Monodromic Systems on Affine Flag Manifolds

Monodromic Systems on Affine Flag ManifoldsAuthor(s): George LusztigSource: Proceedings: Mathematical and Physical Sciences, Vol. 445, No. 1923 (Apr. 8, 1994), pp.231-246Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/52562 .

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Monodromic systems on affine flag manifoldst

BY GEORGE LUSZTIG

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. and Institut Mittag-Leffler, Auravdgen 17, S-18262,

Djursholm, Sweden

There has been recently substantial progress in the programme of expressing the characters of irreducible modular representations of a semisimple group over a field of positive characteristic in terms of combinatorics of affine Hecke algebras (see Andersen et al. 1992; Kazhdan & Lusztig 1979). This paper is a further contribution to this programme: I explain why in the non-simply laced case, the 'dual' affine Weyl group is needed and why in this case it is necessary to use monodromic systems (certain local systems defined on subvarieties of a line

bundle) over an affine flag manifold.

1. Affine Weyl group

1.1. In this section we recall some known results concerning affine Weyl groups (see Bourbaki 1968; Kac 1983).

Let (aij) be an irreducible affine Cartan matrix indexed by I x I (I is a finite set with at least two elements). There are uniquely defined strictly positive integers di,i,, r' (i E I) such that

(a) diaij = djaji for all i,j and di = 1 for some i,

(b) i raij = 0 for all j and ri = 1 for some i, (c) Ej aijr' = 0 for all i and r' = 1 for some i. Let D = maxi di; we have D E {1,2, 3} and for any i, di is equal to 1 or D. Hence we can define di E {1, 2, 3} by didi = D for all i C I. We have diri = r' for all i. We shall assume that (aij) is untwisted, or in other words, that there exists io c I such that rio = r 0 = 1 and dio = D. We fix such io. Let Io = I- {io}.

1.2. Let V be an R-vector space with basis hi (i c I) and let V' be the dual vector space; we denote by (, ) : V x V' -* R the obvious bilinear pairing. For any j I, let wj c V' and h' E V' be defined by (hi, wj) = 6ij and (h, h') = aij for all i. Let c = E. rhi C V. We have (c, h') 0 for all j and Ei r'h' - O. For i E I, let

si : V -- V be the reflection defined by si(y) = y - (y, h')hi and let si : V' -- V' be the reflection defined by s(x) - x-(hi, x)h'. Let W be the subgroup of GL(V) generated by the si; taking contragredients, we identify W with the subgroup of GL(V') generated by the si (in this identification, si : V - V corresponds to

t This paper was produced from the author's disk by using the TIX typesetting system.

Proc. R. Soc. Lond. A (1994) 445, 231-246 () 1994 The Royal Society Printed in Great Britain 231



si : V' - V'). Let WO be the (finite) subgroup of W generated by the reflections si (i c Io). Wo is the Weyl group; W is the affine Weyl group.

1.3. Let R (resp. Ro) be the set of vectors of V of form w(hi) for some i C I (resp. i C I0) and some w c W (resp. w c Wo). Let R' (resp. Rf) be the set of vectors of V' of form w(h') for some i C I (resp. i C lo) and some w C W (resp. w C Wo). The assignment hi - h' (i E I) extends uniquely to a map h - h' from R onto R' such that w(h)' = w(h') for any w C W, h C R. This restricts to a bijection of Ro onto R'. For h C R, we denote by Sh the element of W given by the reflection Sh () y - (y, h')h on V or equivalently by the reflection

sh(x) = x - (h,x)h' on V'. We have shi = Si for i c I and WShW- =- Sw(h) for h C R, w E W. There is a unique function h F dh on R with integral values such that dw(h) = dh for all w C W, h C R and dh = di for all i c I. Note that for any h C R, dh equals 1 or D. Let dh {1, 2, 3} be defined by dhdh = D for all h C R. We have R = {x+xmclx C Ro, m c Z} and R' = R. We have (x+dxmc)' = X' for all X C Ro, m C Z. For any X C Ro and m C Z we set sx,m = sh E W where h - X + dxmc.

1.4. Let 3 (resp. y) be the unique element of Ro such that, for any X C Ro, - 3' (resp. X - y) is a linear combination with coefficients in N of vectors hi

(resp. hi) with i C Io. We have d = D, d, = 1, hio= + c and ho = '. We have

3=--E rihi and ' =-Er:h. i Io icIo

For any X Ro, X - dx/ is a linear combination with coefficients in N of vectors hi (i C Io)

1.5. Let Vz = i Zwi C V'. The set {i i C I} is a Z-basis of Vz. Let Q' be the subgroup of V' generated by R'. The set {h'li E I0} is a Z-basis of Q'. We have Q' c Vz. Let Q" be the subgroup of V' generated by the elements dxx' for various X E Rot. The set {dih'li C Io} is a Z-basis of Q". For z E Q", let 0 : V' -- V' be the transvection given by tz(x) = x + (c, x)z. For X c Ro and m C Z, we have (a) SX,O0X,m - 0xmx Thus 0, C W for any z C Q". The map z v- 09 is an injective homomorphism " -+ W whose image, denoted T, is a normal subgroup of W; moreover, W is a

semidirect product of Wo and T. Let V+ (resp. V-) be the set of all y C V such that all coefficients of y with respect to the basis (hi) are > 0 (resp. < 0). Let R+ =R n V+, R- - Rn V-, R+ - Ro n V+. Then R is the (disjoint) union of R+ and R-. We have R+ -{X + dxmclX Ro, m C Z, m > 0} U R+.

1.6. There is a unique untwisted affine Cartan matrix (aj)i,jc such that aj- aji for all i, j C I0. The corresponding affine Weyl group is denoted by WO.

2. The reflection subgroups Wx

2.1. It is well known that W is a Coxeter group on the set of generators {siJi C I}. We will show that any x E V' defines a subgroup W" of W which is itself a Coxeter group in a natural way.

We consider the affine space V '- {( E V'l(c,) = 1}. Each h E R defines a

Proc. R. Soc. Lond. A (1994)

232 G. Lusztig



Monodromic systems on affine flag manifolds

hyperplane H(h) = { E Vil(h, ) = O} in V1'. Let

C= V'l(h,) > VO i c I}.

For x E V', we set Rx = {h C Rl(h,x) c Z}. It is clear that - -

(a) if h, h E RX, then sh(h) E Rx. Let (Rx)+ = R n R+, (Rx)- - R n R-. Let Wx be the subgroup of W generated by {shlh E RX}. The restrictions of Sh to V;' are orthogonal reflections for a suitable euclidean structure on that affine space. The set of reflecting hyperplanes H(h) (for various h E RX) is stable under the action of Wx, by (a). The action of the discrete group Wx on V1 is properly discontinuous since the action of the larger group W is so. Hence we may apply the general results of Bourbaki (1968, ch. V, ? 3) on groups generated by reflections and we see that the following holds.

Let us call an x-alcove any connected component of the complement in V1' of UhERXH(h). Let Cx be the unique x-alcove containing C. Let P" be the set of all h E (RX)+ such that H(h) is a wall of CX. Then Wx is a Coxeter group on the generators {Sh h E P }.

2.2. We shall denote by 1 : W -- N (resp. Ix : W" -- N) the usual length function on the Coxeter group W (resp. WX) with respect to the set of generators {sili E I} (resp. {shlh E PX}). The following statement is well known: if w E W and h E R+, then (a) w(h) E R+ > 1(wsh) > 1(w); w(h) c R- = l(wsh) < 1(w). Similarly, if w C Wx and h E (RX)+, then (b) w(h) E (RX)+ == lX(wsh) > l(w); w(h) C (Rx)- = x(wsh) < l(W).

Lemma 2.3. Given w E W, there is a unique element wl E wWX such that wi(h) E R+ for all h E (RX)+. We have l(wl) < I(w') for any w' C wWX distinct from wl.

The restriction of 1 : W ?- N to wWx reaches its minimum value at some element w1 E wWX. If h E (RX)+, we have l(wlsh) : 1(W1), by 2.2(a). Since WiSh E wWX, we see from the definition of w1 that l(wlsh) > 1(wi). Hence wl(h) E R+ (see 2.2(a)). Thus the existence part of (a) is proved. Let u E Wx - {1}. There exists h' E (RX)+ such that u(h') C (RX)-. We have wlu(h') C R- and the first part of the argument shows that w1u is not a point of minimum for the restriction of 1 to wWX. Thus (b) holds and the uniqueness in (a) follows.

2.4. Let < (resp. <x) be the standard partial order on the Coxeter group W (resp. WX).

Lemma 2.5. Let u, u' E Wx be such that u' <x u and let wl E wWx be as in Lemma 2.3. Then wlu' < wlu.

We may assume that u' = ush where h E (RX)+ satisfies lx(ush) = 1x(U)- I (hence, by 2.2(b), we have u(h) E (Rx)-). We have wlu(h) E w((Rx)-) C R-. Using 2.2(a), we deduce that l(wiUSh) < 1(w1u). By definition, we then have WiUSh < W1U. The lemma is proved.


233



3. The module M

3.1. Let A = Z[v,v-1]. Let VQ = LijI Qwi C V'. Note that Vz c VQ. The W-action on V' induces a W-action on VQ and on VQ/VZ. Let M be the free A-module with basis {Xxlw C W,x E VQ/VZ}.

In the following lemma we use the following notation: given a property II, we define nI to be 1 if HI is true and 0 if II is false.

Lemma 3.2. (a) The A-linear maps ti : M -. M (i C I) given by ti(Xx)

-

Xsi() + (v- v-1)Sw(hi)eR-6(hi,x)czXx satisfy the relations of the braid group of (aij). (b) The A-linear maps t' : M -> M (i C I) given by t'(Xx) X () - (v -

V-1)6w(hi)eR+6(hi,x)ezXx satisfy the relations of the braid group of (aij). (c) We have tit' = t'ti = 1. (d) We have ti(Xw) - t(X) = (v - V-)6hix)CZXwx

Note that in (a), (b), the meaning of the property '(hi, x) c Z' is that it holds for a representative of x in VQ; then it holds for any other representative. The proof of (a) is by direct computation. Assume, for example, that i, j c I satisfy aij = aji = -1. Then

titjti(XX) - XSssjiS(x + (V -V )6W(hi)R-(hiZXsj

+ (v - v-1)6w(hj)R-6(hj,x)CzXwsi(

+ (v - v-1)26(hi)eR- w(hi+hj)CR- 6hi,x)ez6(hi+hj,x)eZXws

+ (V - v-1)2 (hj)eR-6w(hi+hj)CR- (h,xj,x)ezS(h+h,x)EZXsix

+ (V -V V1)36S (hi)ER- w(hj)ER-6(hi,x)EZ6(hj,x)CZXw

+ (v- v-1)6(hi+hj)eR-6(hi+hj,x)EZXw.

The right-hand side of this equality is symmetric in i, j. This shows that titjti = tjtitj : M -- M, as required. The proof of (c) is trivial. (b) follows easily from (a) and (c); (d) follows from the definitions.

3.3. It follows that for any w E W there are well defined linear maps tw, t' M -- M such that t -= t 1; t,i = ti, t' = t' for i E I and tvtw, = tww,, t'to, - t', if l(w)+l(w') = I(ww'). We have t' - t`l for w E W. By induction on 1(w), we see that X- = t-i (Xw(x)) for all x,w. Let Yw := t' ,(X(x). By induction on 1(w), we see that Yx is equal to Xx plus an A-linear combination of elements XX, with w' < w. Hence, by taking a linear combination of the identities 3.2(d) (for fixed x and variable w) we deduce the identity

(a) ti(Yw) -

t(Ywx) = v - V-1)(hi,x)1zYW.

Lemma 3.4. There is a unique Z-module homomorphism - M -- M such that

(a) ti(m) = t'(m) for all m C M, i c I; (b) Xx = Xx for all x VQ/VZ; (c) vmf = v-mf for all f E M, m C Z.

The homomorphism - has square 1.

We define a Z-module homomorphism -: M -, M by vmXw =- v-mY for


234 G. Lusztig




w E W,x E VQ/V, m E Z. Then (b) and (c) are clearly satisfied. To prove (a), it suffices to verify the identity

(d) tiXx = tiXw for all x, w,i. The left-hand side of (d) is, by definition, Ywi(x) + (v- -v)(w(hi)eR-6(hi,z))ezY while the right-hand side of (d) is t' (Yw). These expressions are equal for wsi > w, since in that case t' , = tt' 1. To check that they are equal for wsi < w, we write t' - = t't' s_ and we are reduced to verifying the equality

Ysi(zx + (V-1 - v)(hit,)EZ'(Yi(x) tt,i(yz)) WSi rsiSi i i wsi

This follows by applying ti to both sides of 3.3(a) (with x, w replaced by si(x), wsi). This proves the existence of -. The uniqueness is immediate. Next we prove that

(e) Yw = Xx for all x, w. We argue by induction on l(w). For w = 1, (e) is immediate. Assume now that l(w) > 1 and let us write w = w'Si where w > w' and i E I. Using 3.3(a), 3.4(a), the induction hypothesis and 3.2(d), we have

y = t (Yi()) - ti(YSi(X)) - (v - v1)6(hi,x)ezY

- t'(Y$,(y)) + (v - v ~)6(h,)EzY)(X)

- ytw(X x)) + (V V- )6(hj,X)EzXwsi()

= t,(Xwi()) -= XV ___ S'Si(X)\ ___ -yx

The equality (e) is proved. It follows that -: M -- M has square 1. The lemma is proved.

Proposition 3.5. Let w E W, x e VQ/VZ. There is a unique element Zw E M such that

(a) Zw = Ew,<wpx;w,,wXx,, wherep;w,,w E v-lZ[v-1] ifw' < w andp;, = 1;

(b) Zw= Zw.

By 3.3, we have Xw = Ew,< wrx;w,,wX , where rx;w,,w E A and r;w, = 1. Since : M -- M has square 1, we have w,;w,,<w< rx;w,,,,r;wr;,, = 6w"= for all w" < w, where the ring involution of A such that vm ~ v-m for all m is denoted by -. Condition (b) can be expressed as a set of equations for the unknown Px;w,,w'

S,w';w" <w' w rx;w" ,w'Px;?w',w = Px;W,,w for all w" < w. This set of equation has a unique solution by Lusztig (1993, ? 24.2.1). The proposition is proved.

For x E VQ/V/ and w' < w in W, let u(x; w', w) E Z be the coefficient of v-1 in px;w,,w E v-1Z[v-1].

Proposition 3.6. Let w E W, x e VQ/VZ and let i E I. Assume that wsi > w.

(a) If (hi, x) Z, we have ti(Z) = Zsis(x) (b) If (hi, x) e Z, we have (ti + v-11)(Zw) - Zs, + Ew,;w,si<w<w IL(x; W, w)Zw,.

We prove (a). Applying - to ti(Zw) gives t'(Zw) and this coincides with ti(ZW),


235



by the assumption of (a). By the definition of ti, we have

ti(Zw) px;w',wti(X,)- = px;w',XWSi(X). w' w w'^ w W'?W W'?W

This is of the form Xs(x) + ,, <,, w)iPwX(x) with pw,, E v- Z[v-1], since w' < w < wsi implies w'si < wsi. Thus, ti(Zw) satisfies the defining property of Zwsi) hence ti(Zx) = Zi(x). We now prove (b). Under the assumption of (b), we see from the definitions that

-(ti +v11)(ZW)- E (X; ', w)Zw, w';w' si<w' <w

is of the form Xwxi + Ew,,<wsi Pw,,Xwxt with pw,, v-1Z[v-1]. On the other hand, we have (- = (t'+v)(Zw)-(ti+v-11)(Zw) and this is equal to zero, by 3.2(d). Thus, C = C. Thus, ( satisfies the defining property of Zw,. hence ( = Zx. and (b) is proved.

3.7. We define RX, Wx C, PXx x, < for x E VQ/Vz in terms of a representative of x in VQ, as in 2.1, 2.2, 2.4. The resulting things are clearly independent of the choice of representative. Given w E W and x E VQ/VZ, we define an element zx E M by

Zw 7 a',aXwloa

a'CWx;a'<Xa

where w1 E W is the element of minimal length in wWx; a E Wx is defined by w = w1a and a,, = v-l (a)+l (a')Pa,a(V2) E Z[v -]. (Pa,,a are the polynomials attached in Kazhdan & Lusztig (1979) to the Coxeter group Wx.) In the following result we use the previous notation as well as the notation u(a', a) E Z for the coefficient of v-1 in Va',a

Proposition 3.8. (a) Assume that (hi,x) E Z and wsi > w. Then hi E Px and si E Wx satisfies lx(si) = 1. We have

(ti +. v-11)(zw) -= ZWi + ZatEWx;a/si<xat<xa l(a', a)Zxila

(b) Assume that (hi, x) Z and wsi > w. Then ti(zx) = zi()

We prove (a). The left-hand side of the identity in (a) is

Wa',a(Xwlasi + i 5wla(hi)ER-VXwXlat + -wla'(hi)ER+V Xwlat), a'/EW

while the right-hand side is

a'as,ilaI + E /(a", a)Tra',a,"Xw ,.

a' W a" E Wx al E WX a'?M _asi aa_ax a'/< asi a" Si < s a" < a a' < a"

By 2.5, we have Swia'(hi)ER- = Sa'si<a' and wi(at(hi)ER+ -= atsi>a'- We see that the desired identity can be expressed purely in terms of the Coxeter group Wx where it is a known property of the 7a',a from Kazhdan & Lusztig (1979). We


G. Lusztig 236




prove (b). The desired identity can be rewritten as follows:

E a Sir() _ a' s XESi(x) 7Ia',a /wla'si bl ,as wla'l

a'EWx b'EWsi(x) a' <a b' <Si() asi

(here 7b',as, is with respect to WSi(x)). This follows immediately from the following statement, whose proof is left to the reader.

If (h, x) ~ Z, then h \-4 si(h) is a bijection of Rx onto Rsi(x) carrying (RX)+ onto (RSi(x))+; w u-4 siwsi is an isomorphism of Coxeter groups and, if w1 is the element of minimal length in wWx, then w1si is the element of minimal length in wsiWsi().

Corollary 3.9. For any w E W and x E VQ/VZ, we have Zw zx.

We argue by induction on l(w). For w = 1 the result is trivial. Assume now that w - 1. We must show that zx satisfies the defining properties of Zx. The only non-trivial property that needs to be verified is that x = zx. We can write w = w'si with 1(w') < 1(w) and i E I. Let x' = si(x). Assume first that (hi, x) E Z. We write the equality of 3.8(a) for x, w' instead of x, w. All terms in the right hand side of that equality except possibly the first one are fixed by - (by the induction hypothesis). To show that the first term zI ,i = z, is fixed by , it is therefore enough to show that the left-hand side is fixed by -. In other

words, we must show that (ti + v-ll)(z,) = (ti+-v-ll)(z,) or, equivalently, that (ti + vl)(Z,)) = (ti + v-1l)(zx,). By the induction hypothesis, we have zx, = Zw, hence zx, = ZW,. Hence we are reduced to verifying (ti-ti+(v -v- )1)(ZW, ) = 0. This follows from 3.2(d). Next we assume that (hi,x) x Z. By 3.8(b), we have z = ti(zx,). It is enough to show that ti(zx) = ti(z,), or, equivalently, that

ti(z',) = ti (z ,). By the induction hypothesis, we have zx,, = Zx, hence zW, = Z, . Hence we are reduced to verifying (t'- ti) (Z',) = O. This follows from 3.2(d). The corollary is proved.

4. Affine flag manifold

4.1. To our affine Cartan matrix (aij) corresponds the (untwisted) affine Lie algebra g (over C) defined by the generators ei, fi, hi (i E I) and the relations [ei,fj] = ijhi, [hi,ej] = aijej, [hi,fj] = -aijfj, [hi,hj] = 0 (for all i,j) and

[ei,[ei, e,]...[e, e...]] 0, [fi,[fi,... [f,fj..] ..]] 0 (for all i 74 j; ei and fi appear 1- aij times). This differs slightly from the definition in Kac (1983) since in the present definition the simple roots are linearly dependent. Let g+ (resp. g-) be the Lie subalgebra of g generated by {eili E I} (resp. {fili E I}). Let g0 be the subspace of g spanned by {hili E I}. The elements {hili E I} form a C-basis of g0 and we will identify V (see 1.2) with the R-subspace of g0 spanned by this basis, and V' (see 1.2) with the R-subspace of Homc(g?, C) spanned by the linear forms which take real values on V. Let Vk =- Ei Nwi C VZ. Given x E VN, let Ax be the quotient space of the enveloping algebra of g- by the left ideal generated by the elements f,hi X)+ (i E I). Let rx E Ax be the image of 1. There is a unique g-module structure on the C-vector space Ax such that the action of g- is induced by left multiplication in the enveloping algebra, and eir]X = 0, hir,x = (hi, x)rlx for all i E I.


237



4.2. To (g, ei, fi, hi) correponds a group G with given homomorphisms ei: C -?

G, fi : C - G, hi: C* -- G (i E I) whose images generate G. For any x E Vk, G acts on Ax as follows. For t E C, we have e(t) - exp(tei), fi(t) = exp(tfi) as maps Ax -? AX; if E E Ax is such that hi = s~ with s E Z, then hi(t)) = tV for t c C*. Moreover, the action of G on (DxAX is faithful. These properties determine G up to a unique isomorphism. Let U be the subgroup of G generated by the subgroups i( C). Let T be the subgroup of G generated by the subgroups hi( C*). The collection of homomorphisms hi define an isomorphism (C*)I A T. Given i E I, let Gi be the subgroup of G generated by the subgroups ei(C), fi(C), hi(C*). Let Ni be the normalizer of hi(C*) in Gi. Let N be the subgroup of G generated by the subgroups Ni (i E I). Then T is normal in N and there is a unique isomorphism of groups W - N/T such that si is mapped to (NiT -T)/T. We use this to identify W = N/T; let wt E N be a representative of w E W. Let B be the subgroup TU = UT of G. An Iwahori subgroup of G is, by definition, a subgroup of G conjugate to B. Let B be the set of all Iwahori subgroups of G. As in Iwahori & Matsumoto (1965) and Tits (1982), the product B x B decomposes under the diagonal action of G into G-orbits O(w) naturally indexed by the elements w E W; O(w) is defined by the requirement that (B, wbB-l1) E O(w). For x E Vz, let Ax: T -+ C* be the homomorphism given by Ax(hi(t)) = t(hi,x)

for all i E I, t E C*. Let LX be the set of T-orbits on (G/U) x C for the T-action T: (gU, z) I- (g--1U, Ax(T)Z). Let px : LX -_ B be defined by px(gU, z) = gBg-.

4.3. Now B and Lx are endowed with natural structures of 'algebraic varieties of infinite dimension' (or rather as inductive limits of ordinary algebraic varieties under closed imbeddings) so that px : LX - B are 'algebraic' line bundles. We sketch the definitions. For any x E VZ such that (hi,x) > 1 for all i, the map gBg-l -t Cgrl from B to the projective space P(Ax) of Ax is injective. The algebraic structure on B is the one induced via this map from that of P(AX); it is independent of the choice of x (see Tits 1982). If x E Vk, we identify Lx with the set of all pairs (B1, ) where B1 E B and ( is a vector in the unique B1-stable line in AX, via (gU, z) -> (gBg-1, zgrl"). Thus, Lx appears as a subset of a pull-back (to B) of the canonical line bundle on P(AX) and thus inherits an algebraic structure. If x E Vz, we write x = x - x2 where x1, x2 E VN. We identify Lx1 0 (LX2)* (tensor product of line bundles over B) with Lx via (gU, z1) () (gU, 2) -4 (gU, Z12). Now Lx1 0 (LX2)* is an algebraic line bundle over B by the previous construction. This gives an algebraic structure on Lx which, as one can check, is independent of the choice of 1, x2.

4.4. For w E W, let

B, = {B,Be E 3(B,Bi) O(w)} and B< = U Bw. w'EW;w'(w

Then BL is an algebraic subvariety of B, isomorphic to Cl(W), and its closure is B<w, an ordinary projective algebraic subvariety of B. For x E Vz, we denote by Lx the complement in Lx of the zero section. Let pX : Lx -> B be the restriction of px. For x E Vz,w C W, we set Lx - (jx)-lBw and L? - (pX)l-B< =

oc oW, c. Lond. A


G. Lusztig 238




5. Monodromic systems

5.1. Let 0 be the group of all roots of 1 in C*. We shall fix an injective homomorphism T': ( -- C* or, equivalently, an isomorphism O -1 O. Assume that x E Vz and n > 1 are given. We consider the action of the group Un = U x C* on Lx given by (u, ) : (gU, z) H* (ugU, Cnz). The orbits of this action are precisely the subvarieties Lx for various w C W. Let On = {z cE Ozn = 1}. Then O,, regarded as a subgroup of U' by z - (1, z), acts trivially on Lx. Let 0 : O( -- C* be the homomorphism defined by +(z) = 1(zd) where d = gcd(n, x) is the largest integer / 1 such that n C dZ and x C dVz. Note that ker b = Od. Let C(x, n) be the category whose objects are the finite direct sums of simple perverse C-sheaves on Lx (with shifts), supported by unions of finitely many U'-orbits, which are U'-equivariant and are such that the action of On (given by equivariance) on the stalks of each cohomology sheaf is through the character fi; the morphisms are those in the derived caregory. (The condition on support is needed so that our perverse sheaves are the usual ones from the case of ordinary algebraic varieties. The equivariance is understood in the following sense: on the support (an ordinary projective variety), U' acts through a quotient of the form U x C*, where U is a unipotent algebraic group, and the equivariance is with respect to this quotient.) Let C(x, n) be the abelian group defined by the generators K, corresponding to the isomorphism classes of objects of C(x, n), and relations K + K' = K" whenever K" is isomorphic to K ? K'. We regard /C(x, n) as an A-module by vmK K[-m] for m E Z.

5.2. There is a C-local system of rank 1 on Lx (unique up to isomorphism) which is U'-equivariant and is such that On acts on its stalks (by the equivariance) through the character /. We denote it by Cz,n. Let Zwx' be the simple perverse sheaf on L<x whose restriction to Lx is Cxn [l(w)+ 1]. It is clear that the elements

Zwxn (for various w E W) form an A-basis of C(x, n). 5.3. Let K be an object of C(x,n). For any j' E Z, the cohomology sheaf

1W' K is a U'-equivariant constructible sheaf with support contained in a finite union of U'-orbits. Hence for any w' E W, the restriction of Hi jK to Lx, is a direct sum of copies of the local system Wc'. The number of copies is denoted by fw',j',+(w,)+l(K) C N. Thus, for any j C Z, we have a homomorphism fw,, : /C(x, n) -? Z. It satisfies the identity fw,,j(K[d]) = fw',j+d(fK) for all d E Z. Note that fw,,j(Zn) is 0 unless w' < w; fw',J(Z'n) is 0 if w' < w and j > 0; it is 0 if w' = w and j 4 0; it is 1 if w' = w and j = O. Hence there is a unique element XW" E /C(x, n) such that fw,,j(X, ' ) is 0 unless w' = w, is 0 if w' = w and j /- 0 and is 1 if w' = w and j = 0. The elements XW (for various w) form again an A-basis of C(x, n); it is related to the basis (Z,'n) by

x,n >_ 3 f j(7x\n)lx, n.

w' w~ ( ') , j<O

The following result gives an explicit formula for the integers fw,j (Zw') above, or equivalently, for the structure of the cohomology sheaves of the simple perverse sheaves Zx'n. It is a common generalization of results in Kazhdan & Lusztig (1980) concerning affine Schubert varieties and results in Lusztig (1984, ? 1) concerning monodromic systems on ordinary flag manifolds.


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Proposition 5.4. Let x C Vz and let n / 1. We write x/n instead of(1/n)x C VQ and also for its image in VQ/VZ. Assume that w' < w in W.

(a) If w'Wx/n wWl/n, then f , j(Z' n) = 0 for all j. (b) If w'Wx/n - wWx/n, let w1 be the element of minimal length in wWx/n and

let us write w = w1a, w' = wia' where a, a' C Wx/n. The sum Ej<0 fw',j(Zw'n)v is equal to Tat,a (notation of 3.7) if a' <x/n a and is zero, otherwise.

The proof can be carried out along lines similar to the proof in Lusztig (1984, ? 1). We will therefore give only a sketch of it, together with references to appropriate parts of Lusztig (1984, ? 1).

Given x E Vz and n 1 we set d = gcd(n, x) (see 5.1), x' = (1/d)x E Vz, n' =

n/d E N, so that x/n = x'/n'. We have a natural map Lx' -- (Lx')?d Ldx' = Lx given by ( -* ?... . X (d factors). This restricts to a d-fold unramified covering Lx' -? Lx. The inverse image of Z~'~ under this covering map is easily seen to be Z" " . Hence we have an isomorphism 1C(x, n) t IC(x', n') carrying each basis element Zw'n to the corresponding basis element Z' ". It also carries X"' to XWX for all w. Thus, 1C(x, n) with its bases (Zw'T), (Xw'n) depends only on the element x/n c VQ and not on x, n separately. Hence for x1 E VQ, we can define IC(xl), ZXl, XW,, as 1C(x, n), Z',n, Xx'n where - x/n with x E Vz, n > 1. As in Lusztig (1984, 1.13), we see that

(c) IC(xi),Zwl1,X ,1, with its bases, depends only on the image of xl E VQ in

Let x E VZ and let i C I. Let x' si(x). We consider the two projections Prl,pr2 : 0(si) -> 3. We denote by Lx the pull-back of the line bundle Lx under pr2 and we denote by Lx the pull-back of the line bundle Lx under prl. Let LX, Lx be the complements of the zero sections in L, Lx . There exists an isomorphism of line bundles (over O(si)) Lx' Lx (unique up to a factor in C*). We choose such an isomorphism. It restricts to an isomorphism t : Lx' - L. Let rl : Lx' - Lx, 7r2: Lx , Lx be the natural projections. If we assume that (hi, x/n) Z then, arguing along the lines of Lusztig (1984, 1.16, 1.22(case 1)) we see that K i> (7rl)!t*7r*(K)[1] is a functor C(x,n) -- C(x',n) which induces a homomorphism IC(x,n) to IC(x',n) that carries Xn to X,Sn. Moreover, it carries Zw'2 to Zx', if w < ws, (the same is true if w > wsi, but we do not need this). Hence, if xl C VQ satisfies (hi, xi) f Z and xl = si(xi), then there is a unique isomorphism ti: /C(xl) -> /C(x) such that ti(Xwl) -= Xi, for all w, and

ti(ZW1) = Zi , for all w such that w < wsi. Similarly, if xl E VQ satisfies (hi, xi) = 0, then as in Lusztig (1984, 1.18, 1.19,

1.22(case 2)), we see that there exists a homomorphism (ti + v-ll): IC(xi) - 1C(xi) such that

(t. + v-l1)(X,l) = Xx, + v-~1Xl, for all w such that w < wsi,

(ti + v-11)(X'w) = Xxi. + vX,w1, for all w such that w > wsi.

Moreover, if w < wsi, then (ti + v-l1)(Zxl) equals Zl,. plus an integer linear combination of elements Zxl for various w' < w. (The homomomorphism (ti + v-1) is induced by the composition of an inverse image functor with a direct


240 G. Lusztig




image functor in P1-bundles; at this point the decomposition theorem must be used.) The same continues to hold if the condition (hi, x) = 0 is replaced by the weaker condition (hi, x1) E Z. Indeed, if the last condition holds, then by adding to x1 a suitable element of Vz, we obtain an element x1 E VQ such that (hi, x) = 0. We then use (c).

Let us identify the A-module DKxC(x) (where x runs over VQ/VZ) with the A-module M in such a way that the basis elements Xx and Xx correspond. Then Z E M. We now prove that

(d) Zw = Zw for all x E VQ/Vz,w E W. We argue by induction on 1(w). When w = 1, the result is obvious: both sides of (d) are equal to X1. Assume now that l(w) > 1. We can find i E I such that w' := wsi < w. Assume first that (hi,x) f Z. Let x' = si(x). Since ti(X' ) = ti(X ') = Xxs for all u E W, we must have

t(i (,) = ti(Zx',). We substitute here ti(Z', ) = Zx and tZW,) - ti((ZW') = Zt (where we have used the induction hypothesis and 3.6(a)). We obtain Zx = Z, as desired. Assume next that (h, x) E Z. Since (ti +v-11)(Xx) = (ti+v-1l)(Xx) for all u E W, we must have (ti+v-1)(Zx,) = (ti+v-1l)(Z,). By the induction hypothesis, we have Zw, = Zx, hence (ti+v-1)(x,) = (ti+v-1)(Zx,). The right- hand side of this equality is equal to Zw plus an integer linear combination of elements Zu for various u < w', (see 3.6(b)) while the left-hand side is equal to Z plus an integer linear combination of elements Zx for various u < w' (for such u we have Zx = Z, by the induction hypothesis). It follows that Zw - Zx = Eu<w, g9uZ where gu E Z. We write both sides of the last equality as Z[v-1]-linear combinations of elements Xx, for various u' E W, and substitute v-1 by 0. We obtain 0 = ZEu<, guX1x. Since the Xxu are linearly independent, we deduce that gu = 0 for all u, hence Zw = Zx. This proves (d). Now the proposition follows by combining (d) with the equality Zw = zx (see 3.9).

6. Highest weight modules of g

6.1. Let p = Ei wi E Vz. Given x E VQ, the Verma module MX is by definition the enveloping algebra of g- with the unique g-module structure such that the action of g- is by left multiplication, and e^l = 0, h1i = (hi, x)1 for all i E I. It is known that, when (c, x + p) ~ 0, the g-module Mx has a unique simple quotient module. We denote it by LX. Assume now that x satisfies (hi, x + p) < 0 for all i E I. Then the elements {w-1(x + p) - plw E W} are all distinct and for any w E W we have (c,(w-(x + p) - p) + p) = (c, x + p) < 0. For any w E W, we have (in the appropriate Grothendieck group):

Lw-l(z+p)-p = gw',wMw'-(x+p)-p, where gw,,w E Z. w'EW;w',w

Let us write x = x'/n where x' C Vz and n > 1. The following is a generalization of the conjecture of Kazhdan & Lusztig (1979) and Kazhdan & Lusztig (1980):

(a) gw,,w = (_)I(w)-1(w') ZEl j0 f,(Zw x'n) for all w' < w (notation of 5.3). In other words, the integers gw',w can be expressed in terms of intersection cohomology. In the case where x = -2p, the conjecture


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(a) has been stated in Lusztig (1990); a proof in this case has been given in Casian (1990), but the validity of the proof is controversial. Very recently a proof of (a) in the integral case (n = 1) has been announced in Kashiwara & Tanisaki (1994). One can expect that the proof of (a) in the general case should not differ substantially from the proof in the case n = 1; this is so in the case of finite dimensional Lie algebras, by the work of Beilinson and Bernstein.

7. The Coxeter group W(a)

7.1. Let u be a non-zero rational number. Let Q"(cf) = Q" fn acQ'. Clearly, Q"(cr) is a W-stable subgroup of Q". Let T(cr) be the (normal) subgroup of W consisting of the transvections O, with z C Q"(a)). Let W(c) be the subgroup (of finite index) WoT(a) = T(()Wo of W, a semidirect product of Wo and T(u7). Let

X(f) {x E V'l(c,x) = a and (hi,x) c Z Vi c Io},

R(c) = {X + dxmclx E Ro, m c Z, cadm e Z} C R, R(o)+ = R(o) n R+

Lemma 7.2. Let x E X(a). (a) We have Rx = R(or). (b) The following three subgroups of W coincide: Wx, W(cr) and

'WX = ({w Wlw(x) - x c Q'}.

We first verify the following statement. (c) Let h E R. We have Sh E 'W" if and only if h E RX.

Indeed, if sh C 'Wx, then Sh(x) - x E Q' so that (h,x)h' E Q'. Since h' is an indivisible element of Q', it follows that (h, x) C Z hence h E RX. The converse is obvious. Next we show that

(d) 'WX = W(a). If j E I0, we have sj(x)- x =-(hj,x)hj E Q'; thus s 'WE It follows that W0 c 'Wx. If z c Q", we have Oz(x) - x = -z; hence O Ec Wx if and only if az C Q', i.e. if z c Q"(or). It follows that 'Wx = W(u), as desired. We prove (a). Let h = X + dxmc c R where X E Ro,m E Z. If h E R(o), then by definition, dxmX' Q"(au), hence Odx,, E T(C), so that sx,0sx,m C T(cr) (see 1.5(a)). Using the fact that sx,0 E Wo C W(a), we see that sx,m C WoT(or) = W(a). Hence, using (d) we have sh E 'WX and, using (c), we have h c Rx. Conversely, assume that h E RX. Then (x+dxmc, x) E Z. Since x C X((), we deduce that ad^Xm C Z, so that h C R(cr) and (a) is proved. We now show that

(e) wx c ''W. It suffices to show that for any h c Rx, we have Sh E 'Wx. This follows from (c). Next we show that

(f) W(a) C W. Let w E W(uo). We write w = w1Oz where w1 E Wo and z C Q"(ro). Now w1 is a product of reflections si = Sh,o (i c Io). We have z -= Eijo dinih' where ni E Z

and cdini c Z for all i E Io. Hence Oz = 1licIo Z where zi = d nih' c Q"(u). We have ,i = 8sh,o0Sh,n, and hi E R(uo), hi + dinic E R(o-) for all i e Io. Using (a), Proc. R. Soc. Lond. A (1994)

242 G. Lusztig




we have hi E RX, hi + dinic e R- for all i E Io. Thus, we have w E Wx and (f) is proved. Now (b) follows by combining (d)-(f).

7.3. Write cr = p/q where p, q are relatively prime integers with p : 0, q > 0. Let P(c) be the following basis of V:

(a) P(u) = {hili E I0} U {/3 + qc}, if D, q are relatively prime; (b) P(r) = {hili E Io} U {7 + qc}, if D divides q.

(In the case where D = 1, the cases (a) and (b) coincide and we have 3 = ry, so that there is no contradiction.)

Lemma 7.4. (a) We have P(cr) C R(ac)+. (b) If h E R(cr)+, then h is a linear combination with coefficients in N of vectors in P((a). (c) If x E X(a), we have Px = P(a).

We prove (a). Since dp = D, the inclusion 3 + qc E R(cr)+ is equivalent to the statement 'q E {1,2,...} and oq E Z', which is obvious. In the case where D divides q, the inclusion y + qc E R(a)+ is equivalent to the statement 'q/D E {1, 2,...} and aq E Z' (since d^ = 1) and this is obvious. Moreover, the inclusions hi E R(u)+ are obvious for i E Io; (a) is proved. We prove (b). Write h = x+dxmc where X E Ro, m E Z, (p/q)dxm E Z. The last condition implies Dm/(d,q) E Z (since p, q are relatively prime). Assume first that m = 0 and that X E Ro. Then h = X is a linear combination with coefficients in N of vectors in {hili E Io}. Assume next that m > 0 and that D, q are relatively prime. We have

+D \ Dm D Dm h -

X-d- (- + c). V=( d\ld dxq d\ dxq

From 1.4, we see that X- (D/dx)/3 and -f are linear combinations with coefficients in N of vectors in {hili E Io}. Since Dm/dxq E Z and D, q are relatively prime, we have m/q E Z, and since m > 0, we have m > q. It follows that Dm/dxq E N and Dm/dxq - D/dx E N. The desired property of h follows. Assume next that m > 0 and that D divides q. We have

( Dm 1)() Dm d q d+ q

,From 1.4, we see that X-y and -y are linear combinations with coefficients in N of vectors in {hili E Io}. Since Dm/dxq E Z and m > 0, we have Dm/dxq - 1 E N. The desired property of h follows; (b) is proved. We prove (c). Let C' = {x E VllI(h,x) > 0 Vh E P(a)}. Clearly, C' is a (non-empty) open simplex in V1 containing C (see 2.1). If x E X(a) then, for any h E R(c)+ = (Rx)+, the affine-linear function -> (h, ~) takes only > 0 values on C' (this follows from (b) and the definition of C'). Hence C' is contained in the x-alcove containing C, that is, C' C Cx. Conversely, let 5 C CX. Then for any h E (RX)+ we have (h, ) > 0. Since P(a) c R(u))+ = (RX)+, it follows that (h,,) > 0 for all h E P(cr); hence

E C'. Thus Cx C C' and Cx = C'. We deduce that the walls of Cx are the same as the walls of C', hence Px = P(cr). The lemma is proved. Proc. R. Soc. Lond. A (1994)

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8. From affine Lie algebras to quantum groups

8.1. Let (7 C Q be such that a + g < 0, where g =- iri. We write c = p/q as in 7.3 and a + g p'/q where p' = p + qg. Note that q , l,p' < -1 and p', q are relatively prime. For technical reasons we shall assume that p' < pO for a certain constant pO < 0 depending on the Cartan matrix. Let us regard V' = {x E V'l(c, x) =- } as an R-vector space with coordinate functions ((hi, ?)li C Io}. Let X((7)+ be the set of all x E V' such that (hi,x) C N for all i E Io. Let Of+g be the category whose objects are the g-modules on which c acts as c times identity and which admit a finite composition series with subquotients of form Lx (x C X(cr)+). The set X(C)+ indexes both the simple objects of 09+g and the 'Weyl modules' (see Kazhdan & Lusztig 1993). Using results of Kac & Kazhdan (1979) together with the computations in ?7, we see that the linkage pattern in O?+g is governed by the Coxeter group generated by the reflections in the walls of the alcove

C={x E V'(h,x+p)<0 Vh P(a)}

in V'. If D, q are relatively prime, then

C = {x Vl(hi, x) <-1 Vi E o; (-i, x) > p' + (, p)}; the corresponding Coxeter group is isomorphic to W. If D divides q, then

C = {x E V'l(hi, x) < -1 Vi e Io; (-I7,x) > p' + (7, p)};

the corresponding Coxeter group is isomorphic to W" (see 1.6). Thus, if D > 1, there is a sharp difference between the linkage pattern in the case where D, q are relatively prime, and that in the case where D divides q.

8.2. For any root of unity ( C C*, let ?(() be the category whose objects are the finite-dimensional integrable representations of the quantum group corresponding to (aij)i,jEiz and to (di)iio,, with parameter (. Let us take ( = exp(7rx/-1/(7+g)), where o is as in 8.1. The simple objects of E(() as well as the 'Weyl modules' are naturally inde--ed by N'O, which we identify with the subset X(U)+ of V' by x - ((hi, x)i C Io). If the order -p' of (2 is relatively prime to D then, according to Andersen, the linkage pattern in 8(() is governed by by the Coxeter group generated by the reflections in the walls of the alcove

C' = {x V 7(hi x)<-1 Vi e Io; (-y, ) > p' + (, p)};

(as we have seen, this is isomorphic to WO). On the other hand, if D divides -p' and D > 1, one can show that the linkage pattern is different from the previous one. Thus, if D > 1, there are two different types of linkage patterns, according to whether D divides p' or not.

8.3. From 8.1 and 8.2 we see that, when the linkage patterns of 0?+g and ?(() are compared, there are three possibilities for a + g = p'/q to consider (assuming D > 1):

(a) D, q are relatively prime and D, p' are relatively prime; (b) D divides q and D,p' are are relatively prime; (c) D, q are relatively prime and D divides p'.

In case (a), C' is strictly contained in C and the linkage patterns are different. In case (b), we have C' = C and the linkage patterns are the same. One can show that in case (c), the linkage patterns are different. Note also that when D = 1, the


244 G. Lusztig




three cases above coalesce into one case; we have C' = C and the linkage patterns coincide.

8.4. The main result of the series Kazhdan & Lusztig (1993, 1994) is the construction of an equivalence of categories , : 0+g Z ?((), under the assumption that (aij)i,jCo is symmetric. Now the construction of Kazhdan & Lusztig 1979 provides a canonical functor a

: ,+g -+ ?(() even without the symmetry assumption. Moreover, the proof given in Kazhdan & Lusztig (1993, 1994) that this is an equivalence is almost independent of the symmetry assumption; the one ex- ception to this is Lemma 37.1 of Kazhdan & Lusztig (1994), which is actually false in the non-symmetric case. (See the remark following Lemma 37.1 in Kazh- dan & Lusztig (1994)). Now Lemma 37.1 in Kazhdan & Lusztig (1994) continues to hold in the non-symmetric case, provided that we restrict ourselves to values of a + g = p'/q such that D divides q, when the linking patterns coincide (see 8.3). Hence in this case, k is an equivalence of categories. On the other hand, if D > 1 and D, q are relatively prime, then 1a is not an equivalence of categories (this is the case, for example, if D > 1 and (7 is an integer).

9. From modular representations to combinatorics

9.1. Let g1 be a simply connected semisimple algebraic group over an algebraic closure of the finite field Ft (1 is a prime number), corresponding to the Cartan matrix (aij)i,jeo. We assume that 1 is sufficiently large. In Lusztig (1980) it has been conjectured that the characters of the finite dimensional irreducible rational

(=modular) representations of g1 can be explicitly computed in terms of the values at 1 of the polynomials Py,, of Kazhdan & Lusztig (1994) corresponding to the affine Weyl group WO (see 1.6). In a lecture at IAS, Princeton (November 1988), the author has proposed a way to divide the hypothetical proof of this conjecture into four steps, assuming that the Cartan matrix is symmetric (see also Lusztig 1990). The first step is to show that the weight structure of the irreducible modular representations (with restricted highest weight) of g1 is the same as the weight structure of the corresponding irreducible object of ?(exp(-r/-1/1)). The second step is to construct an equivalence of categories O-i-g -* ?(exp(--Trv/-1/)) carrying Weyl modules to the corresponding Weyl modules. The third step is to prove formula 6.1 (a) for x E Vz such that (hi, x + p) < 0 for all i E I. The fourth step is to compute the local intersection cohomology of the varieties B<w in terms of the polynomials Py,w of Kazhdan & Lusztig (1979) corresponding to W (which coincides with WO in this case). Now the fourth step is already contained in Kazhdan & Lusztig (1980). For the third step, see ? 6. The second step is established in Kazhdan & Lusztig (1993, 1994). The first step is established in Andersen et al. 1992.

9.2. In the case where the Cartan matrix (aij)i,jEI0 is not symmetric, the previous sequence of steps does not work all the way since the second step is now not true. Instead, we have to use four modified steps as follows. The first step is to show that the weight structure of the irreducible modular representations (with restricted highest weight) of l1 is the same as the weight structure of the

corresponding irreducible object of ?(exp(-7rx/T-D/i)). The second step is to construct an equivalence of categories 0-(/D)-g -- S(exp(-?rV-1D/1)) carrying Weyl modules to the corresponding Weyl modules. The third step is to prove


245



formula 6.1(a) for x E D-1Vz - Vz such that (hi,x) E Z for all i E 1o and (hi,x + p) < 0 for all i e I. The fourth step is to compute (for x as in the

previous step and w E W) the local intersection cohomology of the variety L< with coefficients in the local system Dx,D, in terms of the polynomials Pa,,' of Kazhdan & Lusztig (1979) corresponding to WO. Now the first step is covered by Andersen et al. (1992). The second step is covered by Kazhdan & Lusztig (1993, 1994) and 8.4. For the third step, see ? 6. Step four follows from 5.4. (In our case, the Coxeter group Wx is isomorphic to W~, see 7.2, 7.4, 8.1.)

9.3. A similar succession of steps (without the first step) allows us (assuming the truth of 6.1(a)) to express the characters of the simple objects of ?(() in terms of the Pa',a for WO. Here ( is any root of 1 whose order is not too small and is relatively prime to D. It would be interesting to understand what happens for roots of 1 of order divisible by D, when D > 1.

I am supported in part by the National Science Foundation.

References

Andersen, H. H., Jantzen, J. C. & Soergel, W. 1992 Representations of quantum group at a p-th root of unity and of semisimple groups in characteristic p: independence of p. Preprint.

Bourbaki, N. 1968 Groupes et algebres de Lie, vols IV, V, VI. Paris: Hermann.

Casian, L. 1990 Kazhdan-Lusztig conjecture in the negative level case (Kac-Moody algebras of affine type). Preprint.

Iwahori, N. & Matsumoto, H. 1965 On some Bruhat decomposition and the structure of Hecke rings of p-adic Chevalley groups. Publications Math. I. H. E. S. 25, 5-48.

Kac, V. 1983 Infinite dimensional Lie algebras. Birkhauser.

Kac, V. & Kazhdan, D. 1979 Structure of representations with highest weight of infinite dimensional Lie algebras. Adv. Math. 34, 97-108.

Kashiwara, M. & Tanisaki, T. 1994 Characters of the negative level highest weight modules for affine Lie algebras. Preprint.

Kazhdan, D. & Lusztig, G. 1979 Representations of Coxeter groups and Hecke algebras. Inv. Math.53, 165-184.

Kazhdan, D. & Lusztig, G. 1980 Schubert varieties and Poincare duality. Proc. Symp. Pure Math., vol. 36, pp. 185-203. AMS.

Kazhdan, D. & Lusztig, G. 1993 Tensor structures arising from affine Lie algebras. I, II. J. Am. math. Soc. 6, 905-947 and 949-1011.

Kazhdan, D. & Lusztig, G. 1994 Tensor structures arising from affine Lie algebras. III, IV. J. Am. math. Soc. (In the press.)

Lusztig, G. 1980 Some problems in the representation theory of finite Chevalley groups. Proc. Symp. Pure Math., vol. 37, pp. 313-317. AMS.

Lusztig, G. 1984 Characters of reductive groups over a finite field. Ann. Math. Studies, vol. 107. Princeton University Press.

Lusztig, G. 1990 On quantum groups. J. Algebra 131, 466-475.

Lusztig, G. 1993 Introduction to quantum groups. Progress in Mathematics, vol. 110. Boston: Birkhauser.

Tits, J. 1982 Resume du cours, 1981-82. College de France.

Received 28 September 1993; accepted 7 December 1993


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