma module

M(ψ) with highest weight ψ and its unique irreducible quotient L(ψ) When thehighest weight is given by the homomorphism λ for λ isin Λsl we simply write M(λ)and L(λ) instead of M(λ) and L(λ)

Lemma 22 (Benkart and Witherspoon [5]) We assume that rsminus1 is not a root ofunity and let vλ be a highest weight vector of M(λ) for λ isin Λ+

sl The irreducible

module L(λ) is then given by

L(λ) = M(λ)(nminus1sum

i=1

Uf〈λαi〉+1i vλ

)


Let W be the Weyl group of the root system Φ and let σi isin W denote thereflection corresponding to αi for each 1 i lt n Thus

σi(λ) = λminus 〈λ αi〉αi for λ isin Λ (25)

and σi also acts on Λsl according to the same formulaLet M be a finite-dimensional U -module on which U0 acts semi-simply Then

M =oplus

χ

Mχ

where each χ U0 rarr K is an algebra homomorphism and

Mχ = m isin M | ωim = χ(ωi)m and ωprimeim = χ(ωprime

i)m for all i

For brevity we write Mλ for the weight space Mλ for λ isin Λsl

Proposition 23 Assume that rsminus1 is not a root of unity and that λ isin Λ+sl Then

dimL(λ)micro = dimL(λ)σ(micro)

for all micro isin Λsl and σ isin W

Proof This is an immediate consequence of [5 proposition 28 and the proof oflemma 212]

3 PBW-type bases

From now on we assume that r + s = 0 (or equivalently rminus1 + sminus1 = 0) and theordering (k l) lt (i j) always means relative to the lexicographic ordering

We define inductively

Ejj = ej and Eij = eiEiminus1j minus rminus1Eiminus1jei i gt j (31)

The defining relations for U+ in (R6) can be reformulated as saying

Ei+1iei = sminus1eiEi+1i (32)

ei+1Ei+1i = sminus1Ei+1iei+1 (33)

Even though the relations in the following theorem can be deduced from [21theorem An] we include a self-contained proof in the appendix for the convenienceof the reader

Theorem 31 (Kharchenko [21]) Assume that (i j) gt (k l) in the lexicographicorder Then the following relations hold in the algebra U+

(1) EijEkl minus rminus1EklEij minus Eil = 0 if j = k + 1

(2) EijEkl minus EklEij = 0 if i gt k l gt j or j gt k + 1

(3) EijEkl minus sminus1EklEij = 0 if i = k j gt l or i gt k j = l

(4) EijEkl minus rminus1sminus1EklEij + (rminus1 minus sminus1)EkjEil = 0 if i gt k j gt l


Let E = e1 e2 enminus1 be the set of generators of the algebra U+ We intro-duce a linear ordering ≺ on E by saying ei ≺ ej if and only if i lt j We extend thisordering to the set of monomials in E so that it becomes the degree-lexicographicordering that is for u = u1u2 middot middot middotup and v = v1v2 middot middot middot vq with ui vj isin E we haveu ≺ v if and only if p lt q or p = q and ui ≺ vi for the first i such that ui = viLet AE be the free associative algebra generated by E and S sub AE be the setconsisting of the following elements

EijEkl minus EklEij if i gt k l gt j or j gt k + 1

EijEkl minus sminus1EklEij if i = k j gt l or i gt k j = l

EijEkl minus rminus1sminus1EklEij + (rminus1 minus sminus1)EkjEil if i gt k j gt l

The elements of S just correspond to relations (2)ndash(4) of theorem 31 Note that wemay take S to be the set of defining relations for the algebra U+ since S containsall the (original) defining relations (R5) and (R6) of U+ and the other relations inS are all consequences of (R5) and (R6)

The following theorem is a special case of in [21 theorem An] and its conse-quences Also one can prove it using an argument similar to that in [7] or [3940]

Theorem 32 (Kharchenko [21]) Assume that r s isin Ktimes and r + s = 0 Then

(i) the set S is a GrobnerndashShirshov basis for the algebra U+ with respect to thedegree-lexicographic ordering

(ii) B0 = Ei1j1Ei2j2 middot middot middot Eipjp | (i1 j1) (i2 j2) middot middot middot (ip jp) (lexicographicalordering) is a linear basis of the algebra U+

(iii) B1 = ei1j1ei2j2 middot middot middot eipjp| (i1 j1) (i2 j2) middot middot middot (ip jp) (lexicographical

ordering) is a linear basis of the algebra U+ where eij = eieiminus1 middot middot middot ej fori j

Remark 33 If we define Fij inductively by

Fjj = fj and Fij = fiFiminus1j minus sFiminus1jfi i gt j

and denote by fij the monomial fij = fifiminus1 middot middot middot fj i j then we have linearbases for the algebra Uminus as in theorem 32 Note that U0 and U0 which are groupalgebras have obvious linear bases Combining these bases using the triangulardecomposition U = UminusU0U+ and U = UminusU0U+ we obtain PBW bases for theentire algebras U and U respectively

Now we turn our attention to showing that the algebra U+ is an iterated skewpolynomial ring over K and that any prime ideal P of U+ is completely prime(that is U+P is a domain) when r and s are lsquogenericrsquo (see proposition 36 forthe precise statement) Our approach is similar to that in [33] which treats theone-parameter quantum group case Recall that if ϕ is an automorphism of analgebra R then ϑ isin End(R) is a ϕ-derivation if ϑ(ab) = ϑ(a)b + ϕ(a)ϑ(b) for alla b isin R The skew polynomial ring R[xϕ ϑ] consists of polynomials

sumi aix

i overR where xa = ϕ(a)x+ ϑ(a) for all a isin R


For each (i j) 1 j i lt n we define an algebra automorphism ϕij of U by

ϕij(u) = ωαi+middotmiddotmiddot+αjuωminus1αi+middotmiddotmiddot+αj

for all u isin U

Using lemma 21 one can check that if (k l) lt (i j) then

ϕij(Ekl) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

rminus1Ekl if j = k + 1Ekl if i gt k l gt j or j gt k + 1sminus1Ekl if i = k j gt l or i gt k j = l

rminus1sminus1Ekl if i gt k j gt l

Hence the automorphism ϕij preserves the subalgebra U+ij of U+ generated by

the vectors Ekl for (k l) lt (i j) We denote the induced automorphism of U+ij by

the same symbol ϕij Now we define a ϕij-derivation ϑij on U+

ij by

ϑij(Ekl) = EijEkl minus ϕij(Ekl)Eij =

⎧⎪⎨

⎪⎩

Eil j = k + 1(rminus1 minus sminus1)EkjEil i gt k j gt l

0 otherwise

It is easy to see that ϑij is indeed a ϕij-derivation (cf [33 lemma 3 p 62]) Withϕij and ϑij at hand the next proposition follows immediately

Proposition 34 The algebra U+ is an iterated skew polynomial ring whose struc-ture is given by

U+ = K[E11][E21ϕ21 ϑ21] middot middot middot [Enminus1nminus1ϕnminus1nminus1 ϑnminus1nminus1] (34)

Proof Note that all the relations in theorem 31 can be condensed into a singleexpression

EijEkl = ϕij(Ekl)Eij + ϑij(Ekl) (i j) gt (k l) (35)

The proposition then easily follows from theorem 32

The other result of this section requires an additional lemma

Lemma 35 The automorphism ϕij and the ϕij-derivation ϑij of U+ij satisfy

ϕijϑij = rsminus1ϑijϕij

Proof For (k l) lt (i j) the definitions imply that

(ϕijϑij)(Ekl) =

⎧⎪⎨

⎪⎩

sminus1Eil if j = k + 1(rminus1 minus sminus1)sminus2EkjEil if i gt k j gt l

0 otherwise

On the other hand for (k l) lt (i j)

(ϑijϕij)(Ekl) =

⎧⎪⎨

⎪⎩

rminus1Eil if j = k + 1(rminus1 minus sminus1)rminus1sminus1EkjEil if i gt k j gt l

0 otherwise

Comparing these two calculations we arrive at the result


We now obtain the following proposition

Proposition 36 Assume that the subgroup of Ktimes generated by r and s is torsion-

free Then all prime ideals of U+ are completely prime

Proof The proof follows directly from proposition 34 lemma 35 and [14 theo-rem 23]

4 An invariant bilinear form on U

Assume that B is the subalgebra of U generated by ej ωplusmn1j 1 j lt n and Bprime

is the subalgebra of U generated by fj (ωprimej)

plusmn1 1 j lt n We recall some resultsin [4]

Proposition 41 (Benkart and Witherspoon [4 lemma 22]) There is a Hopfpairing (middot middot) on BprimetimesB such that for x1 x2 isin B y1 y2 isin Bprime the following propertieshold

(i) (1 x1) = ε(x1) (y1 1) = ε(y1)

(ii) (y1 x1x2) = (∆op(y1) x1 otimes x2) (y1y2 x1) = (y1 otimes y2∆(x1))

(iii) (Sminus1(y1) x1) = (y1 S(x1))

(iv) (fi ej) =δijsminus r

(v) (ωprimei ωj) = (ωprimeminus1

i ωjminus1) = r〈εj αi〉s〈εj+1αi〉 = rminus〈εi+1αj〉sminus〈εiαj〉

(ωprimeiminus1 ωj) = (ωprime

i ωjminus1) = rminus〈εj αi〉sminus〈εj+1αi〉 = r〈εi+1αj〉s〈εiαj〉

It is easy to prove for λ isin Q that

λ(ωprimemicro) = (ωprime

micro ωminusλ) and λ(ωmicro) = (ωprimeλ ωmicro) (41)

From the definition of the co-product it is apparent that

∆(x) isinoplus

0νmicro

U+microminusνων otimes U+

ν for any x isin U+micro

where lsquorsquo is the usual partial order on Q ν micro if microminus ν isin Q+ Thus for each i1 i lt n there are elements pi(x) and pprime

i(x) in U+microminusαi

such that the componentof ∆(x) in U+

microminusαiωi otimes U+

αiis equal to pi(x)ωi otimes ei and the component of ∆(x) in

U+αiωmicrominusαi otimes U+

microminusαiis equal to eiωmicrominusαi otimes pprime

i(x) Therefore for x isin U+micro we can write

∆(x) = xotimes 1 +nminus1sum

i=1

pi(x)ωi otimes ei + ς1

= ωmicro otimes x+nminus1sum

i=1

eiωmicrominusαi otimes pprimei(x) + ς2

where ς1 and ς2 are the sums of terms involving products of more than one ej inthe second factor and in the first factor respectively


Lemma 42 (Benkart and Witherspoon [4 lemma 46]) For all x isin U+ζ and all

y isin Uminus the following hold

(i) (fiy x) = (fi ei)(y pprimei(x)) = (sminus r)minus1(y pprime

i(x))

(ii) (yfi x) = (fi ei)(y pi(x)) = (sminus r)minus1(y pi(x))

(iii) fixminus xfi = (sminus r)minus1(pi(x)ωi minus ωprimeip

primei(x))

Corollary 43 If ζ ζ prime isin Q+ with ζ = ζ prime then (y x) = 0 for all x isin U+ζ and

y isin Uminusminusζprime

Lemma 44 Assume that rsminus1 is not a root of unity and ζ isin Q+ is non-zero

(a) If y isin Uminusminusζ and [ei y] = 0 for all i then y = 0

(b) If x isin U+ζ and [fi x] = 0 for all i then x = 0

Proof Assume that y isin Uminusminusζ and that [ei y] = 0 holds for all i From the definition

of M(λ) and lemma 22 we can find a sufficiently large λ isin Λ+sl

such that the map

Uminusminusζ rarr L(λ) u rarr uvλ

is injective where vλ is a highest weight vector of L(λ) Then

Uyvλ = UminusU0U+yvλ = UminusyU0U+vλ = Uminusyvλ L(λ)

so that Uyvλ is a proper submodule of L(λ) which must be 0 by the irreducibilityof L(λ) Thus yvλ = 0 and y = 0 by the injectivity of the map above We can nowapply the anti-automorphism τ of U defined by

τ(ei) = fi τ(fi) = ei τ(ωi) = ωi and τ(ωprimei) = ωprime

i

to obtain the second assertion

Lemma 45 Assume that rsminus1 is not a root of unity For ζ isin Q+ the spaces U+ζ

and Uminusminusζ are non-degenerately paired

Proof We use induction on ζ with respect to the partial order on Q The claimholds for ζ = 0 since Uminus

0 = K1 = U+0 and (1 1) = 1 Assume now that ζ gt 0 and

suppose that the claim holds for all ν with 0 ν lt ζ Let x isin U+ζ with (y x) = 0

for all y isin Uminusminusζ In particular we have for all y isin Uminus

minus(ζminusαi) that

(fiy x) = 0 and (yfi x) = 0 for all 1 i lt n

It follows from lemma 42(i) and (ii) that (y pprimei(x)) = 0 and (y pi(x)) = 0 By the

induction hypothesis we have pprimei(x) = pi(x) = 0 and it follows from lemma 42(iii)

that fix = xfi for all i Lemma 44 now applies to give x = 0 as desired

In what follows ρ will denote the half-sum of the positive roots Thus

ρ = 12

sum

αgt0

α =nminus1sum

i=1

13i = 12 ((nminus1)ε1 +(nminus3)ε2 + middot middot middot+((nminus1)minus2(nminus1))εn) (42)


It is evident from the triangular decomposition that there is a vector-space iso-morphism oplus

microνisinQ+

(Uminusminusνω

primeν

minus1) otimes U0 otimes U+micro

simminusrarr U

This guarantees that the bilinear form which we introduce next is well defined

Definition 46 Set

〈(yωprimeν

minus1)ωprimeηωφx | (y1ωprime

ν1

minus1)ωprimeη1ωφ1x1〉 = (y x1)(y1 x)(ωprime

η ωφ1)(ωprimeη1 ωφ)(rsminus1)〈ρν〉

for all x isin U+micro x1 isin U+

micro1 y isin Uminus

minusν y1 isin Uminusminusν1

micro micro1 ν ν1 isin Q+ and all η η1 φ φ1 isinQ Extend this linearly to a bilinear form 〈middot middot〉 U times U rarr K on all of U

Note that

〈(yωprimeν


ν1

minus1)ωprimeη1ωφ1x1〉

= 〈yωprimeν

minus1 | x1〉 middot 〈ωprimeηωφ | ωprime

η1ωφ1〉 middot 〈x | y1ωprime

ν1

minus1〉 (43)

So the form respects the decompositionoplus

microνisinQ+

(Uminusminusνω

primeν


simminusrarr U

The following lemma is an immediate consequence of the above definition andcorollary 43

Lemma 47 Assume that micro micro1 ν ν1 isin Q+ Then

〈UminusminusνU

0U+micro | Uminus

minusν1U0U+

micro1〉 = 0

unless micro = ν1 and ν = micro1

Since U is a Hopf algebra it acts on itself via the adjoint representation

ad(u)v =sum

(u)

u(1)vS(u(2))

where u v isin U and ∆(u) =sum

(u) u(1) otimes u(2)

Proposition 48 The bilinear form 〈middot | middot〉 is ad-invariant ie

〈ad(u)v | v1〉 = 〈v | ad(S(u))v1〉

for all u v v1 isin U

Proof It suffices to assume u is one of the generators ωi ωprimei ei fi Also without

loss of generality we may suppose that

v = (yωprimeν

minus1)ωprimeηωφx and v1 = (y1ωprime

ν1

minus1)ωprimeη1ωφ1x1

where x isin U+micro y isin Uminus

minusν x1 isin U+micro1

y1 isin Uminusminusν1

and micro ν micro1 ν1 isin Q+


Case 1 (u = ωi) From the definition ad(ωi)v = ωivωminus1i = r〈εimicrominusν〉s〈εi+1microminusν〉v

so that〈ad(ωi)v | v1〉 = r〈εimicrominusν〉s〈εi+1microminusν〉〈v | v1〉

On the other hand we have

ad(S(ωi))v1 = ωminus1i v1ωi = r〈εiν1minusmicro1〉s〈εi+1ν1minusmicro1〉v1

which implies that

〈v | ad(S(ωi))v1〉 = r〈εiν1minusmicro1〉s〈εi+1ν1minusmicro1〉〈v | v1〉

If 〈v | v1〉 = 0 then we must have ν = micro1 and ν1 = micro by lemma 47 Thusmicrominus ν = ν1 minus micro1 and 〈ad(ωi)v | v1〉 = 〈v | ad(S(ωi))v1〉

Case 2 (u = ωprimei) We have only to replace ωi by ωprime

i and interchange εi and εi+1 inthe argument of case 1

Case 3 (u = ei) This case is similar to case 4 below so we omit the calculation

Case 4 (u = fi) Using lemmas 21 and 42(iii) we get

ad(fi)v = vS(fi) + fivS(ωprimei) = minusvfi(ωprime

i)minus1 + fiv(ωprime

i)minus1

= minusy(ωprimeν)minus1ωprime

ηωφxfi(ωprimei)

minus1 + fiy(ωprimeν)minus1ωprime

ηωφx(ωprimei)

minus1


ηωφfix(ωprimei)

minus1 + (sminus r)minus1y(ωprimeν)minus1ωprime

ηωφpi(x)ωi(ωprimei)

minus1

minus (sminus r)minus1y(ωprimeν)minus1ωprime

ηωφωprimeip

primei(x)(ω

primei)


ηωφx(ωprimei)

minus1

= minusr〈εiηminusν〉r〈εi+1φ+micro〉s〈εiφ+micro〉s〈εi+1ηminusν〉yfi(ωprimeν+αi

)minus1ωprimeηωφx

+ r〈εi+1micro〉s〈εimicro〉fiy(ωprimeν+αi


+ (sminus r)minus1rminus〈αimicrominusαi〉s〈αimicrominusαi〉y(ωprimeν)minus1ωprime

ηminusαiωφ+αipi(x)

minus (sminus r)minus1r〈εi+1microminusαi〉s〈εimicrominusαi〉y(ωprimeν)minus1ωprime

ηωφpprimei(x)

Now

ad(S(fi))v1 = ad(minusfi(ωprimei)

minus1)v1 = minusrminus〈εi+1micro1minusν1〉sminus〈εimicro1minusν1〉 ad(fi)v1

We apply the previous calculation of ad(fi)v with v replaced by v1 to see that

ad(S(fi))v1 = r〈εiη1minusν1〉r〈εi+1φ1+ν1〉s〈εiφ1+ν1〉s〈εi+1η1minusν1〉y1fi(ωprimeν1+αi

)minus1ωprimeη1ωφ1x1

minus r〈εi+1ν1〉s〈εiν1〉fiy1(ωprimeν1+αi


minus (sminus r)minus1rminus〈εimicro1minusαi〉r〈εi+1ν1minusαi〉s〈εiν1minusαi〉sminus〈εi+1micro1minusαi〉

times y1(ωprimeν1

)minus1ωprimeη1minusαi

ωφ1+αipi(x1)

+ (sminus r)minus1r〈εi+1ν1minusαi〉s〈εiν1minusαi〉y1(ωprimeν1

)minus1ωprimeη1ωφ1p

primei(x1)

It follows from lemma 47 that 〈ad(fi)v | v1〉 and 〈v | ad(S(fi))v1〉 can be non-zerowhen either (a) ν + αi = micro1 and ν1 = micro or (b) ν = micro1 and ν1 = microminus αi


(a) By lemma 42(i) (ii) we have

〈ad(fi)v | v1〉 = minusr〈εiηminusν〉r〈εi+1φ+micro〉s〈εiφ+micro〉s〈εi+1ηminusν〉

times (yfi x1)(y1 x)(ωprimeη ωφ1)(ω

primeη1 ωφ)(rsminus1)〈ρν+αi〉

+ r〈εi+1micro〉s〈εimicro〉(fiy x1)(y1 x)(ωprimeη ωφ1)(ω


= Atimes (y1 x)(ωprimeη ωφ1)(ω

primeη1 ωφ)(rsminus1)〈ρν〉

where

A = minus(sminus r)minus1r〈εiηminusν〉r〈εi+1φ+micro〉s〈εiφ+micro〉s〈εi+1ηminusν〉rsminus1(y pi(x1))

+ (sminus r)minus1r〈εi+1micro〉s〈εimicro〉rsminus1(y pprimei(x1))

Similarly

〈v | ad(S(fi))v1〉 = B times (y1 x)(ωprimeη ωφ1)(ω


where

B = minus(sminus r)minus1rminus〈εimicro1minusαi〉r〈εi+1ν1minusαi〉s〈εiν1minusαi〉sminus〈εi+1micro1minusαi〉

times (ωprimeη ωi)((ωprime

i)minus1 ωφ)(y pi(x1))

+ (sminus r)minus1r〈εi+1ν1minusαi〉s〈εiν1minusαi〉(y pprimei(x1))

Comparing both sides we conclude that 〈ad(fi)v | v1〉 = 〈v | ad(S(fi))v1〉

(b) An argument analogous to that for (a) can be used in this case

Remark 49 It was shown in [4] that U is isomorphic to the Drinfelrsquod doubleD(B (Bprime)coop) where B is the Hopf subalgebra of U generated by the elementsωplusmn1

j ej 1 j lt n and (Bprime)coop is the subalgebra of U generated by the elements(ωprime

j)plusmn1 fj 1 j lt n but with the opposite co-product This realization of U

allows us to define the Rosso form R on U according to [18 p 77]

R〈aotimes b | aprime otimes bprime〉 = (bprime S(a))(Sminus1(b) aprime) for a aprime isin B and b bprime isin (Bprime)coop

The Rosso form is also an ad-invariant form on U but it does not admit thedecomposition in (43) Rather it has the following factorization (we suppress thetensor symbols in the notation)

R〈xωφωprimeη(ωprime

νminus1y) | x1ωφ1ω

primeη1

(ωprimeν1

minus1y1)〉

= R〈x | ωprimeν1

minus1y1〉 middotR〈ωφω

primeη | ωφ1ω

primeη1

〉 middotR〈ωprimeν

minus1y | x1〉 (44)

That is to say the form R respects the decompositionoplus

microνisinQ+

U+micro otimes U0 otimes (ωprime

νminus1Uminus

minusν) simminusrarr U

For (η φ) isin QtimesQ we define a group homomorphism χηφ QtimesQ rarr Ktimes by

χηφ(η1 φ1) = (ωprimeη ωφ1)(ω

primeη1 ωφ) (η1 φ1) isin QtimesQ (45)


Lemma 410 Assume that rksl = 1 if and only if k = l = 0 If χηφ = χηprimeφprime then(η φ) = (ηprime φprime)

Proof If χηφ = χηprimeφprime then

χηφ(0 αj) = r〈εj η〉s〈εj+1η〉 = χηprimeφprime(0 αj) = r〈εj ηprime〉s〈εj+1ηprime〉

Since r〈εj η〉minus〈εj ηprime〉s〈εj+1η〉minus〈εj+1ηprime〉 = 1 it must be that 〈εj η〉 = 〈εj ηprime〉 for all1 j n From this it is easy to see that η = ηprime Similar considerations withχηφ(αi 0) = χηprimeφprime(αi 0) show that φ = φprime

Proposition 411 Assume that rksl = 1 if and only if k = l = 0 Then thebilinear form 〈middot | middot〉 is non-degenerate on U

Proof It is sufficient to argue that if u isin UminusminusνU

0U+micro and 〈u | v〉 = 0 for all v isin

UminusminusmicroU

0U+ν then u = 0 Choose for each micro isin Q+ a basis umicro

1 umicro2 u

microdmicro

dmicro =dimU+

micro of U+micro Owing to lemma 45 we can take a dual basis vmicro

1 vmicro2 v

microdmicro

ofUminus

minusmicro ie (vmicroi u

microj ) = δij Then the set

(vνi ω

primeν

minus1)ωprimeηωφu

microj | 1 i dν 1 j dmicro and η φ isin Q

is a basis of UminusminusνU

0U+micro From the definition of the bilinear form we obtain

〈(vνi ω

primeν


microj | (vmicro

kωprimemicro

minus1)ωη1ωφ1uνl 〉

= (vνi u

νl )(vmicro

k umicroj )(ωprime


= δilδjk(ωprimeη ωφ1)(ω


Now write u =sum

ijηφ θijηφ(vνi ω

primeν


microj and take v = (vmicro

kωprimemicro

minus1)ωprimeη1ωφ1u

νl

with 1 k dmicro and 1 l dν and η1 φ1 isin Q From the assumption 〈u | v〉 = 0we have sum

ηφ

θlkηφ(ωprimeη ωφ1)(ω

primeη1 ωφ)(rsminus1)〈ρν〉 = 0 (46)

for all 1 k dmicro and 1 l dν and for all η1 φ1 isin Q Equation (46) can bewritten as sum

ηφ

θlkηφ(rsminus1)〈ρν〉χηφ = 0

for each k and l (where 1 k dmicro and 1 l dν) It follows from lemma 410and the linear independence of distinct characters (Dedekindrsquos theorem see forexample [17 p 280]) that θlkηφ = 0 for all η φ isin Q and for all l and k Hencewe have u = 0 as desired

5 The centre of U = Urs(sln)

Throughout this section we make the following assumption

rksl = 1 if and only if k = l = 0 (51)

Under this hypothesis we see that for ζ isin Q

Uζ = z isin U | ωizωminus1i = r〈εiζ〉s〈εi+1ζ〉z and ωprime

iz(ωprimei)

minus1 = r〈εi+1ζ〉s〈εiζ〉z (52)


We denote the centre of U by Z Since any central element of U must commutewith ωi and ωprime

i for all i it follows from (52) that Z sub U0 We define an algebraautomorphism γminusρ U0 rarr U0 by

γminusρ(ai) = rminus〈ρεi〉ai and γminusρ(bi) = sminus〈ρεi〉bi (53)

Thusγminusρ(ωprime

iωminus1i ) = (rsminus1)〈ραi〉ωprime

iωminus1i (54)

Definition 51 The Harish-Chandra homomorphism ξ Z rarr U0 is the restrictionto Z of the map

γminusρ π U0πminusrarr U0 γminusρ

minusminusrarr U0

where π U0 rarr U0 is the canonical projection

Proposition 52 ξ is an injective algebra homomorphism

Proof Note that U0 = U0 oplusK where K =oplus

νgt0 UminusminusνU

0U+ν is the two-sided ideal

in U0 which is the kernel of π and hence of ξ Thus ξ is an algebra homomorphismAssume that z isin Z and ξ(z) = 0 Writing z =

sumνisinQ+ zν with zν isin Uminus

minusνU0U+

ν we have z0 = 0 Fix any ν isin Q+ 0 minimal with the property that zν = 0Also choose bases yk and xl for Uminus

minusν and U+ν respectively We may write

zν =sum

kl yktklxl for some tkl isin U0 Then

0 = eiz minus zei=

sum

γ =ν

(eizγ minus zγei) +sum

kl

(eiyk minus ykei)tklxl +sum

kl

yk(eitklxl minus tklxlei)

Note that eiyk minus ykei isin Uminusminus(νminusαi)

U0 Recalling the minimality of ν we see that onlythe second term belongs to Uminus

minus(νminusαi)U0U+

ν Therefore we havesum

kl

(eiyk minus ykei)tklxl = 0

By the triangular decomposition of U and the fact that xl is a basis of U+ν we

getsum

k eiyktkl =sum

k ykeitkl for each l and for all 1 i lt nNow we fix l and consider the irreducible module L(λ) for λ isin Λ+

sl Let vλ be the

highest weight vector of L(λ) and set m =sum

k yktklvλ Then for each i

eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0

Hence m generates a proper submodule of L(λ) The irreducibility of L(λ) forcesm = 0 Choosing an appropriate λ isin Λ+

slwith lemma 22 in mind we have

sum

k

yktkl = 0

Since yk is a basis for Uminusminusν it must be that tkl = 0 for each k But l can be

arbitrary so we get zν = 0 which is a contradiction


Proposition 53 If n is even set

z = ωprime1ω

prime3 middot middot middotωprime

nminus1ω1ω3 middot middot middotωnminus1 = a1 middot middot middot anb1 middot middot middot bn (55)

Then z is central and ξ(z) = z

Proof We have

eiz = rminus〈ε1+ε2+middotmiddotmiddot+εnαi〉sminus〈ε1+ε2+middotmiddotmiddot+εnαi〉zei = zei for all 1 i lt n

Similarly fiz = zfi for all 1 i lt n so that z is central Finally observe that

ξ(z) = rminus〈ρε1+ε2+middotmiddotmiddot+εn〉sminus〈ρε1+ε2+middotmiddotmiddot+εn〉z = z

By introducing appropriate factors into the definition of the homomorphism λ

in (22) we are able to obtain a duality between U0 and its characters Thus forany λ micro isin Λsl we let λmicro U0 rarr K be the algebra homomorphism defined by

λmicro(ωj) = r〈εj λ〉s〈εj+1λ〉(rsminus1)〈αj micro〉

λmicro(ωprimej) = r〈εj+1λ〉s〈εj λ〉(rsminus1)〈αj micro〉

(56)

In particular λ0 is just the homomorphism λ on U0

Lemma 54 Assume that u = ωprimeηωφ with η φ isin Q If λmicro(u) = 1 for all λ micro isin Λsl

then u = 1

Proof We write η =sum

i ηiαi and φ =sum

i φiαi Then i0(u) = i0(ωprimeηωφ) =

rAisBi = 1 for each 1 i lt n where

Ai = 〈ε2 13i〉η1 + middot middot middot + 〈εn 13i〉ηnminus1 + 〈ε1 13i〉φ1 + middot middot middot + 〈εnminus1 13i〉φnminus1

Bi = 〈ε1 13i〉η1 + middot middot middot + 〈εnminus1 13i〉ηnminus1 + 〈ε2 13i〉φ1 + middot middot middot + 〈εn 13i〉φnminus1

It follows from assumption (51) that Ai = Bi = 0 It is now straightforward to seefrom the definitions that for 1 i lt n

Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

After elementary manipulations we have ηi = φi for all 1 i lt n and η2 = η4 =middot middot middot = 0 and

η1 = η3 = middot middot middot =2n

nminus1sum

j=1

ηj =2nlη1

where l = 12n if n is even and l = 1

2 (nminus 1) if n is odd Therefore u = 1 when n isodd and u = zη1 η1 isin Z when n is even Now when n is even

1 = 01(u) = (01(z))η1 = (rsminus1)2η1

Thus η1 = 0 and u = 1 as desired


Corollary 55 Assume that u isin U0 If λmicro(u) = 0 for all (λ micro) isin Λsl times Λslthen u = 0

Proof Corresponding to each (η φ) isin QtimesQ is the character on the group Λsl times Λsl

defined by(λ micro) rarr λmicro(ωprime

ηωφ)

It follows from lemma 54 that different (η φ) give rise to different charactersSuppose now that u =

sumθηφω

primeηωφ where θηφ isin K By assumption

sumθηφ

λmicro(ωprimeηωφ) = 0

for all (λ micro) isin Λsl times Λsl By the linear independence of different characters θηφ = 0for all (η φ) isin QtimesQ and so u = 0

Set

U0 =

oplus

ηisinQ

Kωprimeηωminusη (57)

U0 =

U0

if n is oddoplus

Kωprimeηωφ if n is even

(58)

where in the even case the sum is over the pairs (η φ) isin QtimesQ which satisfy thefollowing condition if η =

sumnminus1i=1 ηiαi and φ =

sumnminus1i=1 φiαi then

η1 + φ1 = η3 + φ3 = middot middot middot = ηnminus1 + φnminus1

η2 + φ2 = η4 + φ4 = middot middot middot = ηnminus2 + φnminus2 = 0

(59)

Clearly U0 U0

when n is even as z isin U0 U0

There is an action of the Weyl group W on U0 defined by

σ(aλbmicro) = aσ(λ)bσ(micro) (510)

for all λ micro isin Λ and σ isin W We want to know the effect of this action on a prod-uct ωprime

ηωφ where η =sumnminus1

i=1 ηiαi and φ =sumnminus1

i=1 φiαi For this write ωprimeηωφ = amicrobν

where micro =sumn

i=1 microiεi ν =sumn

i=1 νiεi and

microi = ηiminus1 + φi νi = ηi + φiminus1 (511)

for all 1 i n (where η0 = ηn = φ0 = φn = 0) Then for the simple reflection σkwe have

σk(ωprimeηωφ) = σk(amicrobν)

= amicrobνaminus〈microαk〉αk

bminus〈ναk〉αk

= ωprimeηωφ(aka

minus1k+1)

minus〈microαk〉(bkbminus1k+1)

minus〈ναk〉

= ωprimeηωφ(akbk+1)minus〈microαk〉(ak+1bk)〈microαk〉(bminus1

k bk+1)〈micro+ναk〉

= ωprimeηωφ(ωprime

kωminus1k )microkminusmicrok+1(bminus1

k bk+1)microk+νkminusmicrok+1minusνk+1


kωminus1k )ηkminus1minusηk+φkminusφk+1(bminus1

k bk+1)ηkminus1+φkminus1minusηk+1minusφk+1 (512)


From this it is apparent that the subalgebras U0 and U0

of U0 are closed underthe W -action Moreover the W -action on U0

amounts to

σ(ωprimeηωminusη) = ωprime

σ(η)ωminusσ(η) for all σ isin W and η isin Q

Proposition 56 We have

σ(λ)micro(u) = λmicro(σminus1(u)) (513)

for all u isin U0 σ isin W and λ micro isin Λsl

Proof First we show that σ(λ)0(u) = λ0(σminus1(u)) Since

σi(j)0(ak) = r〈εkσi(j)〉 = r〈σi(εk)j〉 = j 0(σi(ak))

and

σi(j)0(bk) = s〈εkσi(j)〉 = s〈σi(εk)j〉 = j 0(σi(bk))

for 1 i j lt n and 1 k n we see that (513) holds in this case Next we arguethat 0micro(u) = 0micro(σminus1(u)) It is sufficient to suppose that u = ωprime

ηωφ and σ = σk

for some k Then (512) shows that

σk(ωprimeηωφ) = ωprime

ηωφ(ωprimekω

minus1k )ηkminus1minusηk+φkminusφk+1

Now using the definition of 0micro we have 0micro(σk(ωprimeηωφ)) = 0micro(ωprime

ηωφ) Finallysince λmicro(u) = λ0(u)0micro(u) the assertion follows

We define

(U0 )W = u isin U0

| σ(u) = u forallσ isin W and (U0 )W = U0

cap (U0 )W (514)

Lemma 57 Assume that u isin U0 and λmicro(u) = σ(λ)micro(u) for all λ micro isin Λsl andσ isin W Then u isin (U0

)W

Proof Suppose that u =sum

(ηφ)θηφωprimeηωφ isin U0 satisfies λmicro(u) = σ(λ)micro(u) for all

λ micro isin Λsl and σ isin W Thensum

(ηφ)

θηφλmicro(ωprime

ηωφ) =sum

(ζψ)

θζψσi(λ)micro(ωprime

ζωψ)

for all λ micro isin Λsl If κηφ and κiζψ are the characters on Λsl times Λsl defined by

κηφ(λ micro) = λmicro(ωprimeηωφ) and κi

ζψ(λ micro) = σi(λ)micro(ωprimeζωψ)

then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)

Each side of (515) is a linear combination of different characters by lemma 54Now if θηφ = 0 then κηφ = κi

ζψ for some (ζ ψ) Moreover for each 1 j lt n

κηφ(0 13j) = 0j (ωprimeηωφ) = (rsminus1)〈η+φj〉

= κiζψ(0 13j) = 0j (ωprime

ζωψ) = (rsminus1)〈ζ+ψj〉


Thus 〈η + φ13j〉 = 〈ζ + ψ13j〉 for all j and so

η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum

j ζjαj and ψ =sum

j ψjαj then the equationκηφ(13i 0) = κi

ζψ(13i 0) along with (516) yields

ηiminus1 + φiminus1 + φi = ζi + ψiminus1 + ψi+1 and ηiminus1 + ηi + φiminus1 = ζiminus1 + ζi+1 + ψi

(with the convention that η0 = ηn = φ0 = φn = ζ0 = ζn = ψ0 = ψn = 0) Thus

ηiminus1 + φiminus1 = ηi+1 + φi+1 1 i lt n (517)

This implies that if θηφ = 0 then ωprimeηωφ isin U0

As a result u isin U0

By proposition 56 λmicro(u) = σ(λ)micro(u) = λmicro(σminus1(u)) for all λ micro isin Λsl andσ isin W But then u = σminus1(u) by corollary 55 so u isin (U0

)W as claimed

Proposition 58 The image of the centre Z of U under the Harish-Chandra homo-morphism satisfies

ξ(Z) sube (U0 )W

Proof Assume that z isin Z Choose micro λ isin Λsl and assume that 〈λ αi〉 0 for some(fixed) value i Let vλmicro isin M(λmicro) be the highest weight vector Then

zvλmicro = π(z)vλmicro = λmicro(π(z))vλmicro = λ+ρmicro(ξ(z))vλmicro

for all z isin Z Thus z acts as the scalar λ+ρmicro(ξ(z)) on M(λmicro)Using [5 lemma 23] it is easy to see that

eif〈λαi〉+1i vλmicro =

(

[〈λ αi〉 + 1]f 〈λαi〉i

rminus〈λαi〉ωi minus sminus〈λαi〉ωprimei

r minus s

)

vλmicro = 0

where for k 1

[k] =rk minus skr minus s (518)

Thus ejf〈λαi〉+1i vλmicro = 0 for all 1 j lt n Note that

zf〈λαi〉+1i vλmicro = π(z)f 〈λαi〉+1

i vλmicro

= σi(λ+ρ)minusρmicro(π(z))f 〈λαi〉+1i vλmicro

= σi(λ+ρ)micro(ξ(z))f 〈λαi〉+1i vλmicro

On the other hand since z acts as the scalar λ+ρmicro(ξ(z)) on M(λmicro)

zf〈λαi〉+1i vλmicro = λ+ρmicro(ξ(z))f 〈λαi〉+1

i vλmicro

Thereforeλ+ρmicro(ξ(z)) = σi(λ+ρ)micro(ξ(z)) (519)

Now we claim that (519) holds for an arbitrary choice of λ isin Λsl Indeed if〈λ αi〉 = minus1 then λ+ ρ = σi(λ+ ρ) and so (519) holds trivially For λ such that〈λ αi〉 lt minus1 we let λprime = σi(λ+ ρ) minus ρ Then 〈λprime αi〉 0 and we may apply (519)


to λprime Substituting λprime = σi(λ + ρ) minus ρ into the result we see that (519) holds forthis case also

Since i can be arbitrary andW is generated by the reflections σi we deduce that

λmicro(ξ(z)) = σ(λ)micro(ξ(z)) (520)

for all λ micro isin Λsl and for all σ isin W The assertion of the proposition then followsimmediately from lemma 57

Lemma 59 z isin Z if and only if ad(x)z = (ı ε)(x)z for all x isin U where ε U rarr K

is the co-unit and ı K rarr U is the unit of U

Proof Let z isin Z Then for all x isin U

ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z

Conversely assume that ad(x)z = (ı ε)(x)z for all x isin U Then

ωizωminus1i = ad(ωi)z = (ı ε)(ωi)z = z

Similarly ωprimeiz(ω

primei)

minus1 = z Furthermore

0 = (ı ε)(ei)z = ad(ei)z = eiz + ωiz(minusωminus1i )ei = eiz minus zei

and

0 = (ı ε)(fi)z = ad(fi)z = z(minusfi(ωprimei)

minus1) + fiz(ωprimei)

minus1 = (minuszfi + fiz)(ωprimei)

minus1

Hence z isin Z

Lemma 510 Assume that Ψ Uminusminusmicro times U+

ν rarr K is a bilinear map and let (η φ) isinQtimesQ There then exists u isin Uminus

minusνU0U+

micro such that

〈u | (yωprimemicro

minus1)ωprimeη1ωφ1x〉 = (ωprime

η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)

for all x isin U+ν y isin Uminus

minusmicro and (η1 φ1) isin QtimesQ

Proof As in the proof of proposition 411 for each micro isin Q+ we choose an arbitrarybasis umicro

1 umicro2 u

microdmicro

(dmicro = dimU+micro ) of U+

micro and a dual basis vmicro1 v

micro2 v

microdmicro

of Uminusminusmicro

such that (vmicroi u

microj ) = δij If we set

u =sum

ij

Ψ(vmicroj u

νi )vν

i (ωprimeν)minus1ωprime

ηωφumicroj (rsminus1)minus〈ρν〉

then it is straightforward to verify that u satisfies equation (521)

We define a U -module structure on the dual space Ulowast by (xmiddotf)(v) = f(ad(S(x))v)for f isin Ulowast and x isin U Also we define a map β U rarr Ulowast by setting

β(u)(v) = 〈u | v〉 for u v isin U (522)

Then β is an injective U -module homomorphism by propositions 48 and 411 wherethe U -module structure on U is given by the adjoint action


Definition 511 Assume that M is a finite-dimensional U -module For each m isinM and f isin Mlowast we define cfm isin Ulowast by cfm(v) = f(v middotm) v isin U

Proposition 512 Assume that M is a finite-dimensional U -module such that

M =oplus

λisinwt(M)

Mλ and wt(M) sub Q

For each f isin Mlowast and m isin M there exists a unique u isin U such that

cfm(v) = 〈u | v〉 for all v isin U

Proof The uniqueness follows immediately from proposition 411 Since cfm de-pends linearly on m we may assume that m isin Mλ for some λ isin Q For

v = (yωprimemicro

minus1)ωprimeη1ωφ1x x isin U+

ν y isin Uminusminusmicro (η1 φ1) isin QtimesQ

we have

cfm(v) = cfm((yωprimemicro

minus1)ωprimeη1ωφ1x)

= f((yωprimemicro

minus1)ωprimeη1ωφ1xm)

= ν+λ(ωprimeη1ωφ1)f((yω

primemicro

minus1)xm)

Note that (y x) rarr f((yωprimemicro

minus1)xm) is bilinear and (41) gives us

(ωprimeη1 ωminusνminusλ) = ν+λ(ωprime

η1) and (ωprime

ν+λ ωφ1) = ν+λ(ωφ1)

Thuscfm(v) = (ωprime

η1 ωminusνminusλ)(ωprime

ν+λ ωφ1)f(y(ωprimemicro)minus1xm)

and lemma 510 enables us to find uνmicro isin UminusminusνU

0U+micro such that cfm(v) = 〈uνmicro | v〉

for all v isin UminusminusmicroU

0U+ν

Now for an arbitrary v isin U we write v =sum

(microν) vmicroν with vmicroν isin UminusminusmicroU

0U+ν

Since M is finite-dimensional there is a finite set F of pairs (micro ν) isin Q times Q suchthat

cfm(v) = cfm

( sum

(microν)isinFvmicroν

)

for all v isin U

Setting u =sum

(microν)isinF uνmicro and using lemma 47 we have

cfm(v) = cfm

( sum


)

=sum

(microν)isinFcfm(vmicroν)

=sum

(microν)isinF〈uνmicro | vmicroν〉 =

sum

(microν)isinF〈uνmicro | v〉 = 〈u | v〉

This completes the proof

The category O of representations of U is naturally defined We refer the readerto [4 sect 4] for the precise definition All highest weight modules with weights in Λslsuch as the Verma modules M(λ) and the irreducible modules L(λ) for λ isin Λslbelong to category O


Assume that M is any U -module in category O and define a linear map Θ M rarr M by

Θ(m) = (rsminus1)minus〈ρλ〉m (523)

for all m isin Mλ λ isin Λsl We claim that

Θu = S2(u)Θ for all u isin U (524)

Indeed we have only to check this holds when u is one of the generators ei fi ωi

or ωprimei and for them the verification of (524) is straightforward

For λ isin Λ+sl

we define fλ isin Ulowast as given by the following trace map

fλ(u) = trL(λ)(uΘ) u isin U

Lemma 513 Assume that λ isin Λ+sl

cap Q Then fλ isin Im(β) where β is defined inequation (522)

Proof Let k = dimL(λ) and fix a basis mi for L(λ) and its dual basis fi forL(λ)lowast We now have

fλ(v) = trL(λ)(vΘ) =ksum

i=1

cfiΘmi(v)

By proposition 512 we can find ui isin U such that cfiΘmi(v) = 〈ui | v〉 for each i

1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)

Thus fλ isin Im(β)

Proposition 514 The element zλ = βminus1(fλ) is contained in the centre Z foreach λ isin Λ+

slcapQ

Proof Using (524) we have for all x isin U

(Sminus1(x)fλ)(u) = fλ(ad(x)u)

= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)

= (ı ε)(x) trL(λ)(uΘ) = (ı ε)(x)fλ(u)


Substituting x for Sminus1(x) in the above we deduce from ε S = ε the relation

xfλ = (ı ε)(x)fλ

We can write

xfλ = xβ(βminus1(fλ)) = β(ad(S(x))βminus1(fλ))

and

(ı ε)(x)fλ = (ı ε)(x)β(βminus1(fλ)) = β((ı ε)(x)βminus1(fλ))

Since β is injective ad(S(x))βminus1(fλ) = (ı ε)(x)βminus1(fλ) Since ε Sminus1 = ε substi-tuting x for S(x) we obtain

ad(x)βminus1(fλ) = (ı ε)(x)βminus1(fλ) for all x isin U

Therefore we may conclude from lemma 59 that βminus1(fλ) isin Z

This brings us to our main result on the centre of U

Theorem 515 Assume that r and s satisfy condition (51)

(i) If n is odd then the map ξ Z rarr (U0 )W = (U0

)W is an isomorphism

(ii) If n is even the centre Z is isomorphic under ξ to a subalgebra of (U0 )W

containing K[z zminus1] otimes (U0 )W ie K[z zminus1] otimes (U0

)W sube ξ(Z) sube (U0 )W where

the element z isin Z is defined in (55)

Proof We set zλ = βminus1(fλ) for λ isin Λ+sl

capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ

where zλν isin UminusminusνU

0U+ν and θηφ isin K Then for (η1 φ1) isin QtimesQ

〈zλ | ωprimeη1ωφ1〉 = 〈zλ0 | ωprime

η1ωφ1〉 =

sum

(ηφ)

θηφ(ωprimeη1 ωφ)(ωprime

η ωφ1)

On the other hand

〈zλ | ωprimeη1ωφ1〉 = β(zλ)(ωprime

η1ωφ1) = fλ(ωprime

η1ωφ1) = trL(λ)(ωprime

η1ωφ1Θ)

=sum

microλ

dim(L(λ)micro)(rsminus1)minus〈ρmicro〉micro(ωprimeη1ωφ1)

=sum

microλ

dim(L(λ)micro)(rsminus1)minus〈ρmicro〉(ωprimeη1 ωminusmicro)(ωprime

micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν

dim(L(λ)micro)(rsminus1)minus〈ρmicro〉χmicrominusmicro


where the characters χηφ are defined in (45) By assumption (51) lemma 410and the linear independence of distinct characters we obtain

θηφ =

dim(L(λ)η)(rsminus1)minus〈ρη〉 if η + φ = 00 otherwise

Hence

zλ0 =sum

microλ

dim(L(λ)micro)(rsminus1)minus〈ρmicro〉ωprimemicroωminusmicro

and by (54)

ξ(zλ) = minusρ(zλ0) =sum

microλ

dim(L(λ)micro)ωprimemicroωminusmicro (525)

Note that z = ξ(z) isin (U0 )W when n is even By propositions 52 and 58 it is

sufficient to show that (U0 )W sube ξ(Z) For λ isin Λ+

slcapQ we define

av(λ) =1

| W |sum

σisinW

σ(ωprimeλωminusλ) =

1|W |

sum

σisinW

ωprimeσ(λ)ωminusσ(λ) (526)

Remembering that for each η isin Q there exists σ isin W such that σ(η) isin Λ+sl

capQwe see that the set av(λ) | λ isin Λ+

slcap Q forms a basis of (U0

)W Thus we haveonly to show that av(λ) isin Im(ξ) for all λ isin Λ+

slcap Q We use induction on λ If

λ = 0 av(0) = 1 = ξ(1) Assume that λ gt 0 Since dimL(λ)micro = dimL(λ)σ(micro) forall σ isin W (proposition 23) and dimL(λ)λ = 1 we can rewrite (525) to obtain

ξ(zλ) = |W | av(λ) + |W |sum

dim(L(λ)micro) av(micro)

where the sum is over micro such that micro lt λ and micro isin Λ+sl

capQ By the induction hypoth-esis we get av(λ) isin Im(ξ) This completes the proof

Example 516 The centre Z of U = Urs(sl2) has a basis of monomials ziCj i isin Zj isin Z0 where z = ωprimeω (we omit the subscript since there is only one of them)and C is the Casimir element

C = ef +sω + rωprime

(r minus s)2 = fe+rω + sωprime

(r minus s)2

Now

ξ(z) = z and ξ(C) =(rs)12

(r minus s)2 (ω + ωprime)

Thus the monomials zicj i isin Z j isin Z0 where c = ω + ωprime give a basis for ξ(Z)The subalgebra (U0

)W consists of polynomials in a = ωprimeωminus1 + (ωprime)minus1ω = 2 av(α)Observe that a + 2 = zminus1c2 isin ξ(Z) but we cannot express c as an element ofK[z zminus1] otimes (U0

)W Since σ((ωprime)ωm) = (ωprime)mω we see that (U0)W has as a basisthe sums (ωprime)ωm + (ωprime)mω for all m isin Z and hence K[z zminus1]otimes(U0

)W ξ(Z) =

(U0 )W = (U0)W as no conditions are imposed by (59)


Acknowledgments

The authors thank the referee for many helpful suggestions GB was supported byNSF Grant no DMS-0245082 NSA Grant no MDA904-03-1-0068 and gratefullyacknowledges the hospitality of Seoul National University S-JK was supportedin part by KOSEF Grant no R01-2003-000-10012-0 and KRF Grant no 2003-070-C00001 K-HL was also supported in part by KOSEF Grant no R01-2003-000-10012-0

Appendix A

Lemma A1 The relations

(i) EijEkl minus EklEij = 0 for i j gt k + 1 l + 1

(ii) EijEkl minus rminus1EklEij minus Eil = 0 for i j = k + 1 l + 1

(iii) Eijej minus sminus1ejEij = 0 for i gt j

hold in U+

Proof The equations in (i) are obviousFor (ii) we fix j and l with j gt l and use induction on i If i = j this is just the

definition of Eil from (31) Assume that i gt j We then have

EijEjminus1l = eiEiminus1jEjminus1l minus rminus1Eiminus1jeiEjminus1l

= rminus1eiEjminus1lEiminus1j + eiEiminus1l minus rminus2Ejminus1lEiminus1jei minus rminus1Eiminus1lei

= rminus1Ejminus1lEij + Eil

by part (i) and the induction hypothesisTo establish (iii) we fix j and use induction on i When i = j + 1 the relation is

simply (32) with j instead of i Assume that i gt j + 1 We then have

Eijej = eiEiminus1jej minus rminus1Eiminus1jejei

= sminus1ejeiEiminus1j minus rminus1sminus1ejEiminus1jei

= sminus1ejEij

by (i) and induction

Lemma A2 In U+

(i) EijEjl minus rminus1sminus1EjlEij + (rminus1 minus sminus1)ejEil = 0 for i gt j gt l

(ii) EijEkl minus EklEij = 0 for i gt k l gt j

Proof The following expression can be easily verified by induction on l

EijEjl minus rminus1sminus1EjlEij + rminus1Eilej minus sminus1ejEil = 0 i gt j gt l (A 1)

We claim thatEj+1jminus1ej minus ejEj+1jminus1 = 0 (A 2)


Indeed we have ejEjjminus1 = sminus1Ejjminus1ej as in (33) and using this we get

Ej+1jEjjminus1 minus rminus1sminus1Ejjminus1Ej+1j

= ej+1ejEjjminus1 minus rminus1ejej+1Ejjminus1 minus rminus1sminus1Ejjminus1ej+1ej + rminus2sminus1Ejjminus1ejej+1

= sminus1ej+1Ejjminus1ej minus rminus1ejej+1Ejjminus1 minus rminus1sminus1Ejjminus1ej+1ej + rminus2ejEjjminus1ej+1

= sminus1Ej+1jminus1ej minus rminus1ejEj+1jminus1

On the other hand we also have from (A 1)

Ej+1jEjjminus1 minus rminus1sminus1Ejjminus1Ej+1j = sminus1ejEj+1jminus1 minus rminus1Ej+1jminus1ej

such that

(rminus1 + sminus1)Ej+1jminus1ej minus (rminus1 + sminus1)ejEj+1jminus1 = 0

Since we have assumed that rminus1 + sminus1 = 0 this implies (A 2)Now to demonstrate that

Eijek minus ekEij = 0 i gt k gt j (A 3)

we fix k and assume first that j = kminus1 The argument proceeds by induction on iIf i = k+ 1 then the expression in (A 3) becomes (A 2) (with k instead of j there)When i gt k + 1

Eikminus1ek = eiEiminus1kminus1ek minus rminus1Eiminus1kminus1ekei

= ekeiEiminus1kminus1 minus rminus1ekEiminus1kminus1ei = ekEikminus1

For the case j lt k minus 1 we have by induction on j

Eijek = Eij+1ejek minus rminus1ejEij+1ek

= ekEij+1ej minus rminus1ekejEij+1

= ekEij

so that (A 3) is verifiedAs a consequence the relations in part (i) follow from (A 1) and (A 3) while

those in (ii) can be derived easily from (A 3) by fixing i j and k and using inductionon l


(i) EijEkj minus sminus1EkjEij = 0 for i gt k gt j

(ii) EijEkl minus rminus1sminus1EklEij + (rminus1 minus sminus1)EkjEil = 0 for i gt k gt j gt l

hold in U+


Proof Part (i) follows from lemmas A1(iii) and A2(ii) For (ii) we apply inductionon l When l = j minus 1 part (i) and lemmas A1(ii) and A2(ii) imply that

EijEkjminus1

= EijEkjejminus1 minus rminus1Eijejminus1Ekj

= sminus1EkjEijejminus1 minus rminus1Eijejminus1Ekj

= rminus1sminus1Ekjejminus1Eij + sminus1EkjEijminus1 minus rminus2ejminus1EijEkj minus rminus1Eijminus1Ekj

= rminus1sminus1Ekjejminus1Eij + sminus1EkjEijminus1 minus rminus2sminus1ejminus1EkjEij minus rminus1EkjEijminus1

= rminus1sminus1Ekjminus1Eij + (sminus1 minus rminus1)EkjEijminus1

Now assume that l lt jminus1 Then Eijel = elEij and Ekjel = elEkj by lemma A1(i)and so by lemma A1(ii) we obtain

EijEkl = EijEkl+1el minus rminus1EijelEkl+1

= rminus1sminus1Ekl+1elEij + (sminus1 minus rminus1)EkjEil+1el

minus rminus2sminus1elEkl+1Eij minus rminus1(sminus1 minus rminus1)elEkjEil+1

= rminus1sminus1EklEij + (sminus1 minus rminus1)EkjEil

by the induction assumption

Lemma A4 In U+

EijEil minus sminus1EilEij = 0 i j gt l (A 4)

Proof First consider the case i = j If l = i minus 1 the above relation is merely thedefining relation in (33) Assume that l lt iminus 1 By induction on l we have

eiEil = eiEil+1el minus rminus1eielEil+1

= sminus1Eil+1elei minus rminus1sminus1elEil+1ei

= sminus1Eilei

When i gt j by induction on j and lemma A2(ii) we get

EijEil = Eij+1ejEil minus rminus1ejEij+1Eil

= Eij+1Eilej minus rminus1sminus1ejEilEij+1

= sminus1EilEij+1ej minus rminus1sminus1EilejEij+1

= sminus1EilEij

The proof of theorem 31 is now complete because we have

(1) lArrrArr lemma A1(ii)(2) lArrrArr lemma A1(i) and lemma A2(ii)(3) lArrrArr lemma A1(iii) lemma A3(i) and lemma A4(4) lArrrArr lemma A2(i) and lemma A3(ii)


References

1 P Baumann On the center of quantized enveloping algebras J Alg 203 (1998) 244ndash2602 G Benkart and T Roby Downndashup algebras J Alg 209 (1998) 305ndash344 (Addendum 213

(1999) 378)3 G Benkart and S Witherspoon A Hopf structure for downndashup algebras Math Z 238

(2001) 523ndash5534 G Benkart and S Witherspoon Two-parameter quantum groups and Drinfelrsquod doubles

Alg Representat Theory 7 (2004) 261ndash2865 G Benkart and S Witherspoon Representations of two-parameter quantum groups and

SchurndashWeyl duality In Hopf algebras (ed J Bergen S Catoiu and W Chin) LectureNotes in Pure and Applied Mathematics vol 237 pp 62ndash95 (New York Marcel Dekker2004)

6 G Benkart and S Witherspoon Restricted two-parameter quantum groups In Finitedimensional algebras and related topics Fields Institute Communications vol 40 pp 293ndash318 (Providence RI American Mathematical Society 2004)

7 L A Bokut and P Malcolmson GrobnerndashShirshov bases for quantum enveloping algebrasIsrael J Math 96 (1996) 97ndash113

8 W Chin and I M Musson Multiparameter quantum enveloping algebras J Pure ApplAlg 107 (1996) 171ndash191

9 R Dipper and S Donkin Quantum GLn Proc Lond Math Soc 63 (1991) 165ndash21110 V K Dobrev and P Parashar Duality for multiparametric quantum GL(n) J Phys A26

(1993) 6991ndash700211 V G Drinfelrsquod Quantum groups In Proc Int Cong of Mathematicians Berkeley 1986

(ed A M Gleason) pp 798ndash820 (Providence RI American Mathematical Society 1987)12 V G Drinfelrsquod Almost cocommutative Hopf algebras Leningrad Math J 1 (1990) 321ndash

34213 P I Etingof Central elements for quantum affine algebras and affine Macdonaldrsquos opera-

tors Math Res Lett 2 (1995) 611ndash62814 K R Goodearl and E S Letzter Prime factor algebras of the coordinate ring of quantum

matrices Proc Am Math Soc 121 (1994) 1017ndash102515 J A Green Hall algebras hereditary algebras and quantum groups Invent Math 120

(1995) 361ndash37716 J Hong Center and universal R-matrix for quantized Borcherds superalgebras J Math

Phys 40 (1999) 3123ndash314517 N Jacobson Basic algebra vol 1 (New York Freeman 1989)18 A Joseph Quantum groups and their primitive ideals Ergebnisse der Mathematik und

ihrer Grenzgebiete series 3 vol 29 (Springer 1995)19 A Joseph and G Letzter Separation of variables for quantized enveloping algebras Am

J Math 116 (1994) 127ndash17720 S-J Kang and T Tanisaki Universal R-matrices and the center of the quantum generalized

KacndashMoody algebras Hiroshima Math J 27 (1997) 347ndash36021 V K Kharchenko A combinatorial approach to the quantification of Lie algebras Pac J

Math 203 (2002) 191ndash23322 S M Khoroshkin and V N Tolstoy Universal R-matrix for quantized (super)algebras

Lett Math Phys 10 (1985) 63ndash6923 S M Khoroshkin and V N Tolstoy The CartanndashWeyl basis and the universal R-matrix

for quantum KacndashMoody algebras and superalgebras In Quantum Symmetries Proc IntSymp on Mathematical Physics Goslar 1991 pp 336ndash351 (River Edge NJ World Sci-entific 1993)

24 S M Khoroshkin and V N Tolstoy Twisting of quantum (super)algebras In General-ized Symmetries in Physics Proc Int Symp on Mathematical Physics Clausthal 1993pp 42ndash54 (River Edge NJ World Scientific 1994)

25 S Levendorskii and Y Soibelman Algebras of functions of compact quantum groups Schu-bert cells and quantum tori Commun Math Phys 139 (1991) 141ndash170

26 G Lusztig Finite dimensional Hopf algebras arising from quantum groups J Am MathSoc 3 (1990) 257ndash296


27 G Lusztig Canonical bases arising from quantized enveloping algebras J Am Math Soc3 (1990) 447ndash498

28 N Yu Reshetikhin Quasitriangular Hopf algebras and invariants of links Leningrad MathJ 1 (1990) 491ndash513

29 N Yu Reshetikhin L A Takhtadzhyan and L D Faddeev Quantization of Lie groupsand Lie algebras Leningrad Math J 1 (1990) 193ndash225

30 C M Ringel Hall algebras Banach Center Publicat 26 (1990) 433ndash44731 C M Ringel Hall algebras and quantum groups Invent Math 101 (1990) 583ndash59132 C M Ringel Hall algebras revisited Israel Math Conf Proc 7 (1993) 171ndash17633 C M Ringel PBW-bases of quantum groups J Reine Angew Math 470 (1996) 51ndash8834 M Rosso Analogues de la forme de Killing et du theoreme drsquoHarish-Chandra pour les

groupes quantiques Annls Scient Ec Norm Sup 23 (1990) 445ndash46735 M Takeuchi A two-parameter quantization of GL(n) Proc Jpn Acad A66 (1990) 112ndash

11436 M Takeuchi The q-bracket product and quantum enveloping algebras of classical types J

Math Soc Jpn 42 (1990) 605ndash62937 T Tanisaki Killing forms Harish-Chandra isomorphisms and universal R-matrices for

quantum algebras Int J Mod Phys A7 (suppl 01B) (1992) 941ndash96138 N Xi Root vectors in quantum groups Comment Math Helv 69 (1994) 612ndash63939 H Yamane A PoincarendashBirkhoffndashWitt theorem for the quantum group of type AN Proc

Jpn Acad A64 (1988) 385ndash38640 H Yamane A PoincarendashBirkhoffndashWitt theorem for quantized universal enveloping algebras

of type AN Publ RIMS Kyoto 25 (1989) 503ndash520

(Issued 9 June 2006 )


groups urs(gln) and urs(sln) are constructed when r and s are roots of unity Theserestricted quantum groups are Drinfelrsquod doubles and are ribbon Hopf algebras undersuitable restrictions on r and s

Much work has been done on the centre of quantum groups for finite-dimensionalsimple Lie algebras [1 12 19 28 29 34 37] and also for (generalized) KacndashMoody(super)algebras [13 16 20] The approach taken in many of these papers (andadopted here as well) is to define a bilinear form on the quantum group whichis invariant under the adjoint action This quantum version of the Killing form isoften referred to in the one-parameter setting as the Rosso form (see [34]) The nextstep involves constructing an analogue ξ of the Harish-Chandra map It is straight-forward to show that the map ξ is an injective algebra homomorphism The maindifficulty lies in determining the image of ξ and in finding enough central elementsto prove that the map ξ is surjective In the two-parameter case a new phenomenonarises the n odd and n even cases behave differently Additional central elementsarise when n is even which complicates the description in that case

Our paper is organized as follows In sect 2 we briefly recall the definition and basicproperties of the two-parameter quantum group U = Urs(sln) In sect 3 we describethe commutation relations which determine a GrobnerndashShirshov basis and allowa PBW basis to be constructed and we prove that the positive part of U has aniterated skew polynomial ring structure The next section is devoted to the con-struction of a bilinear form and the proof of its invariance under the adjoint actionIn the final section we define a Harish-Chandra homomorphism ξ and determinethe centre of U by specifying the image of ξ and constructing central elementsexplicitly

2 Two-parameter quantum groups

Let K be an algebraically closed field of characteristic 0 Assume that Φ is a finiteroot system of type Anminus1 with Π a base of simple roots We regard Φ as a subsetof a Euclidean space R

n with an inner product 〈middot middot〉 We let ε1 εn denote anorthonormal basis of R

n and suppose that Π = αj = εj minus εj+1 | j = 1 nminus 1and that Φ = εi minus εj | 1 i = j n

Fix non-zero elements r s in the field K Here we assume r = s Let U = Urs(gln)be the unital associative algebra over K generated by elements ej fj (1 j lt n)and aplusmn1

i bplusmn1i (1 i n) which satisfy the following relations

(R1) the aplusmn1i bplusmn1

j all commute with one another and aiaminus1i = bjb

minus1j = 1

(R2) aiej = r〈εiαj〉ejai and aifj = rminus〈εiαj〉fjai

(R3) biej = s〈εiαj〉ejbi and bifj = sminus〈εiαj〉fjbi

(R4) [ei fj ] =δijr minus s (aibi+1 minus ai+1bi)

(R5) [ei ej ] = [fi fj ] = 0 if |iminus j| gt 1

(R6) e2i ei+1 minus (r + s)eiei+1ei + rsei+1e2i = 0

eie2i+1 minus (r + s)ei+1eiei+1 + rse2i+1ei = 0



2i = 0


i+1fi = 0


j and (ωprimej)






minus1i = ωprime

j(ωprimej)

minus1 = 1



primei and ωprime



i)


i bplusmn1i









i


i)minus1









ζ =nminus1sum

i=1

ζiαi isin Q

Then



primeζ ωprime






U =oplus

ζisinQ

Uζ

We also haveU+ =

oplus

ζisinQ+

U+ζ and Uminus =

oplus

ζisinQ+

Uminusminusζ






For any λ =sumn




n (23)




n

nsum

j=1

εj

and let

Λ+sl


i=1

li13i









sl The irreducible



i=1


)





M =oplus

χ

Mχ



i)m for all i






3 PBW-type bases





























sumi aix





for all u isin U


ϕij(Ekl) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩






ij by


⎧⎪⎨

⎪⎩


0 otherwise











(ϕijϑij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise


(ϑijϕij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise












(i) (1 x1) = ε(x1) (y1 1) = ε(y1)












∆(x) isinoplus

0νmicro












i=1



i=1







i(x))



primei(x))








such that the map






i














ρ = 12

sum

αgt0

α =nminus1sum

i=1




microνisinQ+

(Uminusminusνω

primeν


simminusrarr U


Definition 46 Set

〈(yωprimeν


ν1







Note that

〈(yωprimeν


ν1


= 〈yωprimeν



ν1

minus1〉 (43)


microνisinQ+

(Uminusminusνω

primeν


simminusrarr U



〈UminusminusνU

0U+micro | Uminus

minusν1U0U+

micro1〉 = 0



ad(u)v =sum

(u)

u(1)vS(u(2))




〈ad(u)v | v1〉 = 〈v | ad(S(u))v1〉




v = (yωprimeν


ν1











which implies that









i)minus1


ηωφxfi(ωprimei)


ηωφx(ωprimei)

minus1


ηωφfix(ωprimei)



minus1


ηωφωprimeip

primei(x)(ω

primei)


ηωφx(ωprimei)

minus1








ηωφpprimei(x)

Now









times y1(ωprimeν1


ωφ1+αipi(x1)



primei(x1)











where



Similarly



where















primeη1

(ωprimeν1

minus1y1)〉



primeη | ωφ1ω

primeη1


minus1y | x1〉 (44)


microνisinQ+


νminus1Uminus













UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References








































2i = 0


i+1fi = 0


j and (ωprimej)






minus1i = ωprime

j(ωprimej)

minus1 = 1



primei and ωprime



i)


i bplusmn1i









i


i)minus1









ζ =nminus1sum

i=1

ζiαi isin Q

Then



primeζ ωprime






U =oplus

ζisinQ

Uζ

We also haveU+ =

oplus

ζisinQ+

U+ζ and Uminus =

oplus

ζisinQ+

Uminusminusζ






For any λ =sumn




n (23)




n

nsum

j=1

εj

and let

Λ+sl


i=1

li13i









sl The irreducible



i=1


)





M =oplus

χ

Mχ



i)m for all i






3 PBW-type bases





























sumi aix





for all u isin U


ϕij(Ekl) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩






ij by


⎧⎪⎨

⎪⎩


0 otherwise











(ϕijϑij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise


(ϑijϕij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise












(i) (1 x1) = ε(x1) (y1 1) = ε(y1)












∆(x) isinoplus

0νmicro












i=1



i=1







i(x))



primei(x))








such that the map






i














ρ = 12

sum

αgt0

α =nminus1sum

i=1




microνisinQ+

(Uminusminusνω

primeν


simminusrarr U


Definition 46 Set

〈(yωprimeν


ν1







Note that

〈(yωprimeν


ν1


= 〈yωprimeν



ν1

minus1〉 (43)


microνisinQ+

(Uminusminusνω

primeν


simminusrarr U



〈UminusminusνU

0U+micro | Uminus

minusν1U0U+

micro1〉 = 0



ad(u)v =sum

(u)

u(1)vS(u(2))




〈ad(u)v | v1〉 = 〈v | ad(S(u))v1〉




v = (yωprimeν


ν1











which implies that









i)minus1


ηωφxfi(ωprimei)


ηωφx(ωprimei)

minus1


ηωφfix(ωprimei)



minus1


ηωφωprimeip

primei(x)(ω

primei)


ηωφx(ωprimei)

minus1








ηωφpprimei(x)

Now









times y1(ωprimeν1


ωφ1+αipi(x1)



primei(x1)











where



Similarly



where















primeη1

(ωprimeν1

minus1y1)〉



primeη | ωφ1ω

primeη1


minus1y | x1〉 (44)


microνisinQ+


νminus1Uminus













UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References










































U =oplus

ζisinQ

Uζ

We also haveU+ =

oplus

ζisinQ+

U+ζ and Uminus =

oplus

ζisinQ+

Uminusminusζ






For any λ =sumn




n (23)




n

nsum

j=1

εj

and let

Λ+sl


i=1

li13i









sl The irreducible



i=1


)





M =oplus

χ

Mχ



i)m for all i






3 PBW-type bases





























sumi aix





for all u isin U


ϕij(Ekl) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩






ij by


⎧⎪⎨

⎪⎩


0 otherwise











(ϕijϑij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise


(ϑijϕij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise












(i) (1 x1) = ε(x1) (y1 1) = ε(y1)












∆(x) isinoplus

0νmicro












i=1



i=1







i(x))



primei(x))








such that the map






i














ρ = 12

sum

αgt0

α =nminus1sum

i=1




microνisinQ+

(Uminusminusνω

primeν


simminusrarr U


Definition 46 Set

〈(yωprimeν


ν1







Note that

〈(yωprimeν


ν1


= 〈yωprimeν



ν1

minus1〉 (43)


microνisinQ+

(Uminusminusνω

primeν


simminusrarr U



〈UminusminusνU

0U+micro | Uminus

minusν1U0U+

micro1〉 = 0



ad(u)v =sum

(u)

u(1)vS(u(2))




〈ad(u)v | v1〉 = 〈v | ad(S(u))v1〉




v = (yωprimeν


ν1











which implies that









i)minus1


ηωφxfi(ωprimei)


ηωφx(ωprimei)

minus1


ηωφfix(ωprimei)



minus1


ηωφωprimeip

primei(x)(ω

primei)


ηωφx(ωprimei)

minus1








ηωφpprimei(x)

Now









times y1(ωprimeν1


ωφ1+αipi(x1)



primei(x1)











where



Similarly



where















primeη1

(ωprimeν1

minus1y1)〉



primeη | ωφ1ω

primeη1


minus1y | x1〉 (44)


microνisinQ+


νminus1Uminus













UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References










































M =oplus

χ

Mχ



i)m for all i






3 PBW-type bases





























sumi aix





for all u isin U


ϕij(Ekl) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩






ij by


⎧⎪⎨

⎪⎩


0 otherwise











(ϕijϑij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise


(ϑijϕij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise












(i) (1 x1) = ε(x1) (y1 1) = ε(y1)












∆(x) isinoplus

0νmicro












i=1



i=1







i(x))



primei(x))








such that the map






i














ρ = 12

sum

αgt0

α =nminus1sum

i=1




microνisinQ+

(Uminusminusνω

primeν


simminusrarr U


Definition 46 Set

〈(yωprimeν


ν1







Note that

〈(yωprimeν


ν1


= 〈yωprimeν



ν1

minus1〉 (43)


microνisinQ+

(Uminusminusνω

primeν


simminusrarr U



〈UminusminusνU

0U+micro | Uminus

minusν1U0U+

micro1〉 = 0



ad(u)v =sum

(u)

u(1)vS(u(2))




〈ad(u)v | v1〉 = 〈v | ad(S(u))v1〉




v = (yωprimeν


ν1











which implies that









i)minus1


ηωφxfi(ωprimei)


ηωφx(ωprimei)

minus1


ηωφfix(ωprimei)



minus1


ηωφωprimeip

primei(x)(ω

primei)


ηωφx(ωprimei)

minus1








ηωφpprimei(x)

Now









times y1(ωprimeν1


ωφ1+αipi(x1)



primei(x1)











where



Similarly



where















primeη1

(ωprimeν1

minus1y1)〉



primeη | ωφ1ω

primeη1


minus1y | x1〉 (44)


microνisinQ+


νminus1Uminus













UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References






















































sumi aix





for all u isin U


ϕij(Ekl) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩






ij by


⎧⎪⎨

⎪⎩


0 otherwise











(ϕijϑij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise


(ϑijϕij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise












(i) (1 x1) = ε(x1) (y1 1) = ε(y1)












∆(x) isinoplus

0νmicro












i=1



i=1







i(x))



primei(x))








such that the map






i














ρ = 12

sum

αgt0

α =nminus1sum

i=1




microνisinQ+

(Uminusminusνω

primeν


simminusrarr U


Definition 46 Set

〈(yωprimeν


ν1







Note that

〈(yωprimeν


ν1


= 〈yωprimeν



ν1

minus1〉 (43)


microνisinQ+

(Uminusminusνω

primeν


simminusrarr U



〈UminusminusνU

0U+micro | Uminus

minusν1U0U+

micro1〉 = 0



ad(u)v =sum

(u)

u(1)vS(u(2))




〈ad(u)v | v1〉 = 〈v | ad(S(u))v1〉




v = (yωprimeν


ν1











which implies that









i)minus1


ηωφxfi(ωprimei)


ηωφx(ωprimei)

minus1


ηωφfix(ωprimei)



minus1


ηωφωprimeip

primei(x)(ω

primei)


ηωφx(ωprimei)

minus1








ηωφpprimei(x)

Now









times y1(ωprimeν1


ωφ1+αipi(x1)



primei(x1)











where



Similarly



where















primeη1

(ωprimeν1

minus1y1)〉



primeη | ωφ1ω

primeη1


minus1y | x1〉 (44)


microνisinQ+


νminus1Uminus













UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References









































for all u isin U


ϕij(Ekl) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩






ij by


⎧⎪⎨

⎪⎩


0 otherwise











(ϕijϑij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise


(ϑijϕij)(Ekl) =

⎧⎪⎨

⎪⎩


0 otherwise












(i) (1 x1) = ε(x1) (y1 1) = ε(y1)












∆(x) isinoplus

0νmicro












i=1



i=1







i(x))



primei(x))








such that the map






i














ρ = 12

sum

αgt0

α =nminus1sum

i=1




microνisinQ+

(Uminusminusνω

primeν


simminusrarr U


Definition 46 Set

〈(yωprimeν


ν1







Note that

〈(yωprimeν


ν1


= 〈yωprimeν



ν1

minus1〉 (43)


microνisinQ+

(Uminusminusνω

primeν


simminusrarr U



〈UminusminusνU

0U+micro | Uminus

minusν1U0U+

micro1〉 = 0



ad(u)v =sum

(u)

u(1)vS(u(2))




〈ad(u)v | v1〉 = 〈v | ad(S(u))v1〉




v = (yωprimeν


ν1











which implies that









i)minus1


ηωφxfi(ωprimei)


ηωφx(ωprimei)

minus1


ηωφfix(ωprimei)



minus1


ηωφωprimeip

primei(x)(ω

primei)


ηωφx(ωprimei)

minus1








ηωφpprimei(x)

Now









times y1(ωprimeν1


ωφ1+αipi(x1)



primei(x1)











where



Similarly



where















primeη1

(ωprimeν1

minus1y1)〉



primeη | ωφ1ω

primeη1


minus1y | x1〉 (44)


microνisinQ+


νminus1Uminus













UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References
















































(i) (1 x1) = ε(x1) (y1 1) = ε(y1)












∆(x) isinoplus

0νmicro












i=1



i=1







i(x))



primei(x))








such that the map






i














ρ = 12

sum

αgt0

α =nminus1sum

i=1




microνisinQ+

(Uminusminusνω

primeν


simminusrarr U


Definition 46 Set

〈(yωprimeν


ν1







Note that

〈(yωprimeν


ν1


= 〈yωprimeν



ν1

minus1〉 (43)


microνisinQ+

(Uminusminusνω

primeν


simminusrarr U



〈UminusminusνU

0U+micro | Uminus

minusν1U0U+

micro1〉 = 0



ad(u)v =sum

(u)

u(1)vS(u(2))




〈ad(u)v | v1〉 = 〈v | ad(S(u))v1〉




v = (yωprimeν


ν1











which implies that









i)minus1


ηωφxfi(ωprimei)


ηωφx(ωprimei)

minus1


ηωφfix(ωprimei)



minus1


ηωφωprimeip

primei(x)(ω

primei)


ηωφx(ωprimei)

minus1








ηωφpprimei(x)

Now









times y1(ωprimeν1


ωφ1+αipi(x1)



primei(x1)











where



Similarly



where















primeη1

(ωprimeν1

minus1y1)〉



primeη | ωφ1ω

primeη1


minus1y | x1〉 (44)


microνisinQ+


νminus1Uminus













UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References










































i(x))



primei(x))








such that the map






i














ρ = 12

sum

αgt0

α =nminus1sum

i=1




microνisinQ+

(Uminusminusνω

primeν


simminusrarr U


Definition 46 Set

〈(yωprimeν


ν1







Note that

〈(yωprimeν


ν1


= 〈yωprimeν



ν1

minus1〉 (43)


microνisinQ+

(Uminusminusνω

primeν


simminusrarr U



〈UminusminusνU

0U+micro | Uminus

minusν1U0U+

micro1〉 = 0



ad(u)v =sum

(u)

u(1)vS(u(2))




〈ad(u)v | v1〉 = 〈v | ad(S(u))v1〉




v = (yωprimeν


ν1











which implies that









i)minus1


ηωφxfi(ωprimei)


ηωφx(ωprimei)

minus1


ηωφfix(ωprimei)



minus1


ηωφωprimeip

primei(x)(ω

primei)


ηωφx(ωprimei)

minus1








ηωφpprimei(x)

Now









times y1(ωprimeν1


ωφ1+αipi(x1)



primei(x1)











where



Similarly



where















primeη1

(ωprimeν1

minus1y1)〉



primeη | ωφ1ω

primeη1


minus1y | x1〉 (44)


microνisinQ+


νminus1Uminus













UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References








































microνisinQ+

(Uminusminusνω

primeν


simminusrarr U


Definition 46 Set

〈(yωprimeν


ν1







Note that

〈(yωprimeν


ν1


= 〈yωprimeν



ν1

minus1〉 (43)


microνisinQ+

(Uminusminusνω

primeν


simminusrarr U



〈UminusminusνU

0U+micro | Uminus

minusν1U0U+

micro1〉 = 0



ad(u)v =sum

(u)

u(1)vS(u(2))




〈ad(u)v | v1〉 = 〈v | ad(S(u))v1〉




v = (yωprimeν


ν1











which implies that









i)minus1


ηωφxfi(ωprimei)


ηωφx(ωprimei)

minus1


ηωφfix(ωprimei)



minus1


ηωφωprimeip

primei(x)(ω

primei)


ηωφx(ωprimei)

minus1








ηωφpprimei(x)

Now









times y1(ωprimeν1


ωφ1+αipi(x1)



primei(x1)











where



Similarly



where















primeη1

(ωprimeν1

minus1y1)〉



primeη | ωφ1ω

primeη1


minus1y | x1〉 (44)


microνisinQ+


νminus1Uminus













UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References











































which implies that









i)minus1


ηωφxfi(ωprimei)


ηωφx(ωprimei)

minus1


ηωφfix(ωprimei)



minus1


ηωφωprimeip

primei(x)(ω

primei)


ηωφx(ωprimei)

minus1








ηωφpprimei(x)

Now









times y1(ωprimeν1


ωφ1+αipi(x1)



primei(x1)











where



Similarly



where















primeη1

(ωprimeν1

minus1y1)〉



primeη | ωφ1ω

primeη1


minus1y | x1〉 (44)


microνisinQ+


νminus1Uminus













UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References















































where



Similarly



where















primeη1

(ωprimeν1

minus1y1)〉



primeη | ωφ1ω

primeη1


minus1y | x1〉 (44)


microνisinQ+


νminus1Uminus













UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References














































UminusminusmicroU


1 umicro2 u

microdmicro

dmicro =dimU+


1 vmicro2 v

microdmicro

ofUminus



(vνi ω

primeν





〈(vνi ω

primeν


microj | (vmicro

kωprimemicro


= (vνi u

νl )(vmicro

k umicroj )(ωprime




Now write u =sum


primeν



kωprimemicro


νl


ηφ




ηφ








iz(ωprimei)








iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References












































iωminus1i (54)



minusminusrarr U0








minusνU0U+



zν =sum


0 = eiz minus zei=

sum

γ =ν


kl


kl






kl



getsum

k eiyktkl =sum


sl Let vλ be the



eim =sum

k

eiyktklvλ =sum

k

ykeitklvλ = 0



sum

k

yktkl = 0





z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References








































z = ωprime1ω




Proof We have








(56)



then u = 1








Ai =iminus1sum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +isum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0

Bi =isum

j=1

ηj minus i

n

nminus1sum

j=1

ηj +iminus1sum

j=1

φj minus i

n

nminus1sum

j=1

φj = 0



nminus1sum

j=1

ηj =2nlη1



1 = 01(u) = (01(z))η1 = (rsminus1)2η1






ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References










































ηωφ)


sumθηφω


sumθηφ



Set

U0 =

oplus

ηisinQ


U0 =

U0

if n is oddoplus


(58)






(59)

Clearly U0 U0








where micro =sumn


i=1 νiεi and






= ωprimeηωφ(aka

minus1k+1)


minus〈ναk〉












amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References









































amounts to








and



ηωφ and σ = σk



ηωφ(ωprimekω




We define

(U0 )W = u isin U0


cap (U0 )W (514)


)W




(ηφ)


ηωφ) =sum

(ζψ)


ζωψ)




then sum

(ηφ)

θηφκηφ =sum

(ζψ)

θζψκiζψ (515)








η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References








































η + φ = ζ + ψ (516)

If η =sum

j ηjαj φ =sum

j φjαj ζ =sum










)W as claimed


ξ(Z) sube (U0 )W





(



r minus s

)

vλmicro = 0

where for k 1




i vλmicro





i vλmicro











ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References














































ad(x)z =sum

(x)

x(1)zS(x(2)) = zsum

(x)

x(1)S(x(2)) = (ı ε)(x)z




primei)



and




minus1

Hence z isin Z



minusνU0U+

micro such that



η1 ωφ)(ωprime

η ωφ1)Ψ(y x) (521)




1 umicro2 u

microdmicro



micro2 v

microdmicro

of Uminusminusmicro



u =sum

ij

Ψ(vmicroj u

νi )vν










M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References









































M =oplus

λisinwt(M)

Mλ and wt(M) sub Q




v = (yωprimemicro



we have



= f((yωprimemicro



primemicro

minus1)xm)




η1) and (ωprime








0U+ν



0U+ν


cfm(v) = cfm

( sum


)

for all v isin U

Setting u =sum


cfm(v) = cfm

( sum


)

=sum


=sum


sum











For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References













































For λ isin Λ+sl







i=1

cfiΘmi(v)


1 i k Set u =sumk

i=1 ui such that

β(u)(v) =ksum

i=1

〈ui | v〉 =ksum

i=1

cfiΘmi(v) = fλ(v)



slcapQ



= trL(λ)

(sum

(x)

x(1)uS(x(2))Θ)

= trL(λ)

(

usum

(x)

S(x(2))Θx(1)

)

= trL(λ)

(

usum

(x)

S(x(2))S2(x(1))Θ)

= trL(λ)

(

uS

(sum

(x)

S(x(1))x(2)

)

Θ

)





We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References









































We can write


and














capQ and write

zλ =sum

ν0

zλν and zλ0 =sum

(ηφ)isinQtimesQ

θηφωprimeηωφ




η1ωφ1〉 =

sum

(ηφ)


η ωφ1)

On the other hand




η1ωφ1Θ)

=sum

microλ


=sum

microλ


micro ωφ1)

Now we may writesum

(ηφ)

θηφχηφ =sum

microν




θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References








































θηφ =


Hence

zλ0 =sum

microλ


and by (54)


microλ




slcapQ we define

av(λ) =1

| W |sum

σisinW


1|W |

sum

σisinW








ξ(zλ) = |W | av(λ) + |W |sum







(r minus s)2

Now






)W ξ(Z) =



Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References







































Acknowledgments


Appendix A





hold in U+










= sminus1ejEij


Lemma A2 In U+














such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References














































such that










= ekEij






hold in U+



EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References








































EijEkjminus1












Lemma A4 In U+





= sminus1Eilei





= sminus1EilEij




References







































References






































on the centre of two-parameter quantum groupskhlee/papers/tqm.pdfthe centre of two-parameter quantum...

Documents