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ON CERTAIN VARIETIES ATTACHED TO A WEYL GROUP ELEMENT 1 1 G. Lusztig 0 2 n 0. Introduction and statement of results a J 9 0.1. Let k be an algebraically closed field. Let G be a connected reductive algebraic group over k. We assume that we are in one of the following two cases. ] T (1): G is the identity component of a reductive group Gˆ with a fixed connected R component D. . h (2): k is an algebraic closure of a finite field F and G has a fixed F -rational q q t a structure with Frobenius map F : G −→ G. m In case (1) we set q = 1 and denote by F : G −→ G the identity map of G so that [ GF = G. Thus when q = 1 we are in case (1) and when q > 1 we are in case (2). 2 Let B be the variety of Borel subgroups of G. Let W be an indexing set for v the set of G-orbits on B × B for the diagonal G-action. Let O be the G-orbit 4 w 7 corresponding to w ∈ W. Note that W is naturally a Coxeter group with length 0 function l(w) = dimO −dimB. 2 w 2. Let I be an indexing set for the set S of simple reflections of W. Let si ∈ S be 1 the simple reflection corresponding to i ∈ I. For B ∈ B we have gBg−1 ∈ B for 0 any g ∈ D (if q = 1) and F(B) ∈ B (if q > 1). There is a unique automorphism of 1 : W (denoted by • or by w 7→ w•) such that Ow• = gOwg−1 for all w ∈ W,g ∈ D v i (if q = 1) and Ow• = F(Ow) for all w ∈ W (if q > 1). We have l(w•) = l(w) for X all w ∈ W. Hence there is a unique bijection i 7→ i• of I such that s•i = si• for all r a i ∈ I. Two elements w,w′ ∈ W are said to be •-conjugate if w′ = a−1wa• for some a ∈ W. The relation of •-conjugacy is an equivalence relation on W; the equivalence classes are said to be •-conjugacy classes. A •-conjugacy class C in W (or an element of it) is said to be •-elliptic if C does not meet any •-stable proper parabolic subgroup of W (see [H1]). (In the case where • = 1 we say ”elliptic, conjugacy class” instead of ”•-elliptic, •-conjugacy class”.) For w ∈ W let B = {(g,B) ∈ D×B;(B,gBg−1) ∈ O } (if q = 1) w w X = {B ∈ B;(B,F(B))∈ O } (if q > 1). w w This is naturally an algebraic variety over k. (The variety X is defined in [DL]. w Supported in part by the National Science Foundation Typeset by AMS-TEX 1 2 G. LUSZTIG The variety B appears in [L3] assuming that D = G and in [L4] in general.) We w shall use the notation X for either B or X . Let ρ : B −→ D be the first w w w w projection. Now GF acts on X by x : (g,B) 7→ (xgx−1,xBx−1) (if q = 1) and by x : B 7→ w xBx−1 (if q > 1). One of the themes of this paper is the analogy between X and B . It seems w w that B is a limit case of X as q → 1. For example it is likely that for any i, w w the multiplicities of various unipotent character sheaves on D in a Jordan-Hölder series of the (i + dimG)-th perverse cohomology sheaf of ρ Q¯ (with q = 1) are ! l the same as the multiplicities of various irreducible unipotent representations of GF in the GF-module Hi(X ,Q¯ ) (with q > 1). Here l is a fixed prime number c w l invertible in k. 0.2. From [DL, 1.11] it is known that if w ∈ W, X has a natural finite covering w X˜ . We now show that (at least if w is •-elliptic and G is semisimple), B has a w w natural finite covering B˜ . w Let B∗ ∈ B and let T∗ be a maximal torus of B∗; if q > 1 we assume in addition that B∗,T∗ are defined over F . Let U∗ be the unipotent radical of B∗. If q = 1 let q d ∈ D besuch thatdT∗d−1 = T∗,dB∗d−1 = B∗. Let N = {n ∈ G;nT∗n−1 = T∗}. We identify N/T∗ = W by nT∗ ↔ w, (B∗,nB∗n−1) ∈ O . According to Tits, for w each w ∈ W we can choose a representative w˙ ∈ N in such a way that w˙ = w˙ w˙ 1 2 whenever w,w ,w in W satisfy w = w w , l(w) = l(w ) + l(w ). We can also 1 2 1 2 1 2 assume that, if w′ = w•, then w˙′ = dw˙d−1 (if q = 1) and w˙′ = F(w˙) (if q > 1). For w ∈ W let U∗ = U∗ ∩w˙U∗w˙−1 and let T∗ = {t ∈ T∗;w˙−1tw˙ = dtd−1} (if w w 1 q = 1), T∗ = {t ∈ T∗;w˙−1tw˙ = F(t)} (if q > 1); let w B˜ = {(g,g′U∗) ∈ D×G/U∗;g′−1gg′ ∈ w˙U∗d} (if q = 1), w w w X˜ = {g′U∗ ∈ G/U∗;g′−1F(g′) ∈ w˙U∗} (if q > 1). w w w We shall use the notation X˜ for either B˜ or X˜ . Now GF acts on X˜ by w w w w x : (g,g′U∗) 7→ (xgx−1,xg′U∗) (if q = 1) and by x : g′U∗ 7→ xg′U∗ (if q > 1). w w w w Also T∗ acts (freely) on X˜ by t : (g,g′U∗) 7→ (g,g′t−1U∗) (if q = 1) and by w w w w t : g′U∗ 7→ g′t−1U∗ (if q > 1); this action commutes with the GF-action. Define w w π : X˜ −→ X by (g,g′U∗) 7→ (g,g′B∗g′−1) (if q = 1) and by g′U∗ 7→ g′B∗g′−1 w w w w w (if q > 1). Note that π is compatible with the T∗ action where T∗ acts on X w w w w trivially. Let F be the fibre of π at a point of X . Then for some t ∈ T∗, w w 0 F can be identified with {t ∈ T∗;Ad(d)(t−1)Ad(w˙−1)(t) = t } (if q = 1) and 0 with {t ∈ T∗;F(t)−1Ad(w˙−1)(t) = t } (if q > 1); hence it is either empty or 0 a principal homogeneous space for T∗. Now if q > 1, T∗ is finite, hence the w w homomorphism T∗ −→ T∗, t 7→ F(t)−1Ad(w˙−1)(t) is surjective and F is a principal homogeneous space for T∗ so that in this case, π is a principal T∗-bundle. If for w w w q = 1 we assume that G is semisimple and w is •-elliptic then T∗ is finite, hence w the homomorphism T∗ −→ T∗, t 7→ Ad(d)(t−1)Ad(w˙−1)(t) is surjective and F is a ON CERTAIN VARIETIES ATTACHED TO A WEYL GROUP ELEMENT 3 principal homogeneous space for T∗ so that in this case, π is again a principal w w T∗-bundle. w Here is another reason why B looks like X when q → 1: assuming that w is w w •-ellipticand G issemisimple, the number |T∗| (in the case q = 1) is obtained from w the number |T∗| (in the case q > 1) viewed as a polynomial in q by substituting w q = 1. The following result gives another instance of analogous behaviour of X ,B . w w Theorem 0.3. Assume that w ∈ W is •-elliptic and that w has minimal length in its •-conjugacy class. If q = 1 assume further that G is semisimple. (a) If q = 1 (resp. q > 1), any isotropy group of the GF action on B˜ (resp. w X˜ ) is {1}. w (b) If q = 1 (resp. q > 1), any isotropy group of the GF action on B (resp. w X ) is isomorphic to a subgroup of T∗; hence it is a finite diagonalizable group. w w (c) If q = 1, the varieties B and B˜ are affine. w w Note that (c) has the following known analogue (see [DL] for sufficiently large q and [OR], [H1] for any q): (d) If q > 1, the varieties X and X˜ are affine. w w The proof of the theorem (given in §3) extends the proof of a weaker form of (b) given in [L5, 5.2]. Let G\B˜ (resp. GF\X˜ ) be the set of orbits of the G-action on B˜ (resp. w w w of the GF-action on X˜ ). Let G\B (resp. GF\X ) be the set of orbits of the w w w G-action on B (resp. of the GF-action on X ). By (a)-(c) above. G\B˜ and w w w G\B are naturally affine varieties (they are the set of orbits of an action of a w reductive group on an affine variety with all orbits being of the same dimension hence closed). Similarly, by (d) above, GF\X˜ and GF\X are naturally affine w w varieties. The affineness properties (c),(d) can be strengthened in certain cases as follows. Theorem 0.4. Assume that G is almost simple of type A ,B ,C or D . We n n n n assume also that • = 1. Let w ∈ W be a •-elliptic element of minimal length in its •-conjugacy class. (a) If q = 1, then G\B˜ is isomorphic to kl(w) and G\B is isomorphic to w w T∗\kl(w) for a T∗-action on kl(w). w w (b) If q > 1, then GF\X˜ is quasi-isomorphic (see 1.1) to kl(w) and GF\X is w w quasi-isomorphic (see 1.1) to T∗\kl(w) for a T∗-action on kl(w). w w This is proved in §4. In a sequel to this paper it is shown that (a),(b) continue to hold without the assumption that • = 1. We conjecture that (a),(b) hold for G of any type. 0.5. Let w ∈ W and let δ be the smallest integer ≥ 1 such that •δ = 1. If q > 1, F : Xw −→ Xw• and Fδ : Xw −→ Xw are well defined. We propose an extension of these maps to the case of Bw namely Ψ : Bw −→ Bw•, (g,B) 7→ (g,gBg−1), 4 G. LUSZTIG see 1.2; we then have Ψδ : B −→ B . In some respects Ψ,Ψδ can be viewed w w as analogues for q = 1 of the Frobenius maps F,Fδ. Assume for example that w is •-elliptic of minimal possible length in W. There is some evidence that, for any i, the (i+ dimG)-th perverse cohomology sheaf of ρ Q¯ is direct sum of ! l mutually nonisomorphic simple character sheaves stable under the map induced by Ψδ and Ψδ acts on each of these summands as multiplication by a root of 1 which is obtained from an eigenvalue of Fδ on Hi(X ,Q¯ ) (described in [L1]) by c w l q → 1. 0.6. Let w ∈ W and let i ∈ I be such that l(w) = l(siw)+1 = l(siwsi•). When q > 1 a quasi-isomorphism (see 1.1) σ : X −→ X was defined in [DL]. In i w siwsi• the late 1970’s and early 1980’s I observed (unpublished but mentioned in [MB, 5A] and [BM]) that by taking compositions of various σ one can obtain nontrivial i quasi-automorphisms of X corresponding to elements in the stabilizer of w for w the •-conjugacy action (see 1.3, 1.4). Further examples of this phenomenon were later found by Digne and Michel [DM]. Additional examples are given in §1,§2. These examples are valid not only for X but also for X˜ , B or B˜ since in w w w w 2.3 and 2.6 we define quasi-isomorphisms analogous to σ in the case when X is i w replaced by X˜ , B or B˜ . w w w 0.7. In §5 we give another example of the close relation between the varieties B ,X by proving (under the assumption that k is asin case 2) a formula relating w w the number of rational points over a finite field of Bw ×D Bw′ and of GF\(Xw × Xw′). 0.8. Notation. For any w ∈ W we set L(w) = {i ∈ I;l(s w) < l(w)}, R(w) = i {i ∈ I;l(ws ) < l(w)}. For k ∈ Z let w 7→ w•k be the k-th power of •. Let w i 0 be the longest element of W. Let Wˆ be the braid group of W with generators sˆ i corresponding to s . If X is a set and f : X −→ X is a map we write Xf instead i of {x ∈ X;f(x) = x}. If X is finite we write |X| for the cardinal of X. 1. Paths 1.1. Let C be a •-elliptic •-conjugacy class in W. Let C be the set of elements min of minimal length of C. If w ∈ C and i ∈ L(w) then w′ := s ws• ∈ C and min i i min i∗ ∈ R(w′); we then write w −i→+ w′ . Conversely if v ∈ C and j• ∈ R(v) then min v′ := s vs• ∈ C and j ∈ L(v′); we then write v −j→− v′. Note that if w,w′ ∈ W j j min then the conditions w −i→+ w′ and w′ −i→− w are equivalent. Let Γ be the graph C whose vertices are the elements of C and whose edges are the triples w ⌣i w′ min with w,w′ in C unordered and i ∈ I such that either w −i→+ w′ or w′ −i→+ w. min The graph Γ has a canonical orientation in which an edge w ⌣i w′ is oriented C from w to w′ if w −i→+ w′ and is oriented from w′ to w if w′ −i→+ w. A path in Γ C is by definition a sequence i of edges of Γ of the form w ⌣i1 w ⌣i2,...,i⌣t−1 w . C 1 2 t ON CERTAIN VARIETIES ATTACHED TO A WEYL GROUP ELEMENT 5 For such i we must have w = z−1w z• where z = s s ...s ∈ W; we shall t i 1 i i i1 i2 it−1 also set (a) z˜ = sˆǫ1sˆǫ2 ...sˆǫt−1 ∈ Wˆ i i1 i2 it−1 i+ i− where ǫ = 1 if w −→r w , ǫ = −1 if w −→r w . We shall sometime specify i r r r+1 r r r+1 by the symbol [w ;∗ ,∗ ,...,∗ ] where ∗ = i if ǫ = 1 and ∗ = i if ǫ = −1 1 1 2 t−1 k k k k k k (ǫ asin (a).) Notethat w ,...,w are uniquely determined by w ,i ,i ,...,i ). k 2 t 1 1 2 t−1 Forw,w′ ∈ Cmin let Pw,w′ betheset ofpathsinΓC such thatthecorresponding sequence w ,w ,...,w satisfies w = w,w = w′. For example if w = s s ...s 1 2 t 1 t i1 i2 ir is a reduced expression in W then [w;i1,i2,...,ir] ∈ Pw,w•. The following result is due to Geck-Pfeiffer [GP, 3.2.7] (in the case where • = 1) and to Geck-Kim-Pfeiffer [GKP] and He [H1] in the remaining cases. (b) For any w,w′ ∈ Cmin, the set Pw,w′ is nonempty. For w,w′ ∈ Cmin we identify a path [w;∗1,∗2,...,∗t−1] ∈ Pw,w′ with the path [w;∗′1,∗′2,...,∗′t′−1] ∈ Pw,w′ in the following cases: (i) t′ = t−2, ∗ = i,∗ = i (for some i ∈ I and some k), and ∗′,∗′,...,∗′ k k+1 1 2 t′−1 is obtained from ∗ ,∗ ,...,∗ by dropping ∗ ,∗ ; 1 2 t−1 k k+1 (ii) t′ = t−2, ∗ = i,∗ = i (for some i ∈ I and some k), and ∗′,∗′,...,∗′ k k+1 1 2 t′−1 is obtained from ∗ ,∗ ,...,∗ by dropping ∗ ,∗ ; 1 2 t−1 k k+1 (iii) t′ = t, ∗ = i,∗ = j,∗ = i,... (m terms), ∗′ = j,∗′ = i,∗′ = k k+1 k+2 k k+1 k+2 j,... (m terms), (for some i 6= j in I withs s of order m and some k) and ∗′ = ∗ i j u u for all other indices; (iv) t′ = t, ∗ = i,∗ = j,∗ = i,... (m terms), ∗′ = j,∗′ = i,∗′ = k k+1 k+2 k k+1 k+2 j,... (m terms), (for some i 6= j in I withs s of order m and some k) and ∗′ = ∗ i j u u for all other indices. This generates an equivalence relation on Pw,w′; we denote by P¯w,w′ the set of equivalence classes. For w,w′,w′′ ∈ Cmin, concatenation Pw,w′×Pw′,w′′ −→ Pw,w′′ induces a map P¯w,w′ × P¯w′,w′′ −→ P¯w,w′′ which makes ⊔w,w′∈C P¯w,w′ into a min groupoid. In particular for w ∈ C , P¯ has a natural group structure. Now min w,w i 7→ z induces a group homomorphism i τ : P¯ −→ W := {z ∈ W;z−1wz• = w} w w,w w and i 7→ z˜ induces a group homomorphism τ˜ : P¯ −→ Wˆ . i w w,w 1.2. Let C be a •-elliptic •-conjugacy class in W and let w ∈ C . We state the min following conjecture. (a) The homomorphism τ : P¯ −→ W is surjective. w w,w w In 1.5, 1.6 we sketch a proof of (a) assuming that W is of classical type and • = 1; in 1.4 we consider in more detail a case arising from D . 4 In any case, if w• = w then w is in the image of τ . In particular, if W is w w generated by w then (a) holds for w. Also from 1.1(b) we see that if (a) holds for some w ∈ C then it holds for any w ∈ C . We say that (a) holds for C if it min min holds for some (or equivalently any) w ∈ C . min 6 G. LUSZTIG 1.3. Assume that w = w and y• = wyw−1 for any y ∈ W. Then the •-conjugacy 0 class of w is C = {w} and is •-elliptic. For any y ∈ W and any reduced expression y = s s ...s for y, we have i := [y;i ,i ,...,i ] ∈ P , z = y. Thus the i1 i2 ik 1 2 k w,w i image of τ is W = W and 1.2(a) holds in this case. w w 1.4. In the remainder of this section we assume that • = 1 on W. We will often denote an element s s s ...s of W as i i i ...i . i1 i2 i3 ik 1 2 3 k The following example appeared in the author’s work (1982, unpublished). As- sume that W is of type D . Let S = {s ,s ,s ,s } with s ,s ,s commuting. Let 4 0 1 2 3 1 2 3 C be the conjugacy class of W consisting of the twelve elements (of length six) 0i0j0k and i0j0k0 (where i,j,k is a permutation of 1,2,3). Note that C = C min is elliptic and any w ∈ C has order 4. We have L(0i0j0k) = {0,i}, R(0i0j0k) = 0+ {j,k}, L(i0j0k0) = {i,j}, R(i0j0k0) = {0,k}. We have 0i0j0k −→ i0j0k0, i+ i+ j+ 0i0j0k −→ 0j0i0k, i0j0k0 −→ 0j0k0i, i0j0k0 −→ i0k0j0 for any i,j,k. Let w = i0j0k0 ∈ C = C . Now W is a nonabelian group of order 16 min w generated by three elements α = 0ij0, β = jk, γ = i0ki0i satisfying (a) γαβ = αβγ = βγα = w. Note that β (resp. α) is the unique element of length 2 (resp. 4) in W : if w n is the number of elements of length i in W and t is an indeterminate, then i w n ti = 1+t2 +t4 +10t6 +t8 +t10 +t12. We have Pi≥0 i i := [w;0,i,j,0] ∈ P ,i′ := [w;j,k] ∈ P ,i′′ := [w;i,0,k,i,0,i] ∈ P , w,w w,w w,w and zi = α, zi′ = β, zi′′ = γ. Thus the image of τw contains the generators α,β,γ of W hence it is equal to W and 1.2(a) holds for C. Note that a relation like w w (a) also holds in the group P¯ : w,w (b) i′′ii′ = ii′i′′ = i′i′′i = [w;i,0,j,0,k,0]. For example, i′′ii′ = [w;i,0,k,i,0,i,0,i,j,0,j,k]= [w;i,0,k,i,i,0,i,i,j,0,j,k]= [w;i,0,k,j,0,k]= [w;i,0,j,0,k,0]. Alsoi,i′,i′′ commutewith[w;i,0,j,0,k,0]inPw,w. Itfollowsthatz˜i′′,z˜i,z˜i′ satisfy a relation like (b) in Wˆ . 1.5. Let n be an integer ≥ 3. Define n ∈ N by n = 2n if n is even , n = 2n+1 if n is odd. Let W be the group of all permutations of [1,n] which commute with the involution i 7→ n−i+1 of [1,n]. For i ∈ [1,n−1] define s ∈ W as a product i of two transpositions i ↔ i + 1, n + 1 − i ↔ n − i; define s ∈ W to be the n transposition n ↔ n − n + 1. Then (W,{s ;i ∈ [1,n]}) is a Weyl group of type i ON CERTAIN VARIETIES ATTACHED TO A WEYL GROUP ELEMENT 7 B . In this subsection we assume that G is almost simple of type C (or B ) and n n n we identify W with W with n = 2n (or n = 2n + 1) as Coxeter groups in the standard way. Let p = (p ≥ p ≥ ··· ≥ p ) be a sequence in Z such that p +···+p = n. ∗ 1 2 σ >0 1 σ Define a partition m +m +···+m = σ by 1 2 e p = p = ··· = p > p = p = ··· = p > .... 1 2 m1 m1+1 m1+2 m1+m2 For any r ∈ [1,σ] we define a permutation w in W by r p +1 7→ p +2 7→ ... 7→ p +p 7→ n−p −1 7→ <r <r <r r <r n−p −2 7→ ... 7→ n−p −p 7→ p +1, <r <r r <r wherep<r = Pr′∈[1,r−1]pr′ andallunspecifiedelementsaremappedtothemselves. Notethatw isa2p -cycleandthatw ,w ,...,w arecommutingwitheachother. r r 1 2 σ Let w = w w ...w and let C be the conjugacy class of w. Note that C is elliptic 1 2 σ and w ∈ C . For every r ∈ [1,σ−1] such that p = p we define an involutive min r r+1 permutation h ∈ W by r p +j 7→ p +j 7→ p +j,n−p −j 7→ n−p −j 7→ n−p −j for j ∈ [1,p ] <r <r+1 <r <r <r+1 <r r (all unspecified elements are mapped to themselves). Note that h w h = w r r+1 r r and h w = w h for all t ∈/ {r,r+1}. Hence h w = wh . The following result is r t t r r r easily verified: (a)The group W is generated by the elements w , w (r ∈ [1,σ−1],p > p ) w σ r r+1 r and h (r ∈ [1,σ−1],p = p ). r r r+1 These generators satisfy the ”braid group relations” of a complex reflection group of type B(2pm1) ×B(2pm1+m2) ×...×B(2pm1+···+me) m1 m2 me (described in [MB, 3A]); the factor B(2pm1+···+mk) is generated by h mk m1+···+mk−1+u (u ∈ [1,m −1]) and by w . k m1+···+mk It is immediate that for r ∈ [1,σ] we have (setting a = n−(p +p +···+ σ σ−1 p )): r+1 i := [w;a,a+1,...,n−1,n,n−1,...,a−p +2,a−p +1] ∈ P . r r r w,w Note that z = w . i r r One can verify that for r ∈ [1,σ −1] such that p = p = p we have (setting r r+1 a = n−(p +p +···+p )): σ σ−1 r+1 i′ := [w;a,a+1,...,a+p−2,a−1,a,a+1...,a+p−4,a−2,a−1,a,..., r a+p−6,...,a−p+2,a+p−1,a+p−3,...,a−p+1,a−p+2,...,a+p−6, ...,a,a−1,a−2,a+p−4,...,a+1,a,a−1,a+p−2,...,a+1,a] ∈ P , w,w 8 G. LUSZTIG Forexampleifp = 1wehavezi′ = [w;a]; ifp = 2wehavezi′ = [w;a,a+1,a−1,a]; r r if p = 3 we have zi′ = [w;a,a+1,a−1,a+2,a,a−2,a−1,a+1,a]. r Note that zi′ = hr. Using (a), we see that the image of τw contains a set of r generators of W hence 1.2(a) holds for C. (In the case where p = p = ··· = p , w 1 2 σ this result is due to Digne and Michel [DM].) Note that any ellipticconjugacy class in W is of the form C as above. We conjecture that (b) the braid group relations satisfied by the generators in (a) remain valid as equations in P¯ if w is replaced by i and h is replaced by i′. w,w r r r r Appplying τ˜ we would get corresponding braid group relations in Wˆ which ac- w tually can be verified. 1.6. In this subsection we assume that G is almost simple of type D . Let W,s n i be as in 1.5 (with n = 2n ≥ 8). Le W′ be the group of even permutations in W (a subgroup of index 2 of W). If i ∈ [1,n − 1] we have s ∈ W′ and we i set s(n−1)′ = snsn−1sn ∈ W′. Then (W′,{s1,s2,...,sn−1,s(n−1)′}) is a Weyl group of type D . We identify W with W′ as Coxeter groups as in [L5, 1.5]. n Let p = (p ≥ p ≥ ··· ≥ p ), w ,w,h be as in 1.5; we assume that σ is ∗ 1 2 σ r r even. Then w ∈ W′. Let C′ be the conjugacy class of w in W′. Then C′ is elliptic and w ∈ C′ . For any r ∈ [1,σ] we have w′ := w w ∈ W′. For any min r r σ r ∈ [1,σ−1] such that r ∈ [1,σ−1],p = p we have h ∈ W′. If p = p we r r+1 r σ−1 σ set h′ = w−1h w . The following result is easily verified: σ−1 σ σ−1 σ (a) If p > p then W is generated by the elements w′ , w′ (r ∈ [1,σ − σ−1 σ w σ r 1],p > p ) and h (r ∈ [1,σ − 2],p = p ). If p = p then W is r+1 r r r r+1 σ−1 σ w generated by the elements w′ , w′ (r ∈ [1,σ − 2],p > p ), h′ and h (r ∈ σ r r+1 r σ−1 r [1,σ−1],p = p ). r r+1 These generators satisfy the ”braid group relations” of a complex reflection group of type B(2pm1) ×B(2pm1+m2) ×...×B(2pm1+···+me−1) ×D(2pm1+···+me) m1 m2 me−1 me (described in [MB, 3A]); the factor B(2pm1+···+mk) (with k < m) is generated by m k h (u ∈ [1,m − 1]) and by w′ ; if m > 1 then the factor m1+···+mk−1+u k m1+···+mk e D(2pm1+···+me) is generated by h (u ∈ [1,m −1]), by h′ me m1+···+me−1+u e m1+···+me−1 and by by w′ ; if m = 1 the factor D(2pm1+···+me) is taken to be a cyclic m1+···+mk e me group of order p .For example the ”braid group relation” σ h w′ h′ = w′ h′ h = h′ h w′ σ−1 σ σ−1 σ σ−1 σ−1 σ−1 σ−1 σ holds if m > 1. (Compare with 1.4(a).) e ON CERTAIN VARIETIES ATTACHED TO A WEYL GROUP ELEMENT 9 Onecanverifythatforr ∈ [1,σ]wehave(settinga = n−(p +p +···+p )): σ σ−1 r+1 i′′ = [w;a,a+1,...,n−1,(n−1)′,n−2,...,a−p +2,a−p +1,n−1,n−2, r r r ...,n−p +1] ∈ P . σ w,w Note that zi′′ = wr′. On the other hand for r ∈ [1,σ − 1],pr = pr+1 we have r hr = zi′ where i′r is given by the same formula as in 1.5 (but viewed in W′); we r have i′ ∈ P . If p = p = p then r w,w σ−1 σ ˜i = [w;(n−1)′,n−2,n−3,...,p+1,p,p+1,...,n−2,p−1,p,...,n−4,..., 3,4,2,(n−1)′,n−3,...,5,3,1,2,4,3,...,n−4,...,p,p−1,n−2,..., p+1,p,p+1,...,n−3,n−2,(n−1)′] ∈ P . w,w For example if n = 10, p = 5 then ˜i := [w;9′,8,7,6,5,6,7,8,4,5,6,3,4,2,9′,7,5,3,1,2,4,3,6,5,4,8,7,6,5,6,7,8,9′]. Note that z = h′ . Using (a), we see that the image of τ contains a set of ˜i σ−1 w generators of W hence 1.2(a) holds for C. Note that any elliptic conjugacy class w in W is of the form C′ as above. We conjecture that (b) the braid group relations satisfied by the generators in (a) remain valid as equations in P¯ if w′ is replaced by i′′, h is replaced by i′ and h′ is replaced w,w r r r r σ−1 by ˜i. Appplying τ˜ we would get corresponding braid group relations in Wˆ which ac- w tually can be verified. 1.7. In this subsection we assume that C is an elliptic conjugacy class in W such that for some w ∈ C we have w = w w ...w where w ,...,w commute min 1 2 r 1 r with each other, l(w) = l(w ) + l(w ) + ··· + l(w ) and the centralizer of w is 1 2 r generated by w ,...,w . (An example of this situation is the case of w in 1.5 with 1 r p > p > ··· > p .) In this case it is immediate that w is in the image of τ 1 2 σ i w hence 1.2(a) holds for C. Another example arises for W of type E (with the elements of I labelled as 8 in [GP]) and with C consisting of elements whose characteristic polynomial in the reflection representation is(X+1)(X7+1). Theelement w = 213423454234565768 belongs to C and l(w) = 18. We have w = s x = xs for some x such that min 2 2 l(x) = 17 and s x7 = w . (This equation holds also in Wˆ .) The centralizer of w is 2 0 a product of a cyclic group of order 2 generated by s and a cyclic group of order 2 14 generated by x. We see that 1.2(a) holds for C. 1.8. Assume that W is of type E . Let C be the elliptic conjugacy class in W 8 consisting of the elements of order 15. We can find w ∈ C such that w = u2 min where u = 12345678so that l(u) = 8, l(w) = 16. Then the centralizer of w consists of the powers of u. We have i := [w;1,2,3,4,5,6,7,8] ∈ P , z = u and we see w,w i that 1.2(a) holds for C. 10 G. LUSZTIG 2. The morphisms σ ,σ˜ i i 2.1. If V,V′ are algebraic varieties over k, we say that a map of sets f : V −→ V′ is a quasi-morphism if: (for q = 1) f is a morphism, or (for q > 1) f is composition V = V −f→1 V −f→2 ... −f−t−−→1 V = V′ where for 1 2 t each i ∈ [1,t − 1], f : V −→ V is either a morphism of algebraic varieties or i i i+1 V = V and f is the inverse of the Frobenius map on V for a rational structure i i+1 i i over a finite subfield of k. We say that f is a quasi-isomorphism if it is a quasi-morphism and has an inverse which is a quasi-morphism. If in addition we have V = V′ we say that f is a quasi-automorphism. 2.2. Let w ∈ W. We define a morphism Ψ : Xw −→ Xw• by (g,B) 7→ (g,gBg−1) if q = 1 and B 7→ F(B) if q > 1. We define a morphism Ψ : X˜w −→ X˜w• by (g,g′U∗) 7→ (g,gg′d−1U∗ ) (if q = 1) and g′U∗ 7→ F(g′)U∗ (if q > 1). w w• w w• Note that each of the morphisms Ψ is a quasi-isomorphism. 2.3. For any w,w′,a,b ∈ W such that w = ab,w′ = ba•, l(w) = l(a)+l(b) = l(w′) we define a morphism σ(a) : Xw −→ Xw′ by (g,B) 7→ (g,B′) where B′ ∈ B is determined by the conditions (B,B′) ∈ O , a (B′,gBg−1) ∈ O (if q = 1); b B 7→ B′ where B′ ∈ B is determined by the conditions (B,B′) ∈ O , a (B′,F(B)) ∈ O (if q > 1). b (If q > 1, the map σ(a) is defined in [DL, p.107,108].) We have a commutative diagram σ(a) Xw −−−−→ Xw′ Ψ Ψ   y y σ(a•) Xw• −−−−→ Xw′• Note that for any w ∈ W we have σ(w) = Ψ : Xw −→ Xw•. If w,w′,a,b are as above then σ(b) : Xw′ −→ Xw• is defined and σ(b)σ(a) : Xw −→ Xw• is equal to Ψ. Interchanging (a,b) with(b•−1,a) we see that σ(a)σ(b•−1) : Xw′•−1 −→ Xw′ is equal to Ψ. Thus σ(a) : Xw −→ Xw′ is a quasi- isomorphism. Let w ∈ W and let i ∈ L(w) be such that, setting w′ = s ws•, we have i i l(w) = l(w′). Then σ(si) : Xw −→ Xw′ is a well defined quasi-isomorphism; we shall often write σ instead of σ(s ). i i 2.4. Assume that w ∈ W and i,j are distinct elements of L(w). Let m be the order of s s and let v = s s s ··· = s s s ... (both products have m factors). i j i j i j i j Let w′ = vwv• and assume that l(w) = l(s ws•) = l(s s ws•s•) = ··· = l(vwv•), i i j i i j