UNIPOTENT REPRESENTATIONS AS A CATEGORICAL CENTRE 4 1 0 G. Lusztig 2 b e F Introduction 7 0.1. Let k be an algebraic closure of the finite field F with p elements. For any 1 p power q of p let F be the subfield of k with q elements. Let G be a reductive q ] connected group over k, assumed to be adjoint. Let B be the variety of Borel T R subgroups of G. . Let W be the Weyl group of G and let c be a two-sided cell of W. Let s ∈ Z h >0 at and let F : G −→ G be the Frobenius map for an Fps-rational structure on G. m Let GF = {g ∈ G;F(g) = g}, a finite group. Let Rep♠(GF) (resp. Repc(GF)) [ be the category of representations of GF over Q¯l which are finite direct sums of unipotent representations in the sense of [DL] (resp. of unipotent representations 2 v whose associated two-sided cell (see 1.3) is c); here l is a fixed prime number 9 invertible in k. 8 8 In the rest of this subsection we assume for simplicity that the Fps-rational 2 structure on G is split. The simple objects of Repc(GF) were classified in [L1]. . 1 The classification turns out to be the same as that [L4] of unipotent character 0 4 sheaves on G whose associated two-sided cell is c. The fact that 1 (a) these two classification problems have the same solution : v has not until now been adequately explained. i X In [L12] we have shown that the category of perverse sheaves on G which are r direct sums of unipotent character sheaves with associated two-sided cell c is a naturally equivalent to the centre of a certain monoidal category CcB2 of sheaves on B2 introduced in [L9] for which the induced ring structure on the Grothendieck group is the J-ring attached to c, see [L10, 18.3]. (The analogous statement for D-modules on a reductive group over C was proved earlier in a quite different way in [BFO].) In this paper we show that Repc(GF) is also naturally equivalent to the centre of CcB2 (see 6.3). This implies in particular that the simple objects of Repc(GF) are naturally in bijection with the unipotent character sheaves with associatedtwo-sided cellc, which explains(a). Italsoimpliesthattheset ofsimple objects Rep (GF) is “independent” of the choice of s; in fact, as we show in 7.1, c Supported in part by National Science Foundation grant 1303060. Typeset by AMS-TEX 1 2 G. LUSZTIG it is also independent of the characteristic of k. It follows that to classify the unipotent representations of GF it is enough to classify the unipotent character sheaves on G in sufficiently large characteristic; for the latter classification one can use the scheme of [L11] which uses the unipotent support of a character sheaf. The methods of this paper are extensions of those of [L12]. We replace RepcGF by an equivalent category consisting of certain G-equivariant perverse sheaves on G , the set of all Frobenius maps G −→ G corresponding to split F -rational s ps structures on G; we view G as an algebraic variety in a natural way. We construct s functors χ ,ζ between this category and the category CcB2 which are q-analogues s s of the truncated induction and truncated restriction χ,ζ of [L12] and we show that most properties of χ,ζ are preserved. We also define a truncated convolution product from our sheaves on Gs and on Gs′ to our sheaves on Gs+s′ which is analogous to the truncated convolution of character sheaves in [L12]; we also give a meaning for this even when s,s′ are arbitrary integers. The main application of this truncated convolution product is in the case where s′ = −s, the result of the product being a direct sum of character sheaves on G; this is used in the proof of a weak form of an adjunction formula between χ ,ζ which is then used to prove s s the main result (Theorem 6.3). 0.2. In this paper we also prove extensions of the results in 0.1 to the case where F : G −→ G is the Frobenius map of a nonsplit F -rational structure. In this case ps the role of unipotent character sheaves on G is taken by the unipotent character sheaves on a connected component of the group of automorphism group of G. Moreover, in this case the centre of CcB2 is replaced by a slight generalization of the centre (the ǫ-centre) which depends on the connected component above. Many arguments in this paper are very similar to arguments in [L12] and are often replaced by references to the corresponding arguments in [L12]. Ourresultscanbeextendedtonon-unipotentrepresentationsandnon-unipotent character sheaves; this will be discussed elsewhere. 0.3. Notation. We assume that we are given a split F -rational structure on G p with Frobenius map F : G −→ G. Let ν = dimB, ∆ = dim(G), ρ = rk(G). We 0 shall view W as an indexing set for the orbits of G acting on B2 := B × B by simultaneous conjugation; let O be the orbit corresponding to w ∈ W and let w O¯ be the closure of O in B2. For w ∈ W we set |w| = dimO −ν (the length w w w of w). Let w be the unique element of W such that |w | = ν. max max As in [L1, 3.1], we say that an automorphism ǫ : W −→ W is ordinary if it leaves stable the set {s ∈ W;|s| = 1} and for any two elements s 6= s′ in that set which are in the same orbit of ǫ, the product ss′ has order ≤ 3. Let A be the group of ordinary automorphisms of W. For B ∈ B, let U be the unipotent radical of B. Then B/U is independent B B of B; it is “the” maximal torus T of G. Let X be the group of characters of T. LetRepW bethecategoryoffinitedimensional representations ofW overQ; let IrrW be a set of representatives for the isomorphism classes of irreducible objects UNIPOTENT REPRESENTATIONS AS A CATEGORICAL CENTRE 3 of RepW. The notation D(X),M(X),D (X),M (X) is as in [L12, 0.2]. (When X is m m G,B,Ow or O¯w, the subscript m refers to the Fps0-structure defined by F0s0 for a sufficiently large s > 0.) For K ∈ D(X), HiK, HiK, Ki, K[[m]] = K[n](n/2)], 0 x D(K) are as in [L12, 0.2]. For K ∈ M (X), gr K is as in [L12, 0.2]. For m j K ∈ D (X), K{i} = gr (Ki)(i/2), is as in [L12, 0.2]. m i If K ∈ M(X) and A is a simple object of M(X) we denote by (A : K) the multiplicity of A in a Jordan-H¨older series of K. The notation C ≎ {C ;i ∈ I} is i as in [L12, 0.2]. If X,X′ are algebraic varieties over k, we say that a map of sets f : X −→ X′ is a quasi-morphism if for some F -rational structure on X and X′ with Frobe- q nius maps F and F′ and some integer t ≥ 0, fFt : X −→ X′ is a morphism equal to F′tf. If, in addition, fF = F′t then we have well defined functors f : D (X) −→ D (X′), f∗ : D (X′) −→ D (X) such that f is the composition ! m m m m ! of usual functors (fFt) (Ft)∗ = (F′t)∗(F′tf) and f∗ is the composition of usual ! ! functors (Ft) (fFt)∗ = (F′tf)∗(F′t) . Theusual properties off ,f∗ for morphisms ! ! ! continue to hold for quasi-morphisms. We will denote by p the variety consisting of one point. For any variety X let L = α Q¯ ∈ D X where α : X × T −→ X is the obvious projection. We X ! l m sometimes write L instead of L . X Let v be an indeterminate. For any φ ∈ Q[v,v−1] and any k ∈ Z we write (k;φ) for the coefficient of vk in φ. Let A = Z[v,v−1]. Contents 1. Truncated induction. 2. Truncated restriction. 3. Truncated convolution from Gǫ,s ×Gǫ′,s′ to Gǫǫ′,s+s′. 4. Analysis of the composition ζ χ . ǫ,s ǫ,s 5. Adjunction formula (weak form). 6. Equivalence of CcG with the ǫ-centre of CcB2. ǫ,s 7. Relation with Soergel bimodules. 1. Truncated induction 1.1. For y ∈ W let L ∈ D (B2) be the constructible sheaf which is Q¯ (with y m l the standard mixed structure of pure weight 0) on O and is 0 on B2 − O ; let y y L♯ ∈ D (B2) be its extension to an intersection cohomology complex of O¯ (equal y m y to 0 on B2 −O¯ ). Let L = L♯[[|y|+ν]] ∈ D (B2). y y y m Let r ≥ 1. For w = (w ,w ,...,w ) ∈ Wr we set |w| = |w |+···+|w |. Let 1 2 r 1 r L[1,r] ∈ D (Br+1) be as in [L12, 1.1]. For any J ⊂ [1,r] let LJ ∈ D (Br+1), w m w m L˙J ∈ D (Br+1) be as in [L12, 1.1]. As in [L12, 1.1(a)], we have a distinguished w m triangle (a) (LJ ,L[1,r],L˙J ) w w w 4 G. LUSZTIG in Dm(Br+1). For any i < i′ in [1,r] let pi,i′ : Br+1 −→ B2 be the projection to the i,i′ factors. For 1L,2L,...,rL in D (B2) we set m 1L•2L•...•rL = p (p∗ 1L⊗p∗ 2L⊗...⊗p∗ rL) ∈ D (B2). 0r! 01 12 r−1,r m 1.2. Let H be the free A-module with basis {T ;w ∈ W}. It is well known that w H has a unique structure of associative A-algebra with 1 = T (Hecke algebra) 1 such that TwTw′ = Tww′ if w,w′ ∈ W, |ww′| = |w|+|w′| and Ts2 = 1+(v−v−1)Ts if s ∈ W, |s| = 1. Let {c ;w ∈ W} be the “new” basis of H defined as in [L10, w 5.2] with L(w) = |w|. For x,y ∈ W, the relations x (cid:22) y, x ∼ y, x ∼ y on W are defined as in [L12, L 1.3]. If c is a two-sided cell of W and w ∈ W, the relations w (cid:22) c, c (cid:22) w, w ≺ c, c ≺ w are defined as in [L12, 1.3]. If c,c′ are two-sided cells of W, the relations c (cid:22) c′, c ≺ c′ are defined as in [L12, 1.3]. Let a : W −→ N be the a-function in [L10, 13.6]. If c is a two-sided cell of W, then for all w ∈ c we have a(w) = a(c) where a(c) is a constant. Let J be the free Z-module with basis {t ;z ∈ W} with the structure of asso- z ciative ring (with 1) as in [L12, 1.3]. For a two-sided cell c of W let Jc be the subgroup of J generated by {t ;z ∈ c}; it is a subring of J with unit element z t where D is the set of distinguished involutions of c. We have J = ⊕ Jc Pd∈Dc d c c as rings. For E ∈ IrrW we define a simple Q⊗J-module E and a simple Q(v)⊗ H- ∞ A module E(v) as in [L12, 1.3]; there is a unique two-sided cell c of W such that E JcEE 6= 0. ∞ Let ǫ ∈ fA. Let E ∈ IrrW. We say that E ∈ Irr W if tr(ǫ(w),E) = tr(w,E) ǫ for any w ∈ W. In this case there exists a linear transformation of finite order ǫ : E −→ E such that e we−1 = ǫ(w) : E −→ E for any w ∈ W; moreover e E E E E is unique up to multiplication by −1. See [L1, 3.2]). For each E ∈ Irr W we ǫ choose e as above. As a Q-vector space we have E = E, E(v) = Q(v)⊗ E E ∞ Q hence, if E ∈ Irr W, e˜: E −→ E can be viewed as a Q-linear map (of finite order) ǫ e˜: E −→ E and as a Q(v)-linear map (of finite order) e˜: E(v) −→ E(v). From ∞ ∞ the definitions we see that e˜t e˜−1 = t : E −→ E and e˜T e˜−1 = T : w ǫ(w) ∞ ∞ w ǫ(w) E(v) −→ E(v) for any w ∈ W. If E ∈ Irr W then ǫ(c ) = c . Let Irr W = {E ∈ Irr W;c = c}. ǫ E E ǫ,c ǫ E 1.3. For any ǫ ∈ A,s ∈ Z let G be the set of bijections F : G −→ G such that ǫ,s (i) if s > 0 then F is the Frobenius map for an F -rational structure on G; ps (ii) if s < 0 then F−1 is the Frobenius map for an F -rational structure on p−s G; (iii) if s = 0 then F is an automorphism of G; moreover in each case (i)–(iii) we require that the following holds: for any w ∈ W and any (B,B′) ∈ O we have (F(B),F(B′)) ∈ O . w ǫ(w) (If ǫ = 1,s = 0 we can identify G and G by g 7→ Ad(g).) Now G acts on G by ǫ,s ǫ,s g : F 7→ Ad(g)FAd(g−1). If s 6= 0, this action is transitive and the stabilizer of a UNIPOTENT REPRESENTATIONS AS A CATEGORICAL CENTRE 5 point F ∈ G is the finite group GF = {g ∈ G;F(g) = g}. For any s ∈ Z and any ǫ,s F˜ ∈ G , the maps λ : G −→ G , g 7→ Ad(g)F˜ and λ′ : G −→ G , g 7→ F˜Ad(g) ǫ,s ǫ,s ǫ,s are bijections (by Lang’s theorem); we use λ (resp. λ′) to view G with s ≥ 0 ǫ,s (resp. s ≤ 0) as an affine algebraic variety isomorphic to G; this algebraic variety structure on G is independent of the choice of F˜. We have dimG = ∆. The ǫ,s ǫ,s G-action above on G is an algebraic group action. When X = G then the ǫ,s ǫ,s subscript in D (X),M (X) refers to the F -structure with Frobenius map m m m ps0 F 7→ Fs0FF−s0 (with F ,s as in 0.3). 0 0 0 0 Note that ⊔ǫ∈A,s∈ZGǫ,s is a group under composition of maps: if F ∈ Gǫ,s,F′ ∈ Gǫ′,s′ then FF′ ∈ Gǫǫ′,s+s′. (It is enough to show that for some F ∈ Gǫ,s,F′ ∈ Gǫ′,s′ we have FF′ ∈ Gǫǫ′,s+s′. We take F = Ad(γ)F0s, F′ = Ad(γ′)F0s′ where γ ∈ Gǫ,1 and γ′ ∈ Gǫ′,1 commute with F0; then FF′ = Ad(γγ′)F0s+s′ and γγ′ ∈ Gǫǫ′,1 commutes with F0 hence FF′ ∈ Gǫǫ′,s+s′.) Note that the composition Gǫ,s × Gǫ′,s′ −→ Gǫǫ′,s+s′ is not in general a morphism of algebraic varieties but only a quasi-morphism (see 0.3), which is good enough for our purposes. Until the end of Section 2 we fix ǫ ∈ A. Let s ∈ Z. We consider the maps B2 ←f− X −→π G where ǫ,s ǫ,s X = {(B,B′,F) ∈ B×B ×G ;F(B) = B′}, ǫ,s ǫ,s f(B,B′,F) = (B,B′),π(B,B′,F) = F. Now L 7→ χ (L) = π f∗L defines a functor D (B2) −→ D (G ). (When ǫ = ǫ,s ! m m ǫ,s 1,s = 0, c coincides with the functor χ defined in [L12, 1.5]). For i ∈ Z,L ∈ ǫ,s D (B2) we write χi (L) instead of (χ (L))i. For any z ∈ W we set R = m ǫ,s ǫ,s ǫ,s,z χ (L♯) ∈ D (G ). (When ǫ = 1,s = 0 this is the same as R in [L12, 1.5].) ǫ,s z m ǫ,s z Let b : G −→∼ G be the isomorphism F 7→ F−1 and let b′ : B2 −→∼ B2 ǫ−1,−s ǫ,s be the isomorphism (B,B′) 7→ (B′,B). From the definitions we see that for L ∈ D (B2) we have χ (b′L) = b χ (L). m ǫ,s ! ! ǫ−1,−s Let CS(G ) be a set of representatives for the isomorphism classes of simple ǫ,s perverse sheaves A ∈ M(G ) such that (A : Rj ) 6= 0 for some z ∈ W,j ∈ Z. ǫ,s ǫ,s,z (When ǫ = 1,s = 0 this agrees with the definition of CS(G) in [L12, 1.5].) Now let A ∈ CS(G ). We associate to A a two-sided cell c as follows. ǫ,s A Assume first that s 6= 0. Since A is G-equivariant and the conjugation action of G on G is transitive, for any F ∈ G we have A| = r [∆] where r is ǫ,s ǫ,s {F} A,F A,F an irreducible GF-module. From the definitions, for any z ∈ W and any F ∈ G ǫ,s we have (A : Rj ) = (r : IHj−∆{(B;(B,FB)∈ O¯ }) s,z A,F z GF where the right hand side is the multiplicity of r in the GF-module A,F IHj−∆{(B;(B,FB)∈ O¯ }; z 6 G. LUSZTIG here IH denotes intersection cohomology with coefficients in Q¯ . In particular, l r is a unipotent representation of GF. By [L1, 3.8], for any A ∈ CS(G ), any A,F ǫ,s F ∈ G , any z ∈ W and any j ∈ Z we have ǫ,s (r : IHj−∆{(B;(B,FB)∈ O¯ }) A,F z GF = (j −∆−|z|;(−1)j−∆ X cA,E,e˜tr(e˜cz,E(v))) E∈IrrǫW or equivalently (a) (A : Rǫj,s,z) = (j −∆−|z|;(−1)j−∆ X cA,E,e˜tr(e˜cz,E(v))) E∈IrrǫW where c are uniquely defined rationalnumbers; now (a) also holds when s = 0, A,E,ǫ see [L12, 1.5(a)] when ǫ = 1 and [L6, 34.19, 35.22], [L8, 44.7(e)] for general ǫ. Moreover, if s 6= 0 then, by [L1, 6.17], given A as above, there is a unique two- sided cell c of W such that ǫ(c ) = c and c = 0 whenever E ∈ Irr W A A A A,E,ǫ ǫ satisfies c 6= c . The same holds when s = 0, see [L12, 1.5] when ǫ = 1 and [L7, E A §41] for general ǫ. When s 6= 0, c differs from the two-sided cell associated to r in [L1, 4.23] A A,F by multiplication on the left or right by w . Similarly, when s = 0, c differs max A from the two-sided cell associated to A in [L7, §41] by multiplication on the left or right by w . max As in [L12, 1.5(b)], for s ∈ Z we have (b) (A : Rj ) 6= 0 for some z ∈ c ,j ∈ Z and conversely, if (A : Rj ) 6= 0 ǫ,s,z A ǫ,s,z for z ∈ W,j ∈ Z, then c (cid:22) z. A For s ∈ Z, A ∈ CS(G ) let a be the value of the a-function on c . If z ∈ ǫ,s A A W,E ∈ Irr W satisfy tr(e˜c ,E(v)) 6= 0 then c (cid:22) z; if in addition we have z ∈ c , ǫ z E E then tr(e˜c ,E(v)) = γ vaE +lower powers of v z z,E,e˜ where γ ∈ Z and a is the value of the a-function on c . Hence from (a) we z,E,ǫ E E see that (c) (A : Rj ) = 0 unless c (cid:22) z and, if z ∈ c , then ǫ,s,z A A (A : Rj ) ǫ,s,z = (−1)j+∆(j −∆−|z|;( X cA,E,e˜γz,E,e˜)vaA + lower powers of v)) E∈IrrǫW;cE=cA which is 0 unless j −∆−|z| ≤ a . A In the remainder of this section we fix a two-sided cell c of W such that ǫ(c) = c; we set a = a(c). For s ∈ Z and Y = G or Y = B2 let M♠Y be the category of perverse sheaves ǫ,s UNIPOTENT REPRESENTATIONS AS A CATEGORICAL CENTRE 7 on Y whose composition factors are all of the form A ∈ CS(G ), when Y = G , ǫ,s ǫ,s or of the form L with z ∈ W, when Y = B2. Let M(cid:22)Y (resp. M≺Y) be the z category of perverse sheaves on Y whose composition factors are all of the form A ∈ CS(G ) with c (cid:22) c (resp. c ≺ c), when Y = G , or of the form L with ǫ,s A A ǫ,s z z (cid:22) c (resp. z ≺ c) when Y = B2. Let D♠Y (resp. D(cid:22)Y or D≺Y) be the category of all K ∈ D(Y) such that Ki ∈ M♠Y (resp. Ki ∈ M≺Y or Ki ∈ M≺Y) for all i ∈ Z. Let M♠Y (or M(cid:22)Y, or M≺Y) be the category of all K ∈ M Y m m m m which are also in M♠Y (or M(cid:22)Y or M≺Y). Let D♠Y (or D(cid:22)Y, or D≺Y) be m m m the category of all K ∈ D Y which are also in D♠Y (or D(cid:22)Y or D≺Y). From (c) m we deduce: (d) If z (cid:22) c, then Rj ∈ M(cid:22)G for all j ∈ Z. If z ∈ c and j > a+∆+|z|, ǫ,s,z ǫ,s then Rj ∈ M≺G . If z ≺ c then Rj ∈ M≺G for all j ∈ Z. ǫ,s,z ǫ,s ǫ,s,z ǫ,s Lemma 1.4. Let s ∈ Z. Let r ≥ 1, J ⊂ [1,r], J 6= ∅ and w = (w ,w ,...,w ) ∈ 1 2 r Wr. Let E = ∆+ra. (a) Assume that w ∈ c for some i ∈ [1,r]. If j ∈ Z (resp. j > E) then i χj (p L[1,r][|w|]) is in M(cid:22)G (resp. M≺G ). ǫ,s 0r! w ǫ,s ǫ,s (b) Assume that w ∈ c for some i ∈ J. If j ∈ Z (resp. j ≥ E) then i χj (p L˙J [|w|]) is in M(cid:22)G (resp. M≺G ). ǫ,s 0r! w ǫ,s ǫ,s (c) Assume that w ∈ c for some i ∈ J. If j ≥ E then the cokernel of the map i χj (p LJ [|w|]) −→ χj (p L[1,r][|w|]) ǫ,s 0r! w ǫ,s 0r! w associated to 1.1(a) is in M≺G . ǫ,s (d) Assume that w ∈ c for some i ∈ J. If j ∈ Z (resp. j > E) then i χj (p LJ [|w|]) is in M(cid:22)G (resp. M≺G ). ǫ,s 0r! w ǫ,s ǫ,s (e) Assume that w ≺ c for some i ∈ J. If j ∈ Z then χj (p L[1,r][|w|]) ∈ i ǫ,s 0r! w M≺G and χj (p LJ [|w|]) ∈ M≺G . ǫ,s ǫ,s 0r! w ǫ,s When ǫ = 1,s = 0 this is just [L12, 1.6]; the proof in the general case is entirely similar (it uses 1.3(b), 1.3(c)). 1.5. Let s ∈ Z. Let CS = {A ∈ CS(G );c = c}. For any z ∈ c we set ǫ,s,c ǫ,s A n = a+∆+|z|. Let A ∈ CS and let z ∈ c. We have: z ǫ,s,c (a) (A : Rsn,zz) = (−1)a+|z| X cA,E,e˜tr(e˜tz,E∞). E∈Irrǫ,cW When ǫ = 1,s = 0 this is just [L12, 1.7(a)]. In the general case, from 1.3(a) we have (A : Rsn,zz) = (−1)a+|z| X cA,E,e˜(a;tr(e˜cz,E(v))) E∈IrrǫW and it remains to use that (a;tr(e˜c ,E(v)) is equal to tr(e˜t ,E ) if E ∈ Irr W z z ∞ ǫ,c and to 0, otherwise. We have: (b) For any A ∈ CSǫ,s,c there exists z ∈ c such that (A : Rǫn,zs,z) 6= 0. The proof, based on (a), is the same as that in the case ǫ = 1,s = 0 given in [L12, 1.7(b)]. 8 G. LUSZTIG Let c0 = {z ∈ c;z ∼ ǫ(z−1)}. If z ∈ c − c0 and E ∈ Irr W, then L ǫ,c tr(e˜t ,E ) = 0. (We can write E = ⊕ t E and e˜t : E −→ E maps the z ∞ ∞ d∈Dc d ∞ z ∞ ∞ summand tdE∞ (where z ∼L d) into tǫ(d′)E∞ (where d′ ∈ Dc, d′ ∼L z−1) and all other summands to 0. If tr(e˜tz,E∞) 6= 0, we must have tdE∞ = tǫ(d′)E∞ 6= 0 and d = ǫ(d′) and z ∼ ǫ(z−1).) From this and (a) we deduce L (c) If z ∈ c−c0, then Rnz = 0. ǫ,s,z 1.6. Let s ∈ Z. For Y = G or B2 let C♠Y be the subcategory of M♠Y ǫ,s consisting of semisimple objects; let C♠Y be the subcategory of M Y consisting 0 m of those K ∈ M Y such that K is pure of weight 0 and such that as an object m of M(Y), K belongs to C♠Y. Let CcY be the subcategory of M♠Y consisting of objects which are direct sums of objects in CS (if Y = G ) or of the form L ǫ,s,c ǫ,s z with z ∈ c (if Y = B2). Let CcY be the subcategory of C♠Y consisting of those 0 0 K ∈ C♠Y such that as an object of C♠Y, K belongs to CcY. For K ∈ C♠Y, let K 0 0 be the largest subobject of K such that, as an object of C♠Y, we have K ∈ CcY. For L ∈ C♠B2 we define ǫL ∈ C♠B2 as follows. We have canonically L = 0 0 ⊕ V ⊗ L where V are finite dimensional Q¯ -vector spaces; we set ǫL = y∈W y y y l ⊕ V ⊗L . We show: y∈W y ǫ−1(y) (a) Let s ∈ N. Define u : G ×B2 −→ G ×B2 by ǫ,s ǫ,s (F,(B ,B )) 7→ (F,F(B ),F(B )) 1 2 1 2 and let L ∈ C♠B2. We have canonically u∗(Q¯ ⊠L) = Q¯ ⊠ǫL. 0 l l We can assume that L = L where y ∈ W; we must show that u∗(Q¯ ⊠ L ) = y l y Q¯ ⊠ L or that u∗(Q¯ ⊠ L♯) = Q¯ ⊠ L♯ . Now Q¯ ⊠ L♯ is the intersec- l ǫ−1(y) l y l ǫ−1(y) l y tion cohomology complex of G ×O¯ with coefficients in Q¯ (extended by 0 on ǫ,s y l G ×(B2 −O¯ )). Hence u∗(Q¯ ⊠L♯) is the intersection cohomology complex of ǫ,s y l y u−1(G × O¯ ) with coefficients in Q¯ (extended by 0 on G × u−1(B2 − O¯ )) ǫ,s y l ǫ,s y that is, the intersection cohomology complex of G ×O¯ with coefficients in ǫ,s ǫ−1(y) Q¯ (extended by 0 on G ×(B2 −O¯ )). This is Q¯ ⊠L♯ , as required. l ǫ,s ǫ−1(y) l ǫ−1(y) Assume that s ∈ Z and let F ∈ G . For any A ∈ C♠G we have A| = >0 ǫ,s ǫ,s {F} r [∆] where r ∈ Rep♠(GF) (see 0.1). Moreover, from the definitions we see A,F A,F that (b) A 7→ r is an equivalence of categories CcG −→∼ Repc(GF) (see 0.1). A,F ǫ,s Proposition 1.7. Let s ∈ Z. (a) If L ∈ D(cid:22)B2 then χ (L) ∈ D(cid:22)G . If L ∈ D≺B2, then χ (L) ∈ D≺G . ǫ,s ǫ,s ǫ,s ǫ,s (b) If L ∈ M(cid:22)B2 and j > a+ν +ρ then χj (L) ∈ M≺G . ǫ,s ǫ,s When ǫ = 1,s = 0 this is just [L12, 1.9]; the proof in the general case is entirely similar (it uses 1.4(a),(e)). UNIPOTENT REPRESENTATIONS AS A CATEGORICAL CENTRE 9 1.8. Let s ∈ Z. For L ∈ CcB2 we set 0 χ (L) = (χa+ν+ρ(L)((a+ν +ρ)/2) = (χ (L)){a+ν+ρ} ∈ CcG . ǫ,s ǫ,s ǫ,s 0 ǫ,s The functor χ : CcB2 −→ CcG is called truncated induction. For z ∈ c we have ǫ,s 0 0 ǫ,s (a) χ (L ) = Rnz (n /2). ǫ,s z ǫ,s,z z When ǫ = 1,s = 0 this is just [L12, 1.10(a)]; the proof in the general case is entirely similar. We shall denote by τ : Jc −→ Z the group homomorphism such that τ(t ) = 1 z if z ∈ D and τ(t ) = 0, otherwise. For z,u ∈ c we have: c z (b) dimHomCcGǫ,s(χǫ,s(Lz),χǫ,s(Lu)) = Xτ(ty−1tztǫ(y)tu−1). y∈c When ǫ = 1,s = 0 this is just [L12, 1.10(b)]. We now consider the general case. Using (a) and the definitions we see that the left hand side of (b) equals X (A : Rǫn,zs,z)(A : Rǫn,us,u), A∈CSǫ,s,c hence, using 1.5(a) it equals X (−1)|z|+|u| X cA,E,e˜cA,E′,e˜tr(e˜tz,E∞)tr(e˜tu,E∞′ ). E,E′∈Irrǫ,cW A∈CSǫ,s,c Replacing in the last sum PA∈CSǫ,s,c cA,E,e˜cA,E′,e˜ by 1 if E = E′ and by 0 if E 6= E′ (see [L1, 3.9] in the case s 6= 0 and [L3, 13.12], [L6, 35.18(g)] in the case s = 0) we obtain X (−1)|z|+|u|tr(e˜tz,E∞)tr(e˜tu,E∞). E∈Irrǫ,cW This is equal to (−1)|z|+|u| times the trace of the operator ξ 7→ t ǫ(ξ)t on z u−1 Q⊗Jc (see [L6, 34.14(a), 34.17]). The last trace is equal to the sum over y ∈ c of the coefficient of t in t t t ; this coefficient is equal to τ(t t t t ) y z ǫ(y) u−1 y−1 z ǫ(y) u−1 since for y,y′ ∈ c, τ(ty′ty) is 1 if y′ = y−1 and is 0 if y′ 6= y−1 (see [L10, 20.1(b)]). Thus we have dimHomCcGǫ,s(χǫ,s(Lz),χǫ,s(Lu)) = (−1)|u|+|z|Xτ(ty−1tztǫ(y)tu−1). y∈c Since dimHomCcGǫ,s(χǫ,s(Lz),χǫ,s(Lu)) ∈ N and Py∈cτ(ty−1tztǫ(y)tu−1) ∈ N, it follows that (b) holds. 10 G. LUSZTIG Lemma 1.9. Let s ∈ N. Let Y ,Y be among G ,B2 and let X ∈ D(cid:22)Y . Let 1 2 ǫ,s m 1 c,c′ be integers and let Φ : D(cid:22)Y −→ D(cid:22)Y be a functor which takes distinguished m 1 m 2 triangles to distinguished triangles, commutes with shifts, maps D≺Y into D≺Y m 1 m 2 and maps complexes of weight ≤ i to complexes of weight ≤ i (for any i). Assume that (a),(b) below hold: (a) (Φ(X ))h ∈ M≺Y for any X ∈ M(cid:22)Y and any h > c; 0 m 2 0 m 1 (b) X has weight ≤ 0 and Xi ∈ M≺Y for any i > c′. 1 Then (c) (Φ(X))j ∈ M≺Y for any j > c+c′, 2 and we have canonically (d) (Φ(X{c′})){c} = (Φ(X)){c+c′}. When ǫ = 1,s = 0 this isjust [L12, 1.12]; the proofin thegeneral case isentirely similar. 1.10. Let s ∈ Z. Let L ∈ CcB2. We have D(L) ∈ CcB2. Moreover we have 0 0 canonically: (a) χ (D(L)) = D(χ (L)). ǫ,s ǫ,s When ǫ = 1,s = 0 this is just [L12, 1.13]; the proof in the general case is entirely similar. 2. Truncated restriction 2.1. Recall that ǫ ∈ A is fixed. In this section we fix s ∈ Z. Let π,f be as in 1.3. Now K 7→ ζ (K) = f π∗K defines a functor D (G ) −→ D (B2). (When ǫ,s ! m ǫ,s m ǫ = 1,s = 0, ζ is the same as ζ of [L12, 2.5].) For i ∈ Z,K ∈ D (G ) we write ǫ,s m ǫ,s ζi (K) instead of (ζ (K))i. ǫ,s ǫ,s Let b : G −→∼ G , b′ : B2 −→∼ B2 be as in 1.3. From the definitions we see ǫ−1,−s ǫ,s that for K ∈ D (G ) we have m ǫ−1,−s (a) ζ (b K) = b′ζ (K). ǫ,s ! ! ǫ−1,−s