Character Sheaves and GGGRs

Transcription

1 and GGGRs Jay Taylor Technische Universität Kaiserslautern Global/Local Conjectures in Representation Theory of Finite Groups Banff, March 2014 Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

2 Introduction The Problem G a connected reductive algebraic group defined over F p. F : G G a Frobenius endomorphism defining an F q -rational structure G F = {g G F (g) = g}. Fix a prime l p and an algebraic closure Q l. Interested in Irr(G F ) Cent(G F ) = {f : G F Q l f (xgx 1 ) = f (x)} Problem Given g G F and χ Irr(G F ) describe χ(g). Two main cases to consider: g G F ss = {x G F p o(x)} g G F uni = {x GF o(x) = p a } Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

3 Introduction The Semisimple Case For any F -stable maximal torus T G and θ Irr(T F ) we have a virtual character R G T (θ) Z Irr(GF ). Theorem (Deligne Lusztig, 1976) For any χ Irr(G F ) and s G F ss we have χ(s) = RT G (θ), χ RG T (θ)(s) (T,θ)/ and R G T (θ)(s) = 1 C G (s)f x G F x 1 sx T F θ(x 1 sx). Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

4 Definition DG := the bounded derived category of Q l -constructible sheaves on G M G := the category of Q l -perverse sheaves on G Can think of an object A DG as a bounded complex A i 1 A i A i+1 of Q l -sheaves on G such that for each i Z the cohomology sheaf H i (A) is constructible. In particular, for each x G, the stalk Hx i (A) is a finite dimensional Q l -vector space. Furthermore we have Hx i (A) 0 for only finitely many i Z. Definition A character sheaf of G is a G-equivariant simple object in M G. We denote by Ĝ the set of character sheaves of G. Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

5 Characteristic Functions The Frobenius endomorphism F : G G induces a functor F : DG DG which preserves Ĝ. We say A DG is F -stable if there exists an isomorphism φ A : F A A DG. We denote by ĜF Ĝ the subset of F -stable character sheaves. Definition Assume now that A ĜF. For each x G F and i Z we have H i x (F A) = H i F (x) (A) = H i x (A) and φ A induces an automorphism φ A : H i x (A) H i x (A). We define the characteristic function of A to be χ A,φA : G F Q l given by χ A,φA (g) = i Z ( 1) i Tr(φ A, H i g (A)). Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

6 Characteristic Functions Theorem (Lusztig, 1986, 2012) There exists a family of isomorphisms {φ A : F A A A ĜF } (unique up to multiplication by roots of unity) such that {χ A,φA A ĜF } is an orthonormal basis for Cent(G F ). Definition We say A u G uni. Ĝ is unipotently supported if H i u (A) 0 for some i Z and Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

7 Induction Assume P G is a parabolic with Levi complement L P. Lusztig has defined a map A 0 L ind G L P (A 0) M G called induction. The complex ind G L P (A 0) satisfies the following properties: ind G L P (A 0) = A 0 if L = P = G. ind G L P (A 0) is semisimple and all indecomposable summands are character sheaves. for any A Ĝ there exists a Levi subgroup L P and a cuspidal character sheaf A 0 L such that (A : ind G L P (A 0)) 0. Furthermore the pair (L, A 0 ) is unique up to G-conjugacy. Definition We say A Ĝ is cuspidal if (A : indg L P (A 0)) 0 implies L = P = G. Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

8 Cuspidal Objects Theorem (Lusztig) If A 0 L is cuspidal and unipotently supported then A 0 = IC(O 0 Z (L), E 0 b L )[dim O 0 + dim Z (L)] where: O 0 G is a unipotent conjugacy class, E 0 is an L-equivariant cuspidal local system on O 0, L is a tame local system on Z (L). Furthermore, the quotient group W G (L) = N G (L)/L is a Weyl group and End DG (ind G L P (A 0)) = Q l W G (L, L ) In particular, we have a bijection {A Ĝ (A : indg L (A 0)) 0} Irr(W G (L, L )) Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

9 The Generalised Springer Correspondence Denote by N G the set of all pairs ι = (O ι, E ι ) where: O ι G is a unipotent class, E ι is a G-equivariant local system on O ι. Theorem (Lusztig, 1984) Denote by ν N L the cuspidal pair (O 0, E 0 ) and assume that L = Q l. Then there is a subset I (L, ν) N G and a natural bijection Hence also a bijection I (L, ν) {A Ĝ (A : indg L (A 0)) 0} ι K ι. I (L, ν) Irr(W G (L)) ι E ι. Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

10 The Generalised Springer Correspondence Let A ĜF be an F -stable summand of ind G L (A 0) then we can assume: F (L) = L F (O 0 ) = O 0 F E 0 = E0 F L = L. In particular we have: F induces an automorphism of W G (L) and W G (L, L ), If A is parameterised by E Irr(W G (L, L )) then this is fixed by F. Proposition Assume we fix an isomorphism ϕ 0 : F E 0 E 0 and an extension Ẽ of E to W G (L, L ) F (similarly an extension Ẽι of E ι ). Then this induces isomorphisms φ A : F A A φ ι : F K ι K ι Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

11 The Generalised Springer Correspondence Theorem (T., 2014) χ A,φA G F uni = Theorem (Lusztig, T.) ι I (L,ν) F Ẽι, Ind WG(L).F W G (L,L ).F (Ẽ) W G (L).F χ Kι,φ ι Let a ι = dim O ι dim Z (L) then we have χ Kι,φ ι = ( 1) aι q (dim G+aι)/2 P ι,ιy ι Theorem (Bonnafé, Shoji, Waldspurger) Assume p is good for G and one of the following holds: Z(G) is connected and G/Z(G) is simple, G is SL n (F p ), Sp 2n (F p ) or SO n (F p ). Then the functions Y ι are explicitly computable. Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

12 Applications Generalised Gelfand Graev Representations Assume now that p is good for G. By Kawanaka (1986) we have a map u G F uni γ u Cent(G) where γ u is the character of a generalised Gelfand Graev representation. These satisfy the following properties: γ u is obtained by inducing a linear character from a p-subgroup of G F, γ u = γ v if xux 1 = v for some x G F, γ 1 is the regular character and γ u is a Gelfand Graev character when u is a regular element. Problem Describe the multiplicities γ u, χ for all χ Irr(G F ). Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

13 Applications The Case of GL n(q) Consider G F = GL n (q) and B the upper triangular matrices then Ind GF B F (1 B F ) = ρ Irr(S n) ρ(1)χ ρ and E(G F, 1) = {χ λ λ n} is the set of unipotent characters. Theorem (Kawanaka) Γ µ, χ λ = { 1 if λ = µ 0 if λ µ Example χ (n) = 1 GLn(q) occurs in the regular representation with multiplicity 1 and in no other GGGR. Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14

14 Applications Multiplicities If p and q are sufficiently large then Lusztig has given an explicit decomposition γ u {χ Kι,φ ι ι N F G } and has conjectured an explicit decomposition χ Irr(G F ) {χ A,φA A ĜF } If we solve this conjecture then the multiplicity γ u, χ can be reduced to the multiplicities χ A,φA, χ Kι,φ ι and these are given by our main theorem! Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March / 14