affine quantum groups and equivariant k

Transformation Groups, Vol. 3, No. 3, 1998, pp. 269-299 C)Birkh~user Boston (1998)

AFFINE QUANTUM GROUPS AND EQUIVARIANT K-THEORY

E. VASSEROT*

D~partement de Math~matiques

Universitd de Cergy-Pontoise 2 Av. A. Chauvin

Cergy-Pontoise Cedex, France

[email protected]

Abstract. We give complete proofs of the K-theoretic construction of the quantized enveloping algebra of affine gl(n) sketched in [GV].

I n t r o d u c t i o n

The purpose of this paper is to give complete proofs of the K-theoretic construction of the quantized enveloping algebra of the affine Lie algebra ~(n) sketched in [GV]. This construction gives a geometric classification of finite dimensional simple modules of ~(n) in terms of intersection cohomology of graded nilpotent orbits of type A. The principal results were obtained jointly with V. Ginzburg. The formula for the Drinfeld polynomials of the simple modules in the last section was not given in [GV]. The basic construction is a rephrasing of the K-theoretic construction of the Iwahori- Hecke algebra given by Ginzburg. Since a detailed proof is available for the case of the Iwahori-Hecke algebra (see [CG]), the present paper deals mainly with the parts of the proof which are different from the affine Hecke algebra one, more precisely Theorem 1 whose proof is inspired by IV] and Theorem 2 whose proof is inspired by [BLM].

The first section of the paper contains some background and some technical lemmas on the K-groups of various flag varieties. We give complete proofs since we were not able to find a proper reference. Similar results in homology may be found in [CG], Chapter 2. The K-theoretic construction of the quantized enveloping algebra is done in Section two (Theorem 1 and Theorem 2). The simple and standard modules are studied in the third section. The reader should be warned that we give a geometric construction of the quantum group of type ~'[(n) while [GV] deals with type ~i(n).

*The author is partially supported by EEC grant no. ERB FMRX-CT97-0100.

Received August 15, 1997. Accepted March 3, 1998.

270 E. VASSEROT

1. T h e c o n v o l u t i o n a l g e b r a in e q u i v a r i a n t K - t h e o r y

1.0. Algebraic equivariant K-theory

Throughout the paper a variety means a complex quasi-projective algebraic variety and subvariety means closed subset for the Zariski topology. Given a complex linear algebraic group G and a G-variety X, let VectG(x) (resp. CohG(X)) denote the set of isomorphism classes of G-equivariant vector bundles (resp. G-equivariant coherent sheaves) on X. Let KG(x) be the complexified Grothendieck group of Coh G (X). The K-group has a natural R(G)-module structure where R(G) is the complexified representation ring of G. Given Jr 6 CohG(X) let [~] denote its class in KG(x). Recall a few properties of the equivariant K-theory (see [CG]for more details).

(a) For any proper map f : X ~ Y between two G-varieties X and Y, there is a derived direct image R f, : KG(x) --+ KG(y). The map R f, is a group homomorphism.

(b) If f : X ~ Y is flat (for instance an open embedding) or is a closed embedding of a smooth G-variety, and Y is smooth, there is an inverse image f* : KG(y) ~ KG(X). The map f* is a ring homomorphism for the product defined in subsection (g).

(c) Given G-varieties X1,X2, Y1, Y2 and a Cartesian square

X1 ~1~ Y1 g$ Sh

X2 I2> Y2

such that fl, f2 are proper and g,h are fiat, we have Rfl,g* = h* Rf2, (proper base change).

(d) Fix a smooth G-variety X and consider a smooth G-subvariety i : Y X. Let T~.X 6 VectG(y) be the conormal bundle of Y in X. Set

A(T~X) = ~(-1)i[AiT,}X] 6 KG(y). i

Then i* Ri, : KG(y) -+ KG(y) is the product by A(T~.X).

(e) Given a G-equivariant vector bundle ~r E : E -~ X with zero section a E : X ~+ E over a smooth G-variety X, the pull-back map r * : KG(x) -+ KG(E) is invertible (Thorn Isomorphism Theorem) and a m = r~ -1

(f) Consider a smooth G-variety X- For any g 6 G, the g-fixed point subvariety, denoted Xg, is smooth. Let i : Xg r X be the inclusion. Suppose moreover that G is abelian. Then Xg is a G-variety. The complexified Grothendieck ring R(G) is identified with the ring of regular functions on G and KG(x), KG(xg), can be viewed as sheaves over G. Then,

AFFINE QUANTUM GROUPS 271

the direct image Ri, induces an isomorphism of the localized K-groups KG(Xg)g-U-+KG(X)g and A(T~cgX ) is invertible in KG(Xg)g. Moreover, i f X is complete and p : X -+ pt and q : Xg --+ pt are the (proper) projections, then for any sheaf ~" �9 CohC(X) the Lefschetz formula holds :

Rp, [~-] = Rq,(A(T~:~X) -1 | [~-]).

L (g) If X is a smooth G-variety, there is a derived tensor product | making

L KG(x ) a ring. Given .~1,.~2 �9 CohG(X), the product is [5vl] | [~'2] -- 5"[5rl [].T2], where [] is the external tensor product and 5 : X ~-+ X x X is the diagonal embedding.

(h) Given two G-subvarieties Z1, Z2 C_ X, the derived tensor product on X induces a map KG(Z1) @ KG(z2) --+ KG(z1 rq Z2). This map depends on the ambient smooth variety X. In particular,

L e m m a 1. Given two smooth G-subvarieties Zt,Z2 C X with conormal bundles T~ X , T~2X , set Z = Z1 rq Z2 and N =- T~ X I z rq T~2XIz �9

VectG(Z). If Z is smooth and TZI l z N TZ21z = TZ, then for any ~1 �9 VectG(z1), .T'2 �9 VectG(z2)

L [5rl] | [~'2] = ~ ( - 1 ) i [ A i N | llz | 21z] e KG(Z).

i

Proof. Consider the deformation to the normal bundle T z X of Z in X. We get a fiat family M z X --+ C whose fiber over z E C is T z X if z = 0 and X otherwise. In particular, MzZ1 and MzZ2 can be viewed as closed subschemes in M z X . Thus, the intersection product can be computed on the special fiber (over z -- 0). We are reduced to the following situation: X is a vector space with a linear G-action and Z1, Z2 are subspaces stable by the action of G. The result follows by using Koszul resolutions. []

1.1. C o m p o s i t i o n o f c o r r e s p o n d e n c e s a n d c o n v o l u t i o n p r o d u c t Let G be a linear complex algebraic group and /141, M2 and M3 smooth G-varieties. Let

q~j : MI x M2 x Ma --+ Mi x Mj

be the projection along the factor not named. The G-action on each factor induces a natural G-action on the Cartesian product such that the projections q~ are G-equivariant. Let Z12 C M1 x M2 and Z23 C M2 x M3 be G-stable closed subvarieties. Assume that the restriction of q13 to q~lZ~2 N q~lZ23 is proper and let Z12 0 Z:3 be its image. The G-subvariety Z12 0 Z23 of M1 x M3 is called the composition of Z~2 and Z~3. Define a convolution map

* : KG(Z,2) | KG(Z 3) Kc(Z o Z23)

272 E. VASSEROT

as follows. Let ~'12, ~'~3 be two equivariant coherent sheaves respectively on ZI~ and Z23. Set

L

[712] . [~,3] = Rqla.(qL [~12] | qL[7~a])-

In this formula, the upper star stands for the pullback morphism, well- defined on smooth maps.

Let us now recall some technical results in intersection theory which will be used in the proof of the surjectivity theorem (Theorem 2). Suppose that F~ is a smooth G-variety for i = 1,2,3. Let G act on Ms = T*Fi in such a way that the projection ~i : Mi ~ Fi is G-equivariant. Let

p~ : FI • F2 x F3 ~ Fi x Fj

be the obvious projection. Fix a locally closed smooth G, subset O~j C Fi • Fj for all (ij) = (12), (23), (13). Denote by Z~ -- TS~ ~ (F ix 5 ) the conormal

bundle and let r i j : Z~ --+ O~ be the projection. Set Z = q~lZl: A q~lZ23 and O -- p~1012 N p~1023 . Suppose that

(a) the map q~3[z is proper, (b) the projections 01, --+/;2 and 023 --+ F2 are smooth fibrations, (c) the restriction p = P~a Io is a smooth and proper fibration over O~a.

As usual, Tp stands for the relative tangent sheaf along the fibers of p.

P r o p o s i t i o n 2. If the hypothesis (a-c) hold, then Z1, o Z,3 = Z13. If moreover the O~j areclosed, then V.%'12 E VectC(O12), Vbr2a E Vecta(O2a),

�9 * 7r* X-"/ l~ilr* ~ [hiTp @ pl2.~ ' Io [o]- ~1,[~1~1 ,~ [5~] = z ~ - J I ~ P * * | i

Proof. Property (a) implies that

z n (F~ • T*-V2 • F3) = (T$,~(F~ • F~) • F3) n (F~ • T ~ 3 (F2 • -v3)) = 0 ,

where 0 is identified with its image via the zero section. Thus,

(d) (TO1, x TF3) + (TF1 x T023 ) = T(F1 x F2 x F3).

On the other hand, the hypothesis (b) implies that the projection O -+ F~ is a smooth fibration. Thus, p~10~2 and p~1023 intersect transversally along O, i.e.,

(e) TO = (TO1, • TF3) ~ (TF~ • TO,a ).

The conormal bundle to q~Z~2 in M~ x M2 x/I//3, say N~, is the pullback of the vector bundle

T01~ @ T512 (El • F2)


by the obvious projection q~1Z12 --+ 0~2. Let 7r : Z --+ 0 be the projection. The formulas (d) and (e) give

Tp=(T012 x F 3 ) r l ( F 1 xT02~) and N121znN231z =.*Tp .

Since the projections O12 --+ F2 and 023 -+ F2 are smooth fibrations, we get smooth fibrations

T*012(F1xF2)C(T*F1)xF2 --+ F2 and T~23(F2xF3)CF2x(T*F3) --+ F2,

and Z is isomorphic to

z T$12(F1 x F2) T*o23(F2 x F3).

- 1 In particular, Z is smooth. Moreover, since p~101~ and P23 023 intersect --1 transversally (see (e)), q~l Z12 and q23 Z23 intersect transversally also. Thus

Lemma 1 implies tha t

L * 7 r * r ~ r - 1 * * q12 12 [J-12 J | q2J23 ['?"23] E ( - 1 ) 'Tr* A iT * * = [ p | |

i

Now O '~ O12 xF2 023 a n d t h u s

TS ( F l X F2 x F3 ) " 012 ( F 1 x F2 ) x T523 ( F2 x F3 ) ' ' Z.

Since the map p is a smooth fibration,

q13(~9(F1 x F2 x F3)) = T~913(F1 x F3).

Thus Z12 o Z23 = Z13. Finally consider the commutat ive square

0 P ) 013

Z q13) Zl 3

Since the vertical maps are vector bundles and q13 ]z is proper and surjective, the square is Cartesian. The result follows by smooth base change. []

As usual, for any locally-closed subset X, let X be the Zariski closure and set OX = -X \ X. Set OT = 0 r3 p~X013.

C o r o l l a r y 3. Suppose in addition to (a-c) that (f) O13 C O12 0 023 is open, (g) O13 r3 (0012 0 O23) = 013 r-] (O12 0 0023) = 0 .

Then Z13 C -Z12 0-Z23 is open and, for all Y12 �9 Vecta(Z12), 5r23 �9 Vecta(Z23), the restriction of [Y12] * [5~23] to Z13 is

~--~( l/i~r * Rp.[AiTp| | x - - ] 13

i

274 E. VASSEROT

Proof. Set ZT = Z N q~l Z~3. The hypothesis (g) implies that the commu- tative square

- - i -- ql s>

ZT q13> Z13

is Cartesian. Moreover, the vertical maps are open because of (f). Thus the restriction of [~'12] * [5r23] to Zla is

Rqls. ((q~2 [U12] ~ q~*~ [.F~])lz~).

Then, apply Proposit ion 2. The square

OT P > 013

ZT q13) Zl 3

is Cartesian (see the proof of Proposit ion 2) and we are done. []

C o r o l l a r y 4. Suppose that F3 = pt and 0 C F1 x F2 is a smooth G- subvariety with conormal bundle Zo C M1 x M2 such that

(h) the projection Zo --+ M1 is proper, (i) both projections Pl,O : 0 -+ F1 and P2,o : 0 --+ F2 are smooth

fibrations and Pl,O is proper. / f~" �9 VectG(F2) then,

[OZo] * 7r~ [~] -- Z ( - 1 ) i T r ~ R P l , O . [AiTpl,o | p~,o jr] �9 KG(M1). i

Proof. It suffices to apply Proposit ion 2 with O1~ = O, 5r1~ = Oo, 053 = F2 and 9r23 -- 9 v (see [V], Lemme 5). []

1.2. P a r t i t i o n s , m a t r i c e s , a n d 3 -a r r ays

For any positive integer n, set [n] = {1, 2 , . . . , n}. Fix two integers n, d _> 1. Let V C 1~ be the set of all compositions v = ( v l , . . . , v n ) of d into n non negative integers. Set ei = (51i,62i, . . . ,6ni) , SO that v = Y]~iviei. For v E V, let [v]i C [d] be the i-th segment of the composition v and put ~i = v l + v 2 + . . . + v i . Thus [v]i = [ l+Vi-l ,Vi] and [v] = ([v]l, [v]2, . . . , [v]n) is a part i t ion of [d] into n subsets.

Let M be the set of (n • n)-matrices A = (aq) with nonnegative integral entries such that Y']~i,j aij = d. For v, w E V, we put

M ( v , w ) = {A e M I ~ a i j =v i , Z a i j = w j } , j i

so that M = Hv,wev M(v , w).


For any v, w, we have a natural map

m = mv,w : Sd ~ M(v , w)

which identifies M(v , w) with the double coset space Sv\Sd/Sw. Namely, m assigns to a permutat ion a E Sd the matrix m a such that

m E. = # { a �9 [v]i I a(a) �9 [w]j}.

We introduce a partial order ~ on M(v , w) as follows. For A = (aij) and B = (bij) in M ( v , w ) , we say that A ~_ B, if for any 1 _< i < j _< n, we have

ars ~ ~ brs

and for any 1 < j < i _~ n, we have

ars _< ~ brs. ~>_i; s<_j r>_i; s<_j

Denote by _< the Bruhat order on Sd. The map m : (Sd, <) ~ ( M ( v , w ) , ~) is monotone (see [BLM], Lemma 3.6), i.e.,

a < T '- m ~ ~ m ~, Va, T E S d.

The order ~ on M = ]_I M(v , w) is the disjoint union of the orders on the components M(v , w), i.e., elements of different components are set to be incomparable. To a matrix A, we associate a partition [A] of the set [d] labelled by the set [n] x [n] as follows: to a pair (i, j ) corresponds the segment

[A]ij = [1 + ~ ahk, ~ ahk], (h,k)<(i,j) (h,k)~_(i,j)

where _< is the right lexicographic order defined by

( i , j )> (h,k) -.' '.. { j > k } or { j = k and i > h } .

Similarly, let A1, A2 be partitions of [d] such that

Ali = U [A]ij and A2j = U [A]ij. j i

Let T be the set of 3-arrays (tijk)l<i,j,k<n of nonnegative integers, such that Y~i,j,k tiJ k ~-- d. For any u, v, w E V, put

T(u , v, w) = {T E T I ~ tijk = ui, ~ tijk = vj, ~ tijk = wk }. j ,k i,k i , j

276 E. VASSEROT

If 1 < i < j _< 3 and T E T, let Tij E M be the matrix obtained by summing the entries of T with respect to the indices not named. Given two matrices A 6 M( t , u) and B 6 M(v , w), put

T ( A , B ) = { T E T IT12 = A, T2a = B}, M ( A , B ) = {T13 I T E T ( A , B ) } .

The set T(A, B) is empty unless u = v. For any partition I = (11, I 2 , . . . , In) of the set [d] into n subsets, let

SI -- SI1 X SI2 X "'" X Sin

be the subgroup of Sa consisting of permutations which preserve each subset. In particular, if v 6 V and A E M, put S,, = S[v] and SA = SIAl.

1.3. T he Flag m a n i f o l d

Fix d, n > 1. The complex linear group of rank d is denoted by GLa. Let (c1, e 2 , . . - , ca) be the canonical basis of C d. Let F be the variety of n-step partial flags in C a , i.e., filtrations of vector spaces

D = {0 = Do C_ D1 C_ D2 C_ . . . ___ On = C d }.

The connected components of F are parametrized by compositions v = (v l , . . . , vn) of d; the component Fv consists of flags D such that dimDi/Di_l = vs. For any composition, let Dv 6 F~ be the flag such that

Dv,i = t ~ Cej. je[1,sd

Denote by P,, the isotropy subgroup of Dv in GLa. The variety Fv is thus identified with the quotient GLd/Pv. Consider the diagonal action of GLd on F x F and F x F x F. The following result is essentially stated in [BLM], Lemma 3.7.

P r o p o s i t i o n 5. (a) Given two partitions v, w E V, the orbits of GLd in F x F w are parametrized by matrices A E M ( v , w ) : the orbit OA corresponding to A consists of pairs of flags (D, D') such that dim(D/M 03) = ~h<~.k<~ ahk. The closure of OA contains OB if and only if B < A with respect to the Bruhat order on M.

(b) For any A , B 6 M, the set p13(p~lOA Mp~IOB) C F 2 is stable with respect to the diagonal GL d action and

-1 U Oc. CeM(A,B)

(c) Moreover, if A, B E M there exists a unique A o B 6 M(A, B) such that M(A, B) C { C I C <_ A o B}.


(d) More precisely, given A, A', B, B ' E M such that A' <_ A and B' <_ B,

(A', B') # (A, B) ~ M(A' , B ' ) C {C I C < A o B}.

Proof. Claims (a) and (b) are immediate. As for the part (c), it suffices to prove that p~3(p~lOA Np~IOB) is irreducible. But P~2 : P~IOA Np~IOB --+ O A is a fibration with an irreducible base and fiber. Thus p~lO A r3 p~lO B is irreducible and so is its image under P~3. Part (d) follows easily. []

For any i -- 1, 2, 3, let p, : F 2 -+ F be the projections along the component not named. For any A, let P~.A be the restriction of p~ to O A. This map is a smooth fibration but is not proper in general. Given a positive integer a < d and a partition v of d - a into n summands and i r j , set

E,~ (v, a) = diag(v) + a . Eij e M ( v + a- ei, v + a- ed)

where 'diag' stands for the diagonal matrix with the prescribed eigenvalues and Eij is the standard n x n-matrix unit (1 at the spot (i,j) and 0 else- where). I f j = i + 1, the corresponding GLd-orbits in F 2 are closed and have the following description

OEi i+ l ( v ,a ) ~- {(D,D ' ) e F 2 ID~ C O k and dim(Dk/D~) = aSki},

OG+~,(v,a ) = {(D,D ' ) e F 2 IDk G D~ and dim(D~k/Dk)= aSki}.

Thus both projections

Fv+a.e~ ~- OE~• (v,a) -~ Fv+a.ei•

are proper with fibers isomorphic respectively to the Grassmannians Gra (a+ vi) and Gra(a + vi+l).

Given a 3-array T E T, put

OT = p~10~1~ n p~lO~3 n p~O~l ~ c F 3.

The set O T is a GLd-variety and may contain several GLd-orbits in general. For any 1 _< i < j _< 3, l e t p ~ : F 3 -+ F 2 be the projection along the component not named and let P~r denote the restriction ofp~r to 0 T C F 3.

Set R C[xq-1, x-b1 • = 1 2 , ' " , x d ], and let C x(d) ---- (Cx)d /s d denote the d-fold symmetric product of C x . For any partition I of the set [d], put

C• = (C •

The rings R (d) = R Sa and R (1) = R & may (and will) naturally be identified with the rings of regular functions on C x (d) and C x (I). In order to simplify notation, for v E V and A E M set

Cx(v) _- Cx([v]) cx(A) -- Cx([A])

R (,') = R([V]), R(A) = R([A]).

278 E. VASSEROT

If J is another part i t ion of [d], consider the symmetrizer G/J and the projection p~ such that

~ J : I tsmsj _~ R(J), f

p~ : (C• n S j -~ C •

X~ o'(f), o'~Sj/SINSj

The pullback map p]* : R (J) --+ R smsJ is the inclusion of rings: As before, if v, w E V, we write | and pW instead of | and o [~] ~ [ v ] "

P r o p o s i t i o n 6. Fix v, v l , v2 E V, A E M(Vl,V2) and i = 1,2. (a) KCLd(F~) ~_ It(v) and KGL~(oA) ~-- It (A) as C-algebras. (b) The map P~,A : OA --+ F is a smooth fibration. The inverse image

morphism P*,A in equivariant K-theory is identified with the pullback map A~ * It(Ai) R(A) PA : -"9" . If 0 A is closed then the direct image morphism

Rp~,A. is well defined and

Rp,.A.(f) = GA'( f 1-[ ( 1 - xs/xt)-~), (~,t)

where the product ranges over all couples

(s, t) e [A]jz x [A]jk, i f i = 1,

(s, t) E [A]lj x [A]kj, i f i = 2,

where l < k < l < n, j E [n].

C d Proof. Fix A C M(v , w). Let L, C (resp. Cj C C d) be the subspaee generated by the et such that t C Uk[A]~k (resp. t E Uk[A]k~). Consider the flags D E Fv and D ~ E Fw such that

D , = L l e L 2 @ . . . e L ~ and D ~ = C I @ C , $ . . . ~ C ~ .

By definition O A is the orbit of the pair (D, D ~) for the diagonal action of GL~ on Fv • Fw. Let PD and PD, be the isotropy groups of D and D'. The induction property in equivariant K-theory gives an isomorphism of the ring K GLd (0 A) with the complexified representations ring of the group PD N PD'. The groups I L GL(L~) and YIj GL(Cj) are isomorphic to Levi subgroups of PD and PD,. Moreover, 1-L,j GL(Li N C~) is isomorphic to the reductive part of PD M PD'. Part (a) follows. By definition of induction, the map

R(PD) ~KGLd(Fv)B;'~KGLd(OA)~--~R(PD M PD')

is the restriction map, i.e., the pullback by pAl. As for the direct image fix, for instance, i -- 1 and suppose that OA is closed. The fibers of the


pro jec t ion Pl,A : OA - ' ~ Fv are isomorphic to the product of partial flag manifolds

n

IIF(ail,ai2 ..... ain)" i = l

Thus the class T~ 6 K GLd (OA) of the relative cotangent bundle to the fibers of Pl,A is

k<l s6[A]ji t6[A]jk 3

where the k-th elementary symmetric polynomial in the xs, s 6 [A]ij, stands for the class in K GLd (OA) of the vector bundle whose fiber at (D, D') 6 0 A is

A Set AT.~ = }-~i(-1)~AiT.~, where A~T.~ is the i-th wedge of T.~. Then, the direct image by Pl,A is the map

K GLd (OA) > K GLd (FA~),

(see [V], Lemme 4). Part (b) follows.

[~] ~ | ([~] @ A(T~)-I)

[]

1.4. The Steinberg variety and the convolution a l g e b r a

Fix a formal variable q and set A - C[q, q- l ] , K = C(q). For any complex vector space V, set VA = V | A and VK = V | Y~ Similarly, if V is an A-module, put VK = V | ~ Set G = GLd • C • and use the notations of the previous sections. In particular, F = ]_Ivev Fv is the n-step flag variety. For any A 6 M, let ZA = T~A (F x F) C T*F 2 be the conormal bundle to O A. Set

Z v , w = U -ZA C T * ( F v • and Z = U ZA C T*F 2. AEM(v,w) AEM

Using the well-known isomorphism,

T * F = { ( D , x ) e F • End(C d) lx(Di+l) c_ Di}

the set Z is identified with the following closed subvariety of T*F 2

Z = { ( D , D ' , x , x ' ) 6 T*F 2 Ix = x'}.

The variety Z is called the Steinberg variety. It is a reducible variety whose irreducible components are the ZA, A 6 M. Let q12, q2~ and q13 be the projections T*F 3 ~ T*F 2 along the component not named. It is known that Z satisfies the following two properties

280 E. VASSEROT

�9 the restriction of the projection q13 to q~lZ N q~lZ is proper, �9 the composition Z o Z = ql~(q~lZ fq q~lZ) is equal to Z.

The first property is obvious, the second one follows from Proposition 2 (see also [CG], Chapter 6). The group G acts naturally on Z: the linear group GLd acts diagonally and z E C • acts by scalar multi )lication by z -2 along the fibers. The convolution product

. : KC(Z) | KG(Z) ~ g a ( z

endows KG(z ) with the structure of an associative A-algebra. For any A 6 M, set Z<_ A --- (.JB<A ZB" The natural maps KG(Z<_A) --+ Ka(Z ) induced by the closed emSeddings Z<A ~ Z are injective and their images form a filtration on the algebra KG(z) indexed by M. Proposition 5 implies that Z<A o Z_< B C Z<_Ao ~. Thus, KG(Z<_A),KG(Z<B) C KG(Z<_AoB). On the other hand, the open immersion Z A ~+ Z<A gives rise to a restriction map KG(Z<A) -+ KG(ZA) and to a short exact sequence

0 -+ KG(z<A) -+ Ka(Z<A) -+ KG(zA) --+ O,

where Z<A stands for [.JB<A Z<B (see [CG], 5.5). The chain of maps

KC(Z<_A) ~ KG(Z<.A)/KG(Z<A) _7+ Ka(ZA) _7+ KG(OA) ~ R(, A)

provides an identification of the associated graded of KG(Z) with the A- module (~)aeM R(A)" Since by Proposition 5 we have Z_<a, o Z_<n, C Z<AoB if A' < A, B' _< B and (A', B') ~ (A, B), the convolution product induces a product still denoted, on {~)~eM R(AA) such that

R~ A) * R~ B) C R (A~ A

E x a m p l e . Given two positive integers a, b < d, and a partition v of d - a - b , put A = E~ ~+i (v + bei, a) and B = E~ ~+~ (v + aei+a, b) (see Section 1.3). The orbits O a and 0 8 are closed. The set T(A, B) reduces to a single element, T, a n d A o B = E ~ + ~ ( v , a + b ) . For simplicity t a k e n = 2, i = 1. Then d = a + b + vl +v2,

, B = A o B = , 0 V 2 V 2 -F a ' v 2

and the 3-array T is defined by

t 121 = t 2 1 2 ~- t211 : t221 ---- 0 a n d t122 = a, t112 = b, t n l = v 1 , t 2 2 2 ----- v 2 .

Set I1 -- [1,vl], 12 = [1 + v l ,b+ Vl], I3 = [1 + b+ v l , d - v2] and /4 = [1 + d - v2, d]. Thus,

SA--~-SIIOI2 • S13 x $14, SB:SII x S12 x Sl3Ui4, SAoB=SII x S12u13 x S14.


The restriction of P13 to O~ = p~lO A M p~lO B is a smooth and proper fibration whose fiber over (D, D") E OAo 8 is

{D' E F IDk C_ D~k c_C_ D~ and dim(D~k/Dk) = a6ki, Vk} - Gra(a + b).

Note that q~lZ A and q[~lZ s are not in general position in T*F 3. Proposi- tion 5 insures that we are in the situation of Corollary 3. Using the isomorphism above the convolution product * : KG(zA) | K G ( z s ) --+ KG(ZAoB) is identified with the map

~A iEI2 j E I 3

(see Proposition 6 and Corollary 3). Now if f E R (I2uI3), then

q2xj - x i ~ QEI-I/2 q2x~--x i~ . . . . .1" I2XI3 H = P ( q ) ' f

I 2 x i 3 \ iEI2 j e l a xj -- X i ,] jE I3 X j -- Xi ,]

where P(q) is a polynomial in q which can be computed explicitly :

p(q) = qab [a + b]! [a]! [b]!

qk _ q-k with [ k ] ! = [ k ] [ k - 1 ] - . - [ 1 ] , [ k ] - q - q - 1 Vk.

Observe that the zeros of P are roots of unity. Since

i~I2 n q2_x_xj _-- xi 1 - q2xj/xi xj-x - H I I '

�9 iEI2 j E I 3

and 1-Iiei2xiERi B), 1-I je i3xjEi t l A) are invertible, the map It(A)|

R(AoB) is surjective, and remains surjective after specialization of q to a nonzero complex number distinct from a root of unity.

P r o p o s i t i o n 7. Fix a partition v E V and suppose that A = diag (v) is a diagonal matrix. Then A o B = B and, given f E R (A), g E I t (B), we have

f * g = f g ,

where f is identified with an element of Ri B) via the map pB1.

Proof. The projection Pls restricts to an isomorphism p~lO a M p~lO B --+ Os. Thus A o B = B and T(A ,B) = {T} where T is the array such that tij k = ~ bjk. The proposition follows from Propositions 5 and 6, and Corollary 3. []

282 E. VASSEROT

P r o p o s i t i o n 8. Fix v 6 V , B �9 M and A = Eh,h+ ~ (V, a). Put l = max{i ] bh+l,i ~ 0} and suppose, in addition, that bh+lj > a and ~']~j bij = ~ j aji.

(a) Then A o B = B + a(Eht - Eh+l,t) = T13 where T = (tijk) is the 3-array in T(A, B) such that

tiik = bik -- 5ih+l(~kla, th,h+l, k -~- (~kla~ tijk = 0 otherwise .

(b) I f bhl = O, then the projection PI3,r is an isomorphism O r ~ O Ao " . Moreover,

�9 SAoB C SB is the subgroup formed by the w ' s which preserve the sets [A o B]h+~,, and [A o B]h,,,

�9 W" SAoB " W -1 C SA i f w E Sd iS such that for all i ~ h the following holds

w([A]hh) = U [ A o B]h~, w([A]hh+~) = [A o B]h~, w([A]~) -- U[A o B]i r jet J

Then, R (A) and I t (B) may be viewed as subrings o f i t (AoB) and, i f f E R (A),

g E R (u), then

: . g = y g e i t[ A~

(c) I r A - ah,h_ 1 �9 Eh,h_ ~ is diagonal, l = min{ i l bh_l,i ~ 0} and bh-x,l ah,h-1, then we have similar formulas with A o B = B + ah,h- l ( Ehl -- E h - l,l ).

L e m m a 9. Let A = Eh,h+ 1 (v, a). For any matr ix B such that ~ j b~j = ~ j aji, the set T(A, B ) is in bijective correspondence with the following set

of n-tuples

S(A,B)={s - - - - ( s l , s2, . . . ,Sn) 10 <_ sk <_ bh+l,k, E s k = a } " k

To an n-tuple s e S(A, B) one assigns the following array T ( s ) = (ti jk):

{ tiik = bik i f i r h + l, th+l,h+l,k --~ bh+l,k -- 8k, th,h+l, k = sk, tijk ---- 0 o therwise .

Moreover, T(s)13 = B + Y~j sj �9 (Ehj - Eh+ld) .

Proof of the lemma. An array T = (tijk) belongs to T(A, B) if and only if

E tijk ---- aij and E t i jk ~ bjk. k i


In particular,

{ tijk = 0 if (i, j ) # (i,i), (h, h + 1), t j jk = bjk if j ~ h + 1, th,h+l,k -b th+l ,h+l ,k : bh+l, k.

Thus T �9 T(A, B) if and only if

t i j k = O i f ( i , j ) # ( i , i ) , ( h , h + l ) , t j jk = bjk if j r h + 1,

th+l ,hTl ,k -~ bh+l,k -- th ,hTl ,k , th,h+l,k ~ bh+l,k,

E k th,h+l, k ~ ah,h+l, E k thT l ,hT l , k : ahTl ,h+l , ~ k bjk = ajj i f j C h + l .

The last two equations follow directly from ~-~j bij = ~ j aji. The condi- tions above mean that there exists an n-tuple s �9 S(A, B) such that

tiik = bik if i ~ h + 1, th+l ,h+l ,k ~ bhTl,k -- 8k, th,h+l, k ~ Sk, t i j k ---- 0 o t h e r w i s e . [ ]

Proof of the proposition. that, if s, s ~ �9 S(A, B), then we have (see Section 1.2)

C(J ) __ C(8)-: :.

For s �9 S(A,B) , we put C(s) = T(s)13. Observe

vs E4 <_ r~_j r~_j

Thus, since ~ k slk = E k Sk (---- a),

C(s ' ) ~ C(s) ~,' ,~

Vj > h,

V j<_h .

and the corresponding array is the array T defined in Proposition 8 (a). We obtain

O r = { ( D , D ' , D " ) �9 p ~ l O ~ I D~ c_ Di and

d im(D/N D~') = dim(D~ N D~') + 6h,6~>~a }.

A o B = B + a(Ehl -- Eh+l j )

Observe on the other hand that bh+l,k = 0 if k > I. Hence sk = 0 if k > l, for any s E S(A,B) . Since bh+l,1 >_ a, the set {C(s) ls E S(A,B)} has a greatest element with respect to _; it is labelled by the n-tuple s = a- ez E S(A,B) . Since the map m is monotone (see Section 1.2), it follows that A o B = C ( a . el). Then the formula in Lemma 9 yields

284 E. VASSEROT

For any (D, D") 6 0 a o B ,

P~,r-1 k'D, D"') = { V I D h _ I C V C Dh and

dim(Dh fq D~:) = dim(V A D~') + a~Sj>_l}.

Thus, the fiber is isomorphic to the Grassmannian of codimension a subspaces in the (a -4- bht)-dimensional vector space

Dh A D~' (Dh-1 fq D~') + (Dh A D ~ l ) "

In particular, if b h l ---- 0, the projection Px~,T induces an isomorphism Or -~ OAo ~. The maps

OAo B -~ 0 T }" 0 A and OAo B ~ O r ~ O~

are GLd-equivariant maps between GLd-orbits. The corresponding pullback morphisms in equivariant K-theory coincide, via the induction, with the restriction of representations of the isotropy subgroups. Then apply Corolla- ry 3. []

1.5. G e n e r a t o r s o f t h e c o n v o l u t i o n a l g e b r a

We use the notations introduced in the previous sections. Recall that if A 6 M is diagonal or is of type E~ ~• (v, a), then the GLd-orbit 0 A is closed in F 2.

P r o p o s i t i o n 10. The convolution ]K-algebra (Kc(Z)• ,*) is generated by the classes of sheaves supported on the irreducible components Z A such that A 6 M is a diagonal matrix or a matrix of type E~ ~• (v, 1).

Proof. Fix a matrix C 6 M(v , w). The proof goes by induction on

2 cij. i#j

If l(C) = 0 or li the matrix C is of the prescribed type. Suppose that l(C) > 1. Then C is not diagonal. Suppose for instance that C is not a lower-triangular matrix. Pu t

( h , l ) = m a x { ( i , j ) l l < i < j < n , c i j r

with respect to the right-lexicographic order (see Section 1.1). Put

B = CA- Ch~(Eh+l, ~ -- Ehz ) and A -- Eh,h+l(V -- Ch~" eh,ch~ ).


Then 1 = max{ i lbh+l , i ~ 0} and bh+l,~ ---- Chl + Ch+l,l >_ Chl = ah,h+l. Moreover,

~ aji = ~-~ cij -- ~ihChl "~- ~i,h+lChl = ~-~ bij J J J

and bhl : O. Thus we are in the situation of Proposition 8 (b). In particular, A o B = C. Given f E R(A) and g e R i B), we get

f . g : f g e c),

modulo the identifications described in Proposition 8 (b). Thus the surjectivity of the map * : R (A) | R(B) ) I t (c) follows from the surjectivity of the map

It(ah h+l) | R(bh+l') A > It(oh') | n(Ch+l') --A , f |

where i is the obvious isomorphism It(ah,h+l) ~__ R(Ch,) and j is the inclusion It(bh+l,,) _+ RiCh,) | R(Ch+~,) (for any partition [d] = 11 t_) 12 the algebra R (l~J:) is generated by It(d) and R(I~)). On the other hand l (B) < l(C). To complete the proof it suffices to remark that any sheaf supported on ZE~_I (v,a) can be obtained by the convolution product of a sheaves supported respectively on

ZEi i• (v~-kei• k = 0, 1 , . . . , a - 1,

using the computations done in Example 1.4. []

2. T h e p o l y n o m i a l r e p r e s e n t a t i o n a n d t h e c o n v o l u t i o n a l g e b r a

Fix n, d > 1.

2.1. T h e Dr in f e ld n e w p r e s e n t a t i o n

The affine quantum group of type ~[ (n) with trivial central charge is a unital associative algebra over ]K = C(q) with generators

Ei,k, Fi,k, Kj,l, K~ 1, i � 9 j � 9 k , l � 9 I r

The relations are expressed in terms of the formal series

E i ( z ) = ~ E i , k . Z -k , F i ( z ) : ~ - ~ F i , k . Z -k , K : ( z ) : g ~ l + ~ g~,• =Fl, kEZ kEZ l>O

as follows

•

(a) K~ -1 .U i = K i - K ~ -1 = 1, [K i ( z ) ,K; (w) ] = [K~(z) ,K~(w)] = 0,

286 E. VASSEROT

(b) 0 1 ( q - l z / ~ , ) . K ~ ( z ) . K ~ : ( w ) = O l ( q - l U ~ o ) . K ~ ( w ) . K ~ : ( z ) if i < j,

(c) •

K ~ (z) . E i ( w ) = Oc~ ( q ~ z / w ) . E j (w) . K i (z) ,

(d) : t :

K ; (z) . Fi (w) = O-c,r (q~J z / w ) . Fi (w) �9 Kj (z),

(e)

(f)

[E,(z), Fj (~)] :

-I + + (q- q ). 5ij. (5(z/w)-Ki+1(w)/K i (w) - 5(z/w)-K~+1(z )/K~(z))~

E~(z) . Ej (w) = Om~ ( r . E j (~) . E i ( z ) ,

(g) Fi (z) . Fj (w) --- 8_~ ( q i - J z / w ) . Fj (w) . Fi(z),

(h)

(i)

{(Ei(zl)" Ei(z~)- Ej(w) - (q + q - l ) . Ei(zl) " Ej(w). Ei(z~)+

+E j (w) .E i ( z l ) .E i ( z= )}+{z l ++z2}=0 i f l i - j l = l ,

{Fi(zl). Fi(z~). Fj(w) - (q + q - l ) . Fi(zl) . Fj(w). Fi(z2)+

+F j (w) -F i ( z l ) -F i ( z2 )}+{z l e+z:}=O, i f l i - j l = l ,

(J) [ F i ( z ) , F j ( w ) ] = [ E i ( z ) , E j ( w ) ] = 0, if l i - j l > 1,

where 5(z) = ~ . ~ _ ~ z n, Ore(z) = ~ and cij, m i j , are the entries of _ z - - q

the following n x n-matrices

C =

O01 1 0 0 -1 1 0 0 -1 --- 0

�9 - ~

0 0 . . . . 1

and M = - C - C t.

See [D] and [DF] for more details. In particular the relations (c), (d), (f) and (g) are written formally, and in practice one should first multiply both sides by the denominator of the function Om since formal currents may have zero divisors.


2.2. T h e p o l y n o m i a l r e p r e s e n t a t i o n

We keep the same notations as in the first section. Thus we have that R C[x~ 1 x~l , . x • and for any composition v �9 V, the algebra = ' " ' ' d l

R (v) is identified with the subalgebra of Sv-invariant polynomials in R. Set K = G , , e v R("). For any subset I C [d], set

e,(z) = I I mEI

Recall that if v �9 V and i �9 [n], then ~ = vl + v2 + . . . + vi and [v]i = [1 + V i - l , V i ] . Let

Ei(z) ,Fi (z ) �9 ~ Hom(R(V),R[v+e~:e~+'))[[z+l]] v E V

be the following operators

E i ( z ) ( f ) =

= ~ (q-q-1)~v+e~-~+~(f'5(xl+v~/z)'O[,,l~(qxl+v~)) if v + e i - e / + l � 9 [ 0 else,

F i ( z ) ( f ) =

= ~ (q_q-1)~v-e~+ei+l (f.j(X~/Z).O[,,h+ 1 (xv~/q)-l) if v - - e i+e i+ l e V , ( 0 else.

Moreover, let ~:~(z) �9 End(KA)[[z+l]] be the operator whose Fourier coefficients act on R(") by multiplication by the corresponding coefficient of the expansion at z = cr 0 respectively, of the rational function

I~i,v(Z) = Oil,~i_ll (qz). Otl+~,dl (z/q).

This rational function differs from the one used in [GV] since we consider here type ~'[(n) instead of ~(n) . Denote by l~i,k, Fi,k, I~i,k and I ~ 1 if k = 0, the Fourier coefficient of z -k in the series ~',i(z), Fi(z), I ~ ( z ) (see Section 2.1). The main result of this section is the following theorem.

P r o p o s i t i o n 11. The map

Ei(z) ~ F,i(z), Fi(z) ~-~ ~'i(z), g ~ ( z ) ~ I ~ ( z ) ,

extends uniquely to a representation of U on K K.

Proof. Our proof is an extension to the affine case of the computations done in IV]. We will prove that the operators Ei, ~'i and I ~ satisfy the relations given in Section 2.1. The first and the second one are immediate. To verify the others, we need more notation. Given a partition I = (I], I2,--. ,In) of

288 E. VASSEROT

the set [a~ into n subsets, for any x E (C x )d let xz C C x(1) be its projection. Next, to any k E [d] we associate two operators, ~-~, on the set of partit ions of [d] in such a way tha t r f f I is the part i t ion with k shifted from one piece of I to the next (resp. the previous) one. Then, for any v E V such that v q= ei + ei+~ ~ V and any f E R (~=~-v€ we get respectively

(E~(z)f)(z t , , ] ) = (q - q-~) ~_, S(x .~+t , , ] )5(xklz)e t~, \ (~(qxk) , ke[v]~

(F i (z ) f ) (x [ , , ] ) = (q - q-i) ~ f(x~.[[v])5(xk/z)Ot,, l,+~ \U~ ( xk /q ) - l . kE[v]~+l

Let us prove (e). First observe that

I ~ + l , , , ( z ) I ~ i , , , ( z ) - 1 = et,,l ' (qz)e[, ,~,+, (z/q) -1.

The values of (q - q-1)-2Ei(z)Fi(w)f and (q - q-1)-2Fi(w)Ei(z)f at x[~ 1 are respectively

E f(xr[ -r+ [v]) 5(xk/z) 5(xl/w) Ot~ h \{k} (qxk)Oi~h+ , u{k}\{,} (xl/q) -1, kE[v]i

/e[vh+ 1 u{k}

E f(x'~+'~E [~]) 5(xk/z) 5(xt/w) Olvl,uUMk } (qxk)Ot,,h+, \U} (XZ/q)-l" ke[v]iu{l}

lE[v]i+l

Their difference is equal to

f(x[v]) E 5(xk/z)5(xk/w)et,,l,\(,~(qx~)et,,l,+, (xk/q) -1- kC[v]~

- f (x [v ] ) kE[v]~+l

5(zk/z) 5(xk/w) e~,,j, (qxk)e�91 vu, j (xk/q) -1

Set

A ( x ) = H (x-xk) and B(x) = 1~ (qx- q-lxk) H (q-lx--qxk)" k~[v]~u[v]i+l ke[v]i kc[v]i+l

Then

([l~.i(z), Fi(w)]f)(xM) = (q _ q-1)2 E 5(xk/z) 5(Xk/W) Xk 1 B(xk) f(xb,]). k~[vJ~u[v],+l A'(xk)

The right hand side in (e) is then obtained by a computa t ion of residues. If i y~ j :t: 1, the preceding formulas for Ei(z) and Fj(w) give immediately


[Ei(z), F j (w)] = 0. Similarly, the values of ( q - q-~)-~Ei(z)Fi+i(w)f and (q- q-~)-2~'i+~(w)Ei(z)f at x M are equal to

f(x,+~[ M) ~(z~/z) ~(x~/w)ei,,~,\~ ~ (qx~)eM,+~ \u ~ (z~/q) -~,

16[v]i+2

whereas the values of (q-q-~ )-2Ei(z)Fi_l (w) f and (q-q-1)-2~'i_~ (w) ~',i ( z) f at x[~] are equal to

f (x ~+~_;-[-t) ~(x~/z) ~(x~/w) O~,,i,\~ (qx,Qet,,~:, ~,~,,} (x~/q) -~. k,/fiN]/

k:gl

Thus [l~i (z), ~'~ (w)] = 0 whenever i # j .

Let us prove (f). The value of (q - q-~)-2Ek(z)Et(w) f at x[~] three cases (k, l) -- (i, i), (i, i + 1), (i + 1, i) is respectively

in the

f(x~+~+N] ) 5(xk/z) 5(xz/w) et,,~,\~ (qxk) O[,,l,\{~,,} (qx~), k,tE[v]i

f(x~t~+[,,] ) ~(Xk/Z) ~(XJW) et,,~,\~k ~ (qxk) O[,.,]~+l~{k}\u } (qxl), kE[v]~

te[vh+ 1 u{k}

E f(x'r+'r+[v] ) 6(Xk/Z) g(XZ/W) Oivl,+,,,.[k } (qxIr @["M,'U} (qxl). kE[v]i+l

ze[,,]~

The first equality gives

(q- lz - qwlF, i(z)Ei(w) = (qz - q- lw)Ei(wlEi(z ) ,

the other two give

(w - z)Ei(z)Ei+l(w) = (qw - q- lz)Ei+l(w)Ei(z)

(note tha t we have used the basic formula (z - w) ~(z) 5(w) = 0). When I i - Jl > 1 the relation (f) is immediate . The proof of (g) is similar.

Let us prove (c). The value of (q - q-1)-2Ek(z)~, l(w)f at x[,,] in the three cases (k, l) = (i, i), (i, i + 1), (i + 1, i) is respectively given by

f(X~.~T~[V] ) rS(xk/z) 5(XI/W) Oivi~X(k}(qXk ) O[v]i, {k,,) (qxl)~ k,IE[v]i

(Ei(z)~,i(w)f)(x[,,1) = (q - q-1)x

x Ef(xr+k[v])g(Xk/Z)OM~\~k} (qXk)Otl._~,_l ~ (qw)O[~+-~.d] (w/q)OI (q--lW/Xk), k~[v]i

290 E. VASSEROT

and

( g i ( ~ ) E i ( ~ ) f ) ( x i ~ 1) =

= (q - q-~) E f(xr+["])6(x~/z)O{"}~\(~} (qx~)O[~,~-l] (qw)O[l+~,,,~] (w/q). kEN]/

Since 6(Xk/Z)O~ (q-~w/xk) ---- 6(Xk/Z)01 (q-~w/z), we thus obtain

E~(~)K~(~) = 01 (q-~, /z)g~(~)E~(~).

Similarly, we have

(E~(:)K~+~(~)f)(x[~]) = (q- q-~)x

• ~ f(z,+[v])~(x~lz)O[vl,\~s (qx~)Oil,~] (qw)Oil+~,+ 1,,q (wlq)O_l (q~l~),

and

(lKi+l (w)Ei(z) f)(x[~]) =

= (q _ q-l) E f(xr+[v])J(Xk/Z)O~vh\{k} (qxk)O[~,~] (qw)O[x+~+ 1,d] (w/q). ~e[v]~

Thus,

E~(z)g~+ ~ (~) = o_1 (q~,/z)if~+~ (~)E~(~).

The other cases are obvious and (d) is proved in the same way. Let us prove formula (j). If [ i - J l > 2 then (q-q-1)-2F, i(z)Ej(w)f(x[v])

equals

E f(x~+~+N]) 6(Xk/Z) 6(xz/w) Oh, li\(k} (qXk) OC,,]j \{,} (qxl). ke[vh leN]j

Switching the factors E/(z) and ]~j (w) yields the same expression. Let us prove the remaining Serre relations. We will only prove the formula

(h) for j = i + 1 since the other cases are similar. First, let notice that (q - q- 1)3 (l~i (zl)]~i (z2)]~i+1 (w) f)(x[v]), (q - q- 1)3 (Ei (zl)Ei+l (w)Ei (z 2) f)(xtv ]) and (q - q - 1 ) a ( E i + l ( W ) E i ( z 1 )~']~i(zl)f)(X[v]) are equal respectively to

E Ak,z,h(Zl,Z2,W) OM~X{k}(qxk) OEv}~\{k.,}(qxz) OM~+1u~k,,}\~}(qXh), k:~/E[v]i

hE[v]~+lU{/e,/}

E Ak,l,h(Zl, z2,w) Oh,h\{k}(qxk) O{,,l~+lu{k}\{h}(qXh) Oivl~\{k,,}(qxt), ~le[v h

hE[v] i+ I u{k].


E Ak,t,h(Zl, z2, w) Otvl,+l X(h} (qXh) Otvl~\(k} (qxk) Otvl,\U.k ~ (qxt),

hq[v]i+l

where Ak,t,h(y, Z, W) = I (X,+,+,+ N] ) 6(Xk/y) 6(Xt/Z) 5(Xh/W). Set I = [v]i+l.

We are thus reduced to prove that, given k # l �9 Iv]i, we have

2 ~ O,~.~k,,,\{h, (qXh) + 2 E O'\'h' (qXh) = hEIU{k,1} hEI

= ( q + q - - 1 ) EO,~<ky\~h}(qXh)+(q+q--1) E O'~UMh}(qXh)" helU{k) helU{l}

For any subset J C [aq with m elements, we have

Oj\{h}(qxh) = [m] hEJ

(see [M] for instance). Thus the equality is a consequence of the (obvious) equality

[ m + 1] + I r a - 1] = [2] [m], Vm �9 N •

which follows from the decomposition of the product of m-dimensional and 2-dimensional simple representations of SL(2). []

2 .3 . K - t h e o r e t i c c o n s t r u c t i o n o f U

Recall that F is the variety of all n-step flags in C d and Z C T*F x T*F is the Steinberg variety (see Sections 1.3 and 1.4). The group G -- GLd x C x acts on Z as in Section 1.4: GLd acts diagonally and z �9 C x acts by multiplication by z -2 along the fibers. Given a partition v of d - 1 into n summands and i r j , we set

Ei~ (v) = Ei~ (v, 1) = diag(v) + Eij E M ( v + ei, v + ej).

I f j = i • 1, the corresponding GLd-orbits in Fv+e~ • F,,+e~• are closed and are given by

OE,,+l(v) = {(D,D') e F 2 ID~ c_ Dk and dim(Dk/D~k) = Ski},

OE,,_l(v) = {(D,D') �9 F 2 IDk ~ D~ and dim(D~/Dk) = 6ki-1}.

Thus both projections

Fv+ei (Pl OEii• ) P2)fv+e,• are smooth and proper with fibers isomorphic respectively to the projective spaces ~v~ and ~v*• Recall that the conormal bundle to OEi,• is denoted by ZE, ,• (~). Given k �9 Z, let Op~ (k) and T'p1 be the relative invertible sheaf O(k) and the relative cotangent sheaf along the fibers of pl respectively. Let Ci,v,k be the pullback to ZE, ,+~ (v) of the sheaf Det(T*pl)| (k)

292 E. VASSEROT

o n OEii+l(v) . Similarly, let ~'i,v,k be the pullback to Z~+I ~(v) of the sheaf Det(T*pl) | Opl(k) on OE~+li(v). Since the orbits OEi• are closed, the varieties ZE~I~(v) are irreducible components of Z. Hence, the sheaves Ei,v,k and Ui,~,k may be viewed as sheaves on Z. They come equipped with the natural G-equivariant structure. We set

~i,k ~" Z(--q)-V~[Ci,v,k] and 2=i,k = ~-~(--q)-V~+x[.~i ,v ,k] v v

in KG(Z). Form the corresponding generating functions

Ci(z) = ~ Ci,kz -k, J~i(z) = ~ , ~ , , k z -k �9 gG(z)[[z,z-~]]. k k

We have introduced in Section 2.2 a generating series I ~ ( z ) whose Fourier coefficients, I~ • (k �9 ~=N), are homogeneous Laurent polynomials in xl i , k . . . , x d. Since T*F is isomorphic to the conormal bundle of the diagonal A C F • F , the cotangent bundle T*F is naturally identified with an irreducible component of Z, say Z~. Under the map

K A ~~ KG(F) ~> KG(T*F) ~~ KG(Z~) > KC(Z)

the I~i,k's can be viewed as classes of equivariant sheaves on Z supported by Za. Let ~i,k denote these classes and write ]Ci(z) = )-~k]Ci,kz -k �9 KG(Z)[[z, z-1]]. The algebra KG(z) acts by convolution on KG(T*F) ~- K~. The following result is stated in [GV], Proposition 7.7.

L e m m a 12. The convolution action of ~i,k, Jzi,~, ]Ci,k �9 KG(Z) on I~(:I* F) ~- K~ is given precisely by the operators Ei,}, Fi,~ and Ki,k.

Proof. See [V]~ Theorem 10. The formula in the proof of Proposition 6 gives

[Tpl]=

Thus, we obtain

H ~ i _ l < ~ _ ~ i X l + ~ i t ,

~-]~i(-1)i[AiTpl] = 1-I~_l<t_<~i (1 - q2xl+~/xt).

The result follows from Corollary 4 and Proposition 6. []

The following result is stated in [GV], Lemma 7.5.

L e m m a 13. The representation of KG(Z) on KG(T*F) by convolution is faithful.

Proof. See [CG], claim 7.6.7. []

The following result is stated in [GV], Theorem 7.9.


T h e o r e m 1. The map Ei,k ~-+ Ei,k, Fi,k ~ Jzi,k, g i ,k ~ ]Ci,k, extends uniquely to a morphism of algebras �9 : U ~ KG(Z)~.

Proof. Immediate since Lemmas 12, 13 and Proposit ion 11 imply that the elements Ei,k, ~-i,k, )Ci,k of the algebra K a ( Z ) satisfy the relations in Section 2.1. []

2.4. S u r j e c t i v i t y o f t h e m a p @

For any t E C X , let Ut and K G ( z ) t be the C X-algebras obtained by spe- cializating U and KG(Z) to q = t. The map @ restricts to an algebra homomorphism

@t : Ut ~ K a ( Z ) t .

T h e o r e m 2. I f t is not a root of unity the map ~t : Ut -+ K a ( Z ) t is surjective.

Fix t E C X which is not a root of unity. For any v E V set

n - 1 d

I =II II i ~ l m ~ 0

w ~ d - - v i

Ki - - t TM

t d - v i _ t m E Ut.

L e m m a 14. @t( Iu )* Ka(Zv ,w) t ~ 0 if and only if u = v. Similarly, g a ( Z v , w ) t * q2t(Iu) r 0 if and only if u = w. []

L e m m a 15. Fix a partition v E V . I f t E C x is not a root of unity, then the Fourier coefficients of the series I~i,v(z) 's generate the algebra R (v).

Proof. Introduce elements Hi,k in U, i E In] and k E Z x, such tha t

K~(z) = K/~1 e x p ( i ( q - q - l ) E Hi,• z=Fk). k > l

Each I-Ii, k can be expressed as an algebraic expression in the Ki,k'S, more precisely

ni,• : • _ q-1)-1 log(g~lK/~ (z)). k___l

A direct computa t ion shows that in the component R (v) of the polynomial representation, the element Hi,• k E l~I• acts via multiplication by the polynomial I:Ic,• such tha t

~~-~ d

- -~-- X/ ).

l= l l= l-[-g i

We are done. []

294 E. VASSEROT

Proof of Theorem 2. The elements Ci,k, .Ti,k and/C~,k of KG(Z)t decompose as the sum of their components in each Kc(Zv,w)t, where v, w E V. Lemma 14 implies that all these components belong to ~ t (Ut ) . On the other hand Lemma 15 implies that

ga(zA)t C ~ t ( U t )

for any diagonal matrix A E M. The surjectivity now follows from Propo- sition 10. []

3. T h e s imple a n d s t a n d a r d m o d u l e s

We keep the previous notation. In particular G = GLd x C • , F -- I J r F,, is the (disconnected) n-step flag manifold, Z is the Steinberg-type variety, M~ = T*F~ the cotangent bundles, M = H~ My, and

N = {x E End(Cd) Ix n = 0}.

Moreover, for any x E N, let Fv,x -- {D E F~ I x(Di) c Fi-1} be the Springer variety. Let T C GLd be the subgroup of diagonal matrices and set A = T • C • . Then, the Bivariant Localization Theorem [CG], Theorem 5.11.10, and t he Bivariant Riemann-Ro ch Theorem [C G], Theorem 5.11.11, give a homomorphism of algebras

KG(Z) ~ H. (Z ~)

for all a = (s, t) E A, where Z a denotes the fixed point subvariety and H . stands for the Borel-Moore homology with complex coefficients (see either [GV]). Moreover, this map is surjective [CG], Theorem 6.2.4. As a consequence, for any a = (s, t) E A, we have a surjective homomorphism

k~a : Vt --~ gG(z) t -~ H.(za).

Fix a = (s, t) E T • C • . Let G(s) be the centralizer of s in GLd. Let M a = Hv M~ be the fixed point subvariety and Ir : M a --4 N a the projection to

N a = { x E N i s x s - l = t - 2 x } .

The map ~r commutes with the action of G(s). For all v E V consider the G(s)-equivariant complex on N a

s = r.CM~ [dim M~]

and set /:a _ SvE~. The equivariant version of the Beilinson-Bernstein- Deligne Decomposition Theorem gives

= G Lo. , i | ICo[i], O C N a

i E Z


where ICo[i] is the shift of the intersection cohomology complex associated to the constant sheaf on O (recall that the isotropy subgroup in G(s) of a point in O is connected). Put Lo,v = @iLo,v,i and Lo = @~Lo,v for all O. The Lo's are simple Ut-modules (see [GV], Theorem 8.4, [CG], Theorem 8.6.12, and the remark after Theorem 4). The total cohomology space H.(F s) is endowed with the structure of a Ut-module (see [CG], Proposition 8.6.15). Since this module only depends on the G(s)-orbit, O, of x in N a, let us denote it by Mo. The module Mo is called a standard module. Then we have the following analogue of the Kazhdan-Lusztig multiplicity formula (see [GV], Theorem 6.6, and [CG], Theorem 8.6.23).

T h e o r e m 3. For any a = (s, t) E A and any G(s)-orbits O, 0 ~ E N a, one has

[Mo, : Lo] = E dimNio'(IC~ [] i

In the formula above dimT-l~,(ICo) is the rank of the i-th local cohomology sheaf of the complex ICo at anypo in t of O ~. We recall that the quantized universal enveloping algebra of s[ (n) (with trivial central charge), denoted by U' , may be viewed as the K-subalgebra of U generated by Fourier coefficients of the series (see [DF])

Ei(z), f i ( z ) , g~+l(tiz)gi~(tiz) -1, i = 1 , 2 , . . . , n - 1.

L e m m a 16. The Ut-modules Lo are simple U~-modules.

Proof. Immediate since Fourier coefficients of I-Iin__l K~ (zt 2i) are central in Ut. []

Fix t E C x , not a root of unity. Then, simple U~-modules are parametrized by their Drinfeld polynomials (the proof is due to Chari and Pressley, see [CP2]). Recall further that a nilpotent matrix y E End(C d) is labelled by the partition A(y) = (),I(Y) _> )~2(Y) _> -..) such that Ai(Y) is the length of the i-th Jordan block of y. If yn = 0 the dual partition, )v (y) = ()~v (y) _> ),~(y) > - . - _> )~nV(y)), is such that

s (y) = dim Ker(y i) - dim Ker(yi-1), i = 1, 2 , . . . , n.

In particular Av (y) may be viewed as a dominant weight of g[ (n) in the obvious way. Denote by )v (O) the dual partition of a representative of any nilpotent orbit O.

T h e o r e m 4. Fix a =- (s,t) E A and fix a G(s)-orbit 0 C N a. Put )~v(O) = v v (~1, A2,...,)~v). Then, the Drinfeld polynomials of the simple U~-module

Lo are given by

I-[ ( z-t -2s-l k J~ i = 1 , 2 , . . . , n - 1,

296 E. VASSEROT

where the sk's are such that the matrix s is conjugate to

A v k - 1

siD(Ai) with D(k) = ~"t-2i~.~ii Yk �9 N • i----1 i = 0

Remark. It follows that the Lo's are non zero for all O. Moreover, this nonvanishing result is still valid if t is a root of unity.

We first mention the following simple result of linear algebra (see [CG], Remark 4.2.2).

L e m m a 17. If x E N and v e V, then

F v , x r ~ v<_AV(x),

where < is the standard order on the weight lattice of 9[(n). []

Proof of Theorem 4. By construction we have

L o , v = { u E L o ] K i . u - - t d - v ~ u , Vi = 1 ,2 , . . . , n} .

r t - -1 In other words, Lo,v C Lo is the subspace of weight ~'~i=1 (vi - vi+l)wi where wi is the i-th fundamental weight of s[(n). Fix x E O and set A v = A v (x). Since F~v,x reduces to the single flag

0 C_ Zer(x) C_ Ker(x 2) C_ . . . c_ Ker(x ~) = C d,

the weight Av has multiplicity one in the U~-module. Moreover, it follows from Lemma 17 that all the weights of Mo are less or equal to )v. By construction

Lo,v # {O} ~ O c r ( M a ) ,

Thus, the weights of Lo are less or equal to A v again. On the other hand Theorem 3 implies that

(i) [Mo : Lo] = 1, (ii) [Mo : Lo,] 7 ~ 0 ~ 0 C 0'.

Fix another orbit O ~ r O containing O in its closure. Then ),v (O ~) < Av (O). So (i) and (ii) imply that the module Lo has highest weight ),v and that Lo,;~v = H* (F~v,x). Since F~vx reduces to a single point, on L o , ~ C Lo we get (see 2.2)

K h l ( t i z ) K ~ (tiz)-I = I I O-l(tl-iskz-1)'

and we are done. []


Let wl ,w2 , . . . ,Wn-1 be the fundamental weights of Mn. For any a E C x and any i = 1, 2 , . . . , n - 1, let V(wi)a denote the fundamental representation of U ~, i.e., the simple finite dimensional U'-module whose Drinfeld polynomials are

Pj(z) = ( z - a ) ~ , V j = l , 2 , . . . , n - 1 .

For any partition A = (A1 > A2 > . . . > As > O) of d and any a = (al , a2 , - - . , at) E (C x )l, consider the block diagonal d • d-matrices

1 k - 1

x ~ = @ J ( A i ) with J ( k ) = E E i , i + l V k E N x, i=1 i----1

and 1

s~,a = @ aiD(Ai), i=1

8 - 1 _-- t 2 (see the previous theorem). In particular s~,a x~ ~,~ x~. Denote by O~,a the G(s~,~)-orbit of x~ in N ax,~, where a~,a = (s~,~, t).

P r o p o s i t i o n 18. I f Ai < n for all i then the classes of Mo~,, and

in the Grothendieck ring of finite dimensional U~-modules are equal.

Proof. First, recall the geometric construction of the Schur functor given in [GRV], Theorem 6.8, between Ut-modules and Ht-modules, where Ht is the affine Hecke algebra of g ig with quantum parameter t. Suppose that n > d. Let Fc be the variety of complete flags in C d and let Mc = T*Fc be the cotangent bundle. Given a E A, let 7r : M a -+ N a be the projection and consider the complex s = ~r.CM2 [dim Mca]. Put

a a Ua = Ex t ( s163 Ha = E x t ( s 1 6 3 and Wa = Ex t ( s163 ~)

(the Yoneda Ext-groups in the bounded derived category of constructible sheaves on N%) The space Wa is a (Ua, Ha)-bimodule. Moreover, since we have surjective algebra homomorphisms

Ut ~ Ua and Ht --~ H a ,

a Ha-module (resp. a Ua-module) can be pulled back to a Ht-module (resp. a Ut-module). The Schur dual of the pullback of a Ha-module V is the pullback of the Ua-module V ~ = W a @Ha V. In particular, fix a nilpotent orbit O C N a and fix a point x E O. Denote by j the embedding {x} ~ N a.

298 E. VASSEROT

The standard module Mo is isomorphic to the cohomology space H* (j!s where Ut acts via the algebra homomorphism

"! a "! a Ut --~ Ua ~ Ext(3'/2 ,3"/2 ).

Mo splits as a vector space into the direct sum

Mo = ( ~ Lo, | Hk(j!ICo,). O',k

Similarly, the underlying vector space of the standard Ht-module corresponding to O is

No = ( ~ Lc,o, | Hk(j!ICo,). O ' , k

Let Aa C Ha be the maximal semi-simple subalgebra of Ha, i.e., Aa =

(]~o End(Lc,o). Recall that we have an isomorphism

Wa = ( ~ Hom(L~,o,Lo) ) | Ha o

(see [GRV], (6.12)). Then,

g~9 = ~O",k(~O' n~ O', LO'))|174174

= ~o,,k(Hom(Lc,o,,Lo,)| Lc,o, @ Hk(j!ICo,)),

= Mo.

Now using the Induction Theorem [KL], Theorem 6.2, one proves that the class [No] of No in the Grothendieck ring is the same as the class of an induced module (see [A]). More precisely, we have

[No~,~ ] = [Ht | Ca ].

Here HA,t C Ht is the affine Hecke algebra of the Young subgroup of ~d associated to the partition )~ and Ca is the one-dimensional module such that

T/F-~ t and Xi ~-~ s~,a,i,

where s~,a,i is the i-th entry in sx,a. In particular if )~ = (d), then x:~ is regular, and thus Nox,~ is the simple Ht-module labelled by O~,a. If moreover d < n, its Schur dual is the simple Ut-module Lo~,~ ~- V(wd)a-lt~-2. Using the behavior of the Schur functor with respect to tensor products (see [CP1] for the case of the affine Hecke algebra) we get the result. []

Remark. Theorem 3 implies in particular that if Ox,a C N ax,~ is open then V(w~)a;~t~l_2 | V(w~2)a;~t~2-2 | | V(w~)a~lt~_2 is irreducible.


[A]

[BLM]

[CC]

[CP1]

[CP2]

[D]

[DF]

[CRV]

[GV]

[KL]

[M]

IV]

References

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A

quantum affine algebra Uq(~[(n)), Comm. Math. Phys. 156 (1994), 277-300. V. Ginzburg, N. Reshetikhin, and E. Vasserot, Quantum groups and flag varieties, Contemp. Math. 175 (1994), 101-130. V. Ginzburg and E. Vasserot, Langlands reciprocity for affine quantum groups of type An, Internat. Math. Res. Notices 3 (1993), 67-85. D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands con- jecture for Hecke algebras, Invent. Math. 87 (1987), 153-215. I. G. MacDonald, Symmetric Functions and Hall Polynomials, Ox- ford University Press, 1995. Russian translation of the edition of 1979: H. Ma~o~a~B~, Cucu~uempuuecrue ~yn~uu u ~uno- ~ounenb~ Xonna, M~p, MocI<Ba, 1985. E. Vasserot, Representations de groupes quantiques et permutations, Annales Sci. ENS 26 (1993), 747-773.

affine quantum groups and equivariant k

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