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D. Kazhdan, G. Lusztig (1987)
Proof of the Deligne-Langlands conjecture for Hecke algebrasInventiones mathematicae, 87
A. Borel, G. Bredon, D. Montgomery, E. Floyd, R. Palais (1961)
Seminar on transformation groups (AM-46)
l®w^A(w) (see 4.3) define an ^module structure on H^^Y,^). Similarly, S^l-^A'^), l®wh>A'(w) define another H-module structure on H^^Y, J^)
V. Ginsburg (1987)
“Lagrangian” construction for representations of Hecke algebrasAdvances in Mathematics, 63
(1982)
Faisceaux pervers
When Y is closed in g^, one uses the case Y = g^? together with 8.11 c) and 3.10. When Y is arbitrary, the corollary follows from the already known results for Y, using 8.11 b) and 3.10. Theorem 8.13
G. Lusztig (1985)
Characters sheaves, I-VAdv. in Math., 56
Let Y be a locally closed sub variety of g^, which is a union of nilpotent orbits. a) H^(Y,^)=0. b) The open embedding i: Y <-^ Y (closure of Y) induces a surjecfive homomorphism f : H° x ^
G. Lusztig (1981)
Green polynomials and singularities of unipotent classesAdvances in Mathematics, 42
Clearly, the isomorphism class of the H-module Eo^^p depends only on râ nd the G-conjugacy class of 0,^5 p. Theorem 8.15. -Any simple H-module ^ is a quotient of a standard H-module
G. Lusztig (1978)
Representations Of Finite Chevalley Groups
G. Lusztig, N. Spaltenstein (1979)
Induced Unipotent ClassesJournal of The London Mathematical Society-second Series
G. Lusztig (1987)
Fourier transforms on a semisimple Lie algebra over Fq
G. Lusztig (1976)
Coxeter orbits and eigenspaces of FrobeniusInventiones mathematicae, 38
A. Borel, G. Bredon (1961)
Seminar on Transformation Groups.
G. Lusztig (1985)
Character sheaves, VAdvances in Mathematics, 61
G. Lusztig (1984)
Intersection cohomology complexes on a reductive groupInventiones mathematicae, 75
M. Goresky, R. Macpherson (1983)
Intersection homology IIInventiones mathematicae, 72
CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I. by G'~ORGE LUSZTIO (1) Introduction The Hecke algebra attached to a finite (resp. affme) Weyl group plays a very important role in the representation theory of a reductive group over a finite (resp. p-adic) field. It is also of considerable interest to consider Hecke algebras in which the parameters attached to the simple reflections s, are powers q"~ of q (depending on s~) where n, are integers/> 1 subject only to the condition that n~ ---- n~ ifs~, sj are conjugate. These more general Heeke algebras arise typically as endomorphism algebras of repre- sentations induced by cuspidal representations of parabolic (resp. parahoric) subgroups, trivial on the " unipotent radical ". (See [7, p. 34, 35].) We would like to understand such Hecke algebras with unequal parameters from a geometric (rather than arithmetic) point of view and to classify their simple modules (in the affine case), extending the known results [5] for equal parameters. I believe that the proper setting for these questions is in equivariant K-homology (as in [3], [5]), mixed with the cuspidal local systems of [9]. This is made very plausible by the results of this paper, in which we replace the affme Hecke algebra by a certain graded version. The connection between an affine Hecke algebra and its graded version is ana- logous to the connection between a reductive group and its Lie algebra or the connection between K-theory and homology. (In fact this is more than an analogy.) 0.1. We shall now define this graded version tt of an affine Hecke algebra. Let t be a C-vector space of finite dimension and let R C t" be a root system, with a set of simple roots FI ---- { al, 9 9 a,, } and Weyl group W with corresponding simple reflections { sl, ..., s,, }. (Thus U has a direct sum decomposition, one summand consis- ting of the W-invariants, the other having II as basis.) Let S be the symmetric algebra oft*~C; we denote r = (0, 1) e t'| C S. Let q, ..., c,, be integers 1> 2 such that q = c i whenever si, s; are conjugate in W. Let ~ ~ *~ be the natural action of W on S and let e be the neutral element of W. (1) Supported in part by the National Science Foundation. 19 146 GEORGE LUSZTIG By definition, It is the C-vector space S | C[W] with a structure of associative C-algebra with unit 1 | e, defined by the rules: a) S--~ I-I, ~ ~. ~ | e, is an algebra homomorphism b) C[W] --~ I-l, w !-~ 1 | w, is an algebra homomorphism c) (~|174174 (~Es, w~w). d) ( l | s,) (~ | e) -- (*'~|174 =c,r--ee, (~S, 14 i,< m). The algebra I-I arises in nature as the graded algebra associated to a certain natural filtration of an affine Hecke algebra (with unequal parameters) ; this can be used to show that the multiplication given by a)-d) is well defined. The variable r appearing in l-I should be thought of as related to q of the Hecke algebra by q = e"'. Thus, just as the Hecke algebra specializes for q ~ 1 to the group algebra of the affine Weyl group, the algebra I-I specializes for r -~ 0 to the " semidirect product" of S(I') and C[W]. (This semidirect product has been considered in recent work of Kostant and Kumar.) 0.2. The main observation of this paper is that I-I can be realized geometrically for many choices of the q in terms of equivariant homology. (The experience of [5] has shown that equivariant K-homology is much better behaved than equivariant K-cohomology; for this reason we use equivariant homology instead of the more familiar equivariant cohomology. See w 1 for the definitions.) Let G be a reductive connected algebraic group over C, with Lie algebra g. We fix a parabolic subgroup P with Levi subgroup L, and unipotent radical U; let p, l, rt be the Lie algebras of P, L, U. We also fix a nilpotent L-orbit ~ in I carrying an irreducible cuspidal L-equivariant local system .f ,~ (in the sense of [9].) Let t be the centre of t. Let R C t" be the set of non zero linear forms on t which appear as eigenvalues in the ad-action oft on g. Then R is a root system with a canonical basis II and with Weyl group W = N(L)/L. We consider the varieties a) gs={(x, gP) ~g � G/P[Ad(g -x) xecg+11} b) gn = {(x, gP, g' P) ~ g x G/P � G/P l (x, gP) ~ aN, (x,g' P) E gn }. We have a natural G � C'-action on gs: c) (g,, X) : (x, gV) ~ (X -2 Ad(g,) x, g, gP) and this induces a G � C*-action on gn. The local system .W on ~' gives rise v/a the function pr~r(Ad(g -1) x) to a local system s on gs. Let .~ be the pull back of .Z ~ [] .~" under gn r gn � gn. This is a G � C'-equivariant local system on gn. CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 147 We consider the vector space d) H.~215 .~) (equivariant homology). Using the two projection gN --* ~ and the cup-product one can regard d) as a module over H~� S(t'*C) = $ in two different ways. We can also define on d) a W x W-action using the method of [8], [9] of constructing Springer representations in terms of intersection cohomology. It turns out that these module structures allow us to regard d) as the two sided regular representation of an algebra l-I as above. This gives a topological realization of the algebra I-I on the vector space d); the constants g (1 <~ i <~ m) can be determined explicitly. 0.8. The case with equal parameters c, = 2 is obtained by taking in the previous setting P to be a Borel subgroup, :g = 0, L# = C. This is the case studied in [5] in K-theoretic terms; even in this special case, the present construction of the W-action is quite different from that of [5] where no intersection cohomology was used. It is likely that in general case, one can realize the affine Hecke algebra (with unequal parameters) as an equivariant K-homology K0 ~ � c'(fi~, s 0.4. Our method leads also to a parametrization of all simple H-modules on which r acts as r 0 ~ C ~ in terms of parameters (x, a, p) (up to G-conjugacy) where x is a nilpotent clement of g, r is a semisimple element of .q such that [a, x] ----- 2r 0 x and p is an irreducible representation of a certain finite group. (The equation [r x] = 2r 0 x is the Lie algebra analogue of the equation Ad(s) x = q0 x appearing in the parame- trization of [5].) The parametrization of simple H-modules will be established in a sequel to this paper. 0.5. Notation. -- All algebraic varieties are assumed to be over C and all alge- braic groups are assumed to be affine. The stalks .Z', of a constructible sheaf .090 (in particular a local system) are assumed to be finite dimensional C-vector spaces. If X is an algebraic variety, we denote by ~X or ~bc(X ) the bounded derived category of com- plexes K of C-sheaves on X whose cohomology sheaves .j~o~ K are constructible. If f: X' -+ X is a morphism, then fo : ~X' -+ ~X, fz : ~X' ~ ~X, f" : ~X ~ ~X' are the usual functors. IfK E ~X, we denote by Hi(X, K), H~c(X, K) the hypercohomology (resp. hypercohomology with compact support) of X with coefficients in K. IfLP is a local system on X, we identify -~ with the complex K ~ ~X such that ~ K = L, ~i K = 0 for i 4 = 0. In particular, H'(X, .o~ o) = ~ Hi(X, s and I-I;(X, s = ~ H'~(X, .La) are well defined. ' ' We shall often denote the inverse image f" .o~ of .~ under a morphism f: X' ~ X again by L,e. 148 GEORGE LUSZTIG We have induced homomorphisms: f" : Hi(X, .oq o) ~ H;(X', .oq ~) (inverse image) f': H~(X, .L~') ~ H~(X', .o90) (inverse image, iff is proper) f~ : H~(X', L#) -+ --c~;-~:'K~--, ..9 ~ (integration along fibres, iffis a locally trivial fibration with all fibres of pure dimension 8) ft : H~(X', .La) ~ H~(X, .'.~) (extension by zero on X -- X', iff is an open embedding). When ~ = C, we write H~(X), H~(X) instead of H~(X, C), HI(X , C). We shall denote by an upper-script* the dual of a vector space or of a local system. If G is a group acting on a vector space V, we denote the space of G-invariant vectors by V ~ We shall denote S(V) = O SJ(V) the symmetric algebra of V; in particular S t V = V; we regard S(V) as a graded algebra: we assign degree 2j to the elements of S ~ V. If G is an algebraic group, we denote by G o its identity component. If H is a sub- group of G we denote by ~#'H or JV" o H the normalizer of H in G. TABLE OF CONTENTS 1. Equivariant homology .................................................................... 148 2. Cuspidal local systems ..................................................................... 155 3. The W � W-action ...................................................................... 161 4. S-module structures ...................................................................... 172 5. The commutation formula ................................................................ 18't- 6. The algebra H .......................................................................... 186 7. Preparatory results ....................................................................... 190 8. Standard H-modules ...................................................................... 192 1. Equivariant homology 1.1. Let G be an algebraic group and let X be a G-variety, that is an algebraic variety with an algebraic action. Let .~ be a G-equivariant local system (or G-local system, for short) on X. We want to define for an integer j, the equivariant cohomology H~(X, .La) and the equivariant homology H,~ .L'a). When G = { e }, then H~(X, .L~ a) Hi(X, Lf) and H?(X, .fe) H2~mX-qX s In the general case, we follow Borel's procedure [2] to define HJo(). For this we choose an integer m >i 1 such that m/> j and a) a smooth irreducible free G-variety F such that H~(I ") = 0 for i = 1, ..., m. (" Free " means that there exists a locally trivial principal G-fibration F -+ G\I".) CUSP1DAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 149 In Borel's definition of H~(), F is not required to be smooth; but the smoothness is important for our definition of H~ ); it implies that H2catmr-((F) = 0 for i -= 1, ..., m. (In any case, a F as in a) exists, as it is well known: embed G as a closed subgroup of GL,(C), then G C GL,(C) � {e }C GL,(C) � GL,,(C)C GL,~_,,(C) hence G acts freely by left translation on F =- { e } � GL,,(I3)\GL,+,,(C); this P has the required properties as soon as 2r'/> m + 2.) If G acts freely on a variety Y and d' is a G-local system on Y, then o a gives rise to a local system ~ on Y = G\Y; the stalk 6~ at .)7 e Y is the space of G-invariant vectors in I-I 6', (where n : Y -+V is the canonical map); G acts naturally on this direct product by the G-equivariance of ~. Applying this to Y = F � X with the diagonal free action of G and to d', the inverse image of .Z ~ under pr~:F � X-~ X, we find a local system 6 7= r'.'~ on rX "= o\(r � X) By definition, H~(X, .if') ---- H~(rX, r.W), H~(X, .Z') = ,,,u~n-;r, rX, r.Z")" where d -= dim(r� ). (When X is empty, d is not defined and we set H,~ .Z') ---- 0.) Hence H~(), H~( ) are zero for j < 0. We shall write H~(X), H~ instead of H~(X, C), H?(X, 13). One has to verify independence of the choice of m, F. Let (m', F') be another choice for (ra, F); then (m + m', F � F') is also such a choice. We have diagrams Hi(rX, rCL.r ) I~ H~(r � r,X, r � r '-~e) ~- H'(r'X, r ''~*a) H~d-J(r x, r Aa~ ~ H~'-'(r � r 'x, r r ,-~e') -~ ~',a'-,t y x ~'r tr '-~, r '-~ where d'=dim r,X, D=dim r� andf:r� 'X-+r X,f':r� 'X-+r 'X are the canonical fibrations with fibres isomorphic to F', I" respectively. The mapsf',/",f~,f~' are isomorphisms; forf', f'* this is well known and forf~,f~' it follows from the lemma below. Then (f")-~ of" and the transpose off~' o (f~)-i establish the independence of the definitions of the choices made. Lemma 1.2. -- Let f: X' ~ X" be a locally trivial fibration such that all fibres f-l(x '') are irreducible of dimension ~' and satisfy H~8'-l(f-l(x"), C,) = O for i = 1, ..., m'. Let d" be a local system on X'. Let a'= dim X', a"= dimX". Then 1-12"'-'(X ' t'" d') ~ H~"-'(X", o a) is an isomorphism for 0 <, i <, m'. Proof. -- We have a canonical spectral sequence = 10 H q * Er ,q H~(X ,.~ ff o a) :> H, +"(X",ff" o a) = H,~+"(X',f" g). 150 GEORGE LUSZTIG We have H~(X',f" ~') o~ Ej_~8,,~8, C ... C E~-2""28'C E~/-2s''2~' = H~-28'(X '', gg). The composition of these maps is by definition f~. It remains to show that in our case, we have a) Ez 2.''-''2v = E~ ''-''2n' = .... E~ ''-''zv = H2c*'-'(X',f" d') for 0.< i.< m'. We have Ef '28'-* = 0 if 1 .< i.< m' and Ef 'q = 0 if p> 2a". From this a) follows. 1.3. Consider the product a) I-~(X) | H~(X, .~) -+ H~ + "(X, .if') defined by the cup-product H~(rX) | Hr(r X, r-~) ~ HJ+r(rX , r.~). where P is as in 1.1 a) with large m. Similarly, the cup-product u2d--j'/ y H'(v x) | H~'-" + '"(r X, r -~') ~"~ ,r--, r -~r gives rise to a pairing HJ(r X) | u2a-~'t y ~a'za-c,+ ~'~l y l'c ~r-*, r-L'~ ~ "'r ~r-*, r-Ls hence to a product b) H~(X) | H~.(X, .2~') --+ o H~+/(X, .~). One verifies that a), b) are independent of (m, F). This makes H~(X) = @ H/~(X) into a graded C-algebra (commutative in the graded sense) with 1 and H~(X, 2') = @ H~(X, .LP), H.G(X, .~) = @ H,~ (-~) into graded H~(X)-modules. 1.4. We discuss the functorial properties of H~(), H~(). Let f: X' -+ X be a G-equivariant morphism between two G-varieties X, X', let s be a G-local system on X' and let .~" =f"-~. We have natural homomorphisms: a) f" : H~(X, c~r ~ H~(X', .~') (in general) b) f : H,+~,,,..x,_d,,x,(X, -~') ~ H,q(X, .o~') (iff is proper) c) f" : H?(X, .~) ~ H?(X', 5r (iff is a locally trivial fibration with irreducible fibres of fixed dimension) G t d) f" : H~(X, .oq ~) ~ H~+~<~x,_da=x~(X, -~') (iff is an open embedding). CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 151 In terms of a r as in I. 1 a) with m large these maps are defined respectively by (or by transposes of): a') (r f)" : H'(rX, r --q~) -~ H'(I'X', r -Z'') H ~a-jt X visa-Jr y, b') (r f)* : e ~r , r -Z'.) --> *-, ~.r-'-, r -Z''*) C') (l'f)t~ : H~n'-'(r x', r c-~r -+ H~J-'(r x, r "W~ Vl2a-~t Y, r~") -+ d') (rf), : --0 u"', H~a-~(rX, r "Z'~ (see 0.5) where rf:r x--+r x' is the map induced by Id � � X--+F � X, d=dimrX, d' =dimr x'. e) Iff is a G-equivariant vector bundle with fibres of tixed dimension then f" in a) and c) are isomorphisms. This follows from the definitions. Now let G' ~ G be a closed subgroup of G. If X, .W are as in 1.1, then X is also a G'-variety and we have a natural homomorphism f) H~(X, .~) --+ H~'(X, .~e). It is defined as follows. Let r be as in 1.1 a) with m large. Then r is also a free G'-variety. Then (f) is the transpose of % : H~ a' -,(o'\(r � x), x x), r.LP*) defined as integration along the fibres (~ G/G') of the canonical fibration : O'\(r x X) G\(r x X). (d = dim(G\(F x X)), d'= dim(G'\(F � X))). Similarly, we have a natural homomorphism g) H~(X, ..~) -+ H~,(X, .Z') defined as r H'(G\(r � X), rL#') -+ H'(G'\(r � X), q)*(r.~')). From the definition off) and g) we see that: h) If G/G' is isomorphic as a variety to an affine space (for example, if G' is a maximal reductive subgroup of G) then the maps f), g) are isomorphisms. 1.5. Now let F be a closed G-stable subvariety of X; let d~ = X- F and let i:F r X, i':O ~ X be the inclusions. We then have a natural long exact sequence G iti i'* 9 _.~ G (] |1 Hj+l_OdtmX+2dlmF(F, -~) ~ ... 152 GEORGE LUSZTIG Indeed, let F be as in I. 1 a) with m large. The partition rX = rF u rO with rF closed, gives rise to a long exact sequence ... ~ _.+ I-~2d--j/ X H~d-;-'(r F, r58") H~n-;(vO, r58") ~ ,,~ ,r , v58") -" H2c~-~(r F, r58") ~... (where d = dim rX). Taking duals we find a portion of the exact sequence a); increasing m, we find a larger and larger portion of a) ; for m ~ oo we find a). If, for example, we have dim F < dim X, then from a) we get an isomorphism $,e b) H0~ 58) ~ Ho~ 58) since H~ ~ ) = 0 for j< 0. 1.6. Let G' be a closed subgroup of G and let X' be a closed G'-stable subvariety of X such that the map G'\(G � X') -- X, (g, x') ~-, gx', is an isomorphism of G-varieties. (G' acts on G � X' by g':(g,x')~-*(gg'-l,g'x') and G acts on G'\(G x X') by g, : (g, x') ~ (g, g, x')). We have natural isomorphisms a) H~(X, 58) ~ Ho(X , 58), H?(X, 58) ~ H,a.(X ', 58). Indeed, choose F as in 1.1 a) with m large. Then the isomorphisms in a) are induced by the natural isomorphism G'\(F x X')--~ GX(F x X). 1.7. We write Hi, H a instead of H~. (point), H. a (point) where the point is regarded as a G-variety in the obvious way. The map X --+ point defines by 1.4 a) a C-algebra homomorphism r : H i -~ HI(X ) preserving the grading. Since H~.(X, ..~), H.~ .~) are H~(X)-modules (1.3) they can be also regarded as Hi-modules , via s. 1.8. Assume now that X has pure dimension. We have a natural homomorphism a) H~(X, 58) --. H(X, 58). It is defined as follows. Choose P as in 1.1 a) with m large. We consider the composition n~(r X, r58) N H~ d-'(r X, v58') ~,~,,~a,~ H~d(I.X) ,,~ C where ~ : r X -. point, and d = dim rX. This defines H'(rX, r58) ~ H an-~(v X, v58")', hence a). b) If X is smooth of pure dimension, then a) is an isomorphism. This follows from Poincard duality for the smooth variety rX. In particular, we have c) H o ~ H. ~ (isomorphism of H~-modules). 1.9. Let G' be a closed normal subgroup of G containing G ~ a) The finite group G/G' acts naturally on H~ .~') and HoJ,(X, 58) and we have H,~ 58) ~ H,~ 58)o,o', H~(X, 58) --% H~.(X, 58)o,Q'. (The maps are given by 1.6 a).) CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 153 Indeed choose P as in 1.1 a) with m large. Then F is also a free G'-variety. Let p:G'\(I' � X) ~ G\(P x X) be the natural map, a finite principal covering with group G'\G. Then p'(r&e) is a G'\G-local system hence G'\G acts naturally on H'(G'\(F x X), p'(r.oq')) and H~a'-J(G'\(F x X), p'(r.~')) (d'= dim G'\(F x X)), and p" : H'(G\(I" x X), r.~) ~ HJ(G'\(P x X), p'(r.2')) ~ p~ : H~a'-~(G'\(F � X),p'(r.~P'))-~ W"-~(G\(r x X),p'(r.~')). Thus, we have a). 1.10. Let T be a torus with Lie algebra t and let X(T) be its character group. For Z ~ X(T) let C x be C with the T-action defined by (t, z) -~ x(t) z. Let i : { 0 }'-+ C and n : C -+ { 0 } be the obvious map. The composition il (~~ 1> H~.({ 0 }) --+ H.T(Cx) H.r({ 0 )) (1 .r of degree 2, hence it must be given by multiplication by an element is H.T-linear and 1.8 c).) Then c:X(T)-+H~(~(-+c(3() ) is a group homomorphism. c(x) E H~. (See unique isomorphism There is a a) 'r" : such that the diagram X(T) t" v _ H~ is commutative, where d?( : f -+ C is the differential of Z : T --~ C ~ at the identity. More generally, let E be a finite dimensional C-vector space with a given linear representation of T. Then E~ Cxl| as a T-module. Let i:{0}~E, 7t : E --~ { 0 } be the obvious maps. Then b) the composition H. r { 0 } i, > H.r(E) (,~,r-~ it T{ 0 } is the multiplication by v(dz,).'v(az,) ... = c(z,) ... c(x,). 1.11. Let R, G be the unipotent radical of G and G, = G/R, G. Let g, be the Lie algebra of G,. Then G acts naturally on gr via the adjoint action. Hence it acts on S#(g;). It is well known that we have natural isomorphisms a) S'(g;) ~ -~ Hg' b) H~ J+l = 0. The map a) is characterized by properties c), d), e) below. 20 154 GEORGE LUSZTIG r If T is a maximal torus of G, with Lie algebra t ~ fl, then the diagram (a) I 1.4 (g) (a) Ss(t') = H~ ~ is commutative. (The left vertical map is induced by the natural map t ~ g.) d) If G = T and j = 1 then the map a) coincides with q' in 1.10 a). e) The product 1.3 a) in ~ corresponds under a) to the natural algebra structure of S(g;) ~ 1.19.. Assume that G O is a central torus in G and that E is an irreducible algebraic representation of G (over C), trivial on G ~ Let g be the Lie algebra of G. We have natural isomorphisms a) H~ ~ H~o b) H.~ E | E') ~- H. G~ This is shown as follows. By our assumption, the adjoint action of G on g is trivial. By 1.11 a), we have H o ~= S(g') ~ = S(fl') =~ H~o hence a). By a), we have dim H~ ---- dim H~o, for all j. Using 1.8 c) for G and G ~ we deduce that dim ~ = dim H~ for all j. Hence from the isomorphism H~ ~ (H~O) ~ we can conclude that G/G ~ acts trivially on H,G. 0. Using 1.9 a) we have G0 H.C(point, E | E') ~ (H. (point, E | E')) ~ _-_ (H. a~ | E | E') ~ (since G O acts trivally on E | E') H. ~176 | (E | E') Q (since G acts trivially on H. G*) H. G~ (since (E | E') ~ = C) hence b). 1.13. Let X, .~a be as in 1.1. a) If HI(X, .Z') ----- 0 then H.G(X, .Z') ---- 0. Indeed let r be as in 1.1 a) with m large. We consider the natural map f: G\(r � X) --~ G\[' with fibres X. Our hypothesis implies that .,~fz(r.Z') ~ 0 for all i. Hence ft(r.L ~') = 0, so that H~(rX, r.L a) = 0, and a) follows. CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 155 9.. Cuspldal local systems 9.. 1. In the remainder of this paper, G denotes a connected reductive algebraic group with Lie algebra g. Then G acts on g by the adjoint action and G x C" acts on g by (gx, X) : x ~)-2 Ad(gl) x. If x e g, we denote by Zo(x ) the stabilizer of x in G and by Mo(x ) the stabilizer of x in G x C'. Thus a) Ma(x ) = {(g,, X) e G x C" I Ad(g,) x = X z x }. Assume now that x is nilpotent. b) ByJacobson-Morozov we can find a homomorphism of algebraic groups q~ : SL2(C) ~ L such that d~[~ 10] --=x. We set Zo(~) = { gx E G [gl ~(A) g~-' = q~(A), V a e SL,(C)} (?0 0 ] [0 0]) MG(~) = t (gx'X)~O xC'[gt~(A)gU'=q~ X-x A , V A e SL,(C) ~" d) It is known that Zo(q~ ) (resp. Mo(~) ) is a maximal reductive subgroup of ZG(x ) (resp. Mo(x)). It is clear that [0 0 ] e) (gx, ~,) ~ (gl ~ X_ 1 , X) defines an isomorphism of algebraic groups Zt~(~ ) � C ~ -~ MG(q~ ). From d) and e) it follows that f) the embedding ZG(x ) ~ Mo(x), gx ~ (gl, 1) induces Za(x)/Zg(x ) -~ Mo(x)/M~ From the existence of ~ it follows that the G-orbit of x is also a G � C~ from f) we see that a G-local system on a nilpotent G-orbit in g is automatically G � C'-equivariant. 9..9.. Let d' be an irreducible G-local system on a nilpotent G-orbit d7 in g. We say that g is cuzpidal if it satisfies the condition a) below. a) For any proper parabolic subalgebra Px of g with nil-radical rh and anyy e Pl we have H'~((y + nx) c~ d~, ~) = 0 for all i. This implies that: b) If./" : 0 ~-~ Ois the inclusion of 0 in its closure then j, 8 =j, g e ~(~). The closely related concept of cuspidal local system on a unipotent class of G is defined and studied in [9], [I0]. The two concepts are related to each other by the expo- 156 GEORGE LUSZTIG nential map; thus the results of loc. cit. can be transferred to Lie algebras. (See [11].) We shall use freely those results (in particular, the classification) in the case of Lie algebras. The implication a) ~ b) follows from [10]. 2.3. We now fix a proper parabolic subgroup P of G with a Levi subgroup L and unipotent radical U. Let p, I, rt be the Lie algebras of P, L, U so that p = [ | n. The concepts in 2.1, 2.2 can be applied to L instead of G. We fix a nilpotent L-orbit ~ in ! and an irreducible L-equivariant (hence L � C'- equivariant) cuspidal local system _9' on (6. We fix an element x 0 e ff and a homomor- phism of algebraic groups %:SL,(C)-~L suchthat dq)0[00 101 =x0. Let T be the identity component of the centre of L and let t be the Lie algebra of T. Let W = No(T)/L. This is a finite group. For w e W, ~b will always denote a repre- sentative of w in N(L); it acts by conjugation on L, T, hence on t. From [9, 9.2 b)] we have, for all w e W, a) zb : L -+ L leaves ~ stable and the inverse image of.s under w : cE -> rE is isomorphic to .Z. From [9, 2.8] and 9.1 d) we have b) Z~(~o ) = T. Using 2.1 e), we see that there is an isomorphism c) T � C'-+M~ (,,X)~ ,% X-' 'X . Using c) and the definition of ML(9o ) we see that d) M~ is contained in the centre of ML(~P0 ). 2.4. For any linear form ~ : t --~ C we set g~ = { x e g [ [y, x] = ~(y) x, Vy e t }. Then g= C) g, since T is a torus. Let R ={0tet*] ~4: 0, g~4: 0}. It is easy to see that n is a sum ofg~'s (~eR). We define R + ={~eR[g~Cn}. Let P1, P2, .-., P,, be the parabolic subgroups of G which contain strictly P and are minimal with this property. Let L~ be the Levi subgroup of P~ which contains L; let p~,l~ be the Lie algebras of P,,L,. Let R + ={aeR + [~[centre(I3 =0}; then linn= @ g~. ~tlE R~ r Proposition 2.5. -- R is a (not necessarily reduced) root system in f'; it spans the Mace of linear forms on t which are zero on the centre of g. Moreover, W acts faithfully on f, t" (trivially on the centre of fl) as the Weyl group of R. The set R~ contains a unique element o~, such that o~J2 r R (1 <~ i <<. m). The set II = ( 0~1, 0~2, ..., O~ m } is a set of simple roots for R. The group W is a CoxeUrgroup on generators si, s2, ..., s,, where s, is the unique non-trivial element of Nq(T)/L. This is easily checked, case by case, using the known classification of cuspidal CUSP1DAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 157 local systems: this forces t0 to be a very special parabolic subalgebra ofg. A slightly weaker statement, namely that the set of indivisible elements of R is a (reduced) root system with Weyl group W is proved in [6, 5.9] and [9, 9.2]. Note that in the generality of [6, 5.9], R is not necessarily a root system. Proposition 2.6. -- a) T is a maximal torus of Z = Z~ b) The natural map Nz(T)/T ~ NQ(T)/L : W is an isomorphism. (Thus, W can be interpreted as the Weyl group of Z.) Proof. -- a) Let S be a torus in Z containing T. Then S is in the centralizer of T, hence S C L. Thus S C Z~ But the last group is T by 2.3 b). Hence S = T. b) To prove injectivity, it is enough to show that Z n L C T. If g e Z c~ L then g commutes with all elements of T (since it is in L). But the centralizer of T in Z is T, by a). Hence g ~ T. To prove surjectivity, it is enough to show that the generators s~ are in the image of our map. This follows from the analogous statement in which G is replaced by L,. So we assume that P is a maximal parabolic subgroup of G. We can also assume that G is simply connected and even almost simple. In that case, W is of order 2 and by a case by case check (using classification of cuspidal local systems) we see that Z is not a torus so Nz(T)/T has also order 2. Hence our map, being injective, is automatically surjective. 2.7. For any ~ e R, 9~ is L-stable for the adjoint action of L on 9, since T is central in L. In particular, 9, can be regarded as an SL2(C)-module via q~0: SLy(C) -+ L. Let M a denote a simple SL~(C)-module of dimension d. Proposition 2.8. -- Let ~ E R. Then the SL2(C)-module g, is isomorphic to: Ml | M3 | . . . Q M2~ + l, for some p, if 2~ r R, Mx if ~/2 E R, M~| M4| ... @M2r , forsomep, /f 2~ ~ R. Proof. -- W permutes R; moreover if zb e N(L) represents w, then Ad(~b) : g, ~-~ 9~,- We can assume that 6} ~ Z~ (see 2.6 b)) so that Ad(th) commutcs with the SL,(C)- action on 9. Hence the SLz(C)-modules 9,, fl~, are isomorphic. Since the W-orbits of R + (1 ~< i~< m) cover R, we see that we can assume ~ E R~ +. We can then replace G by L~ and assume that P is a maximal parabolic subgroup. We may also assume that G is simply connected and even almost simple. In that case we use the classification of cus- pidal local systems. We are in one of the four cases below. (In the following discussion (as well as in 2.10, 2.13) we shall describe nilpotent elements of classical Lie algebras by specifying the sizes of their Jordan blocks; this will be always taken with respect to the standard representation of that Lie algebra: of dimension 2n for sp,,,, of dimension n for ~o,.) 158 GEORGE LUSZTIG , tt t tt Case 1. -- g=~I2,, t=~I,|174 (x0, x 0,0) =x o wherexo, X o are regular nilpotent in sI,, (n>l 1). In this case, R ~ ={a} and 9,-Ma| |174 as an SL2(s , t Case 2.- g = st~s,+s, t = sp~, | C, ~ (Xo, O) = Xo, where x o is a nilpotent element in ~P2, with Jordan blocks of sizes 2, 4, 6, ..., 2p so that 2n = 24- 44- 6 + ... 4- 2p, (n~> 1). In this case, R + ={~,2~} and 9.e,~ M,, 9,~ Me|174174 as an SI~(C)-module. Case 3. -- 9 = st3.+2, I = so. | (x'0, 0) = x o where x o is a nilpotent element in ~o, with Jordan blocks of sizes 1, 3, 5, ..., 2p4- 1 so thatn-- 1 + 3 +... 4-(2P-~-1) (n/> 4). In this case, R + ={a}and 9~ M,| ... | as an SLz(C)-module. , tt p Case 4. -- 9=so,+~, t=so,|174 xo,O), where x o is a nilpotent element in so, with Jordan blocks of sizes 1, 5, 9, ..., 4p + 1 or 3, 7, 11, ..., 4p 4- 3 (so that n=l +5+9+... 4-(4p+1) or n=34-74- 114- ... 4-(4p+3)) and x o' is a regular nilpotent element in ~I.,, (n/> 3). In this case, R + ={~,2a} and g_.~=~ M,, g,= M z| 4@...| , as an SL,.(C)-module, where 2k = i4p + 2, if n = 1 + 5 + 9 + ... + (4p + 1) 14p+4, ifn=3+74- 11+ .+(4/9+3)" 2.9. Let R"Ci" be the set of roots of the reductive Lie algebra 8 = gst~ce~ of Z~(~0 ) with respect to its Cartan subalgebra t. From 2.8, we see that, for ~ e R, dim(g~ns) = ~ 1 if2~ CR, 10 if 2~ e R. a) It follows that R" is precisely the set of non-multipliable roots of R. We define b) ~r = unique nilpotent G-orbit in 9 such that r n (x 0 + rt) is open dense in x 04- It. Proposition 2.10. - - Assume that P is a maximal parabolic subgroup of G and let o~ be the unique simple root of R. Let a : T ~ C" be the character by which T acts on 9~ by the adjoint action, so that o: = da. Let c be the integer >>. 2 defined by the conditions a d(xo)'- 2 : 11 -+ it is 4: O, a d(xo)'- ' : n -+ rt is O. Then a) Ker(a d(xo) ~-z : tt -+ it) is a hyperplane 9ff in n. b) Jt ~ and rt are stable under the action of M~ (restriction of the G � C'-action 2.1 on 9) and the induced action of M[(~?o ) on rt/:,~ is via the character (t, X) , , a(t) 1,-', (t, X) ~ T � 121" s.s (,, MO(%). c) n (Xo + it) = x0 + (n CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 159 Proof.--a) follows from xo=(d~o ) [0 0 10] and the fact the SL2(C)-module la is multiplicity free (cir. 2.8). b) Follows from 2.8 and the following property of the irreducible representation p: SL,((1) --~ Aut(Ma) of dimension d: the largest non-vanishing power of (dp) [~ 10] : Ma ~M a is the (d -- 1)-th power, and p X_ t acts on Ma/ker (dp) as multipli- [0 0 ] ( [00 ~0]) ~-' cation by X -a + 1 To prove c) we again assume that 13 is simply connected, almost simple, so that we are in one of the cases 1-4 in the proof of 2.8. The following results can be verified in each of those cases by simple computations. Case 1. -- If x ~x o + (rt- ~r176 then x E sL~, is a regular nilpotent element; if x e x 0 + ~rt ~, then x is a non-regular nilpotent element. Case 2. -- If x eXo+ (n--.,~), then xesp~,+3 has Jordan blocks of sizes 2, 4, 6, ..., 2p -- 2, 2p + 2; if x e x o + ~, then x has Jordan blocks of sizes ~< 2p + 1. Case 8. -- If x e xo + (II- ~d), then x e sa,+2 has Jordan blocks of sizes 1, 3, 5, ..., 2p -- I, 2p + 3; ifx ex o +.,~F, then x has Jordan blocks of sizes ~ 2p + 2. Case 4. -- Ifx e Xo + (rt -- ~) then x e so,.~ has Jordan blocks of sizes 1, 5, 9, ..., 4p--3,4p+5(resp. 3,7,11,...,4p-- 1,4p+7) ifn= 1 +5+9+ ... + (4p+ 1) (resp. n = 3 + 7 + 11 + ... + (4p + 3)); if x ex 0 +.,~o, then x has Jordan blocks of sizes ~< 4p + 4 (resp. ~< 4p + 6). These results imply c) immediately. 9..11. The proof of 2.8 shows that the integer c in 2. I0 is given explicitly in the cases 1-4 of 2.8 as follows. Case 1. ~ c = 2n. Case 2.- c =2p + 1. Case 3. -- c----2p+2. i4p + 3, if n= 1 +5+ ... + (4p + 1). Case 4. -- c = t4p + 5, ifn=3 +7+ ... + (4p + 3). We now drop the hypothesis in 2.10 that P is maximal. Proposition 8.19.. -- Define integers q >1 2 (1 <~ i <~ m) by the requirement ad(xo)~-~ : ~ r3 n ---> I~ n n is +- O, ad(xo) ~'-1 : I~ c~ rt ~ I~ t3 n is O. Then q = c~ whenever s,, sj are conjugate in W, (1 < i, j ~< m). 160 GEORGE LUSZTIG Proof. -- If ws~w -1 =s t then w(ei) = %. We have I, c~n = g~;| I t n n = g,i| g2~j and if we choose the representative ~b for w in Zg(%) (see 2.6 b)) then Ad(:b): ~ n rt ~ I t n rt is an isomorphism of SL~(C)-modules. Since c~, c~ are determined by the SL2(C)-module structures, they must coincide. 2.13. In this section we assume that G is almost simple, simply connected. We shall indicate in every case that can arise the type of g, t, xo, the type of R, and the values of c i (1 ~< i ~< m) corresponding to the various simple roots of R. We also indicate the type of a nilpotent element xRs e ~/'~s. a) g-simple, I-Cartan subalgebra, x 0 = 0, R-usual root system, c~ = 2, for all i. b) g = ~I~,, I = sl,|174 ... |174 ~-1 (k >/ 2, n >/ 2), xo is regular nilpotent in l, k copies xRs is regular nilpotent in g, R of type A k_ 1, q = 2k for all i. c) g=sp.o,+.ok, I=sl~2,| k(k>/ 1, n>/ 1), s t x o = (x0, 0) e [ where x o is a nilpotent element in ~P2, with Jordan blocks of sizes 2, 4, 6, ..., 2p (so that 2n : 2 -t- 4 + 6 + ... -k- 2p), xas is nilpotent in sP2,+~ with Jordan blocks of sizes 2, 4, 6, ..., 2p -- 2, 2p + 2k, 2 2 2 2 2 2,0+I R of type BCk, c~ : O--O--O'"O--O~O d) g = so,+~,, I = so,| ~ (k>~ 1, n>~ 4), t t , x 0 = (x0, 0) e I where x o is a nilpotent element in so, with Jordan blocks of sizes I, 3, 5, ..., 2p+ 1 (so that n= 1 +3 § ... + (2p+ 1)), XRS is nilpotent in ~O,+2k with Jordan blocks of sizes 1, 3, 5, ..., 2p -- 1, 2p -+- 2k -+- 1, 2 2 2 2 2 2.0+2 R of type Bk, q: O--O--O"'O--O--__O e) g = ~o,+~, [ = so,|174 ... |174 ~, k copies x, = (x0, x~0 ~', X~o ~', ..., x~o ~', 0) e I where x~0 ~' are regular nilpotent in M2, Xo is a nilpotent element in ~o, with Jordan blocks of sizes 1, 5, 9, . .., 4p + 1 (resp. 3, 7, 11, ..., 4p + 3), x~ is nilpotent in so, § ~ with Jordan blocks of sizes 1, 5, 9, ..., 4p -- 3, 4p + 4k § 1 (resp. 3, 7,11, ...,4p-- 1, 4p+4k+3), R of type BCk, 4 4 4 4 4 4p+5 4 4 4 4 4 4p +3 (resp. c,: O--O--O'"O--O~O ) c~: O--O--O'-'O--OZO f) g of type E~, I=~Is(9~13(9C 2, x 0 is regular nilpotent in I, XRN is regular nilpotent in g, 2 6 R of type G2, c~ : O~O CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 161 g) fl of type ET, I = sL2@sLa@sI,| C* corresponds to the subgraph marked with black nodes of the Coxeter graph of g, x0 is regular nilpotent in I, x~s is regular nilpotent in g, 2 2 4 4 R of type F,, c~ : C)~C)~C)~C) We now return to the general case. Proposition 3.14. -- Let x e ~r n (Xo + n) and g e G be such that Ad(g) x E ~g + n" Then g e P. Proof, -- We can assume that G is simply connected, almost simple so that we are in one of the cases in 2.13. From 2.13 we see that a) Zdx0)/Z (x0) where G is the adjoint group of G and L is the image of L in G. Let P be the image of P in G. From [12] we have natural homomorphisms zdx0)/z~ the first of which is surjective and the second injective; from a) it then follows that both are isomorphisms. In particular, Zg(x)C P.Z~ By [12] we have Z~ ]5 so that Zo(x) CP hence Zo(x )CP. Now let geG be such that Ad(g) xe~'+n. Then Ad(g) x E ~r r~ (~ + n). But P acts transitively on r n (~' + n) ([12]) so, replacing g by pg (p ~ P) if necessary, we can assume that Ad(g) x = x, i.e. g e Zc.(X ). Hence g e P, as required. 3. The W � W-actlon 3.1. We shall need the following G � C'-stable subvarieties of g: q/'= U Ad(g) (~+t +n), where ~ is the closure of ~g, o~G ~r U Ad(g) (~'+~+n), where ~,={xetlZG(x ) =L}, o~G r = O Ad(g) (~+n), r (see 2.9 b)), gs = { x E g [ x nilpotent }. If X is a closed subset of p, stable under the adjoint action of P, then O Ad(g) X is a closed subset of g (since G/P is complete), geQ 21 162 GEORGE LUSZTIG Hence a) ~e', ~ _ ~/'rts, "//'s, ~e" _ ~e'RN ' 9S are closed subsets of g. b) ~e'RS is open dense in ~/', ~r is open dense in ~/'s. Clearly, ~r = g~ n ~f. We define c) ~={(x, gP) ~g X G/PlAd(g-~)xECg + t + n}. Note that G � C* acts on ~ as in 0.2 c). For any G � C*-stable subvariety V of g we define a) 9 ={(x, gP) glx V}. In particular, ~P', *P'as, f's, f'xr;, gs are well defined G � C*-stable subvarieties of g. Wc have e) g=r162 Proposition 8.9.. -- a) The map{(x, gL) ~ ~e'Rs � G/L ] Ad(g -a) x e cg + t~}_~ ~ "f'Rs, (x, gL) ~, (x, gP), is an isomorphism. Hence w : (x, gL) ~ (x, grb- 1 L) defines via d/an action of W on f"~s. This makes pra : ~P'~s -+ ~e'a~s into a finite principal W-covering. Both "r ~'~s are smooth, irreducible of dimension 8 = dim(g/I) q- dim(Cg + t). b) ~r is open dense in f"s. Both ~r ~/'~s are smooth, irreducible of dimension 8' = dim(g/I) + dim(if). c) ~P" is smooth and is open dense in "//~ = { (x, gP) e g � G/P ] Ad(g- 1) x e c~ .q_ t -F rt }; ~e', ~ and "t ~ are irreducible of dimension 8. d) prl: f'~.~ ~ ~e'Rs is an isomorphism. Proof. -- a) is a Lie algebra analogue of [9, 4.3 c)] and is proved in the same way. b) is obvious except for the formula for dim ~e'Rs. But it follows from [9] that Pra : ~P'~N ~ ~e'~s is a finite covering, hence dim "//'~s ~- dim f'~s. c) is obvious. By 2.14, the fibre of pr a : ~P'~s ~ ~r at x ~ "//'~s c~ (x 0 + 1t) is (x, P). Since G acts transitively on ~r d) follows. 3.3. We now define a) ~ ={(x, gP, g'P) ~g � G/P � G/PI (x, gP) ~g,(x,g'P) ~}. Then ~ is a closed subvariety of g � g, via (x, gP, g' P) ~ ((x, gP), (x, g' P)), hence it inherits a G � C'-action (from the diagonal G � C* action on g � ~) and two projections pra~ , pr~z : ~ ~ g. CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, 1 163 If V is any G x C'-stable subvariety of g, we define b) ~l={(x, gP, g'P) ~lx~V}. This is a G x C'-stable subvariety off and it has two projections pr12, pros : Q ~ V. c) Let ~ be a locally closed subvariety of G which is a union of P -- P double coset$. We define d) '(t'n={(x, gP, g'P) E'(Zlg-~ g' ef~ }. This is a G x C'-stable subvariety of V. A P -- P double coset in G is said to be good if it is of the form f~(w) = P~bP for some w ~ W. We have f~(w) :t: ~(w') for w 4= w'. All other P -- P double cosets are said to be bad. We have a finite partition t~ into locally coset subsets (f~ runs over the P -- P double cosets in G). From 3.1 e) it follows that f) f = r r g) If w ~W, then fn(,,)= ~n(w) is smooth, irrcducible of dimension 8 (see 3.2 a)), and ~n~wl = ~r is smooth, irreducible of dimension 8' (see 3.2 b)). Indeed, we have a fibration ~n~) ~ G/P n zbPzb -~, (x, gP, g' P) ~ g(P n zbP~b-a), with fibres (~ 4- f 4- rt) n (AdQb) ~ 4- f -- Ad(zb) rt) = (~+f+n) n (~ +t+Ad(6a) n) (see 2.3 a)) = cg -k- t -k- (rt n Ad(~b) n). (The same argument applies to -s .) h) If ~ is a bad P- P double coset, then dim fin = dim ~7-n < 8 and dim fin N = dim ~r ~ 8'. (This can be deduced from [9, 1.2].) i) If f~ is a bad P -- P double coset, then ~J?~s = O. If w ~ W, then --as 4>n ~) is a smooth irreducible variety of dimension 8 and ~/';'~s = is the decomposition IO of ~/;'~ into connected components. Also, 4~n~") is open dense in ~'~') (This follows from 3.2 a) and g).) j) If t2 is a P -- P double coset other than P then ~y-n = O. Thus, -/7" = 4':'~s. This is open dense in ~/'~. (This follows from 3.2 d), b).) U4;'ta"*)--~s 164 GEORGE LUSZTIG From the results above, we deduce: k) dim ~ = dim ~ k~ = dim ~v'Rs = 8, dim gN = dim ~r = dim "r = ~', l) dim(Y;" -- ~/;'~s) < 8, dim($7"~ -- ~/~RS) < 8'. 3.4. We define a local system .~ on g by the requirement f2"-LP =f1".s p in the diagram ~g *-{(x,g) e g x G l Ad(g-1) x e Cg + t -r- n where fl(x, g) = pr~(Ad(g ') x), f~(x, g) = (x, gP). Note that .s is well defined since .2' is L-equivariant. Moreover, .LP is G x C'- equivariant since .if' is L x C'-equivariant and fx, f2 are G x C'-equivariant for the action of G � C" on cg given by (gl, X) :x ~ X-2x and the action (gl, X) : (x, g) ~ (X -2 Ad(gl) x, gig). Similarly, replacing .oqr by .s176 in the definition of .s we get a local system (.s it is the same as the dual &P" of ~. Let K = (prl) ., .~" e ~g, (pr I : g -+ g), = i,. (K,) (i : ~e" ~ .q) where K 1 = (prx) ~ LP" e ~q/', (pr I : ~ = g ~ "//'). We have the following result. a) K[8] is a perverse sheaf on g. More precisely, K 1 is the intersection cohomology complex on ~ defined by the local system (prx), on ~/'Rs, (Prx : r see 3.2 a)). To prove this, we need the following concept. A morphismf: Y -+ Y' is said to be sma/l if Y, Y' are irreducible varieties of the same dimension d, dim(Y x y. Y) = d and any irreducible component of dimension d of Y � y, Y is mapped by f onto a dense subset of Y'. This concept is inspired by (and is more general than) the concept of smallness of Goresky-MacPherson [4]: they require in addition that Y is smooth and f is proper and generically 1 -- 1. In the case where P = B and .if' = C, the proof of a) is based on the observation of [8] that pr 1 : q2- _+ ~e" is small and proper. (In that case, "V" = g.) In the general case, pr 1 : ~ ~ ~ is still small (by 3.2 c), 3.3 g), 3.3 h), 3.3 i)) but is not necessarily proper; this defect will be compensated by the cuspidality of .L ~~ From 2.2 b) we have j~ .~e =j. ~ e!gFg, where j: cg~.~ c~ is the inclusion. It follows immediately that j, .~P =j..o(P e ~-/2 where j:~P" ~ ~ is the inclusion (see 3.2 c)). Let K' = (pr~),.LP~N(g), (pra:g ~g) i.e. K' =i.,Ka, where K; = (pr~)., .%P 9 NiP, (pra : ~P" ~ ~e-). We shall denote the first projection ~ ~ r by l~rt; it is a proper map, hence (l~r~). = (p"ra) t . We have pr~ = p"r x o]: "Or-+ $/" hence K; = (pra);-%P = (l~r~),. (], s = (l~r~). (j, .LP) ---- (l~rl). (L -~) = (pr,)~ s CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 165 We denote by D the Verdier duality. We have DK 1 ----- D((prt), ~q~) = (prl) , (D.~). Here ~q~ is a local system on the smooth irreducible variety ~ of dimension 8 (3.2 c)) hence DL/= .~[2 8]. Thus DK x = (pr,)..~[2 8] ---- K',[2 3]. By the definition of an intersection cohomology complex, to verify a) it is enough to verify that dimsupp~K 1< 8--i for all i>0, and dimsupp~K' x< 8--i for all i>0. But these inequalities follow immediately from the fact that prt:~ ok" -+ ~/" is small. Thus, a) is proved. Our present objective is to define an action of W on K. The definition of such an action was given in [8] in the case where P = B, and in [9] in general; in these references instead of complexes on g, we considered complexes on G. The case of g is entirely similar, but for the sake of completeness we shall explain it here. Let w ~ W. The local system .L~" on r is irreducible (since ~ is irreducible on W) and is isomorphic to its inverse image w'(.o~ ~ under w : ~'Rs ~ ~('Rs in 3.2 a) (due to 2.3 a)). Choose an isomorphism ~o : .o~" --% w ~ .~" (of local systems on ~'Rs). This gives isomorphisms on stalks q~.,,~ : .~,', -% .~," for all r z ~P'as- It is clear that for w, w' ~ W we have o o 0 ~,,r o ~,r = cw, w, ?~,., where c~,~, ~ C" is independent of r Taking direct sum over all v ~ W, we obtain isomorphisms | G -eL, = | -r vE/W YEW teEW or, equivalently, ((pr,)..~'), -% ((prl) " .s (Prx : ~;'Rs ~ ~r e r (See 3.2 a)). This is induced on stalks by an isomorphism (pra) " .L~" -% (pr~)..s of local systems on ]P"]~. Using a), this extends uniquely to an isomorphism ~b ~ : K -% K in ~. We have 0 o 0 +,.q b,,. = c~,.,,. +~,. To normalize q~o and +o, we shall use the following result: b) ~0 K [ "F'R~ is a non-zero, irreducible local system. Now d/~ induces an automorphism of the local system ~0 K [ r which by b) must be the multiplication by a scalar ~t, ~ C'. Replacing o 0 0 = 1. q~, +,, by ~t~ I ~, --- ~, ~t~l q~0, = ~b~, we can assume ~t,, 166 GEORGE LUSZTIG Thus there is a unique normalization ~p., 0 hb ~ (denoted cp~, +.) which induces the identity map on .,~fo K [ Y/'m~. We then have hb~ 47., = 4?,~,0,, so that w ~ q~o is a homomorphism c) W ~ Autg0 K. This is the W-action on K we wanted to construct. 3.5. We fix art integer m t> 1 and a smooth irreducible variety r with a given free G � C'-action such that H*(r) =0 for all ie[1, m]. As in 1.1 (with G � s instead of G) for any G � s Y we shall write rY instead of (G � G')\(F � Y); if g is a G � G'-equivariant local system on Y, we write r # for the corresponding local system on rY; if f: Y -+ Y' is a G � (r-equivariant mor- phism between G � C'-varieties we shall write rf : r Y ~ rY' for the morphism induced by Id � � Y-~r � Y'. In particular, rg, rg, r(g � g), r(g � g), r -Lp, r -L/" are well defined. (G � (I" acts on g � g, g � g diagonally.) Consider the commutative diagram r~ ~ ~ rx~ P~'~ a) I~ prl rg ~ rxg ~- g where ~t, ~' are the canonical maps (principal G � C'-bundles) and rc = prx. Let rK = (r~): (r -~') a ~(rg) = (Id x re), pr;(.i~') ~ x q). It is clear that ~a ~t"(rK ) == K = pr~(K). Now ~t' and pr~:F � g-~ g are smooth morphisms with connected fibres. Using [1, 4.2.5] and the fact that K[~] is perverse we deduce that b) ~ = pr~(K), suitably shifted, is perverse and pr~ defines End~(0~(K ) --% End~(r � ~(pr~ K) = End~t r � ~1(~) ; c) rK, suitably shifted, is perverse and ~z '~ defines End~(r~(rK) -% End~(r � o)(~'" r K) = Endgw � ~(~). Combining b) and c) we find a canonical isomorphism End,o(K ) -~ End~ro,(rK ). CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 167 Composing this with 3.4 c) we obtain a homomorphism d) W ~Autgcr~(rK). We can perform the same construction, replacing .~ by .~; then rK is replaced by rK'= (rTr), (r-~) and d) becomes e) W ~ Autu~rg~(rK' ). Let w, w'e W. We denote by wCl~: rK-+ rK the automorphism of r K corres- ponding to w under d) and by w'C2~ : rK' -+ rK' the automorphism ofrK' corresponding to w' under e). Then w ~l~, w '~2~ define w ~ [] w '~2~ e Auta~r~ � r~(r K [] rK'). It is clear that (w, w')~ w a~ [] w '~2~ is a homomorphism f) W x W -+ Aut~rg � r~(r K [] rK'). We have an embedding i:r(g X g)r rg � rg induced by (Y, x, x') ~ ((V, x), (7, x')), 7 e F, x e g, x' e g'. Let rK [] r K' = i*(rK [] r K') e ~(r(fl � g)); i* induces a homomorphism Enda(r~ � r~(r K :E rK') ~ EndalrC0 � 0,(r K [] rK'). Composing this with f) we obtain a homomorphism g) W � W -~ Auttar~ ~ � ~)(r K [] rK'). Let h) ~ =.r162 .2" =.r [].~. These are G � C*-equivariant local system on g � g hence they give rise to local systems r.LP, r.~" on r(g � g). From the definitions, it follows easily that r K [~ r K' = r(= � ~), (r.o ~') so that g) can be regarded as a homomorphism g') W � W ~Aut,,r~� � ~), (r.~')). 3.6. We shall denote the restrictions of .o@, .L/~' (see 3.5 h)) to ~i, or more generally, to V (for V a G � C*-stable subvariety of g) again by .~, .~~ The diagonal inclusion h:V,-*g � g induces r h:r Vc-~r(g � g) and (rh) ~ defines a homomorphism End~tr,~ � � ~), r-~ ~ -+ End~trv,(rh) ~ (r(= � =), r.~ ~ ---- End~lrV~((rpr~) , r.~ ~ where prz:V-*V is the first projection. Composing with 3.5 g) we find a homomorphism a) W � W -~ Aut~rV~((rprz), (r-~*)). 168 GEORGE LUSZTIG 3.7. We want to give an alternative definition for the restrictions of the homo- morphism 3.6 a) to W � { e } and { e } � W. We have a cartesian diagram 9 P" Pl where Pl, P12, P13 are the obvious projections. This induces a cartesian diagram r~ rP,, =-- rq rPl, l I rh r~ rPt ~-- rV Let h' be the composition rV r~ rV , , rg. Then h"(r K) = (rPls), (rP~(rL/')) and h'" defines a homomorphism a) End~tr~(rK ) ~ Endu~r~}(h"(rK)) End~r+l((rp~), " "" Tensor product with the local system r-oq ~ is a functor ~(r~ r) ~ ~(r 9) so it defines b) EndDtr~,((rp,3), (rp;2(rs ~ End~tr~.,((rpas), (rp;2(r.~')) | s.L/) - | -- End~r~,~((rp18) , " ". rpls(rZp))). 9 = Endg, (r.L~')). Composing a), b) and 3.5 d), we find a homomorphism c) W -,- Aut~tr;O((rp~ ) , (r.~')). Now the functor (riOt), defines a homomorphism d) End~r~,((rp~ ), (r-L~')) -+ Endu,rV,((rp~)l (rP~s), (r---@')) = EndutrV~((rpr~), (r-~')) 9 (We have rPx o rPxa = rprl : r ~ --~ r V.) Composing c) and d) we find a homomorphism e) W ~Aut~trV,((rprx) t (r.~'))- A routine verification shows that e) coincides with the restriction of 3.6 a) to W x{e}. We have the following variants a') -- e') of a) -- e). We have h"(r K') = (rpra~), (rpr~,(r.~)) and h '~ defines a homomorphism a') End~tr~,(rK' ) -+ End~tr(.,(h"(rK')) = End~tr~.,((rp~) , (rp~(r.L~))). r+,((rp~s), CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 169 Tensor product with rs ~ is a functor .~(r'Q) ~ ~(r ~) so it defines b') End~,r+,((rP12) , (rp~s(r.o~))) -+ End~,r;O(r.lP" | ((rPx~), rP~s(r-L/))) = End~lr~)((rp,~) j 9 .. . 9 (rPl2(r "-~ ) ~) i,Pl$(r,~))) = End.cr~,((rP,2), (r s Composing a'), b') and 3.5 e) we find a homomorphism c') W --~ Aut~r+,((rpt2) , (r.~')). The functor (rPx), defines a homomorphism d') End~(r~,((rpl,), (r.L~')) --+ End~(rv,((rpl)t (rPa2), (r-L#')) = End~,rV)((rpr,), (r-L~')) 9 Composing c') and d') we find a homomorphism e') W ~ Autg,rV,((rprl), (r.L~')). Again, a routine verification shows that e') coincides with the restriction of 3.6 a) to {e} � W. 3.8. Our next objective is to define a homomorphism W � W -+ Aut H~ � c.(~, .~), for an integer j I> 0. We choose r and m as in 3.5 with m/> j and apply the functor a) H~ ~- J(rV, ) : ~(rV) -+ C-vector spaces to 3.6 a) (d = dim rV). We obtain a homomorphism W � W -+ Aut H~ d- J(r V, (rpra), (rL;';')) = Aut H~e d- '(r ~, r.~'). Taking duals we obtain a homomorphism W � W --* Aut H~ n- J(rV, s or, equivalently, a homomorphism b) W X W --* Aut H? � 9 LP) as desired. This homomorphism is actually independent of the choice of r; the verification is routine and will be omitted. Similarly applying the functor a) to 3.7 e) and 3.7 e') and then taking duals we find two homomorphisms r W z$: Aut H2o d- ~(rV, -L;~')" = Aut n~ � 0.(~, s which coincide with the restriction of b) to W � { e }, { e } � W respectively. 8.9. Let ie [1, m], P~, L,, I0,, I~ be as in 2.4, let s, be as in 2.5 and let W~ = {e, s~}C W. Let P = ~ r3 P, ~ = ~ c~ n. Let s be a locally closed subset of G which is a union of P, -- P double cosets. Our objective is to define a natural homo- morphism a) W, -+ Aut H. ~ � o.(~n, .IP) which, in the case where fl = G, should coincide with the restriction of 3.8 b) (for V = fl) to the subgroup W, x { e } of W x W. 22 170 GEORGE LUSZTIG We introduce some notation. Let A =- {(x, gP,) e fl � G/P, [ Ad(g -t) x e p, }, ~ix = {(Y, hP) e p, � P,/P I Ad(h-t)Y e rg .[. t + n } Consider the commutative diagram I- 1 A-~ p, xG in which all maps are obvious except for ~li x G --~ g which is (y, hP, y) --~ (Ad (y) y, yhP) and p~ x G ~A which is (y, y) ~ (Ad(y)y, yP,). Now g, ~ I~ is exactly like g ~ g when (G, P, L, if) is replaced by (L,, P, L, ~'). In particular, g, carries a local system .Z~ analogous to the local system .~" on g. More- over the analogue of 3.4 c) (for L~ instead of G) gives a homomorphism b) W, + Aut~tq,((nz), .L~) and the analogue of 3.4 a) shows that c) (~2)t (-~2), suitably shifted, is a perverse sheaf. The inverse image of ~; under the composition gl � G --~ t~1 --~ {Is is the same as the inverse image of .Z ~" under gl � G -+ g. Since our diagram has cartesian squares and the horizontal arrows are smooth, we see as in 3.5 (using [1, 4.2.5] and e)) that d) End~,((~) t .Lf~) = End~c^~((n0), .oM*), canonically. e) End~(m((~ro) , LP') = End~(r^J((r~o) ' rLP'), canonically (for fixed F as in 3.5 with m large). We now consider the cartesian diagram #Is Z n ~ A Pat where [~n=((x, gp,,g,p) [ Ad(g -1) xep,,Ad(g '-t) xe~ + f + n,g-lg'efl}, Pa~ are the obvious projections and ~ is the obvious map. We have natural O x C" actions on the varieties in this diagram, so we can form the cartesian diagram: CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 171 r~n rP~, = rA We denote the inverse image of r.s under rp12:ijn--+ I'g again by r.~'. From the last cartesian diagram we obtain a homomorphism jr) End a,r^,((r~0), (r.Z;')) -+ End~(r~n)(ra,(r c~f')). Consider Pla : ]~n ._+ ft. We denote the inverse image of r.Z; under (rPla)" again by r.Z;, and its inverse image under ~.P13 again by r.L ~. Tensor product by r.Z; is a functor ~(rY~ n) -+ ~(r~ n) hence it defines a homomorphism g) End~(rXn)(r~ ,(r.~')) -+ Endg(rZn)(r~ , (r.~') (D r.LP)) = End~irXn)(ra , (r ~" | rXP)) = End,(rXn)(i,a , (r-~')). Composing b), d), e),f), g) we find a homomorphism h) W, --+ Autg(rr.n)(ra,(r.L~')). Applying the functor ~2a-Jr --o ~r~ x-n , ) (d = dim r~n), j ~< m, and taking duals we find a homomorphism W, ~ Aut H~'- i(r~'a , r.~')" = Aut H? � c*(~n, c L~) which is the desired homomorphism a); it is independent of the choice of r. The following property follows easily from the definitions. Let ~' be a closed subset of ~ which is a union of P~ -- P double cosets. The natural map H~215 n', .~)~ H.~215176 n, .~) induced by the closed embedding ~n' ,_+~n is compatible with the W~ actions a). In particular, taking fl'= P~, ~ = G we see that /) The natural map H. ~215 r .~) _+ H.O� c*(fi, .~) induced by the closed embedding tj vi ~ ~ is compatible with the W~ actions a). Assume now that ~ = G; in this case we write Y, insteadof ~". We want to prove that in this case the action a) is the restriction to W~ � { e } of the action 3.8 b) (for V = g). The key point to be verified is the following: Consider the commutative diagram ,~/ ~,~ with p(x, gP,) =x; A ~ g 172 GEORGE LUSZTIG then the action of s, on (n0) , .~~ (composition of b), d)) is mapped under p, : End~^((~ro) ' c~.) ~ End9 ~ (p,(~ro) ' .L~') = End~0(= , .L~') to the action of s, on =, .L~" given by 3.4 c). The verification of this fact is routine and is omitted. Next, assume that ~ = P,. We introduce the following notation "A ={(x, gP, g' P) ~]x ep,,g e P,,g' eP~}, A = {(~, gP, ~' P) ~ ~ x L,/P x L,/P I Ad(g-') .~ ~ ~' + t + ~, Ad(~'-') s e 9' + t + n }. Note that A is exactly like ~, when (G, P, L, if) is replaced by (L o P, L, cg) ; in particular it is ~ x C'-equivariant and carries an L, x C* equivariant local system .~ analogous to s on ~. Now ~ is a P, � C'-stable closed subvariety of ~x'i and the action of G � C" on ~ defines an isomorphism ((G x C') x A)/P, x C"--% ~ei. Using 1.6 a), 1.4 k), it follows that H?. � c.(~P,, .~) ~ H.P; � c.(X ' .~) ~ H.L, � c.(X ' .~). On the other hand we have an obvious L, x C'-equivariant map & -~ A which is a vector bundle with fibres ,,~ pi[~, hence, by 1.4 e) we have H~ � c'(X ' ~) ~ H. ~ � c'(A ' .~). Combining this with the previous isomorphisms we obtain j) H. ~ � ~ .~) ~ H? � c*(~e,, .o@). It is easy to see that the actions of s, on these two vector spaces, one defined by a) for L i (or equivalently, by the s, x 1 action in 3.8 b) for G -- L,, V = ~), the other defined by a) with ~ = P~, correspond under j). Combining i) and j), we obtain a map ~) H.~ � *'(a, .~) . H. ~ � ~'(g, .~) which is compatible with the actions of s, x 1 in 3.8 b) for ~ and for G. 8.10. Assume that we are given two locally closed G x C'-stable subvarieties V, Vx of ~ such that VI is closed in V. Let Vi = V -- V~. Then V~ (resp. V~) is a closed (resp. open) subvariety of V and the inclusions i a : ~r ~ ~, i~ : 9~ ~ ~' induce a) H. ~ � c'(9,, c~) 'q'k H. ~ � r '.~) '!, H. ~ � c.(~-, .g~). The two maps in a) are compatible with the W X W-actions in 3.8 b). The verification is routine and will be omitted. 4. S-module structures 4.1. Let m be the Lie algebra of M~(~0 ) (see 2.1, 2.3). The differential of the isomorphism T � C'--% M~ (see 2.3 c)) is an isomorphism tO C -+ m. We denote CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 173 by r the linear form on m corresponding to pr2:t 9 C ~ C under that isomorphism. Equivalendy, r is the differential of the restriction of pr 2 : G � C* ~ C" to M~ ~ C'. We define s = s(m'). From the decomposition m'= t*~ C, we can write S = S(t') | C[r]. The natural action of W on t and t* extends to an action of W on S by C[r]-algebra automorphisms; we denote it by w : ~ ~ "~. In particular, "r = r for all w. Proposition 4.2. -- There is a natural isomorphism of graded algebras I-~ x e.({~) ~- S. In particular, t-~ � o.({l) = 0 for odd j. Proof. -- We have an isomorphism (P x C')\((G x C') x (~' + t + u)) --% ~ = {I defined by ((gx,X), x) ~ (X -2 Ad(gx)x, gl P) hence, by 1.6 a), we have I-~� 0.(9) ~ I-I~� c.(~ + t + n). (P x C" acts on rg q_ t q-1t as restriction of the G x C'-action on g. Since prx : ~' + t + rt -+ cg is a P x C'-equivariant vector bundle, we have (1.4 e)) Hvxw(~ + t -b rt) ~ H~,� ). Here P � C" acts on ~' via its quotient L � C'. Using 1.4 h) we have H; � ~.(~e) = Hi � ,.(~e). Now L � C" acts transitively on ~' and the stabilizer of xo is ML(X0). Using 1.6 a) we have H; � ~ Hk~.,,. Using 2.1 d) and 1.4 h), we have Using 2.3 d), 1.12 a) and 1.I1 a) H~ ~ H~,%,,~ H~ � c" = S and the proposition follows. 4.3. Let f: X-+ Y be a G x C'-equivariant morphism between two G � C'- varieties and let La x be a G � C'-local system on X. Then H. ~ � w(X, Laa) can be regarded as a left Hi� e,(Y)-module via its Hi� e,(X)-module structure (1.3 b)) and the algebra homomorphism f" : I-I~� w(Y) -+ Ho� w(X). 174 GEORGE LUSZTIG In particular, for any locally closed subvariety f~ of G which is a union of P -- P double cosets, and V as in 3.3, we consider X = ~ (see 3.3 b)), .~', = .~, Y = g, f:X ~Y the map ~x~:(x, gP, g'P)~(x, gP) or nt~:(x, gP, g'P)~(x,g'P), and we get two S (= H~� structures on H.G�176 n, .L~). The two module struc- tures are denoted as follows: a) ~ e$, heH~ � IA!~)h t in H ~215 by definition A(~) h = g;~(~).h, A'(~) h = g~s(~).h, where ~ is identified with an element of I-I~� by 4.2, so that ~;~(~), g;s(~) ~ H~�162 and then the products axe taken as in 1.3 b). It is easy to see that b) a = h E S). Now I-~. � n, -~) is an I-~� in a natural way (1.7) and it is clear that this is the restriction of either of the two S-module structures above to H~� 0* via the natural algebra homomorphism H~'~ � 0. -~ H~ � c.(g) = $ induced by the map ,l] --* point, or equivalently by the map c) H~ � e* -~ H~h**~ = S induced (I .4f)) by the inclusion M~ ~ G � C'. The homomorphism c) has as image the algebra Sw of W-invariants on $. [An equivalent statement is the following. Let I) be a Cartan subalgebra of fl containing t. Then any W-invariant polynomial t ~ C is the restriction of a polynomial I)--* C invariant under the full Weyl group W' of b; or, equivalently: the natural map t/W ~ b/W' is a (closed) embedding. The verification of this statement is omitted.] It follows that d) A(~) =A'(~) for all ~$w as operators on H. ~ � G-(-~ro, .Lf). We shall write r instead of A(r) = A'(r). e) A(~) = A'(~) on H. ~ � w(~-n, .LP) for all ~ e Sw. Indeed, the two projections ~rv __~ fl coincide. We denote the operators defined in 3.8 b) by (w, e), (e, w) on I-I~. � .~b) by A(w), A'(w) respectively (w ~ W). Proposition 4.4. -- Let w, w' e W, ~ e $, h 9 I~. � o* (~, if,). Thtn a) a(w) A'(~) h = A'(~) a(w) h, b) A'(w) ACt ) h = a(~) a'(w) h, c) aCw) A'(w') h = h. CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 175 Proof. -- We use the following fact. If X is a variety and K e ~X, then we have the usual product H~ + ~t~: I:() : H'(X) | H',(X, K) ~ __, ,__, with the following property: if -~ e End~x K, then 9 induces x ~) : H~(X, K) ~ for all i and d) x('+a'(~(h | B)) = t~(A | x 'q' B) for A e H'(X), B e H~(X, K). We apply this in the following situation. We can assume that ~, h are homogeneous of degrees il, i, with i 1 4- i, =j. Let F be as in 3.5 with m large. We take: X = r~', ---- (rPls),r -~', v=weAut(K-) (see 3.7 r p=il, q=2d--j, d=dim(rV ). We have H~(rV , K) = H~(r~', .~'), so that ~ becomes Hq(r~ r) | H~d-~(rvr , r.i~) --+ H~a-~,(rg, r.i~') or, taking duals, Hq(r~r) | H~-~(rV, r.l@. ). _+ "%u2'-'ttr--, i> r~')" or, equivalently, ~' : I-I~� o.(V) | U~, � o.(~,, .~) ~ H? � r c~). The identity d) implies in this case: (w � 1) ~'(A'| B') = ~'(A'| (w X I) B') for A' eI~� B' eI-I~ � We apply this for A'= image of ~ under I-I~� H~�162 ) (induced by V'--~g), B'= h and we obtain a). The proof of b) is completely similar, and c) is obvious from the definitions. 4,.5. We shall regard I-~.� ) and H?� as g-modules in two different ways as in 4.3. These two S-module structures define two linear maps a) S | ~ � ~ -~) =t H?. � c.(~, ~) and two linear maps b) S | Ho ~ � o.(g, .s :~ ~ � ~.(~, .~). Proposition 4.6. -- The two maps 4.5 a) and the two maps 4.5 b) are isomorphisms. Moreover, dim ~ � ~'(~, .s = dim H~o � ~'(~, .s = # W. The proposition is a special case of the following result. Proposition 4.7. -- Let f~ be a locally closed subvariety of G, which is a union of P -- P doublt cosets. Let n(f~) be the number of good double cosets contained in f~. If V = ~ or ~, then M~ � r ta ~) define isomorphisms the two S-module structures (4.3) on ... ~--, s | H~ � � Moreover, dim H~o � e,(~n, .~) = n(f~). 176 GEORGE LUSZTIG We shall consider only the S-module structure defined by the operators A(~) (see 4.3); the other S-module structure is treated in a similar way. We assume first that ~ is a single P -- P double coset. We choose go e f~ such that L and L'----go Lg~ -1 contain a common maximal torus T O . Let P'----go pg~-X, U' =g0Ug~ -~, p', I', n' be the Lie algebras of P', L', U', ~f'= Ad(go)~Ct', t = Ad(g0) IC l'. Using 1.6 a), 1.4 h), we have H~. � c*(~r "n, .~) ~ H'PnP" � e*(V ', .~) ~ H~nL" � ~ .~), where V' ---- V r~ (~' + t + n) c~ (~f' + t' 4- n') and the inverse image of .L~ under V' ~ ~, x ~ (x, P, go P), is denoted again by .o~. We have a) I~ n I~' = (l n p') @ (n n W); 1 n I~' -- (I n r) (9 (I n n') b) p np'= (p nI')(9(p nrt'); p hi' = (I nI')(9(n nI'). If x E V', we have in particular x E p n p' hence we can write uniquely x = X + v 4- ~t, X ~ I n I','~ = I n rt', tz ~ n (using a)) and x --- X' + v' 4- W, X' s I n I', ~'~rtnI', ~t'ett' (using b)). From X+V+~=X'+~'+~t' we deduce (using a root decomposition of g with respect to To) that X = X'. For fixed y, v, v' we have the equation ~t + v = ~t' 4- ~' for ~, ~t'. Set ~ = ~t -- v', ~' = ~t' -- ~. Then ~ e n, ~' c n' and ~ = ~' ert c~ rt'. Thus we have an (L c~ L') x C*-equivariant vector bundle v' -,v" = (ix, e (I n r) (9 (1 n n') (9 (i' n) Ix + e (e + t) X + ~' e (~' + t') nV} with fibres ~ n n n'. Using 1.4 e), we have H.mnL'~� .~) = H~nL'~� .L~) where -oq 7 denotes the local system on V" obtained as inverse image of ~ [] .~a, under the composition V" -~ cg x W' ~ ~' x W, a(y, v, ,/) ---- (pr~( x 4- ,~), pr~,,(X + v')), f~(y,y') = (y, Ad(go') y'). Assume that f~ is bad. We must show that H~.~nL'~� '', .L~) = 0. By 1.13 it is enough to show that H;(V", .LT) = 0. Let rh = (I rart') (9 (I' c~ n), Ix = (I c~ I') (9 (I' c~ I) ; these are the nil-radical and Levi subalgebra of a proper parabolic subalgebra p~ of I@I'. Let c~ x=~ x c~,Ci(gi,, t~=t(gt'CI(9I'. For (X,v,v') ~V" we define ~et, ~'et' by X4-~+ve~, X+~'4-v'~', ~x=(~,~') eft, Vl=(V,v') er h. If V = gr~, then ~, ~' are necessarily zero. Hence we can identify V"=t{(x'vx'~)e(Ic~I')xn~xt~l(X'X)+~+~ecr ifV=g, I{(X,~) e(Ic~l') � th[ (X,X) 4-'~x ~}, ifV=g~. We define :V"~(Ic~I') x tl, x(Y,'~l,~l) =(Y,~x), ifV=g :V"olnI', x(X,'~l) =Y, ifV=gs. CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 177 From the Leray spectral sequence of v, it suffices to show that c) HI(F, .La[v) = 0 for any fibre F of x. IfV =g, the fibre at (%~1) is{vlenl[ (%y) +~l+vxe~'x};ifV=g~, the fibre at y is {,~x e nt [ (% y) + vte ~x}. Since (% y) and (% y) + ~t are in Pl we see that c) follows from the fact the local system [~'(.La [] .o~*) on c~ 1 = ~ � ~' is cuspidal. (See 2.2 a).) Next, we assume that f~ is good. Then goLg~-~ = L, t = l', t = t', I c~ n' = I' c~ rt ----- 0, hence we can identify V" with{ y ~ I ] y E (c~ + t) n (c~, + t) r3 V }. From 2.3 a) we see that cr cg and Ad(g0): cr ~ carries .oq ~ to .~. Hence V" = (cg + t) c~ V. More precisely, we see that V" = cg + t if V = g and V" = cr if V = g~. We have in both cases H. L � c'(V", .LT) = H. L � c,(~, ~ | .~). (If V = fl, we apply 1.4 e) to the vector bundle prl : cr + t ~ ~,.) Using 1.6 a), 1.4 h) we see that HL�162 La| .2~ H,~'~z~ (.~a |176 ~ H,M"~'({Xo}, (La| .La'),,) The last space is ~ H. ~a~ (by 1.12 b), 2.3 d) and the irreducibility of .W) and hence it is ~ H~a~0 ~ ---- $ (by 1.8 c)). Combining these isomorphisms we get an isomorphism H. o � e* (V "'n , .'.~) ~ S. The proposition follows (in the case where ~ is a single P -- P double coset). We now prove the proposition for general ~, by induction on the number N(f~) of P -- P double cosets contained in ~. The case N(f~) = 1 is already settled. Assume now that N(f~) > 1. Let f~t be a P -- P double coset contained and open in fL We may assume that the proposition is already proved for f2 -- f~x and f~t. We apply 1.5 a) to the partition Vn = ~-nt w ~-n-o~. Assume first that ~1 is bad, so that ~ � 0,(~, ~') = 0. If n(t') -- ~,) = 0 then H.~XC'(V a~, .o<k) = H~�162 .LP) = 0 and 1.5 a) shows that o x e. "" ta H. (V ,.LP) ----0, as required. If n(~ ~)>/ 1 then from 3.3 g), h) we see that dim ~ = dim ~-ta-~h and 1.5 a) shows that H. G � e*(~-a-~h, s ~ H o� e*(~-a, LP). It follows that H. ~ � e*(~, c~') has the required property. Next, we assume that fi~ is good. Then by 3.3 g), h), dim Vn~ = dim ~n. If lq~162 "ta-~a~ _~) 0 and 1.5 a) shows that n(~ -- ~1) = 0 then __. ~_ , -~ � c.(~,,, .L~) ~H.= o � ~.r .~) hence H~215 ha, -~) has the required property. If n(~2 -- ~x) >/ 1 then we also have dim ~)~-n, = dim ~ by 3.3 g), h) and 1.5 a) gives the short exact sequence in the first row of the diagram 0 ~ H.~215 ") --- t t t 0 s H0 x _r 0 S| ,--, 23 178 GEORGE LUSZTIG We take that exact sequence in degree 0 and tensor it with S; we find the short exact sequence in the second row of the diagram. We map the second exact sequence to the first using the S-module structure. We obtain the commutative diagram above. By the induction hypothesis, the first and third vertical maps are isomorphism, hence so is the middle one. We also see that dim H0 ~ � c*(~n, .~) = n(~l) + n(Q -- ~'11) = n(~). The proposition is proved. Corollary 4.8.-- If O in 4.7 is closed in G and V = gs or fl, then the inclusion i : ~l n ~ ~7 induces an injective homomorphism i! : H ~ � c*(~,o, 2) -+ H. a � c*(Q, _~). Proof. -- This follows from the exact sequence 1.5 a) applied to V = Qn t3 ~o-n together with the vanishing of H~ � c*(Qo, .~) and H~, ~ � c.(~-o-n, .~) for odd i (which can be seen from 4.7). 4.9. From 3.3 /) and 1.5 b) we see that the open embedding j : ~7"as ~ -/7" = induces an isomorphism a) j" : Ho ~ � c*(~, .~) ~ .l-lQ.o � c't4~,--as, -~)- Similarly, for any good P -- P double coset s the open embedding induces an isomorphism b) tIo ~ � c*(~n~,}, .~) --% --ol-l~ � o'tpn(-}~__,,s, L;';). In particular, using 4.7 we have dim l-lG� .L~) = 1. Using 3.3 i) we see that the open and closed embedding --Rs4Un~'-+ ~/;'as defines an isomorphism c) 0 H? � , -~) --% H. ~ � ~('/'as, "~). ,oEW This gives a direct sum decomposition d) H?� = | D,,, where D,~ is the image under c) of the summand corresponding to w (a graded space). The W � W-module structure on H. ~215 c'(~r , .~) satisfies d) A(w) D, = D~, A'(w) D, = D,-,, (w EW); this follows immediately from the definitions. Taking components of degree 0 we obtain e) Ho ~� (~r = @ A(w) D., o @ A'(w) D., o t~EW toEW f) A(w) D,, o = A'(w -~) D,.o (D,.o is a line). c't.i?n,*,,,__as CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 179 4,.10. For V = g, gN or ~r we defines a subspace DvC H0 ~215 o*(~, L~) as the Ita� L~) H0~215 induced by the closed image of the homomorphism --0 ~-- , -+ embedding ~i, ,_~ ~. (In each case, we have dim ~i, = dim V, see 3.3.) We shall denote D v in the three cases V -- g, gN, ~r as D, Ds, De.0, respec- tively. (Note that D,. 0 has already becn introduced in 4.9. It is a line. Similarly, D and Ds are lines, by 4.8 and 4.7.) These lines are related as follows: a) Under the isomorphism 4.9 a), D corresponds to Do, 0. ~** .o b) The homomorphism H0 ~215 (gs, ~.") -+ H~215 c~) induced by the closed embed- ding ~NC~ maps the line Ds onto the line rb.D (b =-dimt). We prove a). The cartesian diagram induces a commutative diagram HO � 9 .~/,~) : Ho ~ � c'(~r ' .~) 0 k--R8 o~o... .~) = H o (g, and a) follows. We now prove b). The commutative diagram i~P ~-~ - ~ induces a commutative diagram SO � c'/;~ .~) n0~ � c.(~,~, ~) O. ~N~ Yi Ho � c.~a .~) H~b� 9 l', .I/~ ) 2b k~l 180 GEORGE LUSZTIG and we are reduced to showing that image (ft) = r~.Ho~215176 P, -~). As in the proof of 4.7 we have noa � c.(~i~ ' .s = HoL � e.(c~, .W | .W') = Ho~C**'({ x 0 }, 8) H~2a� P, .LP) ----- H~,� c'(cg + t,p;(-o~| -~ H~*'(xo + t, 6") (where Pl : ~ + t --~ ~' is the projection and 6" denotes (.Z' | .L~')~ o regarded as a local system over { x o ) and also its inverse image under x o + it ~ { x o }). We have only to show that the homomorphism f,' : Xo }, #) + t, 6") induced by the inclusion f' : { x 0 ) ~ x o + t has as image r b Ho~*O'(x0 + t, 6~). But this follows from 1.10 b). 4.11. From 4.9 e),f), 4.10 a), 4.9 a) and 3.10, it follows that a) Ho ~ � = @ A(w) D= @ A'(w) D toEW wEW and b) A(w) D = A'(w -1) D, w e W. We now show that c) the homomorphism Yt :H~� -Lp) -~ H~215 ~a, .~k) induced by the closed embedding T : gl~ C ~, is injective (b = dim t). Indeed, using 4.6 we axe reduced to the case wherej = 0. Let I be the image ofy (forj = 0). By 4.10 b), I contains r b D. But I is clearly W X W-stable (see 3.10) hence it contains 2~ r~A(w)D = rbHo~215 (see a)). This subspace has dimension ~W equal to SW (by 4.6). Thus dim I >/~W. Using 4.6, we have dim Ho~215 e'(~N, ~") = ~W. It follows that dim I ----- g W and y is injective. We have: 0o .. d) Ho ~ � (fiN,-~) = @ A(w) Dr~ @ A'(w) Ds, t~EW toEW e) a(w) D~ = a'(w-') D s. Indeed, by the previous argument, h ~ r -b y, h defines an isomorphism H0 a � ~'(gN, "s ~ H~0 � ~'(g, "~) compatible with the W � W actions and taking D s to D; it remains to apply a), b). 4.19.. We now want to study the S-module structures on I-~. �162 , c~) in the case where P is a maximal parabolic subgroup. In this case, we have W = { e, s } and we have the following result. CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I lal Proposition 4.13. -- a) The S-module structure on H.~215 "~r "~) defined by the operators A(~), ~ ~ S, gives an isomorphism S/(r) | no ~ � c.( "Was, Y) ~ n? � ~'(4~as, ~) where (r) is the ideal of S generated by r. ~o � c'r4~ .#) we have A(U A'(U, ~ ~ S. b) On the image of H? � c~ -~) ,-~ .~. ~--Ra, ---- c) On the image of H? � -~) ~ H? � ~"("/7"~a , .~) we have A(~) = A'('~), ~ ~ S. = 14o � o./~ .2), ~ ~ S. d) A(s) A(~) h A(~ U A(s) h for all h E_. ,--a~, Proof. -- We have ~ ----- t -- t0 where to is the centre of g (a hyperplane in t). In the following calculations we shall often omit writing local systems. Using the oo P � C~ embedding (d + (t -- to) + rt ~-~ ~r , x ~-, (x, P, P), we have UG� ~x" .L/)~ H. F � (t--t0) +n, ) (see 1 6 a)) 9 k--as ) ~ --- HL.� + (t -- t0), ) (see 1.4 h), 1.4 e)). Let it be an affine hyperplane in t, parallel to to but distinct from t0. The stabilizer of~+tl in L � C" is L � (+ 1) and we have H ~215 .W) -_ H.L�177 ' + it, ) (by 1.6 a)) 9 \--RB ) H.~�174 ") (by 1.4 e)) -~ HZ~('o'�177 (.W| (by 1.6 a)) HZa~'�177 },.Z'| (by 1.4 h)) H. ~'*~ H x. (by 1.12 b), 2.3 b), d)). ~*(') (x, P, ~P), Using the L � C" equivariant embedding i : ff + (t -- to) '-~ --as , x we have HO� .2~) ~ H.~� + (t -- to), ) (see 1.6 a)) 9 \--R8 , H. ~ (as above). Using these isomorphisms and 4.9 c), we see that a) holds. Now b) follows from 4.3 e). We now prove c). Note that using i above, we have e) H~ �162176 ~ =~ H~� + (t -- f0)) (by 1.6 a)) H~� + tt) ~ Hz~(~o)� H~ (as above) Consider the automorphisms % a of --as~n(~ defined by 9(x, gP, g' P) = (x, g' P, gP) and e(x, gP, g' P) = (Ad(2) x, ~gP, ig' P). We have a commutative diagram #(s) ~' + (t - to) 9 R S 9oct AdC~) ~r + (t - to) RS e't~i/~"("~--as, 182 GEORGE LUSZTIG This diagram shows that the automorphism of H~ induced by conjugation by g on T " 9 ,#7"n~.)~ (We corresponds under e) to the automorphism r of l-tQ� ~ Rs J induced by % can ignore ~.) Hence, if ply, Ply: --Rs 4;'~~ ~ ~ are the projections, we have cp'(p~(~)) = p~z('~) for ~ eH~�162 We have p~ =p~o~ hence p~(~)= q~'(P~z(~))----P~('~). By 9 , ~e, � c't~n(,~ .s definition, A'(~) ~] =p;s(~).~] =p,~( ~).~ = A(~ ~], ~] ~... ,--as , This completes the proof of c). We now prove d). Assume first that h e D, (notation of 4.9). Then A(s) h e D~ by 4.9 f), hence we have A(U A(s) h = x(.~) A(s) h (by c)) = A(s) A'('~) h (by 4.4 a)) = A(s) A(.~) h (by 4.3 e)) as desired. If now h e D,, then applying the identity d) to A(s)h a D, we obtain A(s) A(~) A(s) h ---- A('~) h. Multiplying both sides by A(s) gives A(~) A(s) h ---- A(s) A('~) h. We now use that H.~215 , 2)= D, + D0 and d) follows. Corollary 4.14. -- In the setup of 4.12, there exists p e C such that the identity A(s) A(~) + A(~) A(s) = p Id holds for all h ~ ~ � c* (~, .~). Proof. -- From 4.6 and 4.13 a) we see that the open embedding j : ~r ~ ~ = gives rise to an exact sequence 9 J" H o x a) 0~H. ~ �162 Q � . r ~0. Since A(s) A(~) + A(~) A(s) _~ 0 on H. c � we see from a) that there exists a G-linear degree-preserving endomorphism r of H~215 s such that b) A(s) A(e) h + A(~) A(s) h = re(h) for all h e H. ~ � r .~). We know that A(s)A(~ ~) h = A(e ~) A(s)h, see 4.3 d). Applying b) twice, we have A(~") A(s) h = A(s) A(~) A(~,) h = -- A(~) a(s) A(~) h + ,r h) = _ A(~) (-- A(~) a(s) h + re(h)) + rr h) = A(a') A(s) h -- r A(~) r + rC(A(~) h). Thus r(r h) -- A(a) O(h)) = 0. Cancelling r, we have c) r h) = a(~) r Since A'(~), h'(s) commute with A(s), A(~) we also have d) r h) = A'(~) r e) O(A'(s) h) = A'(s) (l)(h). If h0 ED, we have A(~) h o = A'(~) h 0 (see 4.3 e)). CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 183 If however, h o eA(s) D, h o 4= 0, then A(0t) h o + A'(ot) h o. Indeed, we have h~ e A(s) ho, ho eD a'(~) h'o = a'(~) A(s) ho = a(s) a'(~) ho = A(s) A(~) ho = -- a(~) A(s) ho + rO(ho), A(~) h; = a(~) A(~) ho. If we had A'(a) h' o --__ A(0t) ho, then 2A(a) A(s) h o = rO(ho). Using 4.6 it would follow that h o = 0 so h' o = 0, a contradiction. Thus, D = Ker(A(a) -- A'(a) : HoG � e'(~i, ZP) ~ ~ � e.(~, Zp)). By c), d), 9 maps this kernel into itself, hence OD C D. If ho is a non-zero element of I) we have therefore @ho = pho for some p e C. Using e) we have O(A'(s) ho) = p A'(s) h o. Since ho, A'(s) h o form a basis of rio~ � c*(~, .L~) we must have O(h) = ph for all h of degree 0. Using 4.6 and d) we have 9 (h) = ph for all h. Proposition 4.15. -- The open embedding Y'RN ~ ~N gives rise to an open embedding j : J?aN "-* ~i;'s = gs and this induces a surjective homomorphism ~o � ~t~ .2) -~ ~G � ~t4~ ~). * : **0 k~N, x't0 k" l~I Proof. --j" preserves degrees by 3.3 k). We have a cartesian diagram jt fin RN It induces a commutative diagram j'* � "" ~o,<r .~) Ho~ c.(~., ~) .---- --0 ,~, � '" HoO � o.( "~;- .~) ~ H0 o (gN, .~1 .~'* and it remains to show that j" is surjective. But j'~ is in fact an isomorphism, by 3.3 l) and 1.5 b). 184 GEORGE LUSZTIG 5. The commutation formula Theorem 5.1. -- Let s~ e W, ~ e R be as in 2.5, let q>>. 2 be as in 2.12 (1 < i~< m), let ~ e S and let hell a . � c~), c V = g or g~. Then a) A(s,) A(~) h -- A("~) A(s,) k = q A(r(~ -- "g)loq) h b) A'(s~) A'(~) h -- A'("~) A'(s,) h = q A'(r(~ -- "~)la,) h. 5.2. The proof will occupy most of this chapter. We shall first make some reductions. Using the injectivity of y, :H.O� ) ~Ha.� in 4.11 c) we see that if the theorem is known for V = g, then it also follows for V ---- gs. (~'x commutes with the operators A(st) , A'(s~), cf. 3.10, and with the operators A(~), A'(~).) Hence we can assume that V -= g. It is enough to prove 5.1 a) ; the proof of 5.1 b) is the same. Let P,, A t be as in 2.4, P, g2, -~ as in 3.9. Assume that 5.1 a) holds on H.~�176 .~) (i.e. when G, P, L, ff are replaced by L~, ]5, L, g'). The map 3.9k) is compatible with the operators A(si) and the operators A(~); moreover, from its definition, its image is the same as the image I of Ha. � ~"(~P', --s ~ H ~ � o*(~, .~) (scc 4.8 with ft = Pt). Hcncc thc idcnti W 5.1 a) holds for h e I; in particular, it holds for h e D (sec 4.10). Hcncc it holds for h in F~ A'(w) D (sincc A'(w) commutes with wEW A(s~), A(',}) (~] eS)), hence, by 4.11 a) it holds for any h ~ HoG� It must then hold for any clement of form h = E A'(~#) h i (~1# e S, hj of dcgrce 0), sincc A'(~h) commutes with A(s~), A(~) (~] e S). But in this way one obtains the most general clements of H~.�162 (by 4.6). Thus, we are rcduced to proving 5.1 a) for Li instead of G. Therefore, in the rest of this chapter we shall assume (as we may) that V = g and P is a maximal parabolic subgroup of G. We shall write s, ~, c instead of si, %, c~. Note that if the identity 5.1 a) holds for ~ = ~a and ~ = ~ (and any h) then it also holds for ~ = ~1~ and ~ =ax~ +a~ (a~,a, ~C). Hence it is enough to prove 5.1 a) when ~ runs through a set of algebra generators of N, for instance { }uS w. If ~ ~ SW we have A(~) = A'(~) (see 4.3 d)) and A(s,) A'(~) = A'(~) A(s,) (see 4.4 a)) hence A(s,) A(~) = A(~) A(s,) so that 5.1 a) holds in this case. It remains to verify 5.1 a) for ~ = 0~. We have '~---- --~ so that in this case, 5.1 a) is equivalent to A(s) ) h + A(s) h = 2 rh. By 4.14, we only have to show that p in 4.14 is equal to 2c. CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 185 Proposition 5.8. -- a) The two S-module structures on H.~215162 defined by A(~), A'(~) coincide; they give an isomorphism of S-modules $/(= -- or) | Ho ~ � c*('/~as , .~) ~ H? � c'(#mo .~) where (or -- cr) is the ideal generated by ~ -- or. In particular, on H ~ � c.(j;- , .~) we have A(,t) = A'(~t) = cr and multiplication by r has kernel O. b) A(s) = A'(s) = Id on H ~ � c'(~k~m~ , .#). c) dim Ho~215 "~/',,s, .~) = I. Proof. -- b) is clear from the definitions of the actions 3.8 b). The equality A(~) = A'(~) on H.~� , .~) follows from 4.3 e) and 3.3 j). In the following computations we shall sometimes omit writing local systems. Ho � r .~) ~ H G � c.({;-~ fL~) (see 3 3 j)) Hr� n (~g + n), ) (by 1.6 a)) _-- Ut�176 c~ (~ + n), ) (by 1.4 h)) H.M~"~ c~ (x o + n), ) (by 1.6 a)) H.Mrr176 + (n--.,~), ) (see 2.10 c)) n.~u*~ + (n --,,'g'), ) (by 1.4 h)). Let M = ML(,o), Mt the kernel of the character X : M~ -+ C" by which M ~ acts on rt/~, rrt = Lie M, d x : m -+ C the tangent map to X, m~ = Lie M~ = ker d x. Let .~t be an affine hyperplane in rt parallel to .,~ and distinct from .,~. Clearly (M 0 x (x 0 + ,,~ffx))/Mx --% x 0 + (1t -- ~) as M~ We have (by 1.9 a)) H. ~ � c*(~sa, .~) ~ H~(x 0 + (n -- ~t~ )~ (by 1.6 a)) (by 1.4 e), 1.9 a)) --- H~(,,o, (.e' | ~'),~)* _ H Mt ~ (see 1.12 b)). It remains to note that H~| ~-%H~, H~=S(m:) =S[(dz) and d x=~-cr (see 2.10 b)). 5.4. Using the injectivity ofyz : H.~215 r176 2) --+ H.G�162 .~) in 4.11 c), we see that the identity in 4.14 which holds on H. G � o.(~, 2) must also hold on H. G � ~~ .LP); using 4.15, it must also hold on arbitrary elements of H~215 c'tg7-~ as, .~). We now take such an element h with h 4= 0 (see 5.3 c)). By 5.3 a), b), we have A(a) h = crh, A(s) h = h. Hence, ifp is as in 4.14, we have 0 = a(s) A(~) h + a(~) a(s) h --prh = A(s) crh + A(a) k --prh = 2crh --prh. By 5.3 a), we have rh 4= 0 in H~�162 "~':'m~,-~). It follows that p = 2c. This completes the proof of Theorem 5.1. 24 186 GEORGE LUSZTIG 6. The algebra H 6.1. Let fl be as in 3.3 c); assume that it is closed in G. If ~l(w)Cfl, then the inclusion i : ~f~JC ~ induces a homomorphism ;,: Ho o � ~'(~,~,, .~) -. HoO ~ ~'(~, .~) (see 3.3 g), h)). Taking the direct sum, we get a homomorphism 9 Ho~215 '', _~) --,- HoQ� wEW ~(w) C This is an isomorphism; the proof is by induction on N(~) as in 4.7. In particular for = G, we have Gx C" "" @ ~ � c'(i~ns'" , ..~) Z H o (g,, if'). w~w X C* "" The image of --o~~215 ,2/~) '--~ H0 ~ (g~, .~) is denoted by E,~. It is a line, since H o � c-(~, ,,,,, ..~) ~ HoG � '~', --@) by the irreducibility of gs"n"~ (3.3 g) and 1.5 b)), and the space H~215 ~, -q;) is one dimensional by 4.7. Hence, we have a decomposition wEW We have clearly b) E, : D s (see 4.10). We have isomorphisms E, =~ H~ � c'(~, .s ="- Ho ~' =~ H~, (see the proof of 4.7). We denote by 1 the element of E corresponding to the unit element of the algebra H~c,o ~ ~ C. = ~o~c'm ~). Lemma6 2.--Forw~W, wehaveA(w) l A'(w-l) I E--o ~uN, Proof. -- Using 6.1 b) and 4.11 e), we see that a) a(w) 1 = a A'(w -1) I for some a 9 C'. From 6.1 a), we have A(w) l - ~,.1 9 | E., ~) A'(w-')l--~'l 9 @ E. IO'~r for some ~, [3' 9 C. c'r;~nt,,~,x CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 187 ~ ** Under the homomorphism j" : H0 ~ � c.(~, .L~) -+ H0 ~ � (Y/'~, -~) in 5.4, the lines E,~, (w'+ e) are mapped to zero. Indeed, we have a cartesian diagram x =-"- @" where X = ~k~s c~'~. By 3.3 j), X is contained in ~,c'~ n g~c.', and hence dim X < 8' (see 3.3 g)). On the other hand, ~ is open in gs hence X is open in ~t"~ which is irreducible of dimension 8'. It follows that X is empty. The cartesian diagram above gives rise to a commutative diagram o ~ Ho ~ � ~'(~i~ '~'', .~) 1 1 X C $ "" HoG ~ ~'( "r d~) n~ (g,,, .~) so that j" E,,, = O, as asserted. Applyingj ~ to b) we obtain therefore A(w)j'(1) = ~j'(1), A'(w-1)j" 1 = [~'j'(1). C~ "" But A(w) = A'(w -1) = Id on H0 ~ (?'Na, c~) (by the definition of the actions 3.8 b)) hence (~ -- 1)j'(1) = (~' -- 1)j'(1) = 0. The surjectivity off* in the proof of 4.15 shows that j'(1) ~ 0. It follows that [3-=[3'= 1. Thus, A(w) l--1 and A'(w -1) 1-1 are in ~) E,,,. Using a) we have tO' :~ e aA'(w-') 1 -- 1 e O E.,. Wa@~ Clearly we have aA'(w -1) l-ale O E,,,. W'~a Subtracting we get(a-- 1) 1 ~ O E,,,.Butl r G E,0, so that a= 1. The lernma is proved. Theorem 6.3. -- Let H : S | C[W]. There is a unique structure of associative C-algebra with unit 1 | e on the C-vector space It such that a) ~ ~ ~ | e is an algebra homomorphism S-+ H, b) w ~ 1 | w is an algebra homomorphism C[W] ~ H, c) (~|174 (~eS, wEW), d) (1 | (~| -- ('it| (1 | = c,r((~ -- ~)/~,) | (~ e $, 1 ~< i~< m). 188 GEORGE LUSZT1G Proof. --Let .,~ = H ~215 c'('ciN, .s Let A be the subalgebra of Endc(Jf a) generated by the endomorphisms A(~) A(w) :,,~ -~.,~, (~ ~S, w ~W). Let A' be the subalgebra of End,(Jf ~) generated by the endomorphisms A'(~) A'(w) :.,'f' ~ (~ e S, w E W). We have C-linear maps l A :H-+A, A(~| =endomorphismh~A(~)A(w)h of e) A' : H -~ A', A'(~ | w) = endomorphism h ~ A'(~) A'(w) h of tz :A ~.~, ~(f) =f(1) Vt' : i ~,,x~', w'(f) =f'(1). From 4.11 d) and 4.6 it follows that f) Ix o A and pt' o A' are isomorphisms H -~.,~'. In particular, pt, [z' are surjective and A, A' are injective. From 4.3 b), 4.4 we see that g) AC Z(A'), A's Z(A) where Z(A) (resp. Z(A')) is the set of all endomorphisms of.,~ which commute with all endomorphisms in A (resp. A'). Let fE Z(A). Then by f), f(1) = A'(X' ) (1) for some X' ~ H. Now let h EJt ~'. We have (by a)) h = A(Z ) (1) for some X ~ H. Hence f(h) =f(A(~) (1)) = A(X ) (f(1)) (since fe Z(A)) = a(z) (A'(z') (1)) = A'(X' ) (A(X) 1) by b) = h. Thus, f = A'(X' ). This shows that Z(A) C image(N). We have obviously image(A') C A' C Z(A) (see g)) hence image A' = A' = Z(A). Similarly, image A = A = Z(A'). It follows that A and A' are isomorphisms. Using f) it follows that ~, ~t' are iso- morphisms. We now define an algebra structure on H by transporting to H via A the algebra structure of A. Then properties a)-d) are satisfied; for example property d) follows from 5.1 a). The uniqueness statement in the theorem ks clear since s~ (1 ~< i ~< m) generate W. This completes the proof. Corollary 6.4. -- a) .,~ = H~ N, .~) can be regarded as a left H-module in two ways: x, h F-, A(z) h Or z, h A'(z) h. CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 189 b) We have A(X ) A'(X' ) h = A'(X' ) A(X ) h for all h 9 ~ X,X' 9 c) The two maps H -+oct ~ X~A(X ) 1 and X~ A'(X) 1 are both isomorphisms. d) There is a unique anti-automorphism X --* X of the algebra H such that A(X ) 1 = A'(~) 1. It satisfies (1 | ^ = 1 | -1, (~| ^ ---- ~| for w 9 ~ 9 Proof. -- a) is clear from the definitions of A(Z), A'(X ) (see 6.3 e)) and from 5.1 ; b) is just 6.3 g); c) is just 6.3f). From c) we see that there is a unique C-linear endomorphism X --* X of H such that A(X ) I = A'(~) 1. From a), b), c) we see that X --* X is an algebra anti-automorpb_ism. We have (4| ^ =~| by 4.3 e) and (l| ^ = l| by 6.2. Theorem 6.5. -- The centre of H is { ~ | e I ~ 9 Sw }. Proof. -- It is clear that ~ | e 9 centre(H) for all ~ 9 8 w. We now prove the converse. Let h 9 We can write uniquely h= Z ~| (~.~s). Assume that ~ # 0 for some w 4: e. Then we can find an integer j >/ 0 such that ~ 9 r j $ for all w + e and ~,~ r r j +l S for some w 4: e (say w = wl). From 6.3 d) we see by induction on the length of w that, for an), ~ 9 $, (l| (~|174174 9174 H, (w 9 W). We have therefore 9 (rJ+t| 1) H if w# 9 =0 if w= 9 We use this in the following calculation O= (~| h -- h(~| X~.| Z~'~Ow + (r~+'| 1) h' tO tO Y~.(~-'~)| (rJ+~| h ', (h' 9 This equality shows that ~.,(~. - "~) e r j +' S. Since ~t r r~ + 1 S, it follows that ~ -- -1~ 9 r$. But this clearly fails for ~ a generic element of t*. This contradiction shows that ~, = 0 for all w 4: e. Thus h----4,| e. We now write the equation (1 | = h(l| Using 6.3 d), we see that (4, -- ,i~,) | s~ = c, r((~, -- "~,)/~) | e. It follows that 4, ----" ~/~,- Since i is arbitrary, it follows that ~, 9 S w. The theorem is proved. 190 GEORGE LUSZTIG 7. Preparatory results 7.1. Let M be a connected algebraic group, let X be an M-variety and let o a be an M-equivariant local system on X. We define a filtration of I-t~. (X, o a) by a) F ~ = F~(H.M(X, 8)) -= the H~-submodule of H.M(X, d') generated by ~[~ H~(X, d'). ~<t ThenF ~ 1C... andF ~=0fori<0.Let b) l-I i ---- H~(X, g)/H~(X, o ~) n Fi_ ~ = component of degree i of Fi/F~_ 1 . We regard l-I i as a graded vector space which is zero in degrees 4: i. We have a natural embedding H i ~ FJF,_ 1 of graded vector spaces (isomorphism in degree i). Since FJFi_ 1 is an H~t-module , this extends to an H~-linear map c) H~ ~c Hi -+ FJF,_ i. The homomorphism d) H~(X, d') --~ H[')(X, d') (see 1.4 f)) is zero on H~(X, d') (~ F~_I hence it factors through a C-linear map e) H i --~H[*~(X, e). Proposition 7.2. -- Recall that M is connected. Assume that H~d(X, ~) = 0. Then: a) The maps 7.1 c), e) are isomorphisms. Hence H~| c H~*)(X, ~') ~ FJF~_ 1 as H~-modules. b) ~' Hood(X , o a) = 0. C) Each F, is a finitely generated projective H~-module and F~s = F.z8 + 1 ..... H.M(X, o~ where ~ = dim X. In particular, H.M(X, o a) is a finitely generated projective H~-module. d) The C-linear map C | H.M( X, ~ -+ H!*~( X, @) defined by 7.1 d) (where C is regarded as an H~-algebra via H~I"*r = C) is an isomorphism. Proof. -- Let m be a large integer and let P be an irreducible, smooth variety with a free M-action such that H~(P) = 0 for 1 ~< i ~< m. We assume, as we may, that M\P is simply connected (since M is connected). Consider the fibrationf: M\(F � X) ~ M\I" with fibres = X. We study the Leray-Serre spectral sequence for ft(rd"). Since M\F is simply connected, this spectral sequence has E~'~= H~(M\F)| g'). This is zero if at least one of the following four conditions is satisfied: podd and p>t 28'--m (8'=dim(M\F)), q odd, q> 2~, p> 28'. (For the first condition, note that Hk = 0 for i odd, Hi(M\F) = H~t for i ,< m and M\F is smooth so it satisfies PoincarE duality.) CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I If Eg,' were zero for all odd p, then the spectral sequence would degenerate at E2. As it is zero only for p odd in the indicated range, the spectral sequence is only degenerate in some range: E~ ,q=E~g ~ ifp +q~> 2(~+~') -- (m--2~-- 1). The spectral sequence converges to H;(M\(P � X), to~ We now define a new spectral sequence: = (v. V-,, whose differentials are duals of the differentials of { E, ~'= }. We have = E~ ifp+q~< m--2~-- 1 and the spectral sequence converges to H;(M\P � X, rd")', which is equal to H.M(X, g) in degrees ,< m. a) follows from these statements, by taking m large. (Recall that H. ~ = H~ by I. 7 c).) Now b), c), d) are immediate consequences of a). 7.3. We want to state a form of Ktirmeth's formula in equivariant homology. Let o a, 6" be local systems on the algebraic varieties X, X'. We have a natural isomor- phism (external cup-product) a) H;(X, 6") | H'.(X', 6") -% H;(X � X', ~" [] oa'). Assume now that X, X' are M-varieties and o a, o a' are M-equivariant. Let m, F be as in the proof of 7.2, with m large. Applying a) to M\(F � X), M\(F � X'), r d', r8 '~ instead of X, X', o a, $" we obtain an isomorphism b) 0 H'o(M\r' x X, r6") | (M\r x X', v6"") ~+$=k H ((M x g)\r x r � x x x', r#" [] r6""); taking duals and assuming k ~< m we obtain an isomorphism c) H~� x X', 6" [] g') _-_ @ (H~"(X, 6")| H~(X', 6")). i+ ~ =k We compose this with the homomorphism 1.4f) MxM t H~ (X � X, 6" [] 6") ~ H~(X � X, o a [] g') induced by the diagonal embedding M C M x M and we obtain a homomorphism n.~(X, d') | H.~( x, 3") ~ H.M( X x x', d' [] #'). factors through a homomorphism of One verifies easily that this homomorphism H~-modules: d) H.M(X, 6") | H.~( X, 6") -+ H.~( x � X', 6" [] oa'). 192 GEORGE LUSZTIG Proposition 7.4. -- Assume that H~~ g') = 0, H~d(X ', 8") = 0 and that M is connected. Then the homomorphism 7.3 d) is an isomorphism. Proof. -- Both factors in the left hand side of 7.3 d) have canonical filtrations F i (see 7.1) hence their tensor product has a canonical product filtration; similarly, the right hand side of 7.3 d) has a canonical filtration F ~ (by 7.1 for X x X'). From the definitions we see easily that the map 7.3 d) is compatible with these filtrations hence it induces a homomorphism on the associated graded spaces for these filtrations: a) (Hgt| H!'~(X, d'))| (H~| H!'~(X ', d")) +H~t| H!'~(X x X', d'~ e'). (We have used 7.2 a) for X, X', X x X'.) It is enough to show that a) is an isomorphism. But a) is the homomorphism 7.3 d) with M replaced by { e } and with scalars extended to H~. Thus, we are reduced to the case where M = { e }. In that case, the result follows from 7.3 a). Proposition 7.5. --- Let M, X, d' be as in 7.2, and let M' be a closed connected subgroup of M. Then the H~t-linear map H.x(X, g) ~ H.w(X, o p) (see 1.4f)) extends to an H~-linear isomorpkism a) H{~, | H-X( x, ~') --% H.M'( x, t). (H~,, is regarded as an H~-algebra, via H~ -+ H~, in 1.4 g).) Proof. -- We consider the filtration F ~ of H~(X, ~') in 7.1. It defines a filtration { H;~, | F~ } of the left hand side of a). (We use 7.2 c).) Similarly, H.~'(X, g) has a canonical filtration, by 7.1. It is clear that the map a) is compatible with these filtrations hence it induces a homomorphism on the associated graded spaces for these filtrations: b) Hiv | (H~,| Hi'}( x, oa)) --~ H~, G o Hi'~(X, @). (We have used 7.2 a) twice.) It is enough to show that b) is an isomorphism. But b) is the identity map. This completes the proos 8. Standard H-modules 8.1. The results in this chapter bear some resemblance to results in chapter 5 of [5]. Let y e g be a nilpotent element, 9 its G-orbit in fl, and a) ~,-----{gPeG/P[Ad(g-1)ye~+n}. Then M(y) = Mo(y ) (see 2.1 a)) acts on ~, by b) (gl, X) : gP ~ gl gP. Let ~'= (G x C')/M~ ~g be defined by (gl,X)~X-~Ad(gt)y; this is a G x C'-equivariant, tinite principal covering of ~ with group c) M(y) = M(y)/M~ (G x C" acts on 0 by left translations). Let d) r = (O x C" x t t t --1 ~t (M(y) acts on G x C" x ~y by (g~, X) : (g~, X, gP) ~ (g, gl , x -~, gl gP)). CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 193 We have a cartesian diagram e) where "r is the projection to the first two factors and/~ is defined by (g~, ~.,, gp) _~ (~.,-2 Ad(gl)y ' g~ gp). We may regard .~, as a closed M(y)-stable subvariety of ~ (see 3.1 d)) by iden- tifying gP E t~ v with (y, gP) ~ d~. The restriction of .LP, .~~ from I~ to ~ will be denoted again .~, ~'. We want to define operators A(w) on H~r*~'~(~,, L/) for any integer j and w e W. Choose m, I ~ as in 3.5 with m/> j and form the cartesian diagram associated to e): f) r~' = rg rh Then rh'(r K) = (rX), (r/f) (r.L p') and rh" defines a homomorphism g) End~w,,(rK) --+ End~,r&(rh'(rK)) ----End~wg,((r~'), (rh') (r-lP')). Composing 3.5 d) with g) we find a homomorphism W --~ Auta~,r~,((rr), (r]f)(r.L,#')). This induces a homomorphism W -, Aut H2c a- '(r ~', (rr): (r]f) (r.~')) = Aut H~' - '(r ~', r]f r.Z ~') (d = dim rd~). Taking duals we find a homomorphism h) A : W ~ Aut ~2d-Jf ~" " " 9 "c ~r v, r h" r-La') . = Aut H ~ j;. -r = Aut H~'~(~,, ..~). (See d) and 1.6 a).) This homomorphism defines the operators A(w) on H~"(..~y, .~) (w E W). Replacing in the previous construction ZP by .LP" we obtain in the same way operators A'(w) on Hj~e~(~, .oq;') (w e W). Replacing in the previous construction .2 ~ by .~ [] .s and the diagram f) by 25 194 GEORGE LUSZTIG 9 ri" 1 1 (h is defined by (/z,/~)), we obtain a homomorphism W X W --* Aut H~215 X8 ~, )~'(_c~ 5~ .s = Aut H~*'(,~, � ~,, .s [] .LP~ the operator corresponding to (w, w') ~ W � W' is denoted by A(w, w'). 8.2. We now define operators A(~) on tl.m")(~v, .~) for ~ ~ S. Let "0 ~ H.M~ .Lf) ---- n. G � c'(0, h" .~), ~ ~ S ---- I~ � 0.(g). Then h"(~) e H~�162 ) and we define A(~) ~ as the product (1.3 b)) h"(~) .B e H. G* r163 = H.~,,(~,, .~). MOI$) Thus, H. (,~y, .~) becomes an S-module 9 Similarly, replacing c~ by .~', we obtain an S-module structure on H.m('~(.~,, ~') with operators A'(~), ~ a S. We now define operators A(~), A'(~) for ~ ~ 8 on H.~"(~, � .~',, .~ [] .L~'). 9 9 Let ~ ~ H.~~ � W,, -~ [] -~') = H.~ ~ � a'(.~ [] .~')), ~ ~ H6.~.(g), 9 ,~, and let p~ : g � g ~ {I be the two projections (i = 1, 2). Then )i'p~(~) ~ I-~ � o,(0' � p,(~).,~, (h as in 8.1) and we define A(~) -~, A'(~) "0 respectively as the products (1.3 b)) "'* ~ 9 9 h" p.~(~).~q ~ H. ~ * r � ]'(._q~ [] ~') = H m','(~, � ~,, .~ [] .~~ Thus __. ,--, � ,~, .L~ [] .Lf') becomes an S-module in two different ways, via A(~) or A'(~). 8.8. The operators A(w), A'(w) in 8.1 are H~,,-linear. (Same proof as for 4.4 a), b).) The operators A(~), A'(~,) in 8.2 are also H~n(,)-linear. 8.4. Consider the homomorphisms A' = H.~"(~,, .~) | -~~ t" A'" = S.~'"(~, � ~,, .Lf [] .~') ,.e ,,, SO � o.(~, .~) CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 195 where 0t is given by 7.3 d), ~ is given by 1.4 f), and 0 is as in 3.3 b). Let w, w' e W, ~, ~' e S. From the definitions one verifies that the operators A(w) | A'(w') (8. l); | (8.2), on A' zx(w) (8. (8.2), on A" A(w) A'(w') (4.3); A(~) A'(~') (4.3), on A'" are compatible with the homomorphisms ~t, ~. 8.5. The finite group M(y) (see 8.1 c)) acts on H~o~,) (by 1.9 a)), by C-algebra HM~ .s H,~"(~,, .~'), H,~'"(# ~, � 0~,, .L~ [] .LP') (by automorphisms and on __. ~_ ,, 1.9 a)) by automorphisms which are compatible with the H~o~vl-module structures (i.e. are semi-linear with respect to the automorphisms of H~t~t ) defined by M(y)). The tensor product of the lql(y) actions on the factors defines an M(y) action on H.~V~(~,,_ .oM)| .~'). The map ~ in 8.4 is compatible with the M(y)-actions. One verifies from the definitions that a) the action of M(y) commutes with the operators A(w), A(~) on H.Z~t'(t~,, .Z~), .r and with A'(w), A'(~) on ... w ,, We have the following " vanishing " result. Proposition 8.6. a) H?d(~,, .~') = H~~ s = 0. = ..o0, ,, = -o0, x o. c) ~ in 8.4 is an isomorphism. d) _.HU~'}(-~,_ v, .Z;), H.M~'~(~y, s are finitely generated projective Hie~,)-modules. e) H~ e'(~r .~) = 0. Proof. --- a) is proved in [10, V, 24.8]. Now b), c), d) follow from a) in view of 7.2 b), c) and 7.4; e) follows from b) and the injectivity of [~ in 8.4. (See 1.9 a).) 8.7. H~) is the coordinate ring of an affine algebraic variety V whose points are the semisimple M~ on a) re(y) = Lie M(y) = Lie M~ = {(x, r0) ~ g | C [ [x,y] = 2roy }. (Let v e V. Iff e H~, we may regardfas a polynomial on the reductive quotient re(y),, invariant under M~ see 1.11 a). Thenf(v) e C is by definition the value of fat any point in the image of v under the canonical map re(y) -+ re(y),.) Then by 8.6 d), c), H.~"(#~, ~q~), H~'"(~'~, .oq~'), H~')(#~ x ~, -~ [] .o q~') may be regarded as spaces of sections of the algebraic vector bundles E, E', F = E | E' (respectively) over V. Thus, the fibre of E at v e V is E, = C_~| ) H.M~ ~) where C, is C regarded as an H~r~)-algebra via the homomorphism H~,~-+ C, f~f(o) as above. 196 GEORGE LUSZTIG Similarly, E; = (I, | H.~"(~,, s F, = E, |162 E;. By 8.3, the operators A(w), A(~) (resp. A'(w), A'(~)) on n.m~"(~,,.~) (resp. H.~*'(.~',, .~')) come from vector bundle maps A(w), A(~):E-+ E (resp. A'(w), A'(~):E'-~ E'), inducing the identity on the base V. Hence these operators act on each fibre E,, E~. By 8.5, ~l(y) acts on H~,,. This corresponds to an action of g-I(y) on V. (It is induced by the adjoint action of M(y) on its Lie algebra.) Moreover, E, E' are natu- rally ~I(y)-equivariant vector bundles over V and the operators A(w), A(~), A'(w), A'(~) on them are ~/(y)-invariant. 8.8. For each v e V we denote the stabilizer of v in M(y) by l~(y, v). (We shall also write IVI(y, 6, r0) instead of M(y, v) where (6, r0) is any element of the orbit v.) -- t Then M(y, v) acts naturally on the fibres E,, E,. Let rep(~l(y, v)) be a set of representatives for the isomorphism classes of irre- ducible M(y, v)-modules. For each 0 e rep(~l(y, v)) we define E,.c0 ~ (resp. ~.~0.,) to be the o-isotypical (resp. p'-isotypical) component of the ~I(y, v)-module E, (resp. E~), and we define a) = (p'| "', E,.0. = (p | From 8.7 it is clear that A(w), A(~) (resp. A'(w), A'(~)) act naturally on E,.~0,, E,.p (resp. on F~.{o. . E' ~, 0*) * 8.9. We have a) E o ~. H!*}(3~,, .~), E' 0 --+ H['}(O~ .~') (by 7.2 d)). Now the action of M(y) on ~'~, .Lfi, .~" induces an action of ~1(y) on H~(,~'~, .s H~(~,, s hence on H!~ .~'), H!'}(~,, -s it is easy to see that these are compa- tible under a) with the actions of ~l(y) on Eo, E' o considered before. Let: b) rep0 M(y, v) be the set of those p ~ rep ~I(y, v) which occur in the restriction of the M(y)-module H!'}(,~,, -~) to ~:I(y, v). Proposition 8.10. -- Let p e rep Kt(y, v). The following conditions are equivalent: a) E,. 0 =r 0; b) E~. o. . O; c) p E repo M(y, v). Proof. -- We restrict the vector bundles E, E' to the subset V' = { tv I t ~ c } of v. (If v is a semisimple M~ in re(y) and t ~ (I then t.v is again a semksimple MO(y)-orbit.) Now iVl(y, v) acts on these vector bundles (as identity on V'). Since V' is connccted and the representations of a finite group do not change by deformation, it follows that the M(.y, v) modules Eo, E o (resp. E',, E0) are isomorphic. Using 8.9 a) we are reduced to showing that the M(y, v)-module H!'}(~,,-~) is isomorphic to the CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 197 dual of the M(y, v)-module H!*)(.~v, .LP'). It is enough to show that the ~i(y)-module H~(~v, .LP) is isomorphic to the dual of the M(y)-module H~(~,, ~."). Since LP has finite monodromy, we can find a fiat, positive definite hermitian form on Lfi; we can assume, by averaging, that this form is invariant under the action of the maximal compact subgroup K of K,I(y). This form gives an isomorphism LP ~ s of local systems/R which is semilinear with respect to complex conjugation and is K-invariant. This induces an isomorphism H;(.~,, c.LP) -~ H;(~,, .LP') of R-vector spaces which is again semilinear with respect to complex conjugation. It is compatible with the action of K hence with that of ~I(y) (which is also the group of components of K). From this, the desired result follows immediately. Proposition 8.11. -- Let Y be a locally closed subvariety of gN, which is a union oJ nilpotent orbits. a) H~ 0.(~z, .~) = 0. b) The open embedding i:Y ~ Y (closure of Y) induces a surjective homomorphism i" : H?. � ze) H? � c) If Y is closed in gN, the closed embedding j : Y ~ g~ induces an injective homomorphism j., : H~215 r s --* H~215 r s Proof. -- a) is proved by induction on the number n(Y) ofnilpotent orbits contained in Y. Ifn(Y) = 1, we use 8.6 e). If n(Y) > 1, we write Y = Yt w Y2 where Y1 is closed in Y, Y, is Y- Yx and n(Yx)< n(Y), n(Y,)< n(Y). We write the long exact sequence 1.5 a) for the partition Y = ~r I w ~r 2. We may assume a) known for Ya, Y2 and we deduce that it is also true for Y. Now b) follows from the long exact sequence 1.5 a) for ~ = (Y -- Y)"" w ~r and from a) for Y, Y and Y -- Y; c) follows from the long exact sequence 1.5 a) for fi.~ = ~r w (gx -- Y)'" and from a) for g_~, Y and gs -- Y. Corollary 8.12. -- If Y is as in 8.1I then ~| I ~ A~.,), r l| !-~ A(w) (see 4.3) define an H-module structure on H ~ � e.(~, if,). Similarly, ~ | 1 -, A'(~), 1 | w ~ A'(w) define another H-module structure on H.G� cj~'). These two H-module structures commute with each other and hence define an H | H-module structure on H. ~ � r c~,). Proof. -- When Y = gN, this follows from 5.1. When Y is closed in gs, one uses the case Y = gs, together with 8.1 1 c) and 3.10. When Y is arbitrary, the corollary follows from the already known results for Y, using 8.11 b) and 3.10. Theorem 8.18. ~ Let y ~ ~) be as in 8.1. Then ~| 1 --~ A(~), 1 | w ~ A(w) (see 8.1, 8.2) define an H-module structure on H.m'u'(~y, LP). Proof. -- Let -- -- c, a(r( -- ~ocv)t~ .~) (1 ~< i~< m). as an operator on ~. ~= y, 198 GEORGE LUSZTIG We may also regard FI~ as an endomorphism of the vector bundle E ~ V (see 8.7) mapping each fibre into itself. We only have to prove that this endomorphism is zero. Consider the endomorphism II~| 1 of E| (see 8.7). Let ~ be a section of the vector bundle F which is M(y)-invariant. Then ~ is in the image of the homomorphism ~ in 8.4 (see 1.9 a)); now l-I~ is zero on the source of ~ by 8.12 for Y----~, and [~ is compatible with the operators A(w), A(~), hence (II~| 1) o ~ = 0 as a section of F. Now let v e V and letfbe an M(y, v) invariant element of the fibre Fo. For each v' in the M(y)-orbit of v, we define f,, ~ Fo, by f,, ---- y(f) where y : F, ~ Fo, is the action on F of an element y 9 M(y) such that yv = v'. (Then jr,, is independent of y by the M(y, v) invariance of v.) Since the M(y)-orbit of v is finite, we see from the Chinese Remainder Theorem that there exists a section ~l of F such that gl(v') =f,, for all v' in that M(y)-orbit. Let ~ = [M(y)[-1 ZVo~ 1 (sum over all y e M(y)). This is then an M(y)-invariant section, still satisfying a(v') ----f,, for all v' in the M(y)-orbit in v. In particular, it satisfies a(v) ----f, ----f. As (II,| 1) o g ---- 0, we must have also (l-I~ | 1) (f) ---- 0. Thus II~ | 1 is zero on F, ~'~ Consider the C-linear map : E,| E;| defined by ,1| r174 ~ r162 It is clearly compatible with the endomorphisms II~ | I | 1 of E, | F_~ | (E~)" and II~ of E,. Its restriction to the subspace (E,| E',)~("')| (E~)" is surjective. (This follows from the equivalence of a) and b) in 8.10.) But as we have seen above, II~ | 1 | 1 is zero on this subspace. It follows that II~ is zero on E~ Since v is arbitrary, l-I, : E -+ E is zero and the theorem is proved. 8.14. The H-module structure on H.~)(~y, .~) in 8.13 is defined by operators which are H~e~,71inear and commute with the action of M(y), hence it defines an H-module structure on each fibre E| (v 9 V) of E ~ V (see 8.7) and on each E,, 0 (p e rep0 M(y, v), see 8.9 b)). The H-modules E~ (p 9 M(y, v)) are called stan- dard. Let (,,r0) 9 We shall write sometimes Eo,,o,~,0 instead of E,. 0. Thus, the standard H-module Eo. ,o. ~, p is defined for any semisimple element ~ 9 g, any nilpotent elementy 9 g, any r 0 9 C such that [,,y] = 2roy and any p 9 rep0 M(y, ,, r0) (see 8.8, 8.9). Clearly, the isomorphism class of the H-module Eo.,o,v,o depends only on r 0 and the G-conjugacy class of ~, y, p. Theorem 8.15. -- Any simple H-module Mr is a quotient of a standard H-module. Proof. -- We can clearly find a non-zero H-linear map H ~ .~', where H is regarded as a left H-module in the obvious way. By 6.3 we can find a non-zero H-linear map CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 199 H.~� .i#) -+.If where H.~� .~) is regarded as an H-module as in the first sentence of 8.12. Let Y be a closed G x C" stable subvariety of g~ such that a) there exists a non-zero H-linear map ?:H. ~215 e'(~/', c.i#) --+.A' (for the H-module structure in the first sentence of 8.12) and b) Y is minimal subject to property a). (" Minimal " refers to the partial order given by inclusion.) Such Y exists since Y = g~ satisfies a) and the number of nilpotent orbits in g is finite. Consider a nilpotent orbit which is open in Y. We may assume that it is 9 of 8.1. We have an exact sequence 0 -+ H. ~ � ~'((Y -- O)"', .i~) _+v H. ~ � r .i~) --+ H. ~ � ~'(0, .ifi) -+ 0 (by 1.5 a) and 8.11 a) applied to Y, Y -- d~, d~). It is an exact sequence of H-modules (8.12, 3.10). By the minimality of Y, q~ must be zero on the image ofy hence it factors through a non-zero H-linear map H.G~e'(O, LP) --+..gr Using 8.6 a) and the fact (1.9 a)) that [~ in 8.4 is an isomorphism onto the M(y)-invariants, we deduce that there exists a non-zero H-linear map (A') i(') -+.~' where A' (see 8.4) is regarded as an H-module using the operators 8.13 on the first factor and the identity on the second factor of the tensor product. Composing this with the averaging map A' --+ A '~') (a surjective H-linear map) we find a non-zero H-linear map A' --+.~(t'. It follows that there exists a non-zero H-linear map ~:~g' --+~r where = H.w~"(~,, .s A well-known argument of Dixmier (applicable since H has countable dimension over C) shows that the centre of H acts on .~r by scalar operators. Hence, by 6.5, there exists a maximal ideal I of S w such that I acts as zero on .At'. Let v:H~� c. --+ H;e~,)be the homomorphism induced (1.4 g)) by the inclusion M~ ,-+ G � C" and let ~': H~� r S w be the (surjective) homomorphism defined by 4.3 c). Then I' = v'-1(I) is a maximal ideal of H i � r From the definitions, it is easy to see that, for h e H~� the action of ,~(h) on ~t ~ (by the H~t~vl-module structure) coincides with the action ofv'(h) (by the H-module structure). It follows that ~(I" .~g') = 0 where I" is the ideal of H~,) generated by ~(I'). Now v corresponds to a finite morphism between the corresponding atrme varieties (a semisimple orbit of G x C" on g ~ C intersects re(y) in a union of finitely many orbits of M~ Hence H~,) is integral over the image of v and I" has finite C-codi- mension in H~t~, ). Note also that I" is a proper ideal (otherwise it would follow that ~ = 0). Since H~e~,)/I" is an artinian C-algebra ~ 0, there exist maximal ideals J,,J2, .-.,J, of H~,(v ) (s~> 1) and integers n~>~ 1, n2~> 1, ...,n,>_. 1 such that I"CJ~, .. I"cJ;', and the natural map HmtJI --+ ~ H~,,~/J, is an algebra isomorphism. It follows that ~[I"~--% O ~/j,~iCg' hence there exists i (1 ~< i ~< s) such 1~<i~<. that ~ defines a non-zero H-linear map ,~[J~ ~ --~,~. Let 0~"= .~/J"a~g ' (J = J~, n = n,). Then ~g" is finite dimensional over C, is 200 GEORGE LUSZTIG a module over the local algebra H~,JJ" with maximal ideal J/J", is an H-module and there exists a non-zero H-linear map .~' ~..~r By a simple lemma [5, 5.14] it follows that there exists a non-zero H-linear map .~']J.,~' = occ'[J&~' --+ ..~'. Hence with the notation of 8.7, there exists a non-zero H-linear map Eo ~.,~' for some v ~ V. Since E, is a direct sum of standard H-modules, the theorem follows. 8.16. As in 8.7, we regard ~ x o* as the coordinate ring of an affine variety U whose points are the semisimple G � C'-orbits in ~ | C. If ~ is a finitely generated H~ � e.-module we define the support of.~ to be the set of all u e U such that the loca- lization of.A' at the maximal ideal of H G � c. corresponding to u is non-zero. This is a closed subvariety of U. From the two descriptions of A'" in 8.4, we see that the support of H. ~ � c.(~, .L~) is contained in the set of semisimple G � C'-orbits on g | C which meet re(y). Using this, and the exact sequences 1.5 a) we see that: a) if Y is as in 8.11 andyl, ...,y, is a set of representatives for the G-orbits in Y, then the support of the Ho� H. ~ ~) is contained in the set of semi- simple G � C'-orbits on ~| which meet re(y1) ~3 re(y2) u... u m(yo). Theorem 8.17. -- Let (r r0) ~ g | C be a semisimple element such that r o + O. Then a) Zo(o ) acts (by the adjoint action) on the vector space { x e g [ [0, x] = 2r 0 x } with finitely many orbits. This vector space consists of nilpotent elements. b) Let y be an dement in the unique open orbit of the action in a). Then (0, to) E re(y). Let p ~ rep0 M(y, o, r0). Then the standard H-module Eo. ,o,,. p (see 8.14) /s simple. c) Let o', ro,y', O' be another set of data satisfying the same assumptions as o, to,y, 0 above, and assume that the standard H-module Eo,, ,~. v'. p' is isomorphic to Eo, ,o,,. P" Then r o = r o and there exists g ~ G which conjugates (o",y', p') to (o,y, p). In preparation for the proot, we state the following elementary result. /_.emma 8.18. -- Let A be an algebra over C with an involutive anti-automorphism a ~ ~, let E,, E~ be finite C-dimert~ional A-modules (1 <~ i <~ p). We regard I~ = ~ (E, G 0 E~) as an A | A-module in a natural way. Assume that there exists r ~ E such that a) (a|162 = (l | ~) r for all a ~ A, b) a ~ (a | 1) , is a surjective map A -+ E. Then E, (1 <~ i <~ p) are simple, mutually non-isomorphic A-modules. The proof is left to the reader. 8.19. Proof of 8.17. -- The second statement of 8.17 a) is obvious. The first statement of 8.17 a) is a consequence of [5, 5.4 c)] and the finiteness of the number of nilpotent orbits in g. We now prove 8.17 b). As in 8.1, we denote by 0 the G-orbit ofy in ft. Let ~ be the union of all nilpotent G-orbits in fl which contain 9 in their closure. Then O is closed in CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I 201 and O is open in g~. From our assumption it follows that the vector space in 8.17 a) has empty intersection with ~- r Hence, by 8.16 a), a) the support of the H~� H.G~c'((~- 0)",-~;) does not contain the G � C'-orbit of (~, r0) in g @ C. Using 1.5 a) for the partition (d3)"" = 0 u (~ -- 0)'" and 8.11 a) we obtain an exact sequence of H | H-modules 0 ~ H?. � ~ .~.") ~ H.~215 c'((r "', .i ;~) --+ H.~ � c'((O -- 0)", .i; ~) -+0 regarded as a H i � c.-algebra via the homomorphism H~ � c. --+ C defined by evaluation of an element of S(g@ C) at (o, r0). We tensor the previous exact sequence with Co. , over H i� r Using a), we see that we obtain an isomorphism: Oxr "" b) co,,0 | H. (0, Co,,0 H. ~ � From 8.11 b), we have a surjectioe H| map r Ha � c.(~s, .i;,~) ,_+ H.O � c.((~)"', ~"). Tensoring the algebra homomorphism H~� c. --+Co,,0 with I-I?.� gives a surjective H | H-linear map d) H. ~ � c.((~)"', .~) __+ Co ` ,0 |215 H~ � c.((~)"', .!;~). The composition of c), d) and the inverse of b) is a surjective H | H-linear map e) H? � c.(.g~, ..~) __.,. Co. ,o | H? � r .~). Using 8.4, 8.6 a) and 1.9 a) we have an isomorphism of H | H-modules f) H?� c'(~, .~) ~_ (A,)R,,,. (A' as in 8.4; we regard A' as an HI| using the H-module structure on H~e~"(O~v, .oM) in 8.13 and the analogous H-module structure on H.W~"(~,, .i~~ Let v be the M~ of (~, r0) in re(y). Taking the value of a section of F at v, defines an H | H-linear A' --,'-F,. This restricts to a surjective H | H-linear map g) (A') R(" ~ V~t"" (see the proof of 8.13). We have n) = (w | = | (t,,0 | E',, 0.) where p runs over rep0(M(y, v)) (see 8.10). Composing f), g), h) we obtain a surjective H| H-linear map i) H?� .~) --+ ~ (E,. , | E;, r The maximal ideal I of H~ � c. corresponding to (~, r0) clearly acts as 0 on the right hand side of i) (via H~ � c. '+ 8w C H | C C H | H) hence i) factors through a surjective H | H-linear map J) Co, ,~ |215 H.~ x c.(~, .g~) ~ 0 (E.,, | E;, r 26 202 GEORGE LUSZTIG Composing e) and j) gives a surjective H | H-linear map k) 14Q� .~) -+ 0 (E,. | Using 8.4 we see that the assumptions of 8.18 are satisfied for i~ the right hand side of k), A = H, r = image of 1 under the map k). We conclude that the H-modules E,, are simple and the H-modules Eo. ~, E,, p, are isomorphic if and only if i~ : P'. This proves 8.17 b). We now prove 8.17 c). The G x C'-orbit of (~, r0) e g | C, or equivalently, the corresponding maximal ideal I of H o � r can be reconstructed from the H-module E~ 0; on~ indeed, I is the set of elements of H~ � c. which act as 0 on E,, p, via H~ � r S w ~ H. Hence (or, ro) and (a', r'0) in c) are G X C'-conjugate. We can thus assume that (r r0) = (a', r0). Now y,y' are in the same Zo(cr)-orbit , by assumption. Hence we can assume y' ----y. But then p ---- ~' by an earlier part of the proof. The theorem is proved. REFERENCES [1] A. A. BEILINSON, J. BZ~a~Sa'EXN, P. DELIGNE, Faisceaux pervcrs, in " Analyse et topologie sur Its espaces singu- liers", AstErisque, 100 (1982), 5-17, Socidt4 Math4matique de France. [2] A. BORZL et al., Seminar on transformation groups, Ann. of Math. Studies, 46 (1960). [3] V. GINZSURO, Lagrangian construction for representations of Hecke algebras, Adv. in Math., I)3 (1987), 100-112. [4] M. Go~xY, R. M_~cPrtsRSON, Intersection homology II, Invent. Math., 79. (1983), 77-129. [5] D. KAZHDAN, G. LUSZTIG, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math., 87 (1987), 153-215. [6] G. LuszTlo, Coxeter orbits and elgenspaces of Frobeniu.~, Invent. Math., 9.8 (1976), 101-159. [7] G. LtrszTIO, Representations of finite Chevalley groups, C.B.M.S. Regional Conference series in Math., 3g, Amer. Math. Soc., 1978. [8] G. Luszvlo, Green polynomials and singularities of urtlpotent classes, Adv. in Math., 42 (1981), 169-178. [9] G. LUSZTIG, Intersection cohomology complexes on a reductive group, Invent. Math., 75 (1984), 205-272. [I0] G. Lusz~rzo, Characters sheaves, I-V, Adv. in Math., 50 (1985), 193-237; 57 (1985), 226-265, 266-315; 5g (1986), 1-63; 61 (1986), 103-155. [11] G. LuszTzo, Fourier transforms on a semisimple Lie algebra over Fq, in " Algebraic Groups--Utrecht 1986 ", Lecture Note~ in Mathematics, 1271, Springer, 1987, 177-188. [12] G. Luszxxo, N. SvAt.-rm, rsreXN, Induced unipotent classes, J. London Math. Sot., lg (1979), 41-52. Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Manuscrit re(u le 16 mai 1988.
Publications mathématiques de l'IHÉS – Springer Journals
Published: Aug 31, 2007
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