Evaluating Characteristic Functions of Character Sheaves at Unipotent Elements

Jay Taylor

Abstract. Assume G is a connected reductive algebraic group de�ned over an algebraic closure = \overline{}_{p} of the �nite �eld of prime order p > 0. Furthermore, assume that F : G → G is a Frobenius endomorphism of G. In this article we give a formula for the value of any F-stable character sheaf of G at a unipotent element. This formula is expressed in terms of class functions of G^{F} which are supported on a single unipotent class of G. In general these functions are not determined, however we give an expression for these functions under the assumption that Z(G) is connected, G/Z(G) is simple and p is a good prime for G. In this case our formula is completely explicit.

1. Introduction

1.1. Throughout this article G will denote a connected reductive algebraic group de�ned over an algebraic closure = \overline{}_{p} of the �nite �eld of prime order p > 0. Furthermore F : G → G will denote a Frobenius endomorphism de�ning an _{q}-rational structure G := G^{F}. If H is a �nite group then we will denote by Cent(H) the space of all class functions f : H → \overline{}_{ℓ} where ℓ ≠ p is a prime and \overline{}_{ℓ} is an algebraic closure of the �eld of ℓ-adic numbers.

1.2. In [Lus85] Lusztig has de�ned a set Ĝ of G-equivariant \overline{}_{ℓ}-perverse sheaves on G known as character sheaves. These geometric objects have a conjectural relationship to the irreducible characters of G. Through this relationship one sees that the character theory of G is intimately related to the geometry of G. Let A ∈ Ĝ be an F-stable character sheaf, i.e. there exists an isomorphism ϕ_{A} : F^{∗}A → A, then Lusztig has associated to A and ϕ_{A} a class function χ_{A,ϕ_{A}} ∈ Cent(G) whose de�nition depends heavily upon the choice of the isomorphism ϕ_{A}. In this article we consider the following problem.

Problem 1.3. Given g ∈ G can we explicitly determine χ_{A,ϕ_{A}}(g)?

1.4. For every character sheaf A ∈ Ĝ we assume �xed an appropriately chosen isomorphism ϕ_{A} : F^{∗}A → A, when one exists, such that the resulting set of functions B = {χ_{A,ϕ_{A}}} is an orthonormal basis of Cent(G). The problem considered above is motivated by Lusztig’s conjecture, mentioned above, which is stated in [Lus84a]. Speci�cally, Lusztig conjectures an explicit description for the change of basis matrix between B and Irr(G) (the set of all irreducible characters of G). Thus, having a solution to Problem 1.3 together with a solution to Lusztig’s conjecture would provide us with a method for determining the values of the irreducible characters of G. In fact, to prove Lusztig’s conjecture one already requires a detailed understanding of the values of the functions in B (see for instance [Lus86; Sho97]).

2010 Mathematics Subject Classi�cation. Primary 20C33; Secondary 20G40.

Keywords and phrases. Finite reductive groups, character sheaves, unipotent elements.

\underline{2}

1.5. In this article we will consider Problem 1.3 when the element g ∈ G is unipotent. Let us denote by N_{G} the set of all pairs (O, ℰ) where O is a unipotent conjugacy class of G and ℰ is a G-equivariant local system on O. This set is partitioned into subsets ℐ [L, ν] called blocks where L ⩽ G is a Levi complement of a parabolic subgroup of G and ν ∈ N_{L} is a cuspidal pair. The map (O, ℰ) ↦ (F^{−1} (O), F^{∗}ℰ) de�nes a natural action of F on the set N_{G}, compatible with the blocks, and we denote by N_{G}^{F} the elements �xed under this action. If ι ∈ ℐ [L, ν] ∩ N_{G}^{F} then we may, and will, assume that F(L) = L and ν ∈ N_{L}^{F}.

For any ι ∈ ℐ [L, ν]^{F} we have a corresponding irreducible character E_{ι}∈ Irr(W_{G}(L)) where W_{G}(L) = N_{G}(L)/L is the relative Weyl group of L. Note that F induces an automorphism of W_{G}(L), hence an action on the set Irr(W_{G}(L)). This correspondence gives a bijection ℐ [L, ν]^{F} → Irr(W_{G}(L))^{F}. Furthermore, if ι = (O, ℰ) then we have a corresponding class function Y_{ι}∈ Cent(G) which is supported only on O^{F}.

Assume now that A ∈ Ĝ is an F-stable unipotently supported character sheaf. Then there exists an F-stable Levi subgroup L, a cuspidal pair ν = (O_{0}, ℰ_{0}) ∈ N_{L}^{F} and a tame F-stable local system ℒ on Z^{∘}(L) such that A is a summand of the induced complex ind_{L}^{G}(A_{ℒ} ) where

A_{ℒ} = IC(\overline{O_{0}}Z^{∘}(L), ℰ_{0} ℒ)[dim O_{0} + dim Z^{∘}(L)] ∈ L̂^{F}.

The summands of the induced complex ind_{L}^{G}(A_{ℒ} ) are in bijective correspondence with the set of irreducible characters Irr(W_{G}(L, ℒ)) where W_{G}(L, ℒ) ⩽ W_{G}(L) is the stabiliser of ℒ. If A ∈ Ĝ corresponds to E ∈ Irr(W_{G}(L, ℒ)) then we have A is F-stable if and only if E is �xed by F. With this we may state our main result (which is stated precisely in Theorem 7.7).

Theorem. For any unipotent element u ∈ G we have

χ_{A,ϕ_{A}}(u) = ∑_{ι′,ι∈ℐ [L,ν]^{F}}⟨Ẽ_{ι}, Ind_{W_{G}(L,ℒ).F}^{W_{G}(L).F} (Ẽ)⟩_{W_{G}(L).F}(−1)^{a_{ι}}q^{(dim G+a_{ι})/2} P_{ι′,ι}Y_{ι′}(u).

Here a_{ι}∈ is an integer associated to ι (c.f. 6.16) and Ẽ_{ι}is the restriction to the coset W_{G}(L).F of an extension of E_{ι}to W_{G}(L) ⋊ ⟨F⟩ (similarly for Ẽ). These extensions are determined by (and determine) the choice of isomorphisms de�ning the functions χ_{A,ϕ_{A}} and Y_{ι′}. Furthermore (P_{ι′,ι}) is a block of a matrix which is computable by a general algorithm (see [Lus85, Theorem 24.4]).

1.6. If the character sheaf A is not unipotently supported then χ_{A,ϕ_{A}}(u) = 0 for all unipotent elements u ∈ G, hence our result covers all F-stable character sheaves. The functions Y_{ι′}are not quite the functions considered in [Lus85, §24]. Here we must scale the functions in [Lus85] by a factor coming from a certain linear character introduced by Bonnaf� e in [Bon04]. This linear character is known in almost all cases, in particular in the case when Z(G) is connected and p is a good prime for G (see 5.7).

All terms in the above formula are explicitly computable, except the functions Y_{ι′}. In Theorem 9.4 we give an explicit formula for these functions whenever Z(G) is connected, G/Z(G) is simple and p is a good prime. Thus under these assumptions our formula is completely explicit and computable. Note that this does not exhaust all cases where these functions are known to be computable. For instance, using Shoji’s results in [Sho07] one could obtain a similar result when G is a special orthogonal or symplectic group (if p ≠ 2 then in this situation Bonnaf� e’s linear character is also known). In the appendix we include results concerning Lusztig�MacDonald�Spaltenstein induction in �nite groups extended by an automorphism, which cyclically permutes isomorphic factors. We hope this to be useful in determining the coset multiplicities occurring in the theorem.

1.7. Our work here is inspired by (and is a generalisation of) that of Lusztig in [Lus86]. In particular, if F is a split Frobenius endomorphism and L is contained in an F-stable parabolic subgroup of G then this result is due to Lusztig (see [Lus86, 2.6(e), 3.2(a), 4.9(a)]). We also note that the work of Bonnaf� e [Bon04] is crucial in dealing with the situation when L is not the complement of an F-stable parabolic subgroup.

Acknowledgments: The author gratefully acknowledges �nancial support from ERC Advanced Grant 291512. He would like to thank Geordie Williamson and Sebastian Herpel for useful conversations and Gunter Malle for his notes on an early version of this paper.

2. Conventions

Perverse Sheaves

2.1. Given a variety X over we denote by 𝒟X = 𝒟_{c}^{b}(X, \overline{}_{ℓ}) the bounded derived category of \overline{}_{ℓ}- constructible sheaves on X. Furthermore we denote by ℳ X the full subcategory of 𝒟X whose objects are the perverse sheaves on X. Assume H is a connected algebraic group acting on X then we take the statement A ∈ 𝒟X is H-equivariant to be de�ned as in [Lus85, §1.9]. If X is itself a connected algebraic group then, unless otherwise explicitly stated, we take X-equivariance to be with respect to the natural conjugation action of X on itself.

We will refer to an element A ∈ 𝒟X as a �complex�. Recall that we may construct for any i ∈ the ith cohomology sheaf ℋ ^{i}A of the complex A, which is a \overline{}_{ℓ}-sheaf. Given any element x ∈ X we then denote by ℋ_{x}^{i}A the corresponding stalk of ℋ ^{i}A. For any A ∈ 𝒟X we call supp(A) := {x ∈ X | ℋ_{x}^{i}A ≠ 0 for some i ∈ } the support of A. If φ : X → Y is a morphism then we denote by φ^{∗} : 𝒟Y → 𝒟X the inverse image functor, φ_{∗} : 𝒟X → 𝒟Y the right derived direct image functor and φ_{!} : 𝒟X → 𝒟Y the right derived direct image functor with compact support. If φ is smooth with connected �bres of dimension d then we denote by φ̃ the shifted inverse image φ^{∗}[d] (c.f. [Lus85, §1.7]).

2.2. Assume X ⊆ Y is a subvariety then for any A ∈ 𝒟Y we denote by A|_{X} the complex i^{∗}A ∈ 𝒟X where i : X ↪ Y is the inclusion map (we call this the restriction of A to X). Assume now that X is a smooth open dense subset of its closure \overline{X} and that ℒ is a local system on X (by which we mean a locally constant \overline{}_{ℓ}-constructible sheaf with �nite dimensional stalks) then ℒ [dim X] ∈ ℳ X is a perverse sheaf on X. We denote by IC(\overline{X}, ℒ)[dim X] ∈ ℳ \overline{X} the intersection cohomology complex determined by ℒ which is an element of ℳ \overline{X} extending ℒ [dim X], i.e. we have IC(\overline{X}, ℒ)|_{X} ≅ ℒ. We may freely consider this as an element of ℳ Y by extending IC(\overline{X}, ℒ)[dim X] to Y by 0 on Y − \overline{X} and we will do so without explicit mention.

2.3. For convenience we recall here the following base change isomorphism (see [Lus85, (1.7.5)]). Assume X, Y, Z and W are varieties over and that we have a commutative diagram of morphisms

X  Y Z  W f ϕ g ψ

such that ϕ and ψ are smooth with connected �bres of common dimension. Then we have an isomorphism f_{!} ∘ ϕ̃ = ψ̃∘ g_{!} of functors 𝒟Y → 𝒟Z. In particular if every �bre of ϕ and ψ is simply a point then we have f_{!} ∘ ϕ^{∗} = ψ^{∗} ∘ g_{!}.

Finite Groups

2.4. Assume A is a \overline{}_{ℓ}-algebra then the statement �M is an A-module� will mean that M is a �nite dimensional left A-module. For any two A-modules M and M′ we will denote by Hom_{A}(M, M′) the space of all A-module homomorphisms f : M → M′. We will denote by Irr(A) a set of representatives for the isomorphism classes of simple A-modules. If G is a �nite group and A is the group algebra \overline{}_{ℓ}G then A-modules will assumed to be left or right modules, as appropriate, and we will write Hom_{G}(M, M′) for Hom_{A}(M, M′). Similarly we will write Irr(G) for Irr(A) which we will also identify with the set of irreducible characters determined by the simple modules.

2.5. Assume now that ϕ : G → G is an automorphism and let us denote by G̃ the semidirect product G ⋊ ⟨ϕ⟩ where ⟨ϕ⟩ ⩽ Aut(G) is the cyclic subgroup generated by the automorphism ϕ. Let H ⩽ G be a subgroup and let g ∈ G be such that ϕ(gHg^{−1} ) = H then H is a normal subgroup of H̃ = H⟨(ϕ(g), ϕ)⟩ ⩽ G̃ and we denote by H.ϕg the set {(hϕ(g), ϕ) ∈ G̃ | h ∈ H} (note that we identify H with its image H × \{1\} in G̃). Furthermore we denote by Cent(H.ϕg) the \overline{}_{ℓ}-vector space of functions f : H.ϕg → \overline{}_{ℓ} which are invariant under conjugation by H. We can de�ne on Cent(H.ϕg) an inner product ⟨−, −⟩_{H.ϕg} : Cent(H.ϕg) × Cent(H.ϕg) → \overline{}_{ℓ} by setting

⟨f, f′⟩_{H.ϕg} = \frac{1}{|H|} ∑_{h∈H} f(hϕ(g), ϕ)\overline{f′(hϕ(g), ϕ)} ,

where : \overline{}_{ℓ} → \overline{}_{ℓ} is a �xed automorphism such that \overline{ω} = ω^{−1} for every root of unity ω ∈ \overline{}_{ℓ}^{×} . We may also de�ne a \overline{}_{ℓ}-linear map Ind_{H.ϕg}^{G.ϕ} : Cent(H.ϕg) → Cent(G.ϕ) by setting

(Ind_{H.ϕg}^{G.ϕ} f)(x, ϕ) = \frac{1}{|H|} ∑_{\begin{matrix}y∈G \\ (y^{−1} xϕ(y),ϕ)∈H.ϕg \end{matrix}} f(y^{−1} xϕ(y), ϕ).

2.6. For any x ∈ G we will denote by ad_{x} : G → G the automorphism given by ad_{x}(h) = xhx^{−1} for all h ∈ G. As the composition ϕ ad_{g} is an automorphism of G we may form, as before, the semidirect product G ⋊ ⟨ϕ ad_{g}⟩. The restriction of ϕ ad_{g} to H is also an automorphism of H so we have H ⋊ ⟨ϕ ad_{g}⟩ is naturally a subgroup of G ⋊ ⟨ϕ ad_{g}⟩. We may now de�ne a surjective homomorphism of groups ψ_{g} : G ⋊ ⟨ϕ ad_{g}⟩ → G̃ given by

ψ_{g}(h, ϕ^{i} ad_{ϕ^{1−i} (g)··· ϕ^{−1} (g)g} ) = (hϕ(g)ϕ^{2}(g) ⋯ϕ^{i}(g), ϕ^{i})

for any i ∈ . The restriction of ψ_{g} de�nes a bijection G.ϕ ad_{g} → G.ϕ (resp. H.ϕ ad_{g} → H.ϕg) which respects the action of G (resp. H) by conjugation. In particular ψ_{g} induces isometries Cent(G.ϕ) → Cent(G.ϕ ad_{g}) and Cent(H.ϕg) → Cent(H.ϕ ad_{g}) and carries the induction map Ind_{H.ϕg}^{G.ϕ} to Ind_{H.ϕ ad_{g}}^{G.ϕ ad_{g}}.

3. Character Sheaves

Recall our assumption that G is any connected reductive algebraic group de�ned over an algebraic closure = \overline{_{p}} of the �nite �eld of prime characteristic p > 0. Also F : G → G is a Frobenius endomorphism of G.

3.1. We assume �xed an F-stable Borel subgroup B_{0} ⩽ G and maximal torus T_{0} ⩽ B_{0} (the assumption of F-stability will not be needed until Section 6). Let Φ_{G} be the roots of G with respect to T_{0} then we will denote by Δ_{G} ⊆ Φ_{G}^{+} ⊆ Φ_{G} the set of simple and positive roots determined by T_{0} ⩽ B_{0}. We will denote by (W_{G}, ) the Coxeter system of G where W_{G} = W_{G}(T_{0}) = N_{G}(T_{0})/T_{0} is the Weyl group with respect to T_{0} and = {s_{α} | α ∈ Δ_{G}} is the set of re�ections determined by Δ_{G}.

If T is a torus then we denote by S(T) the set of (isomorphism classes) of rank 1 local systems ℒ on T such that ℒ^{⊗m} is isomorphic to \overline{}_{ℓ}^{×} for some m coprime to p (we call such a local system tame). The Weyl group W_{G} acts naturally on T_{0} and this in turn gives us an action on S(T_{0}) by ℒ ↦ (w^{−1} )^{∗}ℒ; we denote the corresponding set of orbits by S(T_{0})/W_{G}. Associated to each such local system ℒ Lusztig has de�ned a set of character sheaves Ĝ_{ℒ} (see [Lus85, De�nition 2.10]), which depends only on the W_{G}-orbit of ℒ. The set of character sheaves on G is then de�ned to be

Ĝ = ⨆_{ℒ ∈S(T_{0})/W_{G}} Ĝ_{ℒ} ,

whose elements are irreducible G-equivariant objects in ℳ G. We will denote by 𝒦_{0}(G) the subgroup of the Grothendieck group of ℳ G spanned by the character sheaves of G. We then de�ne a bilinear form (− : −) : (𝒦_{0}(G) ⊗ \overline{}_{ℓ}) × (𝒦_{0}(G) ⊗ \overline{}_{ℓ}) → \overline{}_{ℓ} by setting

(A : A′) = \{ \begin{matrix}1 & if A ≅ A′, \\ 0 & otherwise . \end{matrix}

for all A, A′∈ Ĝ.

3.2. We will denote by Z the set of all pairs (L, P) such that P is a parabolic subgroup of G and L ⩽ P is a Levi complement of P. We then de�ne Z_{std} to be the subset consisting of all standard pairs (L, P), i.e. P contains B_{0} and L is the unique Levi complement of P containing T_{0} (note that every pair in Z is conjugate to a pair in Z_{std}). By projecting onto the �rst factor of Z (resp. Z_{std}) we obtain the set of Levi (resp. standard Levi) subgroups L (resp. L_{std}).

Assume now that (L, P) ∈ Z and let A_{0} ∈ ℳ L be an L-equivariant perverse sheaf on L. In [Lus85, §4.1] Lusztig has associated to A_{0} a complex ind_{L⊂P}^{G} (A_{0}) ∈ 𝒟G, which we call the induced complex. We recall the construction of this complex following [Lus85, §4.1]. Consider the following diagram

\begin{aligned}& L & X̂ & X̃ & G \\ π \\ σ \\ τ \end{aligned}  (3.3)

where we have

X̂ = {(g, h) ∈ G × G | h^{−1} gh ∈ P}  X̃ = {(g, hP) ∈ G × (G/P) | h^{−1} gh ∈ P}

π(g, h) = π̂_{P} (h^{−1} gh)  σ(g, h) = (g, hP)  τ(g, hP) = g

where π̂_{P} : P → L is the canonical projection map. Since A_{0} is L-equivariant there exists a canonical perverse sheaf D on X̃ such that π̃A_{0} = σ̃D (c.f. 2.1). We then de�ne ind_{L⊂P}^{G} (A_{0}) = τ_{!}D.

3.4. We say a character sheaf A ∈ Ĝ is non-cuspidal if there exists a pair (L, P) ∈ Z (with P ≠ G) and a character sheaf A_{0} ∈ L̂ such that A is a direct summand of the induced complex ind_{L⊂P}^{G} (A_{0}); otherwise we say A is cuspidal. Lusztig has shown that if A_{0} ∈ L̂ then ind_{L⊂P}^{G} (A_{0}) is semisimple and contained in ℳ G (see [Lus85, Proposition 4.8(b)]). Furthermore for any A ∈ Ĝ there exists a pair (L, P) ∈ Z and a cuspidal character sheaf A_{0} ∈ L̂ such that A occurs as a direct summand of ind_{L⊂P}^{G} (A_{0}) (see [Lus85, Theorem 4.4]).

Let us now �x a pair (L, P) ∈ Z such that there exists a cuspidal character sheaf A_{0} ∈ L̂. By [Lus85, Proposition 3.12] we have A_{0} is isomorphic to an intersection cohomology complex IC(\overline{Σ}, ℰ)[dim Σ] where Σ ⊂ L is the inverse image under L ↦ L/Z^{∘}(L) of an isolated conjugacy class and ℰ is a local system on Σ. From the proof of this result we know that (Σ, ℰ) is a cuspidal pair in the sense of [Lus84b, De�nition 2.4]. Of particular interest to us will be the special case where supp(A_{0}) ∩ L_{uni} ≠ ∅, where for any connected reductive algebraic group H we denote by H_{uni} the variety of unipotent elements. Assume this is so then there exists a triple (O_{0}, ℰ_{0}, ℒ) consisiting of: a unipotent conjugacy class O_{0} ⊂ L, a cuspidal local system ℰ_{0} on O_{0} and a local system ℒ ∈ S(Z^{∘}(L)) such that A_{0} is isomorphic to

A_{ℒ} := IC(\overline{Σ}, ℰ_{0} ℒ)[dim Σ]  (3.5)

where Σ = O_{0}Z^{∘}(L). Here we consider ℰ_{0} ℒ as a local system on O_{0} × Z^{∘}(L) which we identify with the open subset O_{0}Z^{∘}(L) ⊆ \overline{O_{0}}Z^{∘}(L) under the multiplication morphism in L. With all of this we may now de�ne a map

(L, P, O_{0}, ℰ_{0}, ℒ) −→ ind_{L⊂P}^{G} (A_{ℒ} )  (3.6)

From this point forward Σ will always denote the variety O_{0}Z^{∘}(L) where O_{0} ⊂ L is a unipotent conjugacy class supporting a cuspidal local system.

3.7. Let A ∈ Ĝ be a character sheaf which occurs as a direct summand of the induced complex ind_{L⊂P}^{G} (A_{0}) (where A_{0} ∈ L̂ is still assumed to be cuspidal) then by [Lus86, §2.9] we have

supp A = ⋃_{x∈G} x(supp A_{0})U_{P}x^{−1}  (3.8)

where U_{P} ⩽ P is the unipotent radical of P. In particular we have supp A ∩ G_{uni} ≠ ∅ if and only if supp A_{0} ∩ L_{uni} ≠ ∅. Hence if we are only interested in character sheaves supported by a unipotent element then we need only concern ourselves with those character sheaves occurring in an induced complex of the form given in (3.5). We end our discussion of induced complexes with the following lemma which gives some facts concerning the inverse image of an induced complex.

Lemma 3.9. We denote by H a connected reductive algebraic group and by P a parabolic subgroup of H with Levi complement L. Furthermore we assume that G is a connected reductive algebraic group and i : G → H is a bijective morphism of varieties.

(a) For any A_{0} ∈ L̂ we have

i^{∗} ind_{L⊂P}^{H} (A_{0}) = ind_{i^{−1} (L)⊂i^{−1} (P)}^{G} (i^{∗}A_{0}).

Furthermore any character sheaf A ∈ Ĥ is cuspidal if and only if i^{∗}A ∈ Ĝ is cuspidal.

(b) Let us also assume that G = H and L = i(L) then any isomorphism ϕ : i^{∗}A_{0} → A_{0} induces an isomorphism

ϕ̃: i^{∗} ind_{L⊂i(P)}^{H} (A_{0}) → ind_{L⊂P}^{H} (A_{0}).

Proof. Firstly, by [Hum75, §24.1 - Proposition B], we have i induces a bijection between the Borel subgroups of G and H so this implies that Q := i^{−1} (P) is a parabolic subgroup of G with Levi complement M := i^{−1} (L). For any object □ introduced in 3.2 we write □_{L⊂P}^{H} (resp. □_{M⊂Q}^{G} ) to indicate that it is de�ned with respect to ind_{L⊂P}^{H} (resp. ind_{M⊂Q}^{G} ). We now have a diagram

M  X̂_{M⊂Q}^{G}  X̃_{M⊂Q}^{G}  G

L  X̂_{L⊂P}^{H}  X̃_{L⊂P}^{H}  H

i

π_{M⊂Q}^{G}

i

σ_{M⊂Q}^{G}

i

τ_{M⊂Q}^{G}

i

π_{L⊂P}^{H}

σ_{L⊂P}^{H}

τ_{L⊂P}^{H}

where the vertical maps are the obvious actions of i. The squares of the above diagram are clearly commutative. Let D be the canonical perverse sheaf on X̃_{L⊂P}^{H} satisfying π̃_{L⊂P}^{H} A_{0} = σ̃_{L⊂P}^{H} D. The �bres of π_{L⊂P}^{H} and π_{M⊂Q}^{G} have the same dimension, as do the �bres of σ_{L⊂P}^{H} and σ_{M⊂Q}^{G} , therefore we have i^{∗}D satis�es π̃_{M⊂Q}^{G} i^{∗}A_{0} = σ̃_{M⊂Q}^{G} i^{∗}D because the inverse image is contravariant. By de�nition we have ind_{L⊂P}^{H} (A_{0}) = (τ_{L⊂P}^{H} )_{!}D and ind_{M⊂Q}^{G} (i^{∗}A_{0}) = (τ_{M⊂Q}^{G} )_{!}i^{∗}D so the �rst part of the lemma follows if we can show the equality i^{∗}(τ_{L⊂P}^{H} )_{!}D = (τ_{M⊂Q}^{G} )_{!}i^{∗}D but this is just 2.3. The conclusion concerning cuspidality is an immediate consequence of the �rst part, which proves (a).

We now prove (b). Let Q = i(P) then collapsing the notation □_{L⊂P}^{H} simply to □_{P} we have a diagram

L  X̂_{P}  X̃_{P}  H

L  X̂_{Q}  X̃_{Q}  H i π_{P} i σ_{P} i τ_{P} i π_{Q} σ_{Q} τ_{Q}

with commutative squares. Let K (resp. K′) be the canonical perverse sheaf on X̃_{P} (resp. X̃_{Q}) satisfying π̃_{P} A_{0} = σ̃_{P}K (resp. π̃_{Q} A_{0} = σ̃_{Q}K′). Now π̃_{P} ϕ de�nes an isomorphism π̃_{P} i^{∗}A_{0} → π̃_{P} A_{0} and using the commutativity of the above diagram we may view this as an isomorphism σ̃_{P}i^{∗}K′→ σ̃_{P}K. As σ_{P} is smooth with connected �bres we have σ̃_{P} is a fully faithful functor (see [Lus85, 1.8.3]), hence there exists a unique isomorphism ϕ′ : i^{∗}K′ → K such that π̃_{P} ϕ = σ̃_{P}ϕ′. Using the arguments above we see that ϕ̃ = (τ_{P})_{!}ϕ′ gives the required isomorphism. ■

4. The Space of Unipotently Supported Class Functions

4.1. Let N_{G} denote the set of all pairs ι = (O, ℰ) where O is a unipotent conjugacy class of G and ℰ is an irreducible G-equivariant local system on O. We denote by N_{G}^{0} ⊆ N_{G} the subset consisting of those pairs (O, ℰ) such that ℰ is a cuspidal local system on O (see [Lus84b, De�nition 2.4]); we call the elements of N_{G}^{0} cuspidal pairs. Given a pair ι we will denote the class O by O_{ι}and the local system ℰ by ℰ_{ι}.

Let us denote by M̃_{G} the set of all pairs (L, ν) consisting of a Levi subgroup L ∈ L and a cuspidal pair ν ∈ N_{L}^{0}. We have G acts naturally on M̃_{G} by conjugation and we denote by [L, ν] the orbit containing (L, ν). We also denote by M_{G} the set of all such orbits. Recall that in [Lus84b, Theorem 6.5] Lusztig has associated to every pair ι ∈ N_{G} a unique orbit C_{ι} ∈ M_{G}. This is known as the generalised Springer correspondence.

For each ι ∈ N_{G} we will now choose a representative (L_{ι}, ν_{ι}) ∈ C_{ι} with L_{ι} ∈ L_{std}. For
convenience, we will assume that if ι, ι′∈ N_{G} satisfy C_{ι}= C_{ι′}then (L_{ι}, ν_{ι}) = (L_{ι′}, ν_{ι′}).

In [Lus84b] it was shown that we have a disjoint union

N_{G} = ⨆_{[L,υ]∈M_{G}} ℐ [L, υ]  where  ℐ [L, υ] = {ι ∈ N_{G} | (L_{ι}, υ_{ι}) ∈ [L, υ]}.

We call ℐ [L, υ] a block of N_{G}. If ν ∈ N_{L}^{0} is cuspidal then W_{G}(L) = N_{G}(L)/L is a Coxeter group (see [Lus84b, Theorem 9.2(a)]) and we have a bijection

ℐ [L, ν] ⟷ Irr(W_{G}(L))  (4.2)

for all [L, ν] ∈ M_{G}, which we denote ι ↦ E_{ι}. If L is a torus then ν is simply the pair consisting of the trivial class and the trivial local system, in which case this bijection is the classical Springer correspondence and we call ℐ [L, ν] the principal block.

4.3. To describe the correspondence in (4.2) we will need to recall a description of semisimple objects in ℳ G following [Lus84b, 3.7]. Let K ∈ ℳ G be semisimple and let A = End(K) be the endomorphism algebra of K, by which we mean the algebra Hom_{ℳ G} (K, K) in the category ℳ G. Assume E is any �nite dimensional A-module then we de�ne

K_{E} = Hom_{A}(E, K) ∈ ℳ G.  (4.4)

To see that this is an object of ℳ G we can construct this in the following way. Let us pick a presentation A^{m} →^{φ} A^{n} → E → 0 of the module E. Applying Hom_{A}(−, K) and using the fact that we have an isomorphism K ≅ Hom_{A}(A, K) of A-modules we get a diagram

0  Hom_{A}(E, K)  Hom_{A}(A^{n}, K)  Hom_{A}(A^{m}, K)

0  Ker_{ℳ G} (φ^{∗})  K^{n}  K^{m} ≅ ≅ φ^{∗} ≅ φ^{∗} (4.5)

with exact rows (φ^{∗} is the map induced by φ and the vertical arrows are A-module isomorphisms). The A-module homomorphism φ^{∗} is also a morphism in ℳ G because ℳ G is \overline{}_{ℓ}-linear and the kernel exists because ℳ G is an abelian category. Hence Hom_{A}(E, K) ≅ Ker(φ^{∗}) ∈ ℳ G as desired. Now K_{E} is a simple object of ℳ G if and only if E is a simple A-module and we have a canonical isomorphism

⨁_{E∈Irr(A)} (E ⊗ K_{E}) ≅ K  (4.6)

in ℳ G given by e ⊗ f ↦ f(e) in each summand.

4.7. Let L ∈ L be a Levi subgroup and let ν = (O_{0}, ℰ_{0}) ∈ N_{L}^{0} be a cuspidal pair then following [Lus86, 2.3] we de�ne for any local system ℒ ∈ S(Z^{∘}(L)) a semisimple perverse sheaf K_{ℒ} ∈ ℳ G in the following way. We de�ne an open subset Σ_{reg} = O_{0} · Z^{∘}(L)_{reg} ⊆ Σ where Z^{∘}(L)_{reg} = {z ∈ Z^{∘}(L) | C_{G}^{∘}(z) = L} and we denote by Y the locally closed smooth irreducible subvariety of G given by ∪_{x∈G} xΣ_{reg}x^{−1} . We have the following diagram

\begin{aligned}& Σ & Ŷ & Ỹ & Y \\ α \\ β \\ γ \end{aligned}  (4.8)

where

Ŷ = {(g, x) ∈ G × G | x^{−1} gx ∈ Σ}  Ỹ = {(g, xL) ∈ G × (G/L) | x^{−1} gx ∈ L}
α(g, x) = x^{−1} gx  β(g, x) = (g, xL)  γ(g, xL) = g.

Since the local system ℰ_{0} is L-equivariant there exists a canonical local system ℰ̃_{0} on Ỹ such that β^{∗}ℰ̃_{0} = α^{∗}(ℰ_{0} \overline{}_{ℓ}) (see [Lus85, 1.9.3]). Let δ : Ỹ → Z^{∘}(L) be the map given by δ(g, xL) = (xgx^{−1} )_{ss} (where h_{ss} is the semisimple part of h for all h ∈ G) then ℒ̃ = δ^{∗}ℒ is a local system on Ỹ hence so is ℰ̃_{0} ⊗ ℒ̃. By [Lus84b, 3.2] we have γ is a Galois covering so γ_{∗} = γ_{!} because γ is �nite (hence proper), which means γ_{∗}(ℰ̃_{0} ⊗ ℒ̃) = γ_{!}(ℰ̃_{0} ⊗ ℒ̃) is a semisimple local system on Y. We now de�ne K_{ℒ} ∈ ℳ G to be the complex IC(\overline{Y}, γ_{∗}(ℰ̃_{0} ⊗ ℒ̃))[dim Y]. With this we have de�ned a map

(L, O_{0}, ℰ_{0}, ℒ) −→ K_{ℒ}  (4.9)

4.10. Let us keep the notation of 4.7. We denote by N_{G}(L, ℒ) the set of all elements n ∈ N_{G}(L) such that (ad n)^{∗}ℒ is isomorphic to ℒ, where ad n : G → G is the conjugation morphism given by (ad n)(g) = ngn^{−1} for all g ∈ G. Clearly L is a normal subgroup of N_{G}(L, ℒ) and we denote the quotient group N_{G}(L, ℒ)/L by W_{G}(L, ℒ). Let us denote by A_{ℒ} the endomorphism algebra of the semisimple perverse sheaf K_{ℒ} . By [Lus86, §2.4(a)] there exists a set of basis elements {Θ_{v} | v ∈ W_{G}(L, ℒ)} ⊂ A_{ℒ} such that the map v ↦ Θ_{v} de�nes a canonical algebra isomorphism

\overline{}_{ℓ}W_{G}(L, ℒ) ≅ A_{ℒ} .  (4.11)

Let us now assume that ℒ = \overline{}_{ℓ} then we denote A_{ℒ} simply by A and K_{ℒ} by K. By [Lus84b, Theorem 9.2(b)], if ℒ = \overline{}_{ℓ} we have W_{G}(L, ℒ) = W_{G}(L) hence A is isomorphic to \overline{}_{ℓ}W_{G}(L). Using the description given in 4.3 we have E ↦ K_{E} = Hom_{W_{G}(L)}(E, K) gives a bijection between simple W_{G}(L)-modules and the simple summands of K (up to isomorphism). Furthermore by [Lus84b, Theorem 6.5] we have for each E that there exists a unique pair ι ∈ ℐ [L, ν] satisfying

K_{E}|_{G_{uni}} ≅ IC(\overline{O}_{ι}, ℰ_{ι})[dim O_{ι}+ dim Z^{∘}(L)].

The composition of the maps E ↦ K_{E} ↦ ι then gives the bijection in (4.2) and we will freely denote the object K_{E} by K_{ι}.

Remark 4.12. In [Lus84b, 3.4] the de�nition of N_{G}(L, ℒ) includes the condition that (ad n)(Σ) = Σ. However this is automatically satis�ed in our situation by [Lus84b, Theorem 9.2(b)].

4.13. We now end this section by explicitly describing how K_{ℒ} is related to ind_{L⊂P}^{G} (A_{ℒ} ) (for this we follow [Lus84b]). Here we assume (L, P) ∈ Z and ν ∈ N_{L}^{0}. Recalling the notation of 3.2 we wish to describe the complex D. To do this we �rst observe that

π̃A_{ℒ} = IC(X̂′, π^{∗}(ℰ_{0} ℒ))[dim \overline{Σ} + dim G + dim U_{P}]

where U_{P} is the unipotent radical of P and X̂′ = π^{−1} (\overline{Σ}) = \{(g, h) ∈ G × G | h^{−1} gh ∈ \overline{Σ} · U_{P}\}. Note that for l ∈ L we have π^{−1} (l) = {(g, h) ∈ G × G | h^{−1} gh ∈ l · U_{P}} hence the �bres of π have dimension dim G + dim U_{P}.

Let us �x a set of coset representatives {h_{1}, …, h_{k}} of P in G then we de�ne j : X̃ → X̂ by setting j(g, hP) = (g, h_{i}) where 1 ⩽ i ⩽ k is the unique index such that h ∈ h_{i}P. By the construction in [Lus85, 1.9.3] we see that

D = j^{∗} IC(X̂′, π^{∗}(ℰ_{0} ℒ))[dim \overline{Σ} + dim G + dim U_{P} − dim P] = IC(X̃′, \overline{ℰ_{0} ℒ} )[dim X̃′]

where X̃′ = j^{−1} (X̂′) = \{(g, h_{i}P) ∈ G × G/P | h_{i}^{−1} gh_{i} ∈ \overline{Σ} · U_{P}\} and \overline{ℰ_{0} ℒ} = j^{∗}π^{∗}(ℰ_{0} ℒ). Note that the equality dim X̃′= dim Y = dim G − dim P + dim \overline{Σ} + dim U_{P} is given in the proof of [Lus84b, Lemma 4.3(a)].

Let Y and Ỹ be as in 4.7 then by [Lus84b, Lemma 4.3(b)] we have τ(X̃′) = \overline{Y} hence τ^{−1} (Y) ⊂ X̃′ ⊂ X̃. Furthermore, by [Lus84b, Lemma 4.3(c)], the map (g, hL) → (g, hP) de�nes an isomorphism κ : Ỹ → τ^{−1} (Y). As τ^{−1} (Y) ⊂ X̃′ we have D|_{τ^{−1} (Y)} ≅ \overline{ℰ_{0} ℒ} |_{τ^{−1} (Y)}[dim X̃′] and κ^{∗}(\overline{ℰ_{0} ℒ} |_{τ^{−1} (Y)}) ≅ ℰ̃_{0} ⊗ ℒ̃ (see [Lus84b, 4.4]) hence

(τ_{!}D)|_{Y} = τ_{!}(D_{τ^{−1} (Y)}) ≅ γ_{∗}κ^{∗}(\overline{ℰ_{0} ℒ} |_{τ^{−1} (Y)}[dim X̃′]) ≅ γ_{∗}(ℰ̃_{0} ⊗ ℒ̃)[dim Y].

Here we have applied 2.3 to obtain τ_{!} = id^{∗} ∘τ_{!} = γ_{!} ∘ κ^{∗} = γ_{∗} ∘ κ^{∗} where the last equality follows from the fact that γ is proper. With this we have the following result of Lusztig.

Proposition 4.14 (Lusztig, [Lus84b, Proposition 4.5]). The complex τ_{!}D is a perverse sheaf and is canonically isomorphic to IC(\overline{Y}, γ_{∗}(ℰ̃_{0} ⊗ ℒ̃))[dim Y]. In particular ind_{L⊂P}^{G} (A_{ℒ} ) and K_{ℒ} are canonically isomorphic in ℳ G.

Remark 4.15. Note that Proposition 4.14 implies that ind_{L⊂P}^{G} (A_{ℒ} ) does not depend upon the choice of parabolic P containing L so in this situation we will simply write ind_{L}^{G}(A_{ℒ} ) when convenient.

5. Bases of the Endomorphism Algebra A

5.1. Assume L ∈ L is a Levi subgroup supporting a cuspidal pair (O_{0}, ℰ_{0}) ∈ N_{L}^{0} then we denote by K the image of (L, O_{0}, ℰ_{0}, \overline{}_{ℓ}) under the map in (4.9) and by A the endomorphism algebra of K. As was mentioned in (4.11) Lusztig has de�ned an isomorphism between A and the group algebra \overline{}_{ℓ}W_{G}(L) by specifying a set of basis elements {Θ_{v} | v ∈ W_{G}(L)} ⊂ A. However, in [Bon04, §6.A] Bonnaf� e has de�ned an alternative basis {Θ′_{v} | v ∈ W_{G}(L)} ⊂ A which also de�nes such an isomorphism. In this article we will need to use both bases and we recall here results of Bonnaf� e concerning the relationship between the two.

We now assume that u_{0} ∈ O_{0} is a �xed unipotent element. If L and O_{0} are F-stable then we assume that u_{0} ∈ O_{0}^{F}.

5.2. Assume v ∈ W_{G}(L) then by [Bon04, 5.4] we may �nd a representative v̇ ∈ N_{G}(L) ∩ C_{G}^{∘}(u_{0}) and we will assume that all such representatives are chosen in this way. Let us recall the notation of 4.7. For any v ∈ W_{G}(L) we have by [Lus84b, Theorem 9.2(b)] that there exists an isomorphism θ_{v} : ℰ_{0} \overline{}_{ℓ} → (ad v̇)^{∗}(ℰ_{0} \overline{}_{ℓ}) which we assume �xed. This isomorphism induces an isomorphism θ̂_{v} : ℰ̃_{0} → γ_{v}^{∗}ℰ̃_{0} where γ_{v} : Ỹ → Ỹ is given by γ_{v}(g, xL) = (g, xv̇^{−1} L) (see the proof of [Lus84b, Proposition 3.5]). By de�nition we have γ_{v}γ = γ hence γ_{∗}θ̂_{v} de�nes an endomorphism of γ_{∗}ℰ̃_{0}. As K is the intersection cohomology complex IC(\overline{Y}, γ_{∗}ℰ̃_{0}) we may then de�ne Θ_{v} to be the unique endomorphism of K extending γ_{∗}θ̂_{v}.

By [Lus84b, Theorem 9.2(d)] there is a unique isomorphism θ_{v} for each v ∈ W_{G}(L) such that Θ_{v} induces the identity on ℋ_{u}^{− dim Y} (K) where u is any element of the induced unipotent class Ind_{L}^{G}(O_{0}) (see [Lus84b, Corollary 7.3(a)]). With this choice we have

v ↦ Θ_{v}  (5.3)

de�nes the required algebra isomorphism \overline{}_{ℓ}W_{G}(L) → A. Assume now that ℒ is a local system on Z^{∘}(L). Let K_{ℒ} be the image of (L, O_{0}, ℰ_{0}, ℒ) under the map in (4.9) and let A_{ℒ} be the endomorphism algebra of K_{ℒ} then from the discussion in [Lus86, 2.3] we have an embedding of algebras A_{ℒ} ↪ A which corresponds under (5.3) to the natural embedding \overline{}_{ℓ}W_{G}(L, ℒ) ↪ \overline{}_{ℓ}W_{G}(L). In particular the restriction of (5.3) to \overline{}_{ℓ}W_{G}(L, ℒ) de�nes the isomorphism mentioned in (4.11).

Remark 5.4. Assume L and O_{0} are F-stable. It is clear to see that if v̇∈ N_{G}(L) ∩ C_{G}^{∘}(u_{0}) is a representative for v ∈ W_{G}(L) then v̇^{−1} ∈ N_{G}(L) ∩ C_{G}^{∘}(u_{0}) is a representative of v^{−1} and F(v̇) ∈ N_{G}(L) ∩ C_{G}^{∘}(u_{0}) is a representative of F(v).

5.5. In [Bon04, §6.A] Bonnaf� e shows that for every v ∈ W_{G}(L) there exists an isomorphism θ′_{v} : ℰ_{0} → (ad v̇)^{∗}ℰ_{0} which induces the identity at the stalk of u_{0}. Clearly this also de�nes an isomorphism ℰ_{0} \overline{}_{ℓ} → (ad v̇)^{∗}(ℰ_{0} \overline{}_{ℓ}) which induces the identity at the stalk of any element u_{0}z where z ∈ Z^{∘}(L); we will also denote this by θ′_{v}. As is described in 5.2 this isomorphism determines a unique endomorphism Θ′_{v} of K and by [Bon04, Proposition 6.1] we have the map v ↦ Θ′_{v} de�nes an algebra isomorphism \overline{}_{ℓ}W_{G}(L) → A. The following result describes the relationship between the two sets of basis elements for A.

Proposition 5.6 (Bonnaf�e, [Bon04, Corollary 6.2]). There exists a linear character γ_{L,u_{0}}^{G} ∈ Irr(W_{G}(L)) such that Θ′_{v} = γ_{L,u_{0}}^{G}(v)Θ_{v} and so θ′_{v} = γ_{L,u_{0}}^{G}(v)θ_{v} for all v ∈ W_{G}(L).

5.7. As is remarked in [Bon04, Remark 6.4] the linear character γ_{L,u_{0}}^{G} is known explicitly in almost all cases. Assume that Z(G) is connected and p is a good prime for G. By the reduction arguments given in [Bon04, §4.A, §4.B, Corollary 7.3] it is su�cient to describe this character when G is an adjoint simple group. If G is of type A_{n} or of exceptional type then L is either G or a torus and so γ_{L,u_{0}}^{G} is the trivial character (see [Bon04, Corollary 6.9]). When G is of type B_{n}, C_{n} or D_{n} then in general γ_{L,u_{0}}^{G} is not the identity. However, this character is described by work of Waldspurger [Wal01] as is explained in [Bon04, Example 6.3] (see also [Bon05, Errata]).

6. Rational Structures

6.1. We say a character sheaf A ∈ Ĝ is F-stable if we have an isomorphism ϕ_{A} : F^{∗}A → A in 𝒟G and we denote by Ĝ^{F} ⊆ Ĝ the set of all F-stable character sheaves of G. Clearly this isomorphism induces a map ℋ ^{i}F^{∗}A → ℋ ^{i}A for each i ∈ and hence a map ℋ_{x}^{i}F^{∗}A → ℋ_{x}^{i}A for each x ∈ G; we denote both these maps again by ϕ_{A}. If x ∈ G = G^{F} then ϕ_{A} induces an automorphism of the stalk ℋ_{x}^{i}A because ℋ_{x}^{i}F^{∗}A = ℋ_{F(x)}^{i}A = ℋ_{x}^{i}A. For each such A and ϕ_{A} we then de�ne the characteristic function χ_{A,ϕ_{A}} : G → \overline{}_{ℓ} by setting

χ_{A,ϕ_{A}}(g) = ∑_{i} (−1)^{i} Tr(ϕ_{A}, ℋ_{g}^{i}A)

for all g ∈ G^{F}. Note that this characteristic function depends upon the choice of isomorphism ϕ_{A}. If A is an element of Ĝ^{F} then we will choose ϕ_{A} to satisfy the properties in [Lus85, §25.1]. With such a choice the resulting characteristic function χ_{A,ϕ_{A}} has norm 1; the choice of ϕ_{A} is unique up to scalar multiplication by a root of unity.

6.2. The Frobenius endomorphism acts on the set N_{G} by ι ↦ F^{−1} (ι) := (F^{−1} (O_{ι}), F^{∗}ℰ_{ι}), where F^{∗}ℰ_{ι} is the inverse image of ℰ_{ι} under F. We say ι is F-stable if F^{−1} (O_{ι}) = O_{ι} and F^{∗}ℰ_{ι} is isomorphic to ℰ_{ι} (we also denote this by ι = F^{−1} (ι)); we denote the subset of all F-stable pairs by N_{G}^{F}. The Frobenius also acts on the set M̃_{G} , hence also on M_{G}, by (L, ν) ↦ (F^{−1} (L), F^{−1} (ν)). We say (L, ν) is F-stable if F^{−1} (L) = L and F^{−1} (ν) = ν and we denote by M̃_{G}^{F} the subset of all F-stable pairs (similarly for M_{G}^{F}). Note that the map N_{G} → M_{G} is compatible with the actions of F so that we have an induced map N_{G}^{F} → M_{G}^{F} (c.f. [Lus85, 24.2]). Assume (L, P) ∈ Z_{std} is such that N_{L}^{0} ≠ ∅ then by the classi�cation of cuspidal local systems (see [Lus84b, §10-15]) we have F(L) = L and F(P) = P. In particular any orbit [L, ν] ∈ M_{G}^{F} may be represented by an F-stable standard Levi subgroup.

Twisted Levi Subgroups

6.3. We now wish to choose for each F-stable unipotently supported character sheaf A ∈ Ĝ^{F} a distinguished isomorphism ϕ_{A} : F^{∗}A → A. This was done by Lusztig in [Lus86, §3.3 - §3.4] under the assumption that G is an adjoint simple group and F is a split Frobenius endomorphism. Adapting an idea of Lusztig from [Lus85, §10.3] we will extend this to the case where F is not necessarily split and G has a connected centre. Firstly let us assume that (L, P) ∈ Z_{std} is a standard pair such that N_{L}^{0} ≠ ∅ and A_{0} = A_{ν}^{ℒ} ∈ L̂ is a cuspidal character sheaf satisfying (A : ind_{L}^{G} A_{0}) ≠ 0 (see 3.4 and Lemma 3.9). Using the argument in [Lus85, §10.5] together with Lemma 3.9 we see that there exists x ∈ G such that ^{x}L is F-stable and B_{0} = (ad x^{−1} )^{∗}A_{0} ∈ ^{ x̂}L is an F-stable cuspidal character sheaf satisfying (A : ind_{^{x}L}^{G} B_{0}) ≠ 0 (we may then assume that A is a summand of ind_{^{x}L}^{G} B_{0}).

6.4. Let us assume that (L, P) ∈ Z_{std} is a standard pair such that ν = (O_{0}, ℰ_{0}) ∈ N_{L}^{0} ≠ ∅ and ℒ ∈ S(T_{0}) is a local system. For every v ∈ W_{G}(L) we �x an element g_{v} ∈ G such that g_{v}^{−1} F(g_{v}) = F(v̇^{−1} ) where v̇ ∈ N_{G}(L) is as in 5.2. If v is the identity in W_{G}(L) then we will assume for convenience that v̇= g_{v} is the identity in G (this clearly satis�es the assumption of 5.2). We now de�ne

L_{v} = ^{g_{v}}L  P_{v} = ^{g_{v}}P  O_{v} = ^{g_{v}}O_{0}  Σ_{v} = O_{v}Z^{∘}(L_{v})  ℰ_{v} = (ad g_{v}^{−1} )^{∗}ℰ_{0}  ℒ_{v} = (ad g_{v}^{−1} )^{∗}ℒ.

Through the isomorphism ad g_{v}^{−1} : L_{v} → L we may identify the action of the Frobenius endomorphism F on L_{v} with the action of F_{v} on L de�ned by F_{v}(l) = F(v̇^{−1} lv̇) for all l ∈ L. Clearly we have an isomorphism of abstract groups ad g_{v}^{−1} : L_{v}^{F} → L^{F_{v}} and (ad g_{v}^{−1} )^{∗} induces a bijective correspondence between L̂^{F_{v}} and L̂_{v}^{F}. In particular, assume B ∈ L̂^{F_{v}} and ψ : F_{v}^{∗}B → B is a �xed isomorphism then ψ′ : F^{∗}B′ → B′ is also an isomorphism where ψ′ = (ad g_{v}^{−1} )^{∗}ψ and B′ = (ad g_{v}^{−1} )^{∗}B. From the de�nitions we have the corresponding characteristic functions are related by χ_{B′,ψ′} = χ_{B,ψ} ∘ ad g_{v}^{−1} .

Let us denote by D_{v} the complex IC(\overline{Σ}_{v}, ℰ_{v} ℒ_{v})[dim Σ_{v}] (see (3.5)). From the discussion in 6.3 we need only concern ourselves with summands of ind_{L_{v}}^{G}(D_{v}) where D_{v} is F-stable. Clearly we have

F^{∗}D_{v} = IC(\overline{F^{−1} (Σ)}_{v}, F^{∗}ℰ_{v} F^{∗}ℒ_{v})[dim Σ_{v}]

hence we may, and will, assume that ν ∈ N_{L}^{F} and F^{∗}ℒ_{v} ≅ ℒ_{v} (see [Lus84b, Theorem 9.2(b)]). As we have am equivalence F^{∗}ℒ_{v} ≅ ℒ_{v} ⇔ F_{v}^{∗}ℒ ≅ ℒ we see that the existence of an isomorphism F^{∗}ℒ_{v} ≅ ℒ_{v} is equivalent to v^{−1} being contained in the subset

Z_{G}(L, ℒ) = {n ∈ N_{G}(L) | ad(n)^{∗}F^{∗}ℒ ≅ ℒ }/L ⊂ W_{G}(L).  (6.5)

In particular we will assume that Z_{G}(L, ℒ) is non-empty. Note that Z_{G}(L, ℒ) is in fact a right coset of the group W_{G}(L, ℒ) in W_{G}(L).

If S ⩽ G is any F-stable torus and ℱ ∈ S(S)^{F} is any F-stable local system then we may choose a distinguished isomorphism β : F^{∗}ℱ → ℱ by the condition that the morphism induced by β at the stalk of the identity element 1 ∈ S^{F} is the identity morphism. In particular this de�nes an isomorphism φ_{1}^{v} : F^{∗}ℒ_{v} → ℒ_{v} for each v^{−1} ∈ Z_{G}(L, ℒ).

Relative Weyl Groups

6.6. For the following we refer to [Bon04, Proposition 1.12] and the references therein (in particular [How80 ]). Let us denote by Φ_{L} ⊆ Φ_{G} the root system of L with respect to T_{0}. We will denote by V_{G} the -subspace of ⊗_{}X(T_{0}) spanned by Φ_{G} and by V_{L} ⊆ V_{G} the -subspace spanned by Φ_{L}. Let Δ_{L} ⊂ Δ_{G} be the set of simple roots of L with respect to T_{0} ⩽ B_{0} ∩ L then we denote by \overline{Δ}_{L} the set Δ_{G} − Δ_{L}. For each α ∈ \overline{Δ}_{L} let M_{α} ∈ L_{std} be the standard Levi subgroup of G whose simple roots are Δ_{L} ∪ \{α\} with respect to T_{0} ⩽ B_{0} ∩ M_{α} then the group W_{M_{α}}(L) contains a unique non-trivial element which we denote by s_{L,α}. For every α ∈ \overline{Δ}_{L} we have s_{L,α} acts on the quotient space \overline{V}_{L} = V_{G}/V_{L} as a re�ection sending α + V_{L} to −α + V_{L}. We will also use \overline{Δ}_{L} to denote the image \{α + V_{L} | α ∈ Δ̃_{L} \} of \overline{Δ}_{L} in \overline{V}_{L}. Taking = \{s_{L,α} | α ∈ \overline{Δ}_{L}\} we have (W_{G}(L), ) is a Coxeter system with corresponding root system \overline{Φ} _{L} ⊂ \overline{V}_{L} obtained as the image of \overline{Δ}_{L} under the action of W_{G}(L). Note that if \overline{Φ} _{L}^{+} ⊂ \overline{Φ} _{L} is the positive system of roots determined by \overline{Δ}_{L} then we have \overline{Φ} _{L}^{+} = \overline{Φ} _{L} ∩ \{α + V_{L} | α ∈ Φ^{+} \}.

Now there exists a root system Ψ ⊂ \overline{Φ} _{L} such that W_{G}(L, ℒ) is the re�ection group of Ψ (note that Ψ is not necessarily closed in \overline{Φ} _{L}). Taking Ψ^{+} = Ψ ∩ \overline{Φ} _{L}^{+} we have Ψ^{+} is a system of positive roots in Ψ which determines a unique set of Coxeter generators ⊆ W_{G}(L, ℒ). Assume n ∈ N_{G}(L, ℒ) then for any v^{−1} ∈ Z_{G}(L, ℒ) we have an isomorphism

(ad F_{v}^{−1} (n))^{∗}ℒ ≅ (ad F_{v}^{−1} (n))^{∗}(F_{v})^{∗}ℒ = F_{v}^{∗}(ad n)^{∗}ℒ ≅ F_{v}^{∗}ℒ ≅ ℒ,

in particular as L is F_{v}-stable this shows that F_{v} induces an automorphism F_{v} : W_{G}(L, ℒ) → W_{G}(L, ℒ). By [Lus84a, Lemma 1.9(i)] the coset Z_{G}(L, ℒ) contains a unique element w^{−1} of minimal length (with respect to the length function of (W_{G}(L), )), satisfying the property w^{−1} Ψ^{+} w ⊂ Φ^{+} . In particular F_{w} is an automorphism of the Coxeter system (W_{G}(L, ℒ), ). Taking _{w} to be the image of under ad g_{w} we have (W_{G}(L_{w}, ℒ_{w}), _{w}) is a Coxeter system and F induces an automorphism of this Coxeter system.

Isomorphisms for Local Systems

6.7. We now consider how to choose an isomorphism F^{∗}ℰ_{v} → ℰ_{v} for any v ∈ W_{G}(L). First we will choose an isomorphism φ_{0} : F^{∗}ℰ_{0} → ℰ_{0} such that the induced isomorphism (ℰ_{0})_{u} → (ℰ_{0})_{u} at the stalk of any element u ∈ O_{0}^{F} is q^{(dim(L/Z^{∘}(L))−dim O_{0})/2} times a map of �nite order. Following [Lus90, 9.3] we de�ne an isomorphism φ_{0}^{v} : F^{∗}ℰ_{v} → ℰ_{v} in the following way. Let K be the image of (L, O_{0}, ℰ_{0}, ℒ) under the map in (4.9) and let A be its endomorphism algebra (c.f. 4.10). The basis element θ_{F(v)} of A de�nes an isomorphism θ_{F(v)} : ℰ_{0} → (ad F(v̇))^{∗}ℰ_{0}. Taking the inverse image of θ_{F(v)} along ad g_{v}^{−1} ∘ F we see that we obtain a new isomorphism

(ad g_{v}^{−1} ∘ F)^{∗}θ_{F(v)} : F^{∗}ℰ_{v} → (ad g_{v}^{−1} )^{∗}F^{∗}ℰ_{0}.

Moreover the isomorphism φ_{0} induces an isomorphism

(ad g_{v}^{−1} )^{∗}φ_{0} : (ad g_{v}^{−1} )^{∗}F^{∗}ℰ_{0} → ℰ_{v}.

We will now take φ_{0}^{v} to be the composition (ad g_{v}^{−1} )^{∗}φ_{0} ∘ (ad g_{v}^{−1} ∘ F)^{∗}θ_{F(v)} (note that by our assumption on g_{v} we have φ_{0}^{v} = φ_{0} if v = 1). If v^{−1} ∈ Z_{G}(L, ℒ) then we can de�ne a unique isomorphism ϕ_{D_{v}} : F^{∗}D_{v} → D_{v} with the property that the restriction of ϕ_{D_{v}} to Σ_{v} coincides with φ_{0}^{v} φ_{1}^{v} (c.f. 6.4).

Remark 6.8. Note that the integer dim(L/Z^{∘}(L)) − dim O_{0} is not even in general. Hence our construction above depends upon a choice for q^{1/2} in \overline{}_{ℓ}. However, if Z(G) is connected and p is a good prime then this is an integer by the classi�cation of cuspidal pairs (see [Lus84b]).

Remark 6.9. Here, we have not quite followed the setup of [Lus85] because we have scaled the map of �nite order, chosen in [Lus85, 24.2], by a power of q. When following [Lus85] we must then adjust all terms by this scalar, which the reader may check that we do.

Isomorphisms for Induced Complexes

6.10. Assume v^{−1} ∈ Z_{G}(L, ℒ) and let us denote by K_{v} the image of (L_{v}, O_{v}, ℰ_{v}, ℒ_{v}) under the map in (4.9). We now consider how to choose an isomorphism F^{∗}K_{v} → K_{v}. Recalling the construction in 4.7 we see that the isomorphisms φ_{0}^{v} : F^{∗}ℰ_{v} → ℰ_{v} and φ_{1}^{v} : F^{∗}ℒ_{v} → ℒ_{v} naturally induce isomorphisms φ̃_{0}^{v} : F^{∗}ℰ̃_{v} → ℰ̃_{v} and φ̃_{1}^{v} : F^{∗}ℒ̃_{v} → ℒ̃_{v} hence an isomorphism φ̃_{0}^{v} ⊗ φ̃_{1}^{v} : F^{∗}ℰ̃_{v} ⊗ F^{∗}ℒ̃_{v} → ℰ̃_{v} ⊗ ℒ̃_{v}. Clearly the variety Y of 4.7 is stable under F (because L_{v} is stable under F) so we have

F^{∗}K_{v} = IC(\overline{Y}, F^{∗}γ_{∗}(ℰ̃_{v} ⊗ ℒ̃_{v}))[dim Y] = IC(\overline{Y}, γ_{∗}F^{∗}(ℰ̃_{v} ⊗ ℒ̃_{v}))[dim Y]

because γ_{∗} = γ_{!} and F^{∗}γ_{!} = γ_{!}F^{∗} (see 4.7 and 2.3). We may now de�ne a unique isomorphism ϕ : F^{∗}K_{v} → K_{v} by specifying that ϕ|_{Y} coincides with γ_{∗}(φ̃_{0}^{v} ⊗ φ̃_{1}^{v}).

Isomorphisms for Character Sheaves

6.11. Let K_{v} and ϕ : F^{∗}K_{v} → K_{v} be as in 6.10. By Proposition 4.14 we may assume that any summand A of ind_{L_{v}}^{G}(D_{v}) is a summand of K_{v} ≅ ind_{L_{v}}^{G}(D_{v}). In particular, if A_{v} is the endomorphism algebra of K_{v} then A = K_{v,E} = (K_{v})_{E} for some simple A_{v}-module E (c.f. 4.3). By adapting the construction in [Lus85, §10.3] we will show how ϕ_{A} is determined from ϕ. Let us denote by σ : A_{v} → A_{v} the algebra automorphism given by σ(θ) = ϕ ∘ F^{∗}θ ∘ ϕ^{−1} where F^{∗}θ : F^{∗}K_{v} → F^{∗}K_{v} is the induced map. For any A_{v}-module E we denote by E_{σ} the module obtained from E by twisting the action with σ^{−1} (c.f. A.1). For each such E we then have an induced isomorphism of A_{v}-modules

F^{∗}A = Hom_{A}(E, F^{∗}K_{v}) → Hom_{A}(E_{σ}, K_{v})

given by f ↦ ϕ ∘ f. Here F^{∗}K_{v} is an A-module under the action θ · k = (F^{∗}θ)(k) for all k ∈ F^{∗}K_{v} and the �rst equality is seen to hold by the construction given in (4.5).

Assume we have an A_{v}-module isomorphism ψ_{E} : E → E_{σ} then ϕ_{A} : F^{∗}K_{v,E} → K_{v,E}, given by ϕ_{A}(f) = ϕ ∘ f ∘ ψ_{E}, is an isomorphism of A_{v}-modules which de�nes an isomorphism in 𝒟G under the construction in (4.5). We now need only observe that when K_{v,E} is simple, hence E is simple, all such isomorphisms occur in this way. To see this note that any non-zero f ∈ Hom_{A_{v}}(E, K_{v}) gives an isomorphism E ≅ Im(f), because E is a simple module, so ψ_{E} = f^{−1} ∘ ϕ^{−1} ∘ ϕ_{A} ∘ f is determined by ϕ_{A}. This discussion shows that choosing the isomorphism ϕ_{A} is equivalent to choosing the isomorphism ψ_{E}.

Let us recall Lusztig’s basis {Θ_{y} | y ∈ W_{G}(L_{v}, ℒ_{v})} for the endomorphism algebra A_{v} (c.f. 5.2). From their de�nition one may readily check that the basis elements of A_{v} satisfy σ(Θ_{y}) = Θ_{F^{−1} (y)} for all y ∈ W_{G}(L_{v}, ℒ_{v}). Let W̃_{G}(L_{v}, ℒ_{v}) denote the semidirect product W_{G}(L_{v}, ℒ_{v}) ⋊ ⟨F⟩ where ⟨F⟩ is the �nite cyclic group generated by the automorphism F. Let E be a W_{G}(L_{v}, ℒ_{v})-module then we can extend E to a W̃_{G}(L_{v}, ℒ_{v})-module Ẽ by letting F^{−1} act as ψ_{E} under the isomorphism in (4.11). In this way we see that choosing ϕ_{A} is equivalent to choosing an extension Ẽ ∈ Irr(W̃_{G}(L_{v}, ℒ_{v})) of E.

Assume now that v = w, where w^{−1} ∈ Z_{G}(L, ℒ) is the unique element of minimal length. With this assumption F : W_{G}(L_{w}, ℒ_{w}) → W_{G}(L_{w}, ℒ_{w}) is an automorphism stabilising the set of Coxeter generators _{w} (c.f. 6.6). In this situation we will assume that ψ_{E} is chosen such that Ẽ is Lusztig’s preferred extension of E de�ned as in [Lus85, §17.2].

Passing from Twisted to Split Levis on the Trivial Local System

6.12. Assume, only for this section, that ℒ = \overline{}_{ℓ}. Let K_{v} be the image of (L_{v}, O_{v}, ℰ_{v}, ℒ_{v}) under the map in (4.9) and let ψ^{v} : F^{∗}K_{v} → K_{v} be the isomorphism de�ned in 6.10. We will denote by K the image of (L, O_{0}, ℰ_{0}, ℒ) under the map in (4.9). We wish to de�ne an isomorphism ϕ^{v} : F^{∗}K → K which is related (through Θ_{F(v)}) to the isomorphism ψ^{v}. Recall that in 6.7 and 6.10 we have de�ned isomorphisms ϕ : F^{∗}K → K, φ_{0} : F^{∗}ℰ_{0} → ℰ_{0}, φ_{1} : F^{∗}ℒ → ℒ and φ̃_{0} : F^{∗}ℰ̃_{0} → ℰ̃_{0} (simply take v = 1 in the constructions). With this we obtain isomorphisms

φ_{v,0} := θ_{F(v)}(φ_{0} φ_{1}) : F^{∗}(ℰ_{0} \overline{}_{ℓ}) → (ad v̇)^{∗}(ℰ_{0} \overline{}_{ℓ})
φ̃_{v,0} := θ̃_{F(v)}φ̃_{0} : F^{∗}ℰ̃_{0} → γ_{v}^{∗}ℰ̃_{0}

by composing with the endomorphism de�ned in 5.2. As in 6.10 we de�ne the isomorphism ϕ^{v} to be the unique extension of the isomorphism γ_{∗}φ̃_{v,0}. It is clear from the construction that we have ϕ^{v} = Θ_{F(v)}ϕ and furthermore using the exact same argument as in [Lus85, (10.6.1)] we have for each g ∈ G and i ∈ that

Tr(ϕ^{v}, ℋ_{g}^{i}K) = Tr(Θ_{F(v)}ϕ, ℋ_{g}^{i}K) = Tr(ψ^{v}, ℋ_{g}^{i}K_{v}).  (6.13)

Assume now that K_{ι} is an F-stable summand of K parameterised by the irreducible character E ∈ Irr(W_{G}(L))^{F} (c.f. 4.10). By 6.11 we have an isomorphism ϕ_{ι} : F^{∗}K_{ι} → K_{ι} de�ned by ϕ and Lusztig’s preferred extension of E. Replacing ϕ by ϕ^{v} in the construction of 6.11 we see that the isomorphism ϕ_{ι}is replaced by ϕ_{ι}^{v} = Θ_{F(v)}ϕ_{ι}. Note that we have a bijection

Irr(W_{G}(L))^{F} = Irr(W_{G}(L))^{F_{v}} → Irr(W_{G}(L_{v}))^{F},  (6.14)

induced by the isomorphism ad g_{v}. We will denote by K_{v,ι} the summand of K_{v} which is parameterised by E′ ∈ Irr(W_{G}(L_{v}))^{F} corresponding to E under (6.14). By 6.11 we have an isomorphism ψ_{ι}^{v} : F^{∗}K_{v,ι} → K_{v,ι} whose de�nition depends upon ψ^{v} and an extension Ẽ′ of E′. We may, and will, assume that the restriction of Ẽ′to the coset W_{G}(L_{v}).F coincides with the restriction of Ẽ to the coset W_{G}(L).F under the correspondence in 2.6. Arranging things in this way, and using (6.13), we then have for each g ∈ G and i ∈ that

Tr(ϕ_{ι}^{v}, ℋ_{g}^{i}K_{ι}) = Tr(Θ_{F(v)}ϕ_{ι}, ℋ_{g}^{i}K_{ι}) = Tr(ψ_{ι}^{v}, ℋ_{g}^{i}K_{v,ι}).  (6.15)

Bases of the Space of Unipotently Supported Class Functions

6.16. Before we continue we de�ne here two integer values associated to a pair ι ∈ N_{G}. Recall that [L_{ι}, ν_{ι}] ∈ M_{G} is the orbit such that ι ∈ ℐ [L_{ι}, ν_{ι}] (c.f. 4.1). Let us denote ν_{ι} by (O_{0}, ℰ_{0}) then we attach to ι the integers

a_{ι}= − dim O_{ι}− dim Z^{∘}(L_{ι}),
b_{ι}= (dim G − dim O_{ι}) − (dim L_{ι}− dim O_{0}).

We now consider how the above isomorphisms determine a canonical isomorphism F^{∗}ℰ_{ι} → ℰ_{ι} for an arbitrary pair ι = (O_{ι}, ℰ_{ι}) ∈ N_{G}^{F}. Recall from [Lus84b, Theorem 6.5(c)] that the perverse sheaf K_{ι} is such that ℋ ^{a_{ι}}(K_{ι})|_{O_{ι}} ≅ ℰ_{ι}. In 6.12 we have de�ned an isomorphism ϕ_{ι}^{v} : F^{∗}K_{ι} → K_{ι} (for each v ∈ W_{G}(L_{ι})) and this induces an isomorphism F^{∗}ℋ ^{a_{ι}}(K_{ι})|_{O_{ι}} → ℋ ^{a_{ι}}(K_{ι})|_{O_{ι}} hence an isomorphism φ_{ι}^{v} : F^{∗}ℰ_{ι} → ℰ_{ι}. Setting v = 1 gives the required canonical isomorphism. Following [Lus85, 24.2.2] we de�ne, for every v ∈ W_{G}(L_{ι}), an isomorphism ψ_{ι}^{v} by the condition that φ_{ι}^{v} = q^{(dim G+a_{ι})/2} ψ_{ι}^{v}. Note that we are using Remark 6.9 and the fact that b_{ι} = a_{ι}+ dim supp K_{ι}. This latter statement follows from the fact that dim supp K_{ι}= dim Y where Y is as in 4.13.

6.17. Let us assume that to every element ι ∈ N_{G} we have associated an element v_{ι} ∈ W_{G}(L_{ι}) such that this assignment is constant on blocks (i.e. v_{ι} = v_{ι′}if ι and ι′ are contained in the same block of N_{G}). Following [Lus85, §24.2] we associate to each F-stable pair ι ∈ N_{G}^{F} a unipotently supported class function of G by setting

X_{ι}^{v}(g) = (−1)^{a_{ι}}q^{−(dim G+a_{ι})/2} χ_{K_{ι},ϕ_{ι}^{v}}(g),

for all g ∈ G (where ϕ_{ι}^{v} is as in 6.12). Here we simply write v for v_{ι} with the meaning understood from the context. Furthermore for each ι ∈ N_{G}^{F} we de�ne a second unipotently supported class function of G by setting

Y_{ι}^{v}(g) = \{ \begin{matrix}Tr(ψ_{ι}^{v}, (ℰ_{ι})_{g}) & if g ∈ O_{ι}^{F}, \\ 0 & otherwise , \end{matrix}

for all g ∈ G (where ψ_{ι}^{v} is as in 6.16). The sets Y^{∗} = {Y_{ι}^{v} | ι ∈ N_{G}^{F}} and X^{∗} = {X_{ι}^{v} | ι ∈ N_{G}^{F}} are bases for the subspace Cent_{U}(G) of unipotently supported class functions of G (see [Lus85, 24.2.7]). In particular for each ι, ι′∈ N_{G}^{F} there exists an element P_{ι′,ι}∈ \overline{}_{ℓ} such that

X_{ι}^{v} = ∑_{ι′∈N_{G}^{F}} P_{ι′,ι}Y_{ι′}^{v}.

The matrix (P_{ι′,ι})_{ι′,ι∈N_{G}^{F}} is precisely the matrix considered in [Lus85, §24] and is computable by the algorithm described in the proof of [Lus85, Theorem 24.4]. We recall that the following property of the coe�cients P_{ι′,ι}holds

P_{ι′ι}= 0 if ℐ [L_{ι}, ν_{ι}] ≠ ℐ [L_{ι′}, ν_{ι′}].  (6.18)

From now on the following assumption is in place. If A ∈ Ĝ^{F} is a unipotently supported character sheaf then we assume the isomorphism ϕ_{A} : F^{∗}A → A to be chosen as in 6.11. If ι ∈ N_{G}^{F} is any F-stable pair and v ∈ W_{G}(L_{ι}) then we assume the isomorphism ϕ_{ι}^{v} : F^{∗}K_{ι}→ K_{ι}to be chosen as in 6.12.

7. Restricting Character Sheaves to the Unipotent Variety

7.1. Let us carry forward the setup of the previous section. We will denote by K_{w}^{ℒ} the image of (L_{w}, O_{w}, ℰ_{w}, ℒ_{w}) under the map in (4.9), where w ∈ W_{G}(L) is such that w^{−1} ∈ Z_{G}(L, ℒ) is the unique element of minimal length. Furthermore, we assume A = (K_{w}^{ℒ} )_{E} = K_{w,E}^{ℒ} is the summand of K_{w}^{ℒ} parameterised by E ∈ Irr(W_{G}(L_{w}))^{F} (c.f. 4.3). Setting ℒ = \overline{}_{ℓ} we obtain a new complex which we denote simply by K_{w}. Recall from 6.17 that X^{∗} is a basis for the subspace Cent_{U}(G) of unipotently supported class functions of G. In particular we have

χ_{A,ϕ_{A}}|_{G_{uni}} = ∑_{ι∈N_{G}^{F}} m(A, ι, ϕ_{ι}^{v})χ_{K_{ι},ϕ_{ι}^{v}}

for some coe�cients m(A, ι, ϕ_{ι}^{v}) ∈ \overline{}_{ℓ}. We will assume that the element v_{ι} chosen in 6.17 is w when ι is contained in the block ℐ [L, ν] (ν = (O_{0}, ℰ_{0})). It is the purpose of this section to describe the coe�cients m(A, ι, ϕ_{ι}^{v}). We will do this following the method in [Lus86], in particular we will now recall the sequence of isomorphisms constructed in [Lus86, §2.6].

7.2. Assume (L_{w}, Q) ∈ Z then we will denote by ind_{L_{w}⊂Q}^{G} (A_{ℒ} ) the image of (L_{w}, Q, O_{w}, ℰ_{w}, ℒ_{w}) under the map in (3.6). Let D be the similarly named complex de�ned in 3.2 then by the discussion in 4.13 we have D = IC(X̃′, \overline{ℰ_{w} ℒ_{w}} )[dim X̃′]. Setting ℒ = \overline{}_{ℓ} we obtain new complexes which we respectively denote by D_{0} and ind_{L_{w}⊂Q}^{G} (A_{0}). Recall the notation of 4.13 and let i : \overline{Y}_{uni} ↪ \overline{Y} be the natural inclusion of the unipotent elements contained in \overline{Y}. We have a commutative diagram

\overline{O}_{0}  X̂′_{uni}  X̃′_{uni}  \overline{Y}_{uni}

\overline{Σ}  X̂′  X̃′  \overline{Y} i π i σ i τ i π σ τ

where X̃′_{uni} = τ^{−1} (\overline{Y}_{uni}) and X̂′_{uni} = σ^{−1} (X̃′_{uni}). For X̃′_{uni} and X̂′_{uni} we have the action of i is given by the natural action on the �rst factor. We will denote by Z̃′_{uni} ⊂ X̃′_{uni} the subvariety given by {(g, hQ) ∈ G_{uni} × G/Q | h^{−1} gh ∈ ΣU_{Q}} and by ĩ : Z̃′_{uni} ↪ X̃′_{uni} the inclusion map. It is clear that we have an isomorphism ψ : i^{∗}(ℰ_{0} ℒ) → i^{∗}(ℰ_{0} \overline{}_{ℓ}) of local systems and as π ∘ ĩ∘ j (c.f. 4.13) commutes with i we have an induced isomorphism

ψ′= ĩ^{∗}j^{∗}π^{∗}(ψ) : i^{∗}ĩ^{∗}\overline{ℰ_{0} ℒ} → i^{∗}ĩ^{∗}\overline{ℰ_{0} \overline{}_{ℓ}} ,

which in turn induces an isomorphism

ψ′′: i^{∗}ĩ^{∗}D = IC(X̃′_{uni}, i^{∗}ĩ^{∗}\overline{ℰ_{0} ℒ} )[dim X̃′] → IC(X̃′_{uni}, i^{∗}ĩ^{∗}\overline{ℰ_{0} \overline{}_{ℓ}} )[dim X̃′] = i^{∗}ĩ^{∗}D_{0}.

According to [Lus84b, 6.6] we have (τ_{!}D)|_{\overline{Y}_{uni}} = τ_{!}i^{∗}ĩ^{∗}D and (τ_{!}D_{0})|_{\overline{Y}_{uni}} = τ_{!}i^{∗}ĩ^{∗}D_{0} hence τ_{!}(ψ′′) gives an isomorphism δ_{Q} : ind_{L_{w}⊂Q}^{G} (A_{ℒ} )|_{G_{uni}} → ind_{L_{w}⊂Q}^{G} (A_{0})|_{G_{uni}} in 𝒟G_{uni} (c.f. [Lus86, §2.6(c)]). Note that 2.3 does not apply here because i is a closed immersion.

7.3. For each pair (L_{w}, Q) ∈ Z we will denote by

λ_{ℒ,Q} : ind_{L_{w}⊂Q}^{G} (A_{ℒ} )|_{G_{uni}} → K_{w}^{ℒ} |_{G_{uni}}  and  λ_{Q} : ind_{L_{w}⊂Q}^{G} (A_{0})|_{G_{uni}} → K_{w}|_{G_{uni}}

the restrictions to G_{uni} of the canonical isomorphisms described by Proposition 4.14. Putting these isomorphisms together with δ_{Q} we can now de�ne an isomorphism

ε:= λ_{Q} ∘ δ_{Q} ∘ λ_{ℒ,Q}^{−1} : K_{w}^{ℒ} |_{G_{uni}} → K_{w}|_{G_{uni}},

(c.f. [Lus86, §2.6(a)]). We claim that ε does not depend upon the choice of parabolic subgroup Q used to de�ne it. Firstly, from the construction above, it is clear that δ_{Q} does not depend upon the choice of Q hence we need only show that the same is true of λ_{Q} and λ_{ℒ,Q} . However, using the construction in 4.13 we see that this follows from the commutative diagram

\begin{matrix}Ỹ & τ_{Q}^{−1} (Y) \\ Ỹ & τ_{R}^{−1} (Y) \end{matrix} id κ_{Q} μ κ_{R}

where (L_{w}, R) ∈ Z and μ(g, hQ) = (g, hR). This statement is implicitly used in [Lus86, §2.6].

7.4. Let v ∈ W_{G}(L_{w}, ℒ_{w}) then we denote by v̇∈ N_{G}(L_{w}, ℒ_{w}) a representative of v. Following [Lus86, §2.6(d)] we associate to each v ∈ W_{G}(L_{w}, ℒ_{w}) isomorphisms θ̃_{ℒ,v} : ind_{L_{w}⊂v̇P_{w}}^{G} (A_{ℒ} ) → ind_{L_{w}⊂P_{w}}^{G} (A_{ℒ} ) and θ̃_{v} : ind_{L_{w}⊂v̇P_{w}}^{G} (A_{0}) → ind_{L_{w}⊂P_{w}}^{G} (A_{0}) in the following way. Recall the notational conventions introduced in the proof of Lemma 3.9. Denote by φ_{v} : X̃_{L_{w}⊂v̇P_{w}}^{G} → X̃_{L_{w}⊂P_{w}}^{G} the isomorphism given by φ_{v}(g, hv̇P_{w}) = (g, hv̇P_{w}). By 2.3 we have (τ_{L_{w}⊂v̇P_{w}}^{G} )_{!}φ_{v}^{∗} = id^{∗}(τ_{L_{w}⊂P_{w}}^{G} )_{!} = (τ_{L_{w}⊂P_{w}}^{G} )_{!} so we take θ̃_{ℒ,v} and θ̃_{v} to be the isomorphisms induced by φ_{v}. Using the de�nition of θ_{v} one can check that we have a commutative diagram

K_{w}^{ℒ} |_{G_{uni}}  ind_{L_{w}⊂v̇P_{w}}^{G} (A_{ℒ} )|_{G_{uni}}  ind_{L_{w}⊂v̇P_{w}}^{G} (A_{0})|_{G_{uni}} K_{w}|_{G_{uni}}

K_{w}^{ℒ} |_{G_{uni}}  ind_{L_{w}⊂P_{w}}^{G} (A_{ℒ} )|_{G_{uni}}  ind_{L_{w}⊂P_{w}}^{G} (A_{0})|_{G_{uni}} K_{w}|_{G_{uni}} θ_{v} λ_{ℒ,v}^{−1} θ̃_{ℒ,v} δ_{v} θ̃_{v} λ_{v} θ_{v} λ_{ℒ,1}^{−1} δ_{1} λ_{1}

where λ_{ℒ,v} = λ_{ℒ,v̇P_{w}} and λ_{v} = λ_{v̇P_{w}}. In particular this shows that the isomorphism ε is an isomorphism of W_{G}(L_{w}, ℒ_{w})-modules (recall that ε does not depend upon the choice of parabolic subgroup used to de�ne it).

We now wish to check that the isomorphism ε respects the action of the Frobenius endomorphism. In other words let ϕ_{ℒ}^{w} : F^{∗}K_{w}^{ℒ} → K_{w}^{ℒ} and ϕ_{\overline{}_{ℓ}}^{w} : F^{∗}K_{w} → K_{w} be the isomorphisms de�ned in 6.10 then we wish to show that ε ∘ ϕ_{ℒ}^{w} = ϕ_{\overline{}_{ℓ}}^{w} ∘ F^{∗}ε. Recall that in 6.7 we �xed isomorphisms F^{∗}A_{ℒ} → A_{ℒ} and F^{∗}A_{0} → A_{0} and that by Lemma 3.9 these respectively induce isomorphisms ψ_{ℒ,Q} : F^{∗} ind_{L_{w}⊂F(Q)}^{G} (A_{ℒ} ) → ind_{L_{w}⊂Q}^{G} (A_{ℒ} ) and ψ_{Q} : F^{∗} ind_{L_{w}⊂F(Q)}^{G} (A_{0}) → ind_{L_{w}⊂Q}^{G} (A_{0}). With this we have a commutative diagram

F^{∗}(K_{w}^{ℒ} |_{G_{uni}})  F^{∗}(ind_{L_{w}⊂F(Q)}^{G} (A_{ℒ} )|_{G_{uni}})  F^{∗}(ind_{L_{w}⊂F(Q)}^{G} (A_{0})|_{G_{uni}}) F^{∗}(K_{w}|_{G_{uni}})

K_{w}^{ℒ} |_{G_{uni}}  ind_{L_{w}⊂Q}^{G} (A_{ℒ} )|_{G_{uni}}  ind_{L_{w}⊂Q}^{G} (A_{0})|_{G_{uni}}  K_{w}|_{G_{uni}} ϕ_{ℒ}^{w} F^{∗}λ_{ℒ,F(Q)}^{−1} ψ_{ℒ,Q} F^{∗}δ_{F(Q)} ψ_{Q} F^{∗}λ_{F(Q)} ϕ_{\overline{}ℓ}^{w} λ_{ℒ,Q}^{−1} δ_{Q} λ_{Q}

hence ε commutes with the isomorphisms ϕ_{ℒ}^{w} and ϕ_{\overline{}_{ℓ}}^{w} as desired (note that we have used here that F(G_{uni}) = G_{uni}).

7.5. We now arrive at our ultimate isomorphism (c.f. [Lus86, §2.6(e)]). For any simple W_{G}(L_{w}, ℒ_{w})- module E we will denote by Ê the induced module Ind_{W_{G}(L_{w},ℒ_{w})}^{W_{G}(L_{w})} (E). Now, for any such E we can de�ne a W_{G}(L_{w}, ℒ_{w})-module isomorphism 𝔛 : K_{w,E}^{ℒ}|_{G_{uni}} → K_{w,Ê}|_{G_{uni}} by taking the composition

K_{w,E}^{ℒ}|_{G_{uni}} = Hom_{W_{G}(L_{w},ℒ_{w})} (E, K_{w}^{ℒ} |_{G_{uni}})

≅ Hom_{W_{G}(L_{w},ℒ_{w})} (E, K_{w}|_{G_{uni}})
≅ Hom_{W_{G}(L_{w},ℒ_{w})} (E, Hom_{W_{G}(L_{w})}(\overline{}_{ℓ}W_{G}(L_{w}), K_{w}))|_{G_{uni}}
≅ Hom_{W_{G}(L_{w})}(\overline{}_{ℓ}W_{G}(L_{w}) ⊗ E, K_{w})|_{G_{uni}}
= K_{w,Ê}|_{G_{uni}}.

Here we have used ε and the standard isomorphisms given by (2.6) and (2.19) of [CR81]. Chasing through the isomorphisms we can see that

𝔛 ∘ ϕ_{K_{w,E}^{ℒ}}∘ F^{∗}f = ϕ_{\overline{}_{ℓ}}^{w} ∘ F^{∗}𝔛 ∘ F^{∗}f ∘ (1 ⊗ ψ_{E}).

for all f ∈ K_{w,E}^{ℒ}|_{G_{uni}}.

Using (6.15) and the fact that the modules \overline{}_{ℓ}W_{G}(L_{w}) ⊗ E_{σ} and (\overline{}_{ℓ}W_{G}(L_{w}) ⊗ E)_{σ} are isomorphic as W_{G}(L_{w})-modules we see that we have an equality

χ_{A,ϕ_{A}}|_{G_{uni}} = ∑_{ι∈ℐ (L,ν)^{F}} ⟨Ẽ_{ι}, Ind_{W_{G}(L_{w},ℒ_{w}).F}^{W_{G}(L_{w}).F} (Ẽ)⟩_{W_{G}(L_{w}).F}χ_{K_{ι},ϕ_{ι}^{w}}.  (7.6)

Note that in the above we assume that Ẽ_{ι} and Ẽ are the restrictions of the extensions (assumed to be Lusztig’s preferred extension) to the appropriate coset. In particular we have proved (c.f. 2.6) that

m(A, ι, ϕ_{ι}^{v}) = \{ \begin{matrix}⟨Ẽ_{ι}, Ind_{W_{G}(L,ℒ).Fw^{−1}}^{W_{G}(L).F} (Ẽ)⟩_{W_{G}(L).F} & if ι ∈ ℐ [L, ν] \\ 0 & otherwise , \end{matrix}

for all ι ∈ N_{G}^{F}. Note that, as in (6.14), we identify the sets of irreducible characters Irr(W_{G}(L))^{F} and Irr(W_{G}(L_{w}))^{F}. Combining this with the statements of 6.17 we obtain our main result.

Theorem 7.7. We have the following equality of class functions

χ_{A,ϕ_{A}}|_{G_{uni}} = ∑_{ι′,ι∈ℐ [L,ν]^{F}}⟨Ẽ_{ι}, Ind_{W_{G}(L,ℒ).Fw^{−1}}^{W_{G}(L).F} (Ẽ)⟩_{W_{G}(L).F}(−1)^{a_{ι}}q^{(dim G+a_{ι})/2} P_{ι′,ι}Y_{ι′}^{w}.

Remark 7.8. Note that the equality (7.6) echoes a similar equality which is known to hold for almost characters (see for instance [AA07, Lemme 6.1]).

8. Split Elements and Lusztig’s Preferred Extensions

From now until the end of this article we assume that Z(G) is connected, G/Z(G) is simple and p is a good prime for G.

8.1. To make the formula in Theorem 7.7 computationally explicit we must now show how to compute the functions Y_{ι}^{w}. In [Sho87, Remark 5.1], assuming F is a split Frobenius endomorphism, Shoji has de�ned the notion of a split unipotent element. However, there is also a notion of split unipotent element when F is not split which we would like to recall here. All the statements of this section are due to the combined e�orts of Hotta�Springer, Beynon�Spaltenstein and Shoji. To give the de�nition we must �rst prepare some preliminary notions and notation.

Given a unipotent element u ∈ G we denote by 𝔅_{u}^{G} the variety of Borel subgroups of G containing the unipotent element u and we denote its dimension dim 𝔅_{u}^{G} by d_{u}. The corresponding ℓ-adic cohomology groups with compact support H_{c}^{i}(𝔅_{u}^{G}) := H_{c}^{i}(𝔅_{u}^{G}, \overline{}_{ℓ}) are modules for the direct product A_{G}(u) × W_{G} which are zero unless i ∈ {0, …, 2d_{u}} (in fact i must be even). Note that we take the (A_{G}(u) × W_{G})- module structure to be the one described by Lusztig in [Lus81, §3] (this agrees with the generalised Springer correspondence given in [Lus84b]). This di�ers from the original module structure given by Springer in [Spr78] but one is translated to the other by composing the W_{G}-action with the sign character (see [Hot81, Theorem 1]).

Assume now that u ∈ G is �xed by F then F stabilises 𝔅_{u}^{G} hence we have an induced action of F in the compactly supported cohomology which we denote by F^{•} : H_{c}^{•}(𝔅_{u}^{G}) → H_{c}^{•}(𝔅_{u}^{G}). Let ψ ∈ Irr(A_{G}(u)) be an irreducible character then we denote by H_{c}^{•}(𝔅_{u}^{G})_{ψ} the ψ-isotypic component of the module H_{c}^{•}(𝔅_{u}^{G}). Springer’s main result (see [Spr78, Theorem 1.13]) shows that either H_{c}^{2d_{u}}(𝔅_{u}^{G})_{ψ} is 0 or a simple (A_{G}(u) × W_{G})-module isomorphic to ψ ⊗ E_{u,ψ} (this is the classical formulation of the Springer correspondence).

8.2. Let σ : W_{G} → W_{G} denote the automorphism induced by F; note that σ stabilises the set of Coxeter generators so that σ induces an automorphism of (W_{G}, ). We now de�ne what it means for a unipotent element u ∈ G to be split. This de�nition will be well-de�ned up to G-conjugacy. We will do this on a case by case basis as follows.

(1) G not of type E_{8} and σ the identity. We say u ∈ G^{F} is split if F stabilises every irreducible component of 𝔅_{u}^{G}. [BS84; HS77; Sho82; Sho83]

(2) G of type E_{8}. If q ≡ 1 (mod 3) then we de�ne split elements as in case (1). If q ≡ −1 (mod 3) then for each F-stable class O we de�ne an element u ∈ O^{F} to be split if it satis�es the condition of (1) unless O is the class E_{8}(b_{6}) in the Bala�Carter labelling (see [Car93, pg. 177]). For this class we say u ∈ O^{F} is split if the restriction of F^{•} to H^{2d_{u}}(𝔅_{u})_{ψ} is q^{d_{u}} · id for each irreducible character ψ of A_{G}(u) ≅ 𝔖_{3} which is not the sign character. [BS84, §3 - Case (V)]

(3) G of type A_{n} and σ of order 2. We de�ne any unipotent element u ∈ G^{F} to be split.

(4) G of type D_{n} and σ of order 2. Let G_{ad} denote SO_{2n}( )/Z(SO_{2n}( )), which is an adjoint group of type D_{n}. Let π : SO_{2n}( ) → G_{ad} be the natural projection map and let us �x an adjoint quotient α : G → G_{ad}. We assume that SO_{2n}( ) and G_{ad} are endowed with Frobenius endomorphisms so that the morphisms π and α are de�ned over _{q}. We say u ∈ G is split if (π^{−1} ∘ α)(u) is a split unipotent element as de�ned in [Sho07, 2.10.1]. Note that there can only be one unipotent element in the preimage of α(u) under π.

(5) G of type D_{4} and σ of order 3. We say u ∈ G^{F} is split if it satis�es the condition of (1) with respect to the Frobenius endomorphism F^{3}. [Sho83, §4.24]

(6) G of type E_{6} and σ of order 2. Let F_{0} : G → G be a Frobenius endomorphism which induces the identity on W_{G}. In [BS84, §4] Beynon�Spaltenstein de�ne a bijection between the set of G^{F}- conjugacy classes of unipotent elements and the set of G^{F_{0}}-conjugacy classes of unipotent elements, which is uniquely characterised by three properties. We say a unipotent element u ∈ G^{F} is split if it is contained in a G^{F}-conjugacy class which is in bijective correspondence with a split G^{F_{0}}-conjugacy class under Beynon�Spaltenstein’s bijection.

Remark 8.3. In (4) we have used an adjoint quotient α : G → G_{ad} to de�ne the notion of split unipotent element. We claim that this does not depend upon the choice of α up to G-conjugacy. Assume α′ is another adjoint quotient of G de�ned over _{q} then there exists t ∈ T_{0} such that α′ = α ∘ ad t (see [Ste99, 1.5]). To prove the claim it su�ces to show that we may take t ∈ T_{0}^{F}. Now α′, α, ad t and F are bijections on G_{uni} (the set of unipotent elements), hence the condition F ∘ α′ = α′∘ F implies ad(t^{−1} F(t)) is the identity on G_{uni}. This implies t^{−1} F(t) centralises a regular unipotent element so by [DM91, Lemma 14.15] we must have t^{−1} F(t) ∈ Z(G) as it is semisimple. By the Lang�Steinberg theorem (applied inside the connected group Z(G)) there exists z ∈ Z(G) such that t^{−1} F(t) = z^{−1} F(z) so F(tz^{−1} ) = tz^{−1} . Thus we have α′= α′∘ ad z^{−1} = α ∘ ad tz^{−1} so we are done.

As the irreducible components of 𝔅_{u}^{G} have the same dimension (see [Spa82, I, Proposition 1.12]) they form a basis for H_{c}^{2d_{u}}(𝔅_{u}^{G}). In particular (1)is equivalent to saying that F^{•} acts as q^{d_{u}} times the identity on H_{c}^{2d_{u}}(𝔅_{u}^{G}). With this we see that the references given above show that every F-stable unipotent conjugacy class O ⊂ G contains a split unipotent element. Before continuing we recall the following properties of split elements.

Lemma 8.4. Assume O is an F-stable unipotent conjugacy class of G then the split elements contained in O^{F} form a single G-conjugacy class.

Proof. Assume G is of type A_{n} then this is trivial as A_{G}(u) is trivial. If G is of type B_{n}, C_{n} or D_{n} then this is clear from the de�nition (see [Sho07, §2.7, §2.10]). If G is of exceptional type then this is noted by Benyon�Spaltenstein in [BS84, §3]. The only case not explicitly dealt with is (5), however this is easily checked in this case. ■

Lemma 8.5. If u ∈ G is a split unipotent element then F acts trivially on A_{G}(u).

Proof. If G is of classical type then the action of F on A_{G}(u) is always trivial (see for instance the proof of [Tay13, Proposition 2.4]). Assume now that G is of exceptional type then from [BS84] one sees that Beynon�Spaltenstein start with an element satisfying this property then show that it is split, hence this certainly holds. ■

8.6. Let us denote by W̃_{G} the semidirect product W_{G} ⋊ ⟨σ⟩ where ⟨σ⟩ ⩽ Aut(W_{G}) is the cyclic group generated by σ. Assume u ∈ G^{F} is a split unipotent element and ψ ∈ Irr(A_{G}(u)) then we consider the cohomology group E = H_{c}^{2d_{u}}(𝔅_{u}^{G})_{ψ} as a W_{G}-module. If E is σ-stable then (by Lemma 8.5) we may consider H_{c}^{2d_{u}}(𝔅_{u}^{G})_{ψ} as a W̃_{G}-module which we denote by Ẽ (see [Sho07, §4.1] and [Sho83, 3.9, 3.10]). The Frobenius then acts as q^{d_{u}}σ on H_{c}^{2d_{u}}(𝔅_{u}^{G})_{ψ}. In particular Ẽ is an extension of E and it is our purpose to now describe this extension. Note that there may be several possibilities for such an extension but we recall that in [Lus85, §17.2] Lusztig has given a systematic non-canonical choice for such an extension called the preferred extension.

Proposition 8.7. Assume u ∈ G_{uni}^{F} is a split element then the action of ±σ on Ẽ makes this the preferred extension of E. The sign is always positive unless G is of type E_{8}, q ≡ −1 (mod 3) and u is contained in the class E_{8}(b_{6}).

Proof. The case of type A_{n} (resp. E_{6}) follows from [HS77, Lemma 3.2] (resp. [BS84, §4]) together with [GM00, Proposition 2.23]. Note that the Springer correspondence we have described in (4.2) di�ers from that in [HS77] by tensoring with the sign character. The cases of type D_{n} with σ of order 2 and D_{4} with σ of order 3 are given by [Sho07, Theorem 4.3(ii)] and [Sho83, §4.24] respectively. Finally the case of E_{8} follows from the discussion in [BS84, §3, Case (V)]. ■

9. Computing Functions in Good Characteristic

9.1. We will now show how the functions Y_{ι}^{w} occuring in Theorem 7.7 can be computed using the notion of split unipotent elements. In particular, we must show how to determine the trace of ψ_{ι}^{v} on the stalk (ℰ_{ι})_{u} for a unipotent element u ∈ O_{ι}^{F}. We will start by considering the case where u is a split unipotent element. Note that the isomorphism ψ_{ι}^{v} is determined once we have chosen the isomorphism φ_{0} : F^{∗}ℰ_{0} → ℰ_{0} �xed in 6.7.

We assume that the element u_{0} ∈ O_{0}^{F} chosen in Section 5 is a split unipotent element. Furthermore, we assume that the isomorphism φ_{0} : F^{∗}ℰ_{0} → ℰ_{0} is chosen such that the induced map (ℰ_{0})_{u_{0}} → (ℰ_{0})_{u_{0}} at the stalk of the split element u_{0} is q^{(dim(L/Z^{∘}(L))−dim O_{0})/2} times the identity (note that O_{0} cannot be the class E_{8}(b_{6})).

With these assumptions we may now prove our �nal result, which follows the line of argument given in [Lus86, 3.4] and [Sho97, Lemma 3.6].

Proposition 9.2. If G is of type E_{8} then assume q ≡ 1 (mod 3). For any F-stable pair ι ∈ N_{G}^{F} and v ∈ W_{G}(L_{ι}) the map (ℰ_{ι})_{u} → (ℰ_{ι})_{u} induced by ϕ_{ι}^{v} for some (any) split element u ∈ O_{ι}^{F} is ±γ_{L_{ι},u_{0}}^{G}(F(v))q^{(dim G+a_{ι})/2} times the identity. The sign is positive unless G is of type E_{8}, q ≡ −1 (mod 3) and O_{ι}is the class E_{8}(b_{6}) in which case the sign is negative.

Proof. Let K be the image of (L, O_{0}, ℰ_{0}, \overline{}_{ℓ}) under the map in (4.9) and let ϕ : F^{∗}K → K be the isomorphism de�ned in 6.10. We denote by φ_{u} the isomorphism ℋ_{u}^{a_{ι}}(K) → ℋ_{u}^{a_{ι}}(K) induced by Θ′_{F(v)}ϕ. By Proposition 5.6 it su�ces to show that φ_{u} is ±q^{(dim G+a_{ι})/2} times the identity, where the sign is positive unless G is of type E_{8}, q ≡ −1 (mod 3) and O_{ι}is the class E_{8}(b_{6}).

First of all let us note that under the isomorphism in (4.6) we have

⨁_{E∈Irr(A)} (E ⊗ F^{∗}K_{E}) ≅ F^{∗}K  ⨁_{E∈Irr(A)} (E ⊗ ℋ_{u}^{a_{ι}}(K_{E})) ≅ ℋ_{u}^{a_{ι}}(K).

Both of these isomorphisms are given in the same way as (4.6) by noticing that F^{∗}K_{E} = Hom_{A}(E, F^{∗}K) and ℋ_{u}^{a_{ι}}(K_{E}) = Hom_{A}(E, ℋ_{u}^{a_{ι}}(K)). Assume E ∈ Irr(W_{G}(L))^{F} then under the isomorphism (4.6) the restriction of Θ′_{F(v)}ϕ to the summand isomorphic to E ⊗ K_{E} corresponds to the isomorphism ψ_{E}^{−1} ⊗ (Θ′_{F(v)}ϕ_{A}) (where ψ_{E} and ϕ_{A} are as in 6.11). With this in mind it is enough to show that φ_{u} acts on an A-submodule of ℋ_{u}^{a_{ι}}(K) isomorphic to E as q^{(dim G+a_{ι})/2} times ψ_{E}^{−1} (resp. −ψ_{E}^{−1} if O_{ι}is E_{8}(b_{6}) and q ≡ −1 (mod 3)).

To prove this we follow the argument of [Sho97, Lemma 3.6] which in turn is a modi�cation of the arguments in [Lus86]. Let P ⩽ G be a parabolic subgroup of G such that P = L_{ι}U_{P} is a Levi decomposition of P. As ι ∈ N_{G}^{F} is F-stable we may assume L_{ι} and P are F-stable. We de�ne Z_{u} to be the variety {xP ∈ G/P | x^{−1} ux ∈ O_{0}U_{P}} and similarly we de�ne Ẑ_{u} to be the variety {x ∈ G | x^{−1} ux ∈ O_{0}U_{P}}. We have two natural morphisms ψ_{u} : Ẑ_{u} → Z_{u}, resp. λ_{u} : Ẑ_{u} → O_{0}, given by ψ_{u}(x) = xP and λ_{u}(x) = (O_{0}- component of x^{−1} ux ∈ O_{0}U_{P}). We then de�ne a local system ℰ̂_{ι} on Z_{u} by the condition that ψ_{u}^{∗}ℰ̂_{ι} = λ_{u}^{∗}ℰ_{0}.

As mentioned in 6.16, we have b_{ι}= a_{ι}+ dim supp K_{ι}hence by [Lus85, 24.2.5] we have an isomorphism Φ : ℋ_{u}^{a_{ι}}(K) → H_{c}^{b_{ι}}(Z_{u}, ℰ̂_{ι}) (where this is the cohomology with compact support of Z_{u} with coe�cients in the local system ℰ̂_{ι}). Note that we have a commutative diagram

\begin{matrix}Z_{u} & Ẑ_{u} & O_{0} \\ Z_{u} & Ẑ_{u} & O_{0} \end{matrix} π_{v} π̂_{v} ψ_{u} λ_{u} ad v̇ ψ_{u} λ_{u}

where π_{v} and π̂_{v} are de�ned by π_{v}(x) = xv̇^{−1} and π̂_{v} (xP) = xv̇^{−1} P. Let φ_{0} : F^{∗}ℰ_{0} → ℰ_{0} be as in 6.7 then the composition θ′_{F(v)}φ_{0} : F^{∗}ℰ_{0} → (ad v̇)^{∗}ℰ_{0} is an isomorphism such that the induced isomorphism at the stalk of our �xed split element u_{0} ∈ O_{0}^{F} is the identity (c.f. 5.5). This isomorphism induces an isomorphism φ̂_{0} : F^{∗}ℰ̂_{ι} → π_{v}^{∗}ℰ̂_{ι} which in turn induces a linear map φ̂_{0} of H_{c}^{b_{ι}}(Z_{u}, ℰ̂_{ι}) satisfying Φ ∘ φ_{u} = φ̂_{0} ∘ Φ. With this we see that we are left with showing that φ̂_{0} acts on an A-submodule of H_{c}^{b_{ι}}(Z_{u}, ℰ̂_{ι}) isomorphic to E as q^{(dim G+a_{ι})/2} times ψ_{E}^{−1} (resp. −ψ_{E}^{−1} if O_{ι}is E_{8}(b_{6}) and q ≡ −1 (mod 3)).

Assume L_{ι} is a torus so that ι is in the principal block then (O_{0}, ℰ_{0}) = (\{1\}, \overline{}_{ℓ}) and Z_{u} can be canonically identi�ed with 𝔅_{u}^{G}. Furthermore we have ℰ̂_{ι} is simply the constant sheaf and φ̂_{0} induces the identity at every stalk of ℰ̂_{ι}. Hence the statement is simply that of Proposition 8.7 so we are done in this case. The case where L_{ι} = G (i.e. ι is cuspidal) is trivial so we are left only with the case where L_{ι} is neither G nor T_{0}. For this to be the case we must have G is of type B_{n}, C_{n} or D_{n} but these cases are dealt with by Shoji in [Sho07, Theorem 4.3] (see also the reduction arguments given in [Sho06, 1.5]). Note that to apply this theorem we need the fact that θ′_{F(v)}φ_{0} induces the identity at the stalk of the split element u_{0} ∈ O_{0}^{F}. ■

9.3. With this we may now give a precise description of the function Y_{ι}^{w} for all ι ∈ ℐ [L, ν]^{F}. Let us �x a split element u ∈ O_{ι}^{F} then we denote by A_{G}(u) the �nite component group C_{G}(u)/C_{G}^{∘}(u) of the centraliser of u. Recall that we have a natural bijection between G-equivariant local systems on O_{ι} and Irr(A_{G}(u)) (see [Sho88, pg. 74]). In particular ℰ_{ι} determines a unique irreducible character χ_{ι} ∈ Irr(A_{G}(u)). As F acts trivially on A_{G}(u) (c.f. Lemma 8.5) we have the orbits of G acting on O_{ι}^{F} by conjugation are in bijection with the conjugacy classes of A_{G}(u). For each x ∈ A_{G}(u) we denote by u_{x} ∈ O_{ι}^{F} an element corresponding to x under this bijection. Now, combining Lemma 8.5 and [Sho06, 1.3] we obtain our �nal result.

Theorem 9.4. For any g ∈ G we have

Y_{ι}^{w}(g) = \{ \begin{matrix}±γ_{L_{ι},u_{0}}^{G}(F(w))χ_{ι}(x) & if g ∼_{G} u_{x} ∈ O_{ι}^{F}, \\ 0 & otherise, \end{matrix}

where the sign is positive unless G is of type E_{8}, q ≡ −1 (mod 3) and O_{ι}is the class E_{8}(b_{6}). Note that we write g ∼_{G} u_{x} to denote that g is G-conjugate to u_{x}.

A. Finite Groups and Cosets of Automorphisms

A.1. Assume G is a �nite group and ϕ : G → G is an automorphism then we recall the notation of 2.5 (in particular G̃ = G ⋊ ⟨ϕ⟩). If V is a G-module then we denote by V_{ϕ} the module obtained from V by twisting with ϕ. In other words, if · denotes the original action of \overline{}_{ℓ}G on V then we obtain V_{ϕ} by de�ning a new action ⋆ given by g ⋆ v = ϕ(g) · v for all g ∈ \overline{}_{ℓ}G and v ∈ V (here ϕ is naturally extended to \overline{}_{ℓ}G by linearity). We say V is ϕ-stable if we have an isomorphism σ : V → V_{ϕ} of G-modules. Note that the choice of σ de�nes a G̃-module structure on V by setting ϕ · v = σ(v) for all v ∈ V and all such possible extensions are obtained in this way. Assume χ (resp. χ̃) is the character of G (resp. G̃) a�orded by V then we call χ̃ the σ-extension of χ.

For any two elements x, y ∈ G we say x and y are ϕ-conjugate, and write x ∼_{ϕ} y, if there exists z ∈ G such that x = z^{−1} yϕ(z). We denote by Cl_{G,ϕ}(x) the equivalence class of x under ∼_{ϕ}, called the ϕ-conjugacy class of x, and H^{1}(G, ϕ) the set of all such equivalence classes. Finally we write C_{G,ϕ}(x) = {g ∈ G | g^{−1} xϕ(g) = x} for the ϕ-centraliser of x ∈ G; clearly we have |G| = |C_{G,ϕ}(x)| · | Cl_{G,ϕ}(x)| by the orbit-stabiliser theorem. When ϕ = id then we write Cl_{G}(x) (resp. C_{G}(x)) for Cl_{G,ϕ}(x) (resp. C_{G,ϕ}(x)). We recall here the following lemmas which are left as an exercise to the reader.

Lemma A.2. The natural bijection G → G.ϕ^{i} restricts to an injective map Cl_{G,ϕ^{i}}(x) → ClG̃(x). Furthermore, when i = 1 this map is a bijection.

Lemma A.3. For any f ∈ Cent(H.ϕg) we have an equality

Ind_{H.ϕg}^{G.ϕ}(f) = \frac{1}{[G̃ : G]}· (ResG̃ ; G.ϕ ∘ IndG̃ ; H̃)(f̂)  (A.4)

where

f̂(x) = \{ \begin{matrix}[H̃ : H]f(x) & if x ∈ H.ϕg, \\ 0 & otherwise. \end{matrix}

for all x ∈ H̃ (here [G̃ : G] and [H̃ : H] denote respectively the index of G in G̃ and H in H̃).

Remark A.5. Assume that ϕ is a non-trivial automorphism of G then, in this case, we may allow f̂to take any value on H̃ − H.ϕg. However, if ϕ is trivial then this value must be 0.

A.6. We will denote by Irr(H)^{ϕg} = {χ ∈ Irr(H) | χ ∘ ϕ ∘ ad_{g} = χ} the set of ϕ ad_{g}-invariant characters of H. If χ ∈ Irr(H)^{ϕg} then we �x an arbitrary extension χ̃ of χ to H̃, which exists because H̃/H is cyclic. We may then give an orthonormal basis of Cent(H.ϕg), with respect to ⟨−, −⟩_{H.ϕg}, by setting Irr(H.ϕg) = {χ̃|_{H.ϕg} | χ ∈ Irr(H)^{ϕg}} (this basis depends upon the choice of extension of χ to H̃). Let f = χ̃|_{H.ϕg} ∈ Irr(H.ϕg) then it is easily checked that the function f̂de�ned in Lemma A.3 is given by

f̂= ∑_{ε∈Irr(H̃/H)} ε(ϕ(g), ϕ)^{−1} χ̃ ⊗ ε.  (A.7)

Here we consider any character ε ∈ Irr(H̃/H) as a character of H̃ by in�ation.

Lusztig�Macdonald�Spaltenstein Induction and Cyclic Automorphisms

A.8. Let t be an indeterminate over \overline{}_{ℓ} and let V be a G-module. For any irreducible character χ ∈ Irr(G) we will denote by P_{χ}^{V}(t) the Molien series of χ (see [GP00, §5.2.2] for a de�nition). There then exist unique integers γ_{χ}^{V} ∈ and b_{χ}^{V} ∈ _{0} such that

P_{χ}^{V}(t) = γ_{χ}^{V} · t^{b_{χ}^{V}} + higher powers of t.

We call b_{χ}^{V} the b-invariant of χ. For any natural number d ∈ _{0} we will denote by Irr^{V}(G | d) the set {χ ∈ Irr(G) | γ_{χ}^{V} = 1 and b_{χ}^{V} = d}.

We will denote by U the H-module V / Fix_{H}(V) where Fix_{H}(V) = {v ∈ V | h · v = v for all h ∈ H}. Following [GP00, §5.2.8] we may de�ne a map j_{H}^{G} : Irr^{U}(H | d) → Irr^{V}(G | d), called the j-induction, by setting j_{H}^{G}(χ) to be the unique irreducible character ψ ∈ Irr(G) satisfying:

• ⟨Ind_{H}^{G}(χ), ψ⟩ = 1 and

• b_{ψ}^{V} = b_{χ}^{U}.

(see [GP00, Theorem 5.2.6]). Note that for arbitrary �nite groups this induction map was introduced by Lusztig�Spaltenstein in [LS79]. When there is no confusion concerning the module V we will simply suppress the superscript V in the notation.

A.9. We now consider the special case where G = G_{1} × ⋯× G_{n}, with all factors isomorphic, and ϕ cyclically permutes the factors. In other words ϕ(g_{1}, …, g_{n}) = (ϕ_{n}(g_{n}), ϕ_{1}(g_{1}), …, ϕ_{n−1} (g_{n−1} )) where ϕ_{i} : G_{i} → G_{i+1} , for 1 ⩽ i ⩽ n − 1, and ϕ_{n} : G_{n} → G_{1} are isomorphisms. We then have the homomorphism

ψ_{i} := ϕ_{i−1} ⋯ϕ_{1}ϕ_{n} ⋯ϕ_{i} : G_{i} → G_{i}

is an automorphism of G_{i} satisfying ϕ_{i}ψ_{i} = ψ_{i+1} ϕ_{i} for all 1 ⩽ i ⩽ n − 1 and ϕ_{n}ψ_{n} = ψ_{1}ϕ_{n}. Assume χ ∈ Irr(G)^{ϕ} is a ϕ-stable character then we necessarily have χ = χ_{1} ⋯ χ_{n} where χ_{i} ∈ Irr(G_{i})^{ψ_{i}} is ψ_{i}-stable and χ_{i} = χ_{i+1} ∘ ϕ_{i} for all 1 ⩽ i ⩽ n − 1 and χ_{n} = χ_{1} ∘ ϕ_{n}. In particular, we have a natural bijection between the sets Irr(G)^{ϕ} and Irr(G_{i})^{ψ_{i}}. With this in mind we have the following easy lemma.

Lemma A.10. The natural inclusion map : G_{1} → G (i.e. \overline{g} = (g, 1, …, 1)) induces a bijection H^{1}(G_{1}, ψ_{1}) → H^{1}(G, ϕ) such that |C_{G_{1},ψ_{1}}(g)| = |C_{G,ϕ}(\overline{g})|.

A.11. We now assume that V = V_{1} ⊕ ⋯⊕ V_{n} is a G-module such that V_{i} ≠ 0 is a G_{i}-module and V = V_{i} ⊕ Fix_{G_{i}}(V) for all 1 ⩽ i ⩽ n. Consequently we have Fix_{G}(V) = Fix_{G_{i}}(V_{i}) = 0 for all 1 ⩽ i ⩽ n. Furthermore we assume that V is ϕ-stable and that there exists an isomorphism σ : V → V_{ϕ} with the following properties. There exist \overline{}_{ℓ}-vector space isomorphisms σ_{i} : V_{i} → V_{i+1} , for 1 ⩽ i ⩽ n − 1, and σ_{n} : V_{n} → V_{1} such that

σ(v_{1}, …, v_{n}) = (σ_{n}(v_{n}), σ_{1}(v_{1}), …, σ_{n−1} (v_{n−1} ))  and  σ_{i}(g_{i} · v_{i}) = ϕ_{i}(g_{i}) · σ_{i}(v_{i})

for all g_{i} ∈ G_{i} and v_{i} ∈ V_{i}. Each linear map τ_{i} := σ_{i−1} ⋯σ_{1}σ_{n} ⋯σ_{i} is then an isomorphism V_{i} → (V_{i})_{ψ_{i}} satisfying σ_{i}τ_{i}σ_{i}^{−1} = τ_{i+1} for all 1 ⩽ i ⩽ n − 1 and σ_{n}τ_{n}σ_{n}^{−1} = τ_{1}. The isomorphism σ (resp. τ_{i}) then makes V (resp. V_{i}) a G̃-module (resp. G̃_{i} := G ⋊ ⟨ψ_{i}⟩-module) as in A.1. Finally we will assume that there exists a σ-invariant basis B = B_{1} ⊕ ⋯⊕ B_{n} of V, where B_{i} is a basis of V_{i}. In particular we have σ_{i}(B_{i}) = B_{i+1} for all 1 ⩽ i ⩽ n − 1 and σ_{n}(B_{n}) = B_{1} and τ_{i}(B_{i}) = B_{i} for all 1 ⩽ i ⩽ n.

A.12. Let E = E_{1} ⋯ E_{n} be a ϕ-invariant simple G-module such that the character χ a�orded by E is contained in Irr(G | d)^{ϕ}. Here E_{i} denotes a simple G_{i}-module and we write χ_{i} ∈ Irr(G_{i})^{ψ_{i}} for the character a�orded by E_{i}, which must necessarily be contained in Irr(G_{i} | d/n)^{ψ_{i}} (see [GP00, Exercise 5.2]). As E is ϕ-invariant there exists a family of \overline{}_{ℓ}-vector space isomorphisms δ_{i} : E_{i} → E_{i+1} , for 1 ⩽ i ⩽ n − 1 and δ_{n} : E_{n} → E_{1} such that

δ_{i}(g_{i} · v_{i}) = ϕ_{i}(g_{i}) · δ_{i}(v_{i})

for all 1 ⩽ i ⩽ n. These isomorphisms are uniquely determined up to scalar multiples. As before, each linear map α_{i} := δ_{i−1} ⋯δ_{1}δ_{n} ⋯δ_{i} is an isomorphism E_{i} → (E_{i})_{ψ_{i}} and we denote by χ̃_{i} the α_{i}-extension of χ_{i}. Replacing the δ_{i}’s by scalar multiples we may, and will, assume that χ̃_{i} is the unique extension of χ_{i} satisfying bχ̃_{i} = b_{χ_{i}} (here the b-invariants are taken with respect to the module V_{i} introduced in A.11). In other words jG̃_{i} ; G_{i}(χ_{i}) = χ̃_{i}. We now de�ne an isomorphism α : E → E_{ϕ} by setting

α(v_{1} ⋯ v_{n}) = α_{n}(v_{n}) α_{1}(v_{1}) ⋯ α_{n−1} (v_{n−1} ).

We call the α-extension χ̃ of χ the b-preferred extension of χ. With this we have the following result concerning the j-induction.

Proposition A.13. Assume n is a prime number and χ ∈ Irr(G | d)^{ϕ}. If χ̃ ∈ Irr(G̃) is the b-preferred extension of χ then we have jG̃ ; G(χ) = χ̃.

Proof. It su�ces to show that bV ; χ̃ = b_{χ}^{V}. For this we recall the following result of Molien (see [GP00, Proposition 5.2.4]) which states that

P V ; χ̃(t) = \frac{1}{|G̃|} ∑_{g∈G̃} \frac{χ̃(g^{−1} )}{det_{V}(id_{V} −t · φ_{g})}

where φ_{g} : V → V is the map de�ned by the action of g. The term in the sum is constant on G̃-conjugacy classes. Therefore, using Lemma A.2 and splitting up along cosets we have

P V ; χ̃(t) = \frac{1}{[G̃ : G]} ∑_{ϕ^{i}∈⟨ϕ⟩} ∑_{x∈H^{1}(G,ϕ^{i})} \frac{χ̃((x; ϕ^{i})^{−1} )}{|C_{G,ϕ^{i}}(x)| · det_{V}(id_{V} −t · φ_{(x;ϕ^{i})})} ,  (A.14)

where here the sum is taken over a set of representatives from the equivalence classes H^{1}(G, ϕ^{i}).

When i = 1 it follows from Lemma A.10 that we need only compute this term on elements of the form \overline{x} where x ∈ G_{1}. After suitably ordering the basis B considered in A.11 (and identifying linear maps with matrices) we have

φ_{(\overline{x};1)} = [ \begin{matrix}A \\ & I_{m} \\ & & . & . & . \\ & & & & & I_{m} \end{matrix}]  and  φ_{(1;ϕ)} = [ \begin{matrix}& B \\ & & . & . & . \\ & & & & & B \\ B \end{matrix}]

where I_{m} denotes the m × m identity matrix and m is the common dimension of the V_{j}’s. Here A and B are m × m matrices; in fact B is a permutation matrix. With this we have

det(id_{V} −t · φ_{(\overline{x};ϕ)}) = det [ \begin{matrix}I_{m} & −t · AB \\ & I_{m} & −t · B \\ & & I_{m} & ... \\ & & & ... & −t · B \\ −t · B & & & & I_{m} \end{matrix}] = det(I_{m} − t^{n} · AB^{n}).  (A.15)

Now the right hand side of (A.15) is simply det_{V_{1}}(id_{V_{1}}−t^{n} · φ_{(x;ψ_{1})} ). Hence, using Lemma A.10, we can deduce that

∑_{x∈H^{1}(G_{1},ψ_{1})} \frac{χ̃((\overline{x}; ϕ)^{−1} )}{|C_{G,ϕ}(\overline{x})| · det_{V}(id_{V} −t · φ_{(\overline{x};ϕ)})} = ∑_{x∈H^{1}(G_{1},ψ_{1})} \frac{χ̃_{1}((x; ψ_{1})^{−1} )}{|C_{G_{1},ψ_{1}}(x)| · det_{V_{1}}(id_{V_{1}}−t^{n} · φ_{(x;ψ_{1})} )}  (A.16)

Let us now consider the remaining cosets for the automorphism ϕ^{i}. If i = nk for some k ∈ _{0} then we have ϕ^{i} = ψ_{1}^{k} × ⋯× ψ_{n}^{k}. If this is not the case, as n is prime, we have ϕ^{i} is exactly of the form considered in A.9 and the induced automorphism on G_{k} is precisely ψ_{k}^{i}. In particular we may apply the previous argument to this case to obtain that the statement in (A.16) holds with ϕ replaced by ϕ^{i} and ψ_{1} replaced by ψ_{1}^{i}. Putting this together, and using the fact that {ψ_{1}^{n}, ψ_{1}^{2n} …, ψ_{1}^{nk}} = {id, ψ_{1}, …, ψ_{1}^{k−1} } we get that

P V ; χ̃(t) = \frac{1}{n} [(n − 1)P V_{1} ; χ̃_{1}(t^{n}) + P_{λ}^{V}(t)]

where λ is the restriction of χ̃ to the subgroup G ⋊ ⟨ϕ^{n}⟩ ⩽ G̃. Observe that as λ restricted to G is χ we must have b_{λ}^{V} ⩾ b_{χ}^{V}. From this it follows that bV ; χ̃ = b_{λ}^{V} = b_{χ}^{V} because bV_{1} ; χ̃_{1} = b_{χ_{1}}^{V_{1}} and the b-invariants are compatible with direct products. ■

Remark A.17. The setup here may seem contrived but this situation occurs naturally when G is a Weyl group and V is the natural module. In this case the invariant basis considered in A.11 is given by a set of simple roots for the underlying root system of G.

References

[AA07] P. N. Achar and A.-M. Aubert, Supports unipotents de faisceaux caract�eres, J. Inst. Math. Jussieu 6 (2007), no. 2, 173�207.

[BS84] W. M. Beynon and N. Spaltenstein, Green functions of �nite Chevalley groups of type E_{n} (n = 6, 7, 8), J. Algebra 88 (1984), no. 2, 584�614.

[Bon04] C. Bonnaf� e, Actions of relative Weyl groups. I, J. Group Theory 7 (2004), no. 1, 1�37.

[Bon05] , Actions of relative Weyl groups. II, J. Group Theory 8 (2005), no. 3, 351�387.

[Car93] R. W. Carter, Finite groups of Lie type, Conjugacy classes and complex characters, Wiley Classics Library, Chichester: John Wiley & Sons Ltd., 1993.

[CR81] C. W. Curtis and I. Reiner, Methods of representation theory. Vol. I, With applications to �nite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication, New York: John Wiley & Sons Inc., 1981.

[DM91] F. Digne and J. Michel, Representations of �nite groups of Lie type, vol. 21, London Mathematical Society Student Texts, Cambridge: Cambridge University Press, 1991.

[GM00] M. Geck and G. Malle, On the existence of a unipotent support for the irreducible characters of a �nite group of Lie type, Trans. Amer. Math. Soc. 352 (2000), no. 1, 429�456.

[GP00] M. Geck and G. Pfei�er, Characters of �nite Coxeter groups and Iwahori-Hecke algebras, vol. 21, London Mathematical Society Monographs. New Series, New York: The Clarendon Press Oxford University Press, 2000.

[HS77] R. Hotta and T. A. Springer, A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups, Invent. Math. 41 (1977), no. 2, 113�127.

[Hot81] R. Hotta, On Springer’s representations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 863�876 (1982).

[How80] R. B. Howlett, Normalizers of parabolic subgroups of re�ection groups, J. London Math. Soc. (2) 21 (1980), no. 1, 62�80.

[Hum75] J. E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, New York: Springer-Verlag, 1975.

[LS79] G. Lusztig and N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. (2) 19 (1979), no. 1, 41�52.

[Lus81] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), no. 2, 169�178.

[Lus84a] , Characters of reductive groups over a �nite �eld, vol. 107, Annals of Mathematics Studies, Princeton, NJ: Princeton University Press, 1984.

[Lus84b] , Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205�272.

[Lus85] , Character sheaves. I, Adv. in Math. 56 (1985), no. 3, 193�237; Character sheaves. II, Adv. in Math. 57 (1985), no. 3, 226�265; Character sheaves. III, Adv. in Math. 57 (1985), no. 3, 266�315; Character sheaves. IV, Adv. in Math. 59 (1986), no. 1, 1�63; Character sheaves. V, Adv. in Math. 61 (1986), no. 2, 103�155.

[Lus86] , On the character values of �nite Chevalley groups at unipotent elements, J. Algebra 104 (1986), no. 1, 146�194.

[Lus90] , Green functions and character sheaves, Ann. of Math. (2) 131 (1990), no. 2, 355�408.

[Sho82] T. Shoji, On the Green polynomials of a Chevalley group of type F_{4}, Comm. Algebra 10 (1982), no. 5, 505�543.

[Sho83] , On the Green polynomials of classical groups, Invent. Math. 74 (1983), no. 2, 239�267.

[Sho87] , Green functions of reductive groups over a �nite �eld, in: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), vol. 47, Proc. Sympos. Pure Math. Providence, RI: Amer. Math. Soc., 1987, 289�301.

[Sho88] , Geometry of orbits and Springer correspondence, Ast� erisque (1988), no. 168, Orbites unipotentes et repr� esentations, I, 9, 61�140.

[Sho97] , Unipotent characters of �nite classical groups, in: Finite reductive groups (Luminy, 1994), vol. 141, Progr. Math. Boston, MA: Birkh� auser Boston, 1997, 373�413.

[Sho06] , Generalized Green functions and unipotent classes for �nite reductive groups. I, Nagoya Math. J. 184 (2006), 155�198.

[Sho07] , Generalized Green functions and unipotent classes for �nite reductive groups. II, Nagoya Math. J. 188 (2007), 133�170.

[Spa82] N. Spaltenstein, Classes unipotentes et sous-groupes de Borel, vol. 946, Lecture Notes in Mathematics, Berlin: Springer-Verlag, 1982.

[Spr78] T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279�293.

[Ste99] R. Steinberg, The isomorphism and isogeny theorems for reductive algebraic groups, J. Algebra 216 (1999), no. 1, 366�383.

[Tay13] J. Taylor, On unipotent supports of reductive groups with a disconnected centre, J. Algebra 391 (2013), 41�61.

[Wal01] J.-L. Waldspurger, Int�egrales orbitales nilpotentes et endoscopie pour les groupes classiques non rami��es, Ast� erisque (2001), no. 269.

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