Transformation Groups, Vol. 1, Nos.1 & 2,1996, pp. 83-97 (~)Birkh~iuser Boston (1996)

A F F I N E W E Y L G R O U P S A N D C O N J U G A C Y CLASSES IN W E Y L G R O U P S

GEORGE LUSZTIG *

Department of Mathematics M. I. T.

Cambridge, MA 02139, USA

gyuri~math.mit .edu

A b s t r a c t . We define a map from an affine Weyl group to the set of conjugacy

classes of an ordinary Weyl group.

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

0.1. Let G be a semisimple, simply connected algebraic group over C whose Lie algebra g is simple. Let W be the Weyl group of G. Steinberg [St] (and independently, Spaltenstein and Cross) defined a map a : W ---* 11 (the set of nilpotent G-orbits in g) as follows. Let w 6 W. Choose Borel subalgebras /3,/3' of g such that (/3,/3') are in relative position w. Let n~, nZ, be the nil- radicals of/3,/3'. Since 11 is a finite set and nz M n~, is an irreducible variety consisting of nilpotent elements, there is a unique nilpotent G-orbit C on g such that C n n~ M n~, is open dense in n n N n~,. By definition, a(w) = C.

For any nilpotent G-orbit C in g, the fiber a - z ( C ) is the image of a map

(a) D c x D c ~ W

which is defined as follows. D c is the set of irreducible components of the variety of Borel subalgebras of g that contain a fixed element x E C; the map (a) associates to (X, X ' ) E D e x D c the relative position (in W) of a generic point of X with a generic point of X ~.

The fibers of a form a partition of W into nonempty subsets which we call the S-cells of W. (The letter S stands for Steinberg, Spaltenstein and Springer; the partition of W into S-cells is closely related to Springer's representations of W.)

0.2. The partition of W into S-cells is somewhat similar to the known

partition of W into (two-sided) cells defined in terms of Hecke algebras or perverse sheaves. (The two pictures coincide in type A but diverge for other

* Supported in part by the National Science Foundation. Received February 6, 1996. Accepted April 30, 1996.

Typeset by .AA/IS-TEX

84 G. LUSZTIG

types.) Since the Hecke algebra type cells have a meaning also for affine Weyl groups, one can ask whether there is a reasonable generalization of the theory of S-cells to the case of the affine Weyl group Wa (attached to W).

The purpose of the present paper is to propose such a generalization. In the same way as the set of special nilpotent orbits (which parametrizes

the usual two-sided cells of W) should be replaced in the affine case by the larger set 11 of all nilpotent orbits, we can expect that the set H (which parametrizes the S-cells of W) should be replaced in the affine case by a lar- ger set. Such a larger set is provided by W., the set of conjugacy classes in W.

Note that, by [KL2, w there is a canonical map H --* W. which is bijective in type A [loc. cit.] and injective, at least for classical types [S1], and probably in general [$3]. Our hope is then that the S-cells of Wa should be parametrized by W.. In Section 1 we will indeed construct a map W~ --. W. and we call its fibers the S-cells of W~. Various experiments suggest that these S-cells behave very nicely, although much remains to be verified.

An alternative definition of cells in W~ (conjecturally the same as that of S-cells) could be given by developing an affine analogue of the theory of Springer representations. This will be done elsewhere; however, in Section 5 we will give a definition of a natural representation of W~ on the homology of the space of Iwahori subaigebras containing a given element of a loop algebra.

0.3. N o t a t i o n . Let F = C((e)) where e is an indeterminate; then the field ~' = t_Jn>lC((el/n)) is an algebraic closure of F. Let v :/~ --* Q u co be the valuation defined by v(aoe m/~ +higher powers of e) = m / n for a0 �9 C* and v(0) -- oe. Let A-- C[[e]] C F. Let A ' = C[e -1] C F. We set G = G ( F ) . Let

0A = A | g, ~A' : A' | g, g = F | 9, 0,~ -- F | g-

Then g is a semisimple Lie algebra over F, I~A is a Lie algebra over A, 0A' is a Lie algebra over A' and gp is a semisimple Lie algebra over F. Let P : ~A " - 4 ~ be the projection induced by A --* C (evaluation at e = 0). Let P' : gA, -~ g be the projection induced by A --+ C (evaluation at e -1 -- 0).

Let Ad denote the adjoint action of G on g. Let 2u be the number of roots of g. Let rk(f 0 be the rank of 0-

An Iwahori (resp. anti-Iwa.hori) subalgebra is, by definition, a C-subspace of g which is in the Ad(G)-orbit of a subspace of the form p-l(/~) (resp. p,-l(/~)) where/~ is a Borel subalgebra of g. Let B (resp. B') be the set of Iwahori (resp. anti-Iwahori) subalgebras of g; note that G acts transitively by the A d action on B (resp. on B').

If b �9 B (resp. b �9 B ~) we define nb C b as follows. We choose g �9 G such that b -- A d ( g ) ( p - l ( f l ) ) (resp. b = Ad(g ) (p ' - l ( /~ ) ) ) where 15 is as above,

AFFINE WEYL GROUPS AND CONJUGACY CLASSES 85

and we set n~ = Ad(g)(p-l(n~)) (resp. nb -~ Ad(g)(p'-l(n~))) where nf~ is the nilpotent radical of ft.

A Cartan subalgebra of g is, by definition, an F-Lie subalgebra h of g such that _P | h is a Cartan subalgebra of the ~'-Lie algebra gp. The set of Cartan subalgebras of g falls into finitely many orbits under the Ad-action of G. The set of orbits is naturally in 1-1 correspondence with the set W. of conjugacy classes in W. (See [KL2,w

Let A : g - . F be the polynomial function given by A(x) = tr(A2~ad(x)). An element x E g is regular semisimple (or RS) if its centralizer in g is a Cartan subalgebra or, equivalently, if A(x) ~ 0. To each RS element x E g, we associate a conjugacy class 7 of W, namely the conjugacy class that corresponds to the centralizer of x in g. We then say that x is of type 7-

0.4. I wish to thank Kirsten Bremke for her help with executing the draw- ings.

1. T h e m a p ~- : Wa --, W.

1,1. Let W~ be the set of all G-orbits on B x B (with G acting simulta- neously on the two factors by the Ad-action). Then Wa may be regarded in the well known way as a group. In fact it is an affine Weyl group whose length function l : W~ -* N attaches to the G-orbit of (b, b') E B x B the number

codimb(bNb') = codimv(bMb') -= codimnb(n~Mn~) = codimnb, (n~Mn~) e IN.

Here codimensions are relative to C. The unit element 1 of W is the orbit of the diagonal in B x B. We denote by {s~[i E I} the set of elements of length 1 in W~.

Let W be the set of G-orbits on the set of ordered pairs of Borel subal- gebras of g. Again, W may be regarded in the well known way as a group. In fact it is a (finite) Weyl group. We may regard W as a subgroup of W~: to the G-orbit of a pair (fl, fl') of Borel subalgebras of g, we associate the G-orbit of ~-l(fl) ,p-!(fl ')) in B x B.

1.2. Let V be a finite dimensional F-vector space. A lattice in V is an A-submodule of V generated by a basis of V. A t-lattice of V is a subset of V of the form x + L where x E V and L is a lattice in V.

Let X be a t-lattice in V. If L is a lattice in V such that L+X-=X, then we may consider the space of orbits X/L of the L-action on X given by translation. Then X/L is a finite dimensional affine space over C. A subset Y c X is said to be open in X if there exists a lattice L in V such that L + X = X and such that Y is the inverse image of a Zariski open set of X/L under the canonical map X --~ X/L. (This agrees with a definition in [KL2,w

86 G. LUSZTIG

1.3. We shall need the following result ([KL2, Prop. 6.1]).

I f X is a t-lattice in g then there is a unique "yEW. (with W. as in 0:2)

such that some nonempty open subset of X consists of RS elements of type 7.

1.4. Let w E Wa. We choose (b, b') in the G-orbit on B x B indexed by

w. Since rib,rib, are lattices in g, the intersection nb n nb, is a lattice in g. By 1.3, there is a unique ~/E W* such that some nonempty open subset of

nb n nb, consists of RS elements of type 7- Note tha t 7 depends only on w;

the correspondence w ~-~ 7 is a map zr : Wa --+ W.. The sets lr-1(~/) for various 7 E W. are called the S-cells of Wa. For example, if g is s[2, then ~r-1(1) -- W~ - {1} and the fiber of ~r over the element ~ 1 of W. is {1}.

If it is ~[3, then the S-cells of Wa coincide with the usual two-sided cells of

Wa, which are described by a picture at the end of [L3].

Assume now that g = sp4.

.~

: : oO"

~;;]~" 7 = ;."i ..... ,t

U.... ~176176176

i . . . .

A ..: �9 ..-",s i A ~

. , . . . -

* * *-. ,.o-

.... .i,..:.i...i. . . . . ;:.... - " : ' " . A : A :

" -~ f / / I I : " " mm: mn

/ i - �9 = / i " -

~. . . - ... ."~.

0 : 0

;:.:i.-" ~ .:-7-:.~ .."A i ,r

y. "'.. �9

mm'...

' . . o o ~ cs... ~ * i

,a. o . : ..-"mm i

A .." ! .-"a i

i "~ i

" . A

A ~ ~ �9 ~

"'. ~ i ~ . . " ' " . . . . . i . . .

- .'~

A - " i . A �9 .-":A i A'"......-'"

A F F I N E W E Y L G R O U P S AND C O N J U G A C Y CLASSES 87

We identify W~ in the usual way with the set of alcoves in the following picture, so that the unit element corresponds to the alcove labelled by x. The S-cells are the sets of alcoves with a fixed label. Thus the alcove labelled with x is Ir-1(~/) where q, is the class of the Coxeter element of W.

The 10 alcoves labelled with O form 7r -1 (7) where 7 is the longest element of W (a conjugacy class by itself). The alcoves labelled with ~- form ~r-l(7) where ~/is the class of one of the simple reflections of W.

The alcoves labelled with [] form ~r-l(~/) where 7 is the class of the other simple reflections of W. The alcoves labelled with A form 7r-a('y) where -y i s l .

The union of all (closed) alcoves with a fixed label is a union of con- nected components which are bounded by the thick lines. These connected components axe the analogues of left cells in this case.

This picture is very similar (although not identical) to the picture of usual cells of affine B2 at the end of [L3].

We return to the general case.

P r o p o s i t i o n 1.5. The following diagram is commutative

W a ~ ~j

1 "1 W.

Here a is as in 0.I and the left vertical map is as in 1.1; T is as in [KL2, w

Proof. Let w E W. Let (/3,/~1) be a pair of Borel subalgebras of g in the G- orbit defined by w. Let b = p-l(f~), D' = p-l(j3 ') . Let/5 :nb Mn~, --* n~ Mn~,

be the restriction of the canonical map p : 9A --* g- The kernel of 15 is egA. We can find a nonempty open subset U of nb M nb, such that any y E U is RS

of a fixed type 7 E W.. Let C = a(w) E 11. We must show that T(C) = 7.

It is clear that t5 maps open sets in nbnnb, to Zariski open sets in n~ nn~,. Thus, iS(U) is a (nonempty) Zariski open set in n~ n n~,. By the definition of a(w), we see that there exists x e 15(V) n C. Now p - l ( x ) n U = ~ - l ( x ) M U

is a nonempty open subset of the t-lattice p - l ( x ) . By the definition of T, there exists x' E p - l ( x ) n U such that x' is RS of type r (C) . Since x' E U,

x' is ItS of type 7. Hence ~-(C) = 7.

A similar commutative diagram holds when 11 is replaced by the set of nilpotent classes in the reductive quotient of a maximal paxahoric subalge- bra, W by the Weyl group of that reductive quotient and ~- by the map in [KL2, 9.13].

88 G. LUSZTIG

1.6. The map r has been described in some cases in [KL2, 9.8, 9.12], and in almost all remaining cases in [S1], [$2], [$3]. Using in particular [KL2, 9.12] together with 1.5, we can deduce the following examples.

lr(1) = class of Coxeter element of W; ~r(si) -- class of element of order 24 of W (type Es); Ir(sisj) = class of element of order 20 of W (type Es); here i ~ j and

sisj = s~s~.

2. T h e se t g~

2.1. We say that an element x E g is topologically nilpotent (or TN) if ad(x) ~ --* 0 in E n d f ( g ) for r --~ cx~, or equivalently, if x E nb for some b E B. (The equivalence of these two conditions follows from [KL2, Lemma

2.1(a)1.) Given 7 E W., let gx be the set of elements x E g that are TN and

RS of type 7. This set is nonempty since, if y E g is RS of type 7, then eny E gx for sufficiently large n. Let v(7) be the minimum value of the function x ~-* v ( A ( x ) ) on g~ and let

g:, = { x e g lv(a(x)) =

The values of v(7) have been computed explicitly by SpaJtenstein [$3] for all "r in the ease of classical Lie algebras and for many "r in the ease of exceptional Lie algebras.

2.2. If 7 E W. we denote by def(7) the number of eigenvalues distinct from 1 of w (a representative of "y) in the reflection representation of W. For any x E g we set B~ = {b E BIx E b}. This is a union of an increasing sequence of projective C-subvarieties of B.

Assume now that x is RS of type "r and TN; then B~ is nonempty and a locally finite union of ordinary irreducible projective algebraic varieties over C, all of the same dimension (see [KL2,w Moreover, by [KL2] and [B] we have

(a) dim(B,) = ( v (A(x ) ) -- def(7))/2.

Let Dx be the set of irreducible components of Bx (a possibly infinite set). Given X, X ' in Dx, we can define an element (X : X t) E Wa by the following condition: there exists non-empty open subsets U, U' of X, X ~ such that for any b E U and any b' E U', the pair (b, b') belongs to (X : X ' ) . Thus we have a map

(b) Dx x Dx ~ W

given by X, Xt ~-* (X : X') .

AFFINE WEYL GROUPS AND CONJUGACY CLASSES 8g

C o n j e c t u r e 2.3. I f x E g~, then the image of the map 2.2(b) /s exactly

Thus, D~ is (conjecturally) the affine analogue of the set D c in 0.1(a).

2.4. From 2.3 it would follow that 1r-i(7) is nonempty for any 3". It would also follow that 7r-i(3') is finite whenever 7 is the class of an element of W which has no eigenvalue 1 in the reflection representation of W.

3. T h e se t n~

3.1. We fix b E B; let n = nb. A subset X of n is said to be locally closed if there exists k > 1 and a locally closed subset Xk of the finite dimensional C-vector space n/ekn such that X is the inverse image of Xk under the canonical map q~ : n --~ n/ekn. Then the irreducible components of X can be defined as the inverse images under qk of the irreducible components of Xk. (They are independent of the choice of k). For each irreducible component of X we may define its codimension in n as the codimension in n/ekrt of the corresponding irreducible component of Xk. (This is again independent of the choice of k.)

* * This is clearly a nonempty set. For 3' E W. we set n~ = n f] g~. The following conjecture is probably necessary for a proof of Conjecture

2.3. Its classical (nonaffine) analogue is known [$4].

C o n j e c t u r e 3.2. n~ is a locally closed subset of n such that each irreducible component of n~ has codimension (v(3") - def(7))/2 in n.

By 2.2(a), the quantity (v(7) - clef(3"))/2 is equal to dimBx for x E g ; . Let us illustrate the conjecture in the simplest case, g = st2. We may assume that n consists of all matrices

ea b ) e c - - e a '

where a, b, c E A. We have

a = a 0 modeA, b = b 0 + e b i mode2A, c = c 0 + e c i mode2A,

where a0, b0, bi, co, ci are in (3. If 3' is the class of the nontrivial element of W, the set n~ is defined by the conditions v(e2a + ebc) = 1 or equivalently by the conditions b0co r 0, so n~ is open (hence of codimension 0 = v(3")).

If 3' is the class of 1, the set n~ is defined by the conditions v(e2a+ebc) = 2 or, equivalently, by the conditions

b0c0 = 0, boCl q- bl co + a 2 ~ 0,

90 G. LUSZTIG

so n~ is locally closed with two irreducible components of codimension 1 --

v(9"). In the general case, it is easy to see that, when 9" is the class of the Coxeter

element, the n~ is open in n (hence it is irreducible of codimension 0) and we can expect that, for 9" = 1, the number of irreducible components of n7

is the order of W.

C o n j e c t u r e 3.3. Let w E 7r-1(~/) and let (b, b') E 13 x ~ be in the G- orbit defined by w. Let 6(w) E N be the minimum value of the function x ~-* v (A(x) ) on nb N rib,. We have 6(w) = v(7).

Note that the inequality

(a) > v(9")

is trivially satisfied. The conjecture above implies in particular that the function 6 : Wa --+ N

is bounded above. We can verify this fact directly. Namely, we will show that ~f(w) < 2u.

Let (b, b ~) be as above. We may assume that b M b ~ contains some Cartan subalgebra [j of g. Then n~ M nb, contains e[j and clearly, v(A(x) ) = 2v for all x in a nonempty open subset of el). Thus, v(A(x) ) ~- 2v for some x E nb N nb,, so that ~f(w) .< 2~,, as asserted. Combining this with (a) and with the equality v(1) -- 2~, (which is easily verified) we see that the

conjecture above is true for 9' = 1.

4. S-cells

4.1. In this section, we propose a second generalization of the picture in 0.1 to the affine case. (This is closely related to a construction in [L1].)

Let W~ be the set of all G-orbits on B x B ~ (with G acting simultaneously on the two factors by the Ad-action). Then W~ may be identified in the well known way with Wa (as a set) in such a way that the orbit of (b, b') E B • B' corresponds to w E Wa with l(w) = dimc(nb N nb,).

The following result was stated without proof in [L1].

Proposition 4.2. For any (b, b ~) E 13 • 13', the intersection nb Mnb, consists of nilpotent elements of gp.

Proof. We argue by induction on I = dimc(nb Mnb,). When I = 0, the result is obvious.

Assume that l > 1. First note that nbMne, is a direct sum of 1-dimensional

C-subspaces each of which is contained in a root subspace of I~p with respect to a Cartan subalgebra. Hence these one dimensional subspaces consist of nilpotent elements in g~.

AFFINE WEYL GROUPS AND CONJUGACY CLASSES 91

We can find f~ E B ~ such that nbNn~ is a codimension I subspace of nbnn~, and n~ M n~, is a codimension 1 subspace of n~ and of rib,. The induction

hypothesis is applicable to (b, b); it shows that, for any x E u~ N n~, some power of ad(x) : i~$" --* gR is zero; hence some power of ad(x) : na M n~ ---, nb M n~ is zero. By Engel's theorem, the C-Lie algebra nb ;1 ni is nilpotent. Let r be the radical of the C-Lie algebra nb n rib,. The image of nb N ni

in Cab N n~,)/r is a nilpotent Lie subalgebra of (nb n nb,)/r of codimension 1. This forces (rib M nb,)/r (a semisimple C-Lie algebra) to be 0. Thus,

nb ;1 rib, ---- r so that nb A rib, is a solvable C-Lie algebra. Using a corollary of Lie's theorem, we see that the set of all x E nb M rib, such that some power of ad(x) : nb N n b, --~ n~ N nb, is zero, is a vector subspace V of nb M nb, over C. As we have seen earlier, the C-vector space is spanned by elements that are nilpotent in gp. It follows that V = nb M nb,. Using again Engel's theorem, we see that nb M rib, is a nilpotent C-Lie algebra. Then, clearly, the/o-subspace of 95. spanned by n~ g) n~, is a nilpotent/o-subalgebra. Thus any element of this subalgebra (and in particular any element of nb M rib, is a nilpotent element of 9R. (Since the/0-Lie algebra {~ is semisimple, any nilpotent/o-Lie subalgebra of it consists of nilpotent elements of 9~.) The proposition is proved.

4.3. Let Gad be the adjoint group of G and let Gad = Gad(F). Let l~l be the set of orbits of Gad on the set of nilpotent elements of g. The set ~[ is finite since the number of nilpotent orbits of Gad(/o) on gp is finite and the set of F-rational points of any of these orbits decomposes into a collection of Gad-orbits in natural bijection with the (finite) set of conjugacy classes of the group of components of the centralizer in Gad(/o) of a representative of the orbit. (This is shown by an argument similar to that in [KL2, w

It follows that, if V is a finite dimensional C-subspace of g consisting of nilpotent elements in g, then there is a unique Gad-orbit C E 1~ such that C n V contains a nonempty (Zariski) open subset of C M V. (This is seen by an argument of the same type, as the one in the proof of [KL2, Prop. 8.1].)

4.4. Let w E Wa. We choose (b, b') in the G-orbit on B x B' indexed by w.

Since nbAnb, is a finite dimensional C-subspace of g consisting of nilpotent elements in g (see 4.1, 4.2) we may use 4.3 and we see that there is a unique C E ~ such that C N V contains a nonempty (Zariski) open subset of C N V. Note that C depends only on w; the correspondence w ~-~ C is a map ~ : Wa -- , l] .

The sets ~- I (C) for various C E 11 are called the S-cells of Wa. For

example, if g is s[2, then ~- l ( regular nilpotent orbit) = Wa - {1} and ~-1({0}) = {1}. Thus, in this case, the S-cells are the same as the S-cells. The same holds for s[3.

92 G. LUSZTIG

Assume now that g = 5p4. We identify W~ in the usual way with the set of alcoves in the following picture, so that the unit element corresponds to the alcove labelled by x. The S-cells are the sets of alcoves with a fixed label. Thus the alcove labelled with x is 7r -1 ({0}).

The 2 alcoves labelled with O form 7r-1(C) where d is the orbit of transvections.

The alcoves labelled with -k form lr-l(C) where C is the Gad-orbit of subregular nilpotent elements in g which meets 9-

The alcoves labelled with �9 form 7r-1(C) where C is the G~d-orbit of subregular nilpotent elements in g which does not meet 9-

The alcoves labelled with A form 7r-1(C) where C is the Gaa-orbit of regular nilpotent elements.

.... i.. ~..-" ~ " - . . . - " --...i...- i " . . .~ i

I * " - - ! :.::-~ii-.::- ~ ~ n . : - - - : .~i ~/i--:: I i,~::":.:: :.--i...~.~.:::,.i..,N ' * ;":Z.,.!-.,:::/.~.i...: �9 . : . . 'A ! A i ... I ~ ' . i , ! A ! A'-. ! .. �9 .. ! ..- ~ ! ~ . . - - I I . . . . . . : . ~ / . * _ ~ . - " i " . . . i . . . - !

i A.-"i"-.. A i . . i i . . - "A i A".. i

�9 . '" " - i ~ " i " ; g'-.. i ~ .... "

* * i ~ i ** Z~ J " : ' ~ ON A �9 . - ' : \

A

�9 " I i

. . i I . . - l \ i i i ...

,-..j ...--, i .~ \ . . . , : " - " : " / 1 " *1 " " inn"-, ] ~ A n ."-. I . " A

...... .... i ....

... A i A . - " i \ n

�9 i l k - i : ! . . . . . . . ~ . . . . i..:-,i

" . . . . . . . .! '-. . . . . . . - . i "

",%~

. . ."~i ~-.. i . . .

AFFINE WEYL GROUPS AND CONJUGACY CLASSES 93

Again, this picture is very similar (although not identical) to the picture of usual cells of affine B2 at the end of [L3].

5. Affine Springer representations

5.1. Let x E g be an element which is both RS and TN. The idea tha t Wa should act naturally on the homology of B~ (generalizing the Springer representations of ordinary Weyl groups) was formulated in [KL2, p. 141, 142], where the action of the simple reflections was considered following the method of [KL1]. But with that method there are difficulties in checking tha t the relations of Wa are satisfied so tha t we actually have representa- tions of Wa. (For the purposes of [KL21 it was not necessary to have those relations.) Here we will define the W~-action on the homology of B~ by using, instead of the method of [KL1], the method of [L2] (see also [L4]), which is based on the theory of perverse sheaves. The present method is in fact applicable to any x E g such that B~ r O.

5.2. In the remainder of this paper, all algebraic varieties are over C. Let __G be a connected reductive algebraic group with Lie algebra _~. Let __W be the Weyl group of _G. L e t / ~ c be the variety of all Borel subalgebras of _0 and let ~ = {(x, fl) E g_ x BaG_ [ x E ~}. Note that G acts on g by adjoint action and on ~ by adjoint action on the first factor and conjugation on the second factor; clearly the first projection Pl : ~ --+ g_ is compatible with the G actions.

Let K e :D(_g) (bounded derived category of constructible C-sheaves) be the direct image of the constant sheaf C under Pl- By [L4, 7.12], K[dimg] is a perverse sheaf on g. This perverse sheaf is clearly __G-equivariant (for the adjoint action of G on It) hence, by [IA, 1.16], we may regard K as an object of 79_G(_g). (79c_ denotes the equivariant derived category in the sense of Bernstein-Lunts [BL].) By [L2], [L4, 7.14, 1.16(a)], we have canonically

(a) Endg~(~) (K) = End~(~) (K) = C[W].

5.3. Let X be an algebraic variety and let E --* X be an algebraic lo- cally trivial principal G--bundle. Using the G--actions on ~, g and on BG_ (by conjugation), we can form the associated bundles

C \ ( E • x, _G\(E • _g) - , x.

The projection Pl is compatible with the G-action hence induces a morphism

p~: _GG\(E • _g) --, __G\(E x g).

94 G. LUSZTIG

Assume that we are given a morphism j : X --* _G\(E x -9) such that the

following composition is the identity map: X 3_~ __Gk(E x _g) ---* X (the second map is induced by E --* X). We have a diagram

X ' , _C\(E• , E • , _g

u q X J , G_\(Ex_8) , Ex_9 , -9

where X' is defined by the condition that the left diagram is cartesian; the other diagrams are also cartesian. The vertical maps are proper. All the maps in the diagram are compatible with the natural G_-actions, where the action of__G on the four spaces on the left of the diagram is trivial. It follows that

u*(p~)f.(C) = (1 x pi)f.(C) = (pl).~(C) = g a n d / . ( C ) = j*(p~)v(C).

Here

/!(C) e Z)(X), (p~)! (C) e :D(_G_G\(E • -9)), (I x pl)!(C) e Y)G_(E x -9), g e Z)G_(-9)

and

j * : :D(__G\(E x 9_)) --+ Z)(X),

Z)(_G\(E x _9)) V c ( E x _9), q*:/)c_(g_) - + / ) R ( E x _9),

are inverse image functors. Hence j*, u*, q* induce algebra homomorphisms

(a) Endv(~\(Sx_g)) ((Pl), (C)) -+ nndv(x) ( f , (C)),

(b) End~(ak(Ex_~))((pl).~(C)) -+ Endvc_(Ex_g)((1 x pi) ,(C)) ,

(c) Endz~(~) ~ Endv~(Ex~)((1 x pl)~.(C)),

and (b) is an isomorphism. (As __G acts freely on E x _9, u* is an equivalence of categories, see [BL, Prop. 2.2.5]). Hence combining (a),(b),(c) we obtain an algebra homomorphism

End~c_(_g ) (K) --* Endz)(x)(f! (C)).

Combining this with 5.2(a), we obtain an algebra homomorphism

C[W__] ~ Endz)(x)(f! (C)).

We compose this with the natural algebra homomorphism

Endz)(x) (f! (C)) ---, EndcH~ (X, f! (C)) * !

= EndcH~ (X, C) = Endcg.~176 ', C)

(where H* is cohomology with compact support and H~ is Borel-Moore ho- mology), and we obtain an algebra homomorphism C[W] --* EndcH.C~X',C~ that is, a representation of W on the homology space H.~176 ', (3).

AFFINE WEYL GROUPS AND CONJUGACY CLASSES 95

5.4. Let x �9 g be an element such that Bx r o .

We fix b z �9 B. For any i �9 I (see 1.1), we set b i = b z + b where b �9 B

is such that (b z, b) is in the G-orbit on B x B indexed by s~ �9 W (see 1.1).

Then b i is a C-Lie subalgebra of g independent of the choice of b. Given a subset J of I , distinct from ~ and from I, we denote by b J the C-Lie subalgebra of g generated by )-]ieg hi- Note that b {i} = b i. Thus, b J is defined for any subset J of I , distinct from I.

For such J , let G J = {g �9 G}Ad(g)(b J) = b J}. This is a proalgebraic

group over C whose prounipotent radical is denoted by G J. The quotient G J / G ~ is a connected reductive algebraic group over C, denoted by (~J. Let ~J be the Lie algebra of (~J. Then we have a canonical quotient map gJ --* ~J whose kernel is denoted by nzJ.

Let B J be the set of C-Lie subalgebras of g that are in the Ad(G)-orbi t of gg. For any p �9 B J we set np = Ad(g)ngj where g �9 G is such that p = Ad(g)p. Let ~ = p/np. Then ~ is naturally a reductive C-Lie algebra (a quotient of p). Let 7rJf : B --~ 13 J be the map defined by rJ(D) = p, where b c p .

For any I _> 1, let

B~(l) = {p �9 ~JIx �9 P, dnb~ c p}.

Let Y = ( ' f fJ)- lBJ(l) C B. N o w 7{" g restricts to a map Y --~ B~J(I); this

is naturally a morphism between two projective algebraic varieties over C, with all fibers isomorphic to the flag manifold of (~J. More precisely, this is the flag bundle associated to a principal fibration E ~ B~(1) with group (~J, where

E = {G J .g �9 GJ\GIAd(g)b J �9 B~(/)},

and the map E --* B~(l) is G J . g ~ Ad(g)b J. The Lie algebra bundle associated to this principal bundle is the bundle

whose fiber at p E B~(1) is the Lie algebra ~. This bundle has a natural section j , which associates to any p �9 B J(1) the image of x under the canonical map p --* ~.

We are in the setup of 5.3 with X = B~(l) and _G = GJ . The space X ' in 5.3 becomes in our case the variety

B (0 = {b �9 Bxl %, c

The Weyl group of (~J may be canonically identified with the subgroup W J of Wa generated by {sill E J}. By the construction of 5.3, we obtain an action of W J on the ordinary homology space H.(Bx(1), C) (which is the same as the Borel-Moore homology since Bx(l) is projective).

96 G. LUSZTIG

From the definitions we see that the closed imbedding B~(l) C B~(l + 1) induces in homology a map H.(B~(I), C) --* H.(13~(l + 1), C) which is compatible with the W J-actions. Hence we obtain a representation of W J on the direct limit lira I H.(Bx(I), C) = H.(Bz, C). (Note that Ut>lB~:(l) = B~:.)

5.5. Assume now that J, J ' are two subsets of I, distinct from I, such that J C J ' . Then W J C W J' and one checks from the definitions that the W J- action on H.(Bx, C) in 5.5 is the restriction of the analogous WJ'-action on H. (Bx, C).

But to give a representation of W~ on a vector space is the same as

to give representations of W J on that vector space, for any subset J of I,

distinct from I, so that these representations are compatible with each other

under the various inclusions W J C W J' as above. (This follows from the Coxeter presentation of W~.) Hence there is a unique representation of W~ on H.(Bx, C) whose restriction to each W J as above is the representation constructed in 5.4.

References

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R. Bezrukavnikov, Dimension of the fized point set on the aJ~ne flag manifold, preprint. J. Bernstein and V. Lunts, Equivariant sheaves and functors, LNM 1578, Springer Verlag, 1994. D. Kazhdan and (3. Lusztig, A topological approach to Springer's representations, Adv. in Math. 38 (1980), 222-228. D. Kazhdan and (3. Lusztig, Fixed point varieties on a]flne flag ma- nifolds, Israel J. of Math. 62 (1988), 129-168. (3. Lusztig, The two sided cells of the aJ~ne Weyl group of type A, Infinite Dimensional Groups with Applications, MSRI Publ. 4, Springer Verlag, 1985, pp. 275-283. (3. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), 169-178. (3. Lusztig, Cells in a.O~ne Weyl groups, Algebraic Groups and Re- lated Topics, Adv. Stud. Pure Math. 6, North-Holland and Kino- kuniya, Tokyo and Amsterdam, 1985, pp. 255-287. (3. Lnsztig, Cuspidal local systems and graded Hecke algebras II, Representations of Groups, B. Allison and (3. Cliff editors, Canad. Math. Soc. Conf. Proc. 16, Amer. Math. Soc., 1995, pp. 217-275.

N. Spaltenstein, Polynomials over local fields, nilpotent orbits and conjugacy classes in Weyl groups, Astdrisque 168 (1988), 191-217. N. Spaltenstein, A note on the Kazhdan-Lusztig map for even or- thogonal Lie algebras, Arch. Math. 55 (1990), 431-437.

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N. Spaltenstein, On the Kazhdan-Lusztig map for exceptional Lie algebras, Adv. in Math. 83 (1990), 48-74. N. Spaltenstein, Classes unipotentes et sous-groupes de Borel, LNM 946, Springer Verlag, 1982. R. Steinberg, On the desingularization of the unipotent variety, In- vent. Math. 36 (1976), 209-224.