Generalised Gelfand�Graev Representations in Small Characteristics

Jay Taylor

Abstract. Let G be a connected reductive algebraic group over an algebraic closure \overline{_{p}} of the �nite �eld of prime order p and let F : G → G be a Frobenius endomorphism with G = G^{F} the corresponding _{q}-rational structure. One of the strongest links we have between the representation theory of G and the geometry of the unipotent conjugacy classes of G is a formula, due to Lusztig [Lus92], which decomposes Kawanaka’s Generalised Gelfand�Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes de�ned on the closure of a unipotent class. Unfortunately, the results in [Lus92] are only valid under the assumption that both p and q are large enough. In this article we show that Lusztig’s formula for GGGRs holds under the much milder assumption that p is an acceptable prime for G (p very good is su�cient but not necessary). As an application we show that every irreducible character of G (resp. character sheaf of G) has a unique wave front set (resp. unipotent support) whenever p is good for G.

1. Introduction

1.1. Let G be a connected reductive algebraic group de�ned over an algebraic closure = \overline{_{p}} of the �nite �eld of prime order p and let F : G → G be a Frobenius endomorphism de�ning an _{q}- rational structure G = G^{F} on G. Assuming p is a good prime for G a theory of generalised Gelfand� Graev representations (GGGRs) was developed by Kawanaka in [Kaw86] building on his investigations in [Kaw85]. These are certain unipotently supported representations Γ_{u} of G which are de�ned for any unipotent element u ∈ G. Note that, identifying Γ_{u} with its character, we have Γ_{u} = Γ_{v} whenever u, v ∈ G are G-conjugate so the GGGRs are naturally indexed by the unipotent conjugacy classes of G.

1.2. Let ρ ∈ Irr(G) be an irreducible character and O an F-stable unipotent conjugacy class of G. We say Õ is a unipotent support of ρ if for any other F-stable unipotent conjugacy O of G we have

∑_{g∈O^{F}} ρ(g) ≠ 0 ⇒ dim O ⩽ dim Õ.

We say that Õ is a wave front set of ρ if for any unipotent element u ∈ G we have

⟨Γ_{u}, ρ⟩ ≠ 0 ⇒ dim O_{u} ⩽ dim Õ,

\underline{2}

where O_{u} is the G-conjugacy class of u. Every irreducible character of G admits a unipotent support and a wave-front set. However, it was conjectured by Lusztig [Lus80] (resp. Kawanaka [Kaw85]) that each irreducible character ρ ∈ Irr(G) admits a unique unipotent support O_{ρ} (resp. wave front set O_{ρ}^{∗}). If this conjecture is satis�ed then we say the unipotent support/wave front set of ρ is well de�ned.

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

Keywords and phrases. Finite reductive groups, character sheaves, generalised Gelfand�Graev representations, wave front sets.

1.3. Assuming p and q are su�ciently large then Lusztig gave a decomposition of the GGGRs of G as an explicit linear combination of IC complexes de�ned on the closure of a unipotent class (see [Lus92]). Using these results Lusztig was able to show that, indeed, the unipotent support and wave front set of an irreducible character is well de�ned. He also gave a de�nition for the unipotent support of a character sheaf and showed that each character sheaf has a well-de�ned unipotent support. These results provide one of the most profound relationships between irreducible characters of G and the geometry of the algebraic group G. They also highlight the central role that character sheaves play in the representation theory of �nite reductive groups.

1.4. Using Lusztig’s results Geck was able to show that each irreducible character has a unique unipotent support whenever p is a good prime for G (see [Gec96]). In turn, using Geck’s result together with ideas developed in [Lus86] Aubert was able to prove, in certain special cases, that character sheaves admit a unique unipotent support whenever p is good (see [Aub03]). The following completes this picture in good characteristic (see Theorems 13.8 and 14.10).

Theorem. Assume p is a good prime for G then any irreducible character of G has a unique wave front set and any character sheaf of G has a unique unipotent support.

1.5. In particular, if p is a good prime for G then we have two well-de�ned maps

Irr(G) → {F-stable unipotent conjugacy classes of G}

given by ρ ↦ O_{ρ} and ρ ↦ O_{ρ}^{∗}. These turn out to be dual in the following sense. Let ρ^{∗} ∈ Irr(G) be the unique irreducible character such that ρ^{∗} = ±D_{G}(ρ) where D_{G}(ρ) is the Alvis�Curtis dual of ρ then O_{ρ^{∗}} = O_{ρ}^{∗}. In other words, the unipotent support of the Alvis�Curtis dual of ρ is the wave front set of ρ.

In [Lus12] Lusztig has obtained a re�nement of the notion of a unipotent support for a character sheaf in good characteristic. There it is stated that the uniqueness of unipotent supports in good characteristic may be deduced from the case of large characteristic. In Proposition 13.10 we give an example of such a reduction in the special case where G is SL_{n}( ).

1.6. As in [Lus92] our main tool is to show that the formula relating GGGRs and IC complexes on unipotent classes from [Lus92] remains true whenever p is an acceptable prime for G (see Theorem 11.13). It is su�cient to assume that p is very good for G but this is not necessary; for instance every prime is acceptable for GL_{n}( ). The assumption that p is an acceptable prime comes from Letellier’s work on the Fourier transform (see [Let05]), which we use proli�cally throughout this article. This is based heavily on Lusztig’s work in [Lus87].

It is likely that the statement of Theorem 11.13 holds whenever p is a good prime. However, one cannot use the methods of this article to deduce this. For example, if p is not an acceptable prime for G then there is no non-degenerate G-invariant symmetric bilinear form on the Lie algebra of G. In particular, the Fourier transform is not well de�ned in this case so Letellier’s results [Let05] are no longer available to us.

1.7. The layout of this paper is as follows. In sections §2 and §3 we review, and slightly generalise, results of Premet on the classi�cation of nilpotent orbits for algebraic groups in good characteristic (see [Pre03]). In §4 and §5 we prove the existence of a Springer isomorphism which is suitable for de�ning GGGRs and then we recap the construction and main results concerning GGGRs (all these ideas are based on the work of Kawanaka [Kaw85]). Sections §6 - §11 are concerned with proving our main result. Here we follow [Lus92] to the letter, simply �nding alternative arguments when either the exp or log maps were used or when the theory of 𝔰𝔩_{2}-triples was used. We have tried not to unnecessarily repeat arguments from [Lus92] but some things are repeated to improve the quality of the exposition. Having said this, we have chosen to give most of the arguments from [Lus92, §6] as this was originally proved under the assumption that F is split (but later remarked that this assumption is unnecessary [Lus92, 8.7]). Finally §12 - §14 concern the uniqueness of unipotent supports and wave front sets.

Acknowledgments: The author would like to thank Pramod Achar, Olivier Dudas, Simon Goodwin, Sebastian Herpel, Gunter Malle, Iulian Simion and Britta Sp� ath for useful conversations/correspondence. We especially thank Simon Goodwin for pointing us towards [GG02] and [BR85], which formed important ingredients in this paper. Finally the author gratefully acknowledges the �nancial support of ERC Advanced Grant 291512 awarded to Prof. Gunter Malle.

2. Proximate Algebraic Groups

We will assume that G is a connected reductive algebraic group de�ned over (an algebraic closure of a �nite �eld _{p} of prime order p) and that F : G → G is a generalised Frobenius endomorphism de�ning an _{q}-rational structure on G.

2.1. Throughout this article we maintain the notation that G_{der} is the derived subgroup [G, G] of G and ϕ_{sc} : G_{sc} → G_{der} is a simply connected covering. The corresponding Lie algebras will be denoted by 𝔤 = Lie(G), 𝔤_{der} = Lie(G_{der}) and 𝔤_{sc} = Lie(G_{sc}). We will also denote by _{a} the �eld viewed as an additive algebraic group and by _{m} the group of units of viewed as a multiplicative algebraic group. To make it possible to pass between the unipotent elements of G and the nilpotent elements in 𝔤 we will need to place some mild restrictions on ϕ_{sc}. Namely, we will need the following concept.

De�nition 2.2. We say G is proximate if ϕ_{sc} is a separable morphism.

2.3. Note that this de�nition does not depend upon the choice of the isogeny ϕ_{sc}. The upshot of a proximate algebraic group is the following well known fact concerning separable morphisms.

Proposition 2.4. If G is proximate then the di�erential dϕ_{sc} : 𝔤_{sc} → 𝔤_{der} is an isomorphism.

2.5. We now wish to show that we can replace our algebraic group G by a proximate algebraic group without a�ecting the �nite reductive group G^{F} (up to isomorphism). Let (X, Φ, Y, Φ ) be the root datum of G and ⟨−, −⟩ : X × Y → be the usual perfect pairing. For any subset A ⊆ X we set

A^{⊥} = {y ∈ Y | ⟨x, y⟩ = 0 for all x ∈ A}
A^{⊤} = {x ∈ X | ⟨x⟩_{}∩ A ≠ {0}}

where ⟨x⟩_{}⊆ X is the submodule generated by x (and similarly for subsets of Y). Note that A^{⊤} is the rational closure of A in X so A^{⊤} /A is the torsion subgroup of X/A.

Now, according to [Spr09, 8.1.9] the root datum of G_{der} is given by (X_{der}, Φ, Y_{der}, Φ ) where X_{der} = X/{ \atop Φ^{⊥}} and Y_{der} = { \atop Φ^{⊤}} . As X_{der} is a free (hence projective) -module there exists a submodule X̃_{der} of X so that X = X̃_{der} ⊕ { \atop Φ^{⊥}} . Taking duals, this implies that

Hom(X, ) = Hom(X̃_{der}, ) ⊕ Hom({ \atop Φ^{⊥}} , ).

Using the perfect pairing we may identify Hom(X, ) with Y and Hom(X̃_{der}, ) = Hom(X_{der}, ) with Y_{der}. In particular, this shows that there exists a submodule Y_{der}^{⊥} ⊆ Y such that Y = Y_{der} ⊕ Y_{der}^{⊥}.

Remark 2.6. One must be very careful when using these direct sum decompositions! Although the roots Φ ⊂ X embed naturally in the quotient X_{der} they are not necessarily contained in the submodule X̃_{der}. However, the coroots Φ are clearly contained in the submodule Y_{der} ⊆ Y.

Lemma 2.7. The group G is proximate if and only if Y/ Φ has no p-torsion.

Proof. Given the roots Φ we may form the -vector space Φ = ⊗_{}Φ, of which the root lattice is a natural subset. In fact, both Φ ⊆ X_{der} may naturally be considered as subgroups of the weight lattice

Λ = {ω ∈ Φ | ⟨ω, α⟩ ∈ for all α ∈ Φ } ⊂ Φ.

The root datum (Λ, Φ, Φ , Φ ) is the root datum of the simply connected group G_{sc}. Furthermore the simply connected covering ϕ_{sc} is determined by the isogeny of root data given by the natural inclusion X_{der} → Λ whose transpose is the inclusion Φ → Y_{der}. The groups Λ/X_{der} and Y_{der}/ Φ are �nite groups in duality (see [GP11, XXI, 6.2.3]) so Y_{der}/ Φ has no p-torsion if and only if Λ/X_{der} has no p-torsion. However, this latter condition is precisely equivalent to ϕ_{sc} being separable (see [Let05, Proposition 2.4.4]). It now su�ces to observe that the torsion subgroup Tor(Y/ Φ ) ⩽ Y/ Φ is naturally isomorphic to the quotient Y_{der}/ Φ because Y_{der}^{⊥} is a free module. ■

Remark 2.8. If p is a good prime for G then this shows that G is proximate so long as G has no connected normal simple subgroups of type A_{pr−1} with r ⩾ 1.

Proposition 2.9. There exists a proximate algebraic group \overline{G}, de�ned over , which is abstractly isomorphic to G.

Proof. Consider the unique subgroup Φ ⊆ \overline{Y}_{der} ⊆ Y_{der} such that

\overline{Y}_{der}/ Φ = (Y_{der}/ Φ )_{p′}

is the subset of all elements whose order is coprime to p. We then form a new free abelian group \overline{Y} = \overline{Y}_{der} ⊕ Y_{der}^{⊥} so that we have a natural short exact sequence

0  \overline{Y}  Y  Y/\overline{Y} 0

The quotient Y/\overline{Y} is a �nite group, hence dualising we obtain an exact sequence

0  0  Hom(Y, )  Hom(\overline{Y}, )

Set \overline{X} = Hom(\overline{Y}, ) then \overline{X} and \overline{Y} are naturally free abelian groups in duality. Furthermore, identifying X with Hom(Y, ) we have a natural embedding f : X → \overline{X} which is (by de�nition) the transpose of the embedding \overline{Y} → Y. With this we can construct the root datum (\overline{X}, Φ, \overline{Y}, Φ ) induced by \overline{Y} (see [GP11, XXI, 6.5]). Let us take \overline{G} to be the connected reductive algebraic group over determined by this root datum. The map f is an isogeny of root data which determines an isogeny of algebraic groups ϕ : \overline{G} → G. By design the quotient Y/\overline{Y} is a p-group, so according to [Bon06, 1.11] we have Ker(ϕ) = \{1\}. In particular, this implies that ϕ is an isomorphism of abstract groups. Finally, it remains only to show that \overline{G} is proximate but this follows from Lemma 2.7. ■

2.10. Assume ϕ : \overline{G} → G is the isomorphism of abstract groups constructed in the proof of Proposition 2.9. Setting \overline{F} = ϕ^{−1} ∘ F ∘ ϕ we have \overline{F} is a generalised Frobenius endomorphism of \overline{G} and \overline{G}\overline{F} ≅ G^{F} as �nite groups. In particular, this shows that any �nite reductive group is obtained as the �xed point subgroup of a proximate algebraic group under a generalised Frobenius endomorphism. Hence, to deduce statements about the �nite group G^{F} we may simply replace the pair (G, F) by (\overline{G}, \overline{F}). We will not need this here but the following lemma can be deduced from Lemma 2.7 in the same way as [DM91, Lemma 13.14].

Lemma 2.11. If G is proximate then so is any Levi subgroup of G.

3. Unipotent and Nilpotent Elements

From this point forward we assume that p is a good prime for G and that F : G → G is a Frobenius endomorphism.

Springer Morphisms

3.1. Let us �x an F-stable maximal torus and Borel subgroup T_{0} ⩽ B_{0} ⩽ G. The choice of a Borel subgroup uniquely determines a set of simple and positive roots Δ ⊂ Φ^{+} ⊂ Φ where Φ ⊂ X(T_{0}) are the roots of G with respect to T_{0}. As G is equipped with a Frobenius endomorphism so is 𝔤 and we denote this again by F : 𝔤 → 𝔤. Our main interest of study will be the variety of unipotent elements U ⊂ G and the nilpotent cone N ⊂ 𝔤. To relate these two objects we will need the existence of a Springer morphism, which is a G-equivariant bijective morphism of varieties ϕ_{spr} : U → N compatible with the Frobenius endomorphisms on G and 𝔤 (this is then a homeomorphism of topological spaces). Note that this is G-equivariant in the sense that

ϕ_{spr} ∘ Inng = Ad g ∘ ϕ_{spr}

for all g ∈ G, where Inng : G → G is the inner automorphism given by Inng(x) = gxg^{−1} for all x ∈ G and Ad g = d(Inng) : 𝔤 → 𝔤 is its corresponding di�erential.

Lemma 3.2. There exists a Springer morphism ϕ_{spr} : U → N. Furthermore, if G is proximate then ϕ_{spr} may be chosen to be an isomorphism of varieties.

Proof. Firstly let us note that U ⊆ G_{der} and N ⊆ 𝔤_{der} (e.g. see [Pre95, pg. 2966]). We denote by U_{sc} ⊂ G_{sc} and N_{sc} ⊂ 𝔤_{sc} the unipotent variety and nilpotent cone then ϕ_{sc} restricts to a bijective morphism U_{sc} → U and dϕ_{sc} restricts to a bijective morphism N_{sc} → N. Let us denote by F_{sc} : G_{sc} → G_{sc} a Frobenius endomorphism such that F ∘ ϕ_{sc} = ϕ_{sc} ∘ F_{sc} (this exists by [MT11, 22.7]). By [SS70, III, 3.12] there exists an isomorphism ϕ̃_{spr} : U_{sc} → N_{sc} which is G_{sc}-equivariant and respects the Frobenius endomorphism F_{sc}. Then ϕ_{spr} = dϕ_{sc} ∘ ϕ̃_{spr} ∘ ϕ_{sc}^{−1} : U → N is also a bijective morphism which is G_{der}-equivariant (see [Slo80, 3.12, Remark]) and respects the Frobenius endomorphism F. As G = G_{der}Z(G) this morphism is also G-equivariant as Z(G) must act trivially on both U and N. The statement concerning proximate algebraic groups follows immediately from the de�nition and the construction of ϕ_{spr}. ■

3.3. Note that Springer isomorphisms are by no means unique. In fact, in the appendix to [McN05] Serre has shown that the Springer isomorphisms form a variety whose dimension is given by the rank of G. We recall the following properties of Springer isomorphisms.

Proposition 3.4. The following hold:

(i) For any Springer isomorphism ϕ_{spr} and any parabolic subgroup P ⩽ G with unipotent radical U ⩽ P we have the restriction of ϕ_{spr} to U is an isomorphism U → Lie(U).

(ii) The bijections U/G → N /G between unipotent conjugacy classes and nilpotent orbits induced by any two Springer isomorphisms are the same.

Proof. The �rst part is given in [McN05, Remark 10] and the second part is a result of Serre given in [McN05, §10. Appendix]. ■

Smoothness of Centralisers

3.5. An issue for us in this article will also be the so-called smoothness of centralisers. For any subset 𝔥 ⊆ 𝔤 we de�ne

C_{G}(𝔥) = {g ∈ G | Ad g(x) = x for all x ∈ 𝔥},
𝔠_{𝔤}(𝔥) = {y ∈ 𝔤 | [y, x] = 0 for all x ∈ 𝔥}.

If 𝔥 = {x} then we simply write C_{G}(x) (resp. 𝔠_{𝔤}(x)) for C_{G}(𝔥) (resp. C_{𝔤}(𝔥)). We say the centraliser C_{G}(𝔥) is smooth if Lie(C_{G}(𝔥)) = 𝔠_{𝔤}(𝔥) (or equivalently that dim C_{G}(𝔥) = dim 𝔠_{𝔤}(𝔥)). In [Her13] Herpel has considered the smoothness of subgroup schemes of connected reductive algebraic groups. To apply his results we will need to recall his elegant notion of a pretty good prime.

De�nition 3.6 (Herpel, [Her13, De�nition 2.11]). Assume (X, Φ, Y, Φ ) is the root datum of G then we say p is a pretty good prime for G if X/ Ψ and Y/ Ψ have no p-torsion for any subsets Ψ ⊆ Φ and Ψ ⊆ Φ .

Proposition 3.7 (Herpel). Assume p is a pretty good prime for G then C_{G}(x) is smooth for all x ∈ 𝔤.

Proof. Let 𝔥 ⊆ 𝔤 be the 1-dimensional subalgebra generated by x. According to the proof of [Her13, Lemma 3.1(ii)] we have C_{G}(𝔥) is smooth if and only if 𝔥 is separable in 𝔤. Noting that C_{G}(x) = C_{G}(𝔥) and 𝔠_{𝔤}(x) = 𝔠_{𝔤}(𝔥) we see that the result follows from Theorem 1.1 and Lemma 3.1(ii) of [Her13]. ■

Remark 3.8. In [Her13, Lemma 2.12] Herpel gives some characterisations of pretty good primes in special cases. For instance, he shows that any very good prime is a pretty good prime. Consequently, if G is simple or GL_{n}( ) then this result is classical and contained in [SS70, I, 5.6] (see also [Slo80, 3.13, Theorem]).

Cocharacters

3.9. We now recall some results from [Pre03]. Assume λ ∈ Y(G) = Hom( _{m}, G) is a cocharacter of G then λ de�nes a -grading 𝔤 = ⨁ _{i∈} 𝔤(λ, i) on the Lie algebra by setting

𝔤(λ, i) = {x ∈ 𝔤 | (Ad λ(k))(x) = k^{i}x for all k ∈ _{m}}.

In particular, note that we have

[𝔤(λ, i), 𝔤(λ, j)] ⊆ 𝔤(λ, i + j)  (3.10)

for all i, j ∈ . We note the following useful observation concerning the weight spaces. Let 𝔷 = Lie(Z(G)) then we clearly have 𝔤 = 𝔤_{der} + 𝔷. Furthermore, it is also obvious that 𝔷 ⊆ 𝔤(λ, 0) so we must have 𝔤(λ, i) ⊆ 𝔤_{der} for any i ≠ 0. In particular, if λ ∈ Y(G_{der}) ⊆ Y(G) then we have 𝔤(λ, i) = 𝔤_{der}(λ, i) for all i ≠ 0.

3.11. To any cocharacter λ ∈ Y(G) we assign a parabolic subgroup P(λ) ⩽ G with unipotent radical U(λ) given by

P(λ) = {x ∈ G | lim_{k→0} λ(k)xλ(k)^{−1} exists} U(λ) = {x ∈ G | lim_{k→0} λ(k)xλ(k)^{−1} = 1}

(see [Spr09, 3.2.15, 8.4.5]). We then have a Levi decomposition P(λ) = L(λ)U(λ) by setting L(λ) = C_{G}(λ( _{m})). The Lie algebras of these subgroups are given by

𝔭(λ) = ⨁_{i⩾0} 𝔤(λ, i)  𝔩(λ) = ⨁_{i=0} 𝔤(λ, i)  𝔲(λ) = ⨁_{i>0} 𝔤(λ, i).

For any i ∈ _{>0} we will also need the subspace 𝔲(λ, i) = ⨁ _{j⩾i} 𝔤(λ, j) of nilpotent elements and also its corresponding closed connected unipotent subgroup U(λ, i) ⩽ U(λ).

Assume now that λ ∈ Y(T_{0}) and let X_{α} ⩽ G be the 1-dimensional unipotent root subgroup whose Lie algebra is the root space 𝔤_{α} for any α ∈ Φ. If P(λ) contains T_{0} ⩽ B_{0} then we may also write the above as

P(λ) = ⟨T_{0}, X_{α} | ⟨α, λ⟩ ⩾ 0⟩  L(λ) = ⟨T_{0}, X_{α} | ⟨α, λ⟩ = 0⟩  U(λ) = ⟨X_{α} | ⟨α, λ⟩ > 0⟩,
𝔭(λ) = 𝔱_{0} ⊕ ⨁_{⟨α,λ⟩⩾0}𝔤_{α}  𝔩(λ) = 𝔱_{0} ⊕ ⨁_{⟨α,λ⟩=0} 𝔤_{α}  𝔲(λ) = ⨁_{⟨α,λ⟩>0} 𝔤_{α},

as 𝔱_{0} ⊆ 𝔤(λ, 0). With this we see that we have an analogue of (3.10) which follows immediately from Chevalley’s commutator relation. Namely,

[U(λ, i), U(λ, j)] ⊆ U(λ, i + j)  (3.12)

for any i, j ⩾ 1. Note this clearly holds for all cocharacters.

Weighted Dynkin Diagrams

3.13. Let us assume temporarily that G is semisimple and simply connected. We will denote by G_{} an algebraic group over with the same root datum as G and by T_{}⩽ G_{}a maximal torus of G_{}. Fix a Chevalley basis B_{}= {X_{α} | α ∈ Φ} ∪ {H_{α} | α ∈ Δ} of 𝔤_{}and denote by 𝔤_{}⊂ 𝔤_{}the -span of B_{}. As G is simply connected we may identify 𝔤 with 𝔤_{}⊗_{}as Lie algebras. Setting e_{α} = X_{α} ⊗ 1 and h_{α} = H_{α} ⊗ 1 for each α ∈ Φ we have B = {e_{α} | α ∈ Φ} ∪ {h_{α} | α ∈ Δ} is a basis for 𝔤 and the corresponding root space 𝔤_{α} of 𝔤 is simply e_{α}.

By the Jacobson�Morozov theorem (see [Car93, 5.3.2]) any nilpotent element e ∈ 𝔤_{}= Lie(G_{}) is contained in an 𝔰𝔩_{2}-triple {e, h, f} ⊂ 𝔤_{}. Arguing as in [Pre03, pg. 344] we may assume, after possibly replacing {e, h, f} by {Ad g(e), Ad g(h), Ad g(f)} for some g ∈ G, that h = ∑ _{α∈Δ} q_{α}H_{α} with q_{α} ∈ and α(h) ⩾ 0 for all α ∈ Δ. The function d : Φ → given by α ↦ α(h) is called the weighted Dynkin diagram of the nilpotent orbit O ⊂ 𝔤_{}containing e. We denote by D_{Φ} the set of weighted Dynkin diagrams.

Note that, for each weighted Dynkin diagram d ∈ D_{Φ} , there exists a cocharacter λ_{d} ∈ Y(T_{0}) ⊆ Y(G) such that Ad λ_{d}(k)(e_{α}) = k^{d(α)}e_{α} and Ad λ_{d}(k)(x) = x for all α ∈ ±Δ, x ∈ 𝔱_{0} and k ∈ _{m}. In fact this can be constructed by setting λ_{d} = ∑ _{α∈Δ} q_{α}α ∈ Φ where q_{α} ∈ is as above and α ∈ Φ is the coroot corresponding to α ∈ Δ.

3.14. We now drop our assumption that G is semisimple and simply connected. The isogeny ϕ_{sc} induces a natural injection Y(G_{sc}) → Y(G) so we may consider the cocharacter λ_{d} ∈ Y(G_{sc}), constructed as above, as a cocharacter of G for any weighted Dynkin diagram d ∈ D_{Φ} . We will denote by D(G) the set

{Inng ∘ λ_{d} | d ∈ D_{Φ} and g ∈ G}.

If G is a proximate algebraic group then through the isomorphism ϕ_{sc} we will identify the Chevalley basis considered in 3.13 with a basis of 𝔤_{der}.

Classi�cation of Nilpotent Orbits

3.15. We now have the following cheap generalisation of [Pre03, Theorem 2.3], which gives the classi�cation of nilpotent orbits in good characteristic by weighted Dynkin diagrams (for historical remarks concerning this theorem see [Pre03, Remark 2]). In fact, we will give a stronger statement than simply the classi�cation of nilpotent orbits but for this we will need the assumption that G is proximate. This assumption is necessary as is evident by the example given at the end of [Pre95, Introduction].

Theorem 3.16 (Kawanaka, Premet). Assume G is proximate.

(i) For any cocharacter λ ∈ D(G) ⊆ Y(G) let 𝔤(λ, 2)_{reg} be the unique open dense orbit of L(λ) acting on 𝔤(λ, 2) (c.f. [Car93, pg. 132]). Then for any nilpotent element e ∈ 𝔤(λ, 2)_{reg} ⊆ N the following hold:

(a) 𝔠_{𝔤}(e) ⊆ 𝔭(λ) and C_{G}(e) ⊆ P(λ),

(b) [e, 𝔭(λ)] = 𝔲(λ, 2),

(c) (Ad P(λ))e is dense in 𝔲(λ, 2).

(ii) For any λ ∈ D(G) denote by e_{λ} an element of 𝔤(λ, 2)_{reg} then the map D(G) → N given by λ ↦ e_{λ} induces a bijection D(G) → N /G.

Proof. It’s clear that we only have to prove (i) for the set of cocharacters {λ_{d} | d ∈ D_{Φ} } ⊆ D(G), so assume λ is such a cocharacter. The simply connected cover G_{sc} of G_{der} is a direct product G_{1} × ⋯× G_{r} of simple simply connected algebraic groups. Setting 𝔤_{i} = Lie(G_{i}) we have 𝔤_{der} ≅ 𝔤_{sc} = 𝔤_{1} ⊕ ⋯⊕ 𝔤_{r} by Proposition 2.4, in particular the nilpotent element may be written uniquely as e = e_{1} + ⋯+ e_{r}. If pr_{i} : G_{sc} → G_{i} denotes the projection onto the ith factor then we set λ_{i} := pr_{i} ∘λ, where we identify λ as a cocharacter of G_{sc}. Assume now that G_{i} is not of type A then (a) and (b) are given by [Pre03, Theorem 2.3]. As G_{i} is not of type A we have p is a pretty good prime for G_{i} so the centraliser C_{G_{i}}(e_{i}) is smooth by Proposition 3.7. In particular, by (a) we have

dim P(λ_{i}) − dim C_{P(λ_{i})}(e_{i}) = dim 𝔭(λ_{i}) − dim 𝔠_{𝔤_{i}}(e_{i}) = dim[e, 𝔭(λ_{i})]

and combining this with (b) gives (c).

Assume now that G_{i} is of type A_{n−1} then G_{i} ≅ SL_{n}( ) and we may consider G_{i} as the derived subgroup of G̃_{i} = GL_{n}( ). We may now apply [Pre03, Theorem 2.3] to the group G̃_{i} to deduce that (a) � (c) hold in G̃_{i}. We wish to show that this implies (a) � (c) hold for λ in G_{i}. Using the observations in 3.9 we must have e_{i} ∈ 𝔤_{i}(λ_{i}, 2). Let P̃(λ_{i}) = L̃(λ_{i})Ũ(λ_{i}) ⩽ G̃_{i} be the canonical parabolic subgroup with Lie algebra 𝔭̃(λ_{i}) then we have P̃(λ_{i}) = P(λ_{i})Z(G̃_{i}) and 𝔭̃(λ_{i}) = 𝔭(λ_{i}) + 𝔷̃_{i} where 𝔷̃_{i} = Lie(Z(G̃_{i})). In particular, we see that

𝔠𝔤̃_{i}(e_{i}) ⊆ 𝔭̃(λ_{i}) ⇒ 𝔠_{𝔤_{i}}(e_{i}) +𝔷̃_{i} ⊆ 𝔭(λ_{i}) +𝔷̃_{i} ⇒ 𝔠_{𝔤_{i}}(e_{i}) ⊆ 𝔭(λ_{i})

and similarly that

CG̃_{i}(e_{i}) ⊆ P̃(λ_{i}) ⇒ C_{G_{i}}(e_{i})Z(G̃_{i}) ⊆ P(λ_{i})Z(G̃_{i}) ⇒ C_{G_{i}}(e_{i}) ⊆ P(λ_{i})

so (a) holds. Now (b) holds in G_{i} by the observations in 3.9because [e,𝔭̃(λ_{i})] = [e, 𝔭(λ_{i})]. By Proposition 3.7 we can apply the previous argument to deduce that the P̃(λ_{i})-orbit is dense but this implies the P(λ_{i})-orbit is dense as the centre acts trivially, which gives (c).

We have thus shown that the �rst part of the theorem holds for the simply connected cover G_{sc}. As the isogeny ϕ_{sc} : G_{sc} → G_{der} has central kernel we easily deduce that (i) holds for G_{der}. Now arguing as in the case of SL_{n}( ) we see that the result holds in G. We leave the details of this to the reader. We now just need to show that the map λ ↦ e_{λ} induces a bijection. However, this is precisely given by [Pre03, Theorems 2.6, 2.7] and the previous reduction techniques. ■

Corollary 3.17. Assume G is proximate and e ∈ N is a nilpotent element such that C_{G}(e) is smooth. For any cocharacter λ ∈ D(G) and e ∈ 𝔤(λ, 2)_{reg} there exists a subspace 𝔰 ⊂ 𝔤 such that the following hold:

• 𝔤 = 𝔰 ⊕ T_{e}(Ad G(e)),

• 𝔰 is stable under the _{m}-action (k, x) ↦ (Ad λ(k))(x),

• 𝔰 ⊆ ⨁ _{i⩽0} 𝔤(λ, i).

Furthermore the action map π : G × (e + 𝔰) → 𝔤, given by π(g, x) = Ad g(x), is a smooth morphism so e + 𝔰 is a transverse slice in 𝔤 at e to the nilpotent orbit Ad G(e) (in the sense of [Slo80, pg. 60]). Consequently e + 𝔰 intersects every G-orbit on 𝔤 transversally.

Proof. Firstly, as C_{G}(e) is smooth we have the tangent space T_{e}(Ad G(e)) is simply [𝔤, e], which is stable under the _{m}-action. Hence, there exists a graded (i.e. _{m}-invariant) complement 𝔰 of T_{e}(Ad G(e)) in 𝔤. We need only show that 𝔰 ⊆ ⨁ _{i⩽0} 𝔤(λ, i). However, this follows from the fact that 𝔲(λ, 2) = [𝔭(λ), e] ⊆ [𝔤, e] and 𝔤(λ, 1) = [𝔤(λ, −1), e] ⊆ [𝔤, e] (see Theorem 3.16).

Recall that for a morphism of varieties f : X → Y we say x ∈ X is a smooth point of f if x and f(x) are non-singular and the di�erential d_{x}f : T_{x}X → T_{f(x)} Y is surjective. Now clearly (1, e) is a smooth point of the action map π because 𝔤 = 𝔰 ⊕ [𝔤, e]. Hence there exists an open neighbourhood O of (1, e) consisting of smooth points of π (see the proof of [Gec03, 1.4.15]) so π is a smooth morphism on O by [Har77, III, §10, 10.4]. We now need only show the morphism is smooth everywhere.

We de�ne a _{m}-action ρ : _{m} → End(𝔤) on 𝔤 by setting ρ(k)(x) = k^{2}(Ad λ(k^{−1} ))(x) for all x ∈ 𝔤 and k ∈ _{m}. Note that the _{m}-action ρ preserves 𝔰 and as e ∈ 𝔤(λ, 2) we have ρ(k)(e + x) = e + ρ(k)(x) for all x ∈ 𝔰. In particular, we have ρ restricts to a _{m}-action on e + 𝔰. Furthermore, we note that for any x ∈ 𝔰 we have lim_{k→0} ρ(k)(e + x) = e so the _{m}-action is a contraction to e. From the existence of this contraction it follows that π is smooth everywhere (see the proof of Corollary 1 in [Slo80, pg. 111]). ■

4. Springer Isomorphisms and Kawanaka Isomorphisms

De�nition 4.1. Given any cocharacter λ ∈ Y(G) we say an isomorphism ψ : U(λ) → 𝔲(λ) is a Kawanaka isomorphism if it commutes with the action of the Frobenius endomorphism and furthermore the following hold:

(K1) ψ(U(λ, 2)) ⊆ 𝔲(λ, 2)

(K2) ψ(uv) − ψ(u) − ψ(v) ∈ 𝔲(λ, i + 1) for any u, v ∈ U(λ, i) and i ∈ {1, 2},

(K3) ψ([u, v]) − c[ψ(u), ψ(v)] ∈ 𝔲(λ, 2i + 1) for any u, v ∈ U(λ, i) and i ∈ {1, 2} where c ∈ ^{×} is a constant not depending on u or v; it may depend on i.

Note that [u, v] denotes the commutator uvu^{−1} v^{−1} of u and v.

4.2. Kawanaka isomorphisms will be the crucial ingredient for the de�nition of generalised Gelfand� Graev representations. In [Kaw86, §3] Kawanaka gave a general construction for a Kawanaka isomorphism. However, the construction he gives is not in general G-equivariant so cannot be obtained as the restriction of a Springer isomorphism (which is sensible to expect from Proposition 3.4). In this section we wish to show that there always exists a Springer isomorphism whose restriction to U(λ) is a Kawanaka isomorphism for all cocharacters λ ∈ Y(G).

Lemma 4.3. Assume G is an adjoint simple group of exceptional type then there exists a Springer isomorphism ϕ_{spr} : U → N whose restriction to U(λ) is a Kawanaka isomorphism for every cocharacter λ ∈ Y(G).

Proof. As G is adjoint we have the adjoint representation Ad : G → GL(V) (with V = 𝔤) is faithful and according to [SS70, I, §5, 5.3] we have

𝔤𝔩(V) = ad(𝔤) ⊕ 𝔪,

where 𝔪 is an Ad G-invariant subspace of 𝔤𝔩(V) containing id_{V}. Let π : 𝔤𝔩(V) → ad(𝔤) be the natural projection map then according to Bardsley and Richardson the composition ϕ_{spr} = ad^{−1} ∘π ∘ Ad is a Springer isomorphism whose di�erential is the identity (see [BR85, 9.3.4]).

We may and will assume that T_{0} ⩽ B_{0} are contained in P(λ). For each root α ∈ Φ we choose an isomorphism x_{α} : _{a} → X_{α}, where X_{α} ⩽ G is the root subgroup corresponding to α, normalised such that dx_{α}(k) = ke_{α} ∈ 𝔤_{α} for all k ∈ _{a} (c.f. 3.14). We claim that for any α ∈ Φ^{+} and k ∈ _{a} we have

Ad x_{α}(t) = 1 + t ad e_{α} + t^{2} \frac{(ad e_{α})^{2}}{2} + t^{3} \frac{(ad e_{α})^{3}}{6} ,  (4.4)

in fact (ad e_{α})^{3} = 0 unless G is of type G_{2}. If G is of type E_{n} then this follows from the general argument given in [Spr09, 10.2.7]. If G is of type G_{2} or F_{4} then one can use the implementation of the adjoint representation in [Gap] to check that this holds. Indeed, one can check the order of ad e_{α} and can check that for j ∈ {2, 3} the matrix (ad e_{α})^{j}/j! is integer valued, the result then follows as in [Car72, §11.3].

Now Ad g(ad x) = ad(Ad g(x)) for all g ∈ G and x ∈ 𝔤 so we have x ∈ 𝔤(λ, i) if and only if Ad λ(k)(ad x) = k^{i} ad x for all k ∈ _{m}. Let us assume now that x ∈ 𝔤(λ, i) and y ∈ 𝔤(λ, j). As the action of Ad λ(k) on ad x is given simply by conjugation we have

Ad λ(k)(ad x ad y) = Ad λ(k)(ad x) Ad λ(k)(ad y) = k^{i+j} ad x ad y.

In particular, this implies that

(ad^{−1} ∘π)(ad x ad y) ∈ 𝔤(λ, i + j).  (4.5)

Note this could of course simply be 0.

We now wish to show that the restriction of ϕ_{spr} to U(λ) satis�es the properties (K1) to (K3). For this we �x a total ordering α_{1}, …, α_{m} on the set of positive roots Φ^{+} then, as a variety, we may identify U(λ) with the product ∏ _{i=1}^{m} X_{α_{i}}. In particular, any element u ∈ U(λ) may be written uniquely as

u = x_{α_{1}}(t_{1}) ⋯x_{α_{m}}(t_{m})

for some t_{i} ∈ _{a}. To understand ϕ_{spr}(u) we note that

Ad u = Ad(x_{α_{1}}(t_{1}) ⋯x_{α_{m}}(t_{m})) = Ad x_{α_{1}}(t_{1}) ⋯Ad x_{α_{m}}(t_{m}).  (4.6)

Noting that u^{−1} = x_{α_{m}}(−t_{m}) ⋯x_{α_{1}}(−t_{1}) we see that properties (K1) to (K3) can now be shown to hold by repeatedly applying (4.4), (4.5) and (4.6). We leave it to the reader to �ll in the details. ■

Proposition 4.7. Assume G is a proximate algebraic group then there exists a Springer isomorphism ϕ_{spr} : U → N whose restriction to U(λ) is a Kawanaka isomorphism for every cocharacter λ ∈ Y(G).

Proof. Assume G is SL(V), Sp(V) or SO(V) then Kawanaka already observed in [Kaw85, 1.2] that such a Springer isomorphism exists. If G is SL(V) then one simply takes the map f ↦ f − id_{V}, for which the statement is easy to deduce. If G is Sp(V) or SO(V) then one can use the Cayley map x ↦ (f − id_{V})(f + id_{V})^{−1} . In fact, if G is Sp(V) or SO(V) then one could argue as in the proof of Lemma 4.3 to deduce the existence of ϕ_{spr} but instead replacing the adjoint representation with the natural representation.

Let G_{sc} be a simple simply connected algebraic group de�ned over a �eld of good characteristic. Then there is a natural surjective separable morphism π : G_{sc} → G of algebraic groups where G is either SL(V), Sp(V), SO(V) or an adjoint exceptional group. According to Lemma 4.3 and the above remarks there exists a Springer isomorphism ϕ_{spr} of G whose restriction to each U(λ) ⩽ G is a Kawanaka isomorphism. Arguing as in Lemma 3.2 we see this lifts to a Springer isomorphism on G_{sc} which also has this property. One may now argue as in Lemma 3.2 to show that such a Springer isomorphism exists for all proximate algebriac groups. We leave the details to the reader. ■

Remark 4.8. It seems likely that the properties de�ned in De�nition 4.1 hold for most Springer isomorphisms. It would be interesting to �nd a case free proof of Proposition 4.7.

5. Generalised Gelfand�Graev Representations

From this point forward we will assume that G is proximate and we maintain our assumption that p is a good prime for G. We will denote by:

• \overline{}_{ℓ} a �xed algebraic closure of the �eld of ℓ-adic numbers where ℓ ≠ p is a prime,

• G = G^{F} the _{q}-rational structure determined by F,

• e ∈ N^{F} a �xed nilpotent element and λ ∈ D(G) a cocharacter such that e ∈ 𝔤(λ, 2)_{reg} (c.f. Theorem 3.16),

• ϕ_{spr} : U → N a �xed Springer isomorphism satisfying the property of Proposition 4.7,

• u ∈ U^{F} the unique element satisfying ϕ_{spr}(u) = e.

De�nition 5.1 (Kawanaka, [Kaw82, 3.1.4]). An involutive homomorphism ^{†} : 𝔤 → 𝔤 is called an _{q}- opposition automorphism if the following holds:

(i) 𝔱_{0}^{†} = 𝔱_{0},
(ii) e_{α}^{†} ∈ _{q}e_{−α} for all α ∈ Φ where e_{α} ∈ 𝔤_{α} is as in 3.13.

Lemma 5.2. The map de�ned by t^{†} = −t if t ∈ 𝔱_{0} and e_{α}^{†} = −e_{−α} for all α ∈ Φ is an _{q}-opposition automorphism of 𝔤.

Proof. We only have to show ^{†} is a Lie algebra homomorphism. Recall from 3.14 that 𝔤_{der} has a Chevalley basis then by [Car72, pg. 56] we see that the restriction of ^{†} to 𝔤_{der} is a homomorphism. However this easily implies that ^{†} is a homomorphism as 𝔤 = 𝔤_{der} + 𝔷(𝔤) (where 𝔷(𝔤) is the centre of the Lie algebra) and 𝔷(𝔤)^{†} = 𝔷(𝔤). ■

We now assume that ^{†} : 𝔤 → 𝔤 is a �xed _{q}-opposition automorphism, which exists by Lemma 5.2.

Proposition 5.3. The elements e and −e^{†} are contained in the same G-orbit.

Proof. For any α ∈ Φ and t ∈ _{a} let us denote by \overline{x}_{α}(t) the element Ad x_{α}(t) then \{\overline{x}_{α}(t) | α ∈ Φ, t ∈ _{a}\} is a generating set for Ad G (here x_{α}(t) is de�ned as in the proof of Lemma 4.3). Furthermore, for each α ∈ Φ we denote by γ_{α} ∈ _{q}^{×} the scalar such that e_{α}^{†} = γ_{α}e_{−α} . As in the proof of [Car72, Proposition 12.2.3] let θ : Ad G → Ad G be the automorphism de�ned by θ(\overline{x}_{α}(t)) = \overline{x}_{−α} (γ_{α}t) for all α ∈ Φ and t ∈ _{a} then we have

Ad g(x^{†}) = θ(Ad g)(x)^{†}  (5.4)

for all g ∈ G and x ∈ 𝔤.

By [Jan04, 2.10, Lemma] we see that it is su�cient to show that e and e^{†} are in the same G-orbit. Let us denote by ẇ_{0} ∈ N_{G}(T_{0}) a representative for the longest element w_{0} ∈ W_{G}(T_{0}). The action of −w_{0} on Φ induces a permutation ρ : Φ → Φ on the roots which is known to satisfy ⟨α, λ⟩ = ⟨ρ(α), λ⟩ for all α ∈ Φ. It su�ces to observe that the weighted Dynkin diagrams of A_{n}, D_{2n+1} and E_{6} are invariant under the graph automorphism induced by w_{0}, which is easily checked (see [Car93, §13.1]). In particular, it follows that Ad ẇ_{0}(e^{†}) ∈ 𝔤(λ, 2), θ(Ad L(λ)) = Ad L(λ) and Ad ẇ_{0} normalises Ad L(λ). Combining this with (5.4) we see that

Ad L(λ)(Ad ẇ_{0}(e^{†})) = Ad ẇ_{0}[Ad L(λ)(e)^{†}].

As the orbit Ad L(λ)(e) is dense in 𝔤(λ, 2) we must have Ad L(λ)(Ad ẇ_{0}(e^{†})) is dense in 𝔤(λ, 2) so Ad ẇ_{0}(e^{†}) ∈ 𝔤(λ, 2)_{reg}. The statement now follows from Theorem 3.16. ■

5.5. We now proceed to de�ne GGGRs following [Kaw86, §3]. Note that all results in this section are due to Kawanaka. Recall that the de�nition of GGGRs requires the choice of a G-invariant symmetric bilinear form κ : 𝔤 × 𝔤 → . Here, the G-equivariance means κ(Ad g(x), Ad g(y)) = κ(x, y) for all x, y ∈ 𝔤 and g ∈ G. Such a form can be obtained as a trace form

(x, y) ↦ Tr(dτ(x) ∘ dτ(y))

where τ : G → GL_{n}( ) is a �nite dimensional rational representation. For convenience we recall that such a form satis�es the property

κ(x, [y, z]) = κ([x, y], z)

for all x, y, z ∈ 𝔤, which we will use without explicit mention. It is important to note that, even with the assumption that G is proximate, we cannot always choose κ to be non-degenerate (see [Let05, Proposition 2.5.10])! However, we can choose it so that it is not too degenerate.

Lemma 5.6. There exists a form κ on 𝔤 de�ned over _{q} such that, for all α ∈ Φ, we have

𝔤_{α}^{⊥} = 𝔱_{0} ⊕ ⨁_{β∈Φ\{−α}} 𝔤_{β}  (5.7)

where 𝔤_{α}^{⊥} = {x ∈ 𝔤 | κ(x, y) = 0 for all y ∈ 𝔤_{α}}.

Proof. Let ϕ : G → G_{ad} be an adjoint quotient of G. Now G_{ad} is a direct product of simple groups G_{1} × ⋯× G_{r} so 𝔤_{ad} = 𝔤_{1} ⊕ ⋯⊕ 𝔤_{r} where 𝔤_{i} = Lie(G_{i}). If G_{i} is not of type A then there exists a nondegenerate G_{i}-invariant symmetric bilinear form κ_{i} on 𝔤_{i} de�ned over _{q} (see [SS70, I, 5.3]). Using [Let05, 2.5.1(2)] we can deduce that the appropriate version of (5.7) holds for κ_{i} by dimension counting.

Now assume G_{i} is of type A then G_{i} ≅ PGL_{n}( ) for some n and we set G̃_{i} = GL_{n}( ) and 𝔤̃_{i} = 𝔤𝔩_{n}( ) = Lie(G̃_{i}). As the natural trace form κ̃_{i}(x, y) = Tr(xy) is non-degenerate on 𝔤̃_{i} and de�ned over _{q} we have (5.7) holds in 𝔤̃_{i}. According to [Let05, 2.3.1] we have an isomorphism of Lie algebras 𝔤̃_{i} ≅ Lie(Z^{∘}(G̃_{i})) ⊕ 𝔤_{i}. Through this isomorphism we may de�ne a G_{i}-invariant symmetric bilinear form κ_{i} on 𝔤_{i} by restricting κ̃_{i} (this is not necessarily non-degenerate). As the image of each root space under this isomorphism must be contained in 𝔤_{i} we see that (5.7) holds for κ_{i}.

We now set κ_{ad} = κ_{1} + ⋯+ κ_{r} and de�ne κ by setting κ(x, y) = κ_{ad}(dϕ(x), dϕ(y)). Clearly this is G-invariant and we see that (5.7) holds by noticing that Ker(dϕ) (i.e. the centre of 𝔤) is contained in 𝔱_{0}. ■

We now assume that κ : 𝔤 × 𝔤 → is a �xed G-invariant symmetric bilinear form, de�ned over _{q}, satisfying (5.7).

5.8. If 𝔥 is a Lie algebra over then we will denote by 𝔥^{∗} = Hom(𝔥, ) the dual space. Assume now that i ≠ 0 and 𝔤_{α} ⊆ 𝔤(λ, i) then clearly we have 𝔤(λ, i)^{⊥} ⊆ 𝔤_{α}^{⊥} . In particular, applying (5.7) we see that 𝔤(λ, i)^{⊥} ∩ 𝔤(λ, −i) = {0} because 𝔤_{α} ⊆ 𝔤(λ, i) if and only if 𝔤_{−α} ⊆ 𝔤(λ, −i). With this we see that the map x ↦ κ(x, −) gives an identi�cation of 𝔤(λ, −i) = 𝔤(λ, i)^{†} with the dual space 𝔤(λ, i)^{∗}.

Lemma 5.9 (Kawnaka). The skew-symmetric bilinear form 𝔤(λ, 1) × 𝔤(λ, 1) → given by (x, y) ↦ κ(e^{†}, [x, y]) is non-degenerate.

Proof. Assume x ∈ 𝔤(λ, 1) and κ(e^{†}, [x, y]) = κ([e^{†}, x], y) = 0 for all y ∈ 𝔤(λ, 1). As [e^{†}, x] ∈ 𝔤(λ, −1) we must have [e^{†}, x] = 0 by 5.8 but this implies x = 0 because x ∈ 𝔠_{𝔲(λ)}(e^{†}) = 𝔠_{𝔤}(e)^{†} ∩ 𝔲(λ) = {0} (see Theorem 3.16). ■

5.10. We now choose once and for all a non-trivial additive character χ_{p} : _{p}^{+} → \overline{}_{ℓ}^{×} of the additive subgroup of the prime �eld. If r ∈ _{>1} is an integral power of p then we denote by _{r} ⊆ the sub�eld �xed by x ↦ x^{r}. The choice of χ_{p} gives rise to a character χ_{r} : _{r}^{+} → \overline{}_{ℓ}^{×} by setting χ_{r} = χ_{p} ∘ Tr_{_{r}/ _{p}} where Tr_{_{r}/ _{p}} is the �eld trace. As ϕ_{spr} satis�es (K2) we obtain a linear character φ_{u} : U(λ, 2) → \overline{}_{ℓ}^{×} by setting

φ_{u}(x) = χ_{q}(κ(e^{†}, ϕ_{spr}(x))).

5.11. We could now induce the character φ_{u} to G to obtain a character of G but this turns out not to be the right idea. Instead we construct an intermediary subgroup as follows. The non-degeneracy of the form (c.f. Lemma 5.9) implies that there exists a Lagrangian subspace 𝔪 = 𝔪^{⊥} ⊆ 𝔤(λ, 1), i.e. we have

κ(e^{†}, [x, y]) = 0  (5.12)

for all x, y ∈ 𝔪. This subspace necessarily has dimension dim 𝔤(λ, 1)/2. Note that 𝔪 is not necessarily unique so we must choose some such subspace. We then de�ne U(λ, 1.5) to be the variety

U(λ, 1.5) = {x ∈ U(λ) | ϕ_{spr}(x) ∈ 𝔪 + 𝔲(λ, 2)},

which is a closed connected F-stable subgroup of U(λ) containing U(λ, 2). This follows from the properties of a Kawanaka isomorphism and the fact that κ is de�ned over _{q}.

Lemma 5.13 (Kawanaka, [Kaw86, 3.1.9]).

(i) U(λ, 1.5) is a subgroup of G containing U(λ, 2) such that [U(λ) : U(λ, 1.5)] = [U(λ, 1.5) : U(λ, 2)].

(ii) φ_{u} extends to a linear character φ̃_{u} of U(λ, 1.5).

Proof. (i) The �rst statement is obvious and the second follows from a theorem of Rosenlicht (see [Gec03, 4.2.4]).

(ii) We �rst observe that U(λ)/ Ker(φ_{u}) is abelian which follows from 5.8, (K3), (5.12) and the definition of U(λ, 1.5). The result now follows from the following general fact: If H is a �nite group and X ⩽ H is a subgroup with linear character χ ∈ Irr(X) such that H/ Ker(φ) is abelian then χ extends to H. Indeed, passing to the quotient we may assume that H is abelian and χ is a faithful linear character of X, in particular X must be cyclic. We can then write H as a direct product X̃ × Y such that X ⩽ X̃ and X̃ is cyclic. The result then follows from [Isa06, Corollary 11.22]. ■

De�nition 5.14. We call the induced representation Γ_{u} = Ind_{U(λ,1.5)}^{G}(φ̃_{u}) the Generalised Gelfand�Graev Representation (GGGR) associated with u. We identify Γ_{u} with its character.

Lemma 5.15 (Kawanaka, [Kaw86, 3.1.12]). We have Γ_{u} = q^{− dim 𝔤(λ,1)/2} Ind_{U(λ,2)}^{G}(φ_{u}). In particular, the construction of Γ_{u} does not depend upon the choice of Lagrangian 𝔪 or extension φ̃_{u}.

Proof. First of all let us note that by (3.12) we have U(λ, 2) contains the derived subgroup of U(λ) so both U(λ, 2) and U(λ, 1.5) are normal subgroups of U(λ). Assume v ∈ U(λ, 1.5) then by de�nition we have

Ind_{U(λ,1.5)}^{U(λ)}(φ̃_{u})(v) = q^{− dim 𝔤(λ,1)/2−dim 𝔲(λ,2)} ∑_{g∈U(λ)} φ̃_{u} (gvg^{−1} ).

As [g, v] ∈ U(λ, 2) (again by (3.12)) we may rewrite the sum on the right as

∑_{g∈U(λ)} φ̃_{u} (gvg^{−1} ) = ∑_{g∈U(λ)} φ̃_{u} (v)φ_{u}([g, v]) = φ̃_{u} (v) ∑_{x∈𝔲(λ)^{F}} ζ_{v}(x).

where ζ_{v} is the linear character x ↦ χ_{q}(κ(e^{†}, c[x, ϕ_{spr}(v)])) of the abelian group 𝔲(λ)^{F} (here c is as in (K3)). Now the character sum is 0 unless ζ_{v} is identically 1, in which case it is simply q^{dim 𝔲(λ)}. However, we have ζ_{v} = 1 if and only if v ∈ U(λ, 2) which shows that

Ind_{U(λ,1.5)}^{U(λ)}(φ̃_{u})(v) = \{ \begin{matrix}q^{dim 𝔤(λ,1)/2} φ_{u}(v) & if v ∈ U(λ, 2), \\ 0 & if v ∉ U(λ, 2). \end{matrix}

Applying the exact same argument as above to Ind_{U(λ,2)}^{U(λ)}(φ_{u})(v) we see that

Ind_{U(λ,1.5)}^{U(λ)}(φ̃_{u}) = q^{− dim 𝔤(λ,1)/2} Ind_{U(λ,2)}^{U(λ)}(φ_{u}),  (5.16)

from which the result follows immediately. ■

Remark 5.17. In [Kaw86] GGGRs are de�ned with no assumption on the algebraic group G. However, it seems to be necessary to have some assumption on G to de�ne GGGRs (see for instance the proof of Lemma 5.9). Note that assumptions similar to G being proximate were made in [Kaw85, 1.1.1] but not in [Kaw86]. In any case, we have by 2.10 that our assumption on G is not restrictive in de�ning GGGRs in good characteristic.

6. Fourier Transforms on the Lie Algebra

De�nition 6.1 (Lettelier, [Let05, 5.0.14]). We say the prime p is acceptable for G if:

(i) p is a pretty good prime for G (c.f. De�nition 3.6 )

(ii) p is a very good prime for any Levi subgroup L of G supporting a cuspidal pair (S, ℰ) (in the sense of [Lus84b, 2.4]) such that S contains a unipotent conjugacy class of L.

(iii) there exists a non-degenerate G-invariant bilinear form on 𝔤.

6.2. This de�nition of an acceptable prime is slightly di�erent than that given in [Let05, 5.0.14]. However we note that if p is acceptable in the sense of De�nition 6.1 then it is acceptable in the sense of [Let05, 5.0.14]. Indeed, we need only observe that if p is an acceptable prime in the sense of De�nition 6.1 then: p is a good prime for G (see [Her13, Lemma 2.12(b)]), p does not divide | Tor(X(T_{0})/ Φ)| and there exists a Springer isomorphism (see Lemma 3.2). To show that these de�nitions are actually equivalent it would be su�cient to prove the converse to Lemma 3.2, namely that the existence of a Springer isomorphism implies that G is proximate. With this we note the following observation (see [Let05, 5.0.16]).

Lemma 6.3. The following hold:

(i) p is very good for G ⇒ p is acceptable for G,

(ii) if G = GL_{n}( ) then all primes are acceptable for G.

From now on we assume that p is an acceptable prime for G and consequently that the bilinear form κ is non-degenerate. In particular, G is a proximate algebraic group (c.f. Lemma 2.7) and the centraliser C_{G}(x) is smooth for all x ∈ 𝔤 (c.f. De�nition 3.6 ).

6.4. Let us denote by Γ_{u} : 𝔤^{F} → \overline{}_{ℓ} the Ad G-invariant function obtained as the extension of Γ_{u}|_{U^{F}}∘ ϕ_{spr}^{−1} by 0 outside 𝔤^{F} \ N^{F}. As Γ_{u} is zero outside U^{F} we see that Γ_{u} contains the same information as that of Γ_{u}. The upshot of working with Γ_{u} is that we have the Fourier transform at our disposal, which is de�ned as follows. For any function f : 𝔤^{F} → \overline{}_{ℓ} we de�ne the Fourier transform of f to be the function F(f) : 𝔤^{F} → \overline{}_{ℓ} given by

F(f)(y) = ∑_{x∈𝔤^{F}} χ_{q}(κ(y, x))f(x).

We recall the following property of the Fourier transform (see [Let05, 3.1.9, 3.1.10]).

Lemma 6.5. The Fourier transform admits an inverse F^{−} given by q^{− dim 𝔤} (F ∘ Ψ) where Ψ(f)(x) = χ(−x) for all x ∈ 𝔤^{F} and f : 𝔤^{F} → \overline{}_{ℓ}. In other words, we have (F^{−} ∘ F)(f) = (F ∘ F^{−} )(f) = f for all functions f : 𝔤^{F} → \overline{}_{ℓ}.

6.6. Following [Lus92, Proposition 2.5] we would like to obtain an expression for the value of the Fourier transform F(Γ_{u}) at an element of 𝔤^{F}. The argument used in [Lus92] can be applied verbatim to our situation once we know that [Lus92, Lemma 2.2] holds. In fact we will prove a stronger statement than that of [Lus92, Lemma 2.2]. The proof of this stronger statement is due to Gan�Ginzburg who considered the corresponding statement over (see [GG02, Lemma 2.1]). To obtain the desired result we will need the following (general) lemma.

Lemma 6.7. Assume, for i ∈ {1, 2}, that X_{i} is an a�ne variety with a contracting _{m} action such that the �xed point x_{i} is non-singular. Let us denote by T_{i} the tangent space of X_{i} at the point x_{i}. If α : X_{1} → X_{2} is a m-equivariant morphism such that the di�erential d_{x_{1}}α : T_{1} → T_{2} is an isomorphism then α is an isomorphism.

Proof. It su�ces to show that the induced map α^{∗} : [X_{2}] → [X_{1}] is bijective. Firstly, as α induces an isomorphism between the tangent spaces of non-singular points we must have α^{∗} is injective.

We now consider surjectivity, for which we will use the contracting _{m}-actions. Let 𝔪_{x_{i}} ⊂ [X_{i}] be the (maximal) vanishing ideal of x_{i}. Clearly the _{m}-action on X_{i} induces an action on [X_{i}] which preserves the vanishing ideal 𝔪_{x_{i}} because x_{i} is �xed by _{m}. We denote by gr [X_{i}] the associated graded ring with respect to 𝔪_{x_{i}}, i.e.

gr [X_{i}] = ⨁_{j⩾0} 𝔪_{x_{i}}^{j}/𝔪_{x_{i}}^{j+1} ,

where 𝔪_{x_{i}}^{0} = [X_{i}]. This clearly also inherits an action of _{m}. We then have decompositions into weight spaces for all the algebras with _{m} actions, i.e.

[T_{i}] = ⨁_{n∈} [T_{i}]_{n}  [X_{i}] = ⨁_{n∈} [X_{i}]_{n}  gr [X_{i}] = ⨁_{n∈} (gr [X_{i}])_{n}.

Let us assume that the following equality holds

dim [X_{1}]_{n} = dim [T_{1}]_{n} = dim [T_{2}]_{n} = dim [X_{2}]_{n}  (6.8)

for all n ∈ . Then, as α^{∗} is injective and _{m}-equivariant we must have α^{∗} is surjective. Thus we need only prove (6.8).

By assumption, the point x_{i} is non-singular so the tangent space T_{i} coincides with the tangent cone of X_{i} at x_{i}. This implies that [T_{i}] is isomorphic to the associated graded algebra gr [X_{i}] and this isomorphism is in fact _{m}-equivariant (see [Mum99, III, §3, §4]). In particular, we have an equality

dim [T_{i}]_{n} = dim(gr [X_{i}])_{n} = dim [X_{i}]_{n}

for all n ∈ , where the latter equality is obvious. However, we must have dim [T_{1}]_{n} = dim [T_{2}]_{n} for all n ∈ by assumption, which proves (6.8). ■

Proposition 6.9. Let Σ = −e^{†} + 𝔰 where 𝔰 ⊆ 𝔤_{⩾0} is as in Corollary 3.17with respect to −e^{†}. Then the action map α : U(λ, 2) × Σ → −e^{†} + 𝔭(λ) given by (u, x) ↦ Ad u(x) is an isomorphism.

Proof. We de�ne a _{m}-action ρ : _{m} → End(𝔤) on 𝔤 by setting ρ(k)(x) = k^{−2} (Ad λ(k^{−1} ))(x) for all x ∈ 𝔤 and k ∈ _{m}. Note that the _{m}-action preserves both 𝔭(λ) and 𝔰 (c.f. Corollary 3.17) and as −e^{†} ∈ 𝔤(λ, −2) we have ρ(k)(−e^{†} + x) = −e^{†} + ρ(k)(x) for all x ∈ 𝔭(λ). In particular, we have ρ restricts to a _{m}-action on −e^{†} + 𝔭(λ) and Σ. Furthermore, we note that for any x ∈ 𝔭(λ) we have lim_{k→0} ρ(k)(−e^{†} + x) = −e^{†} so both _{m}-actions are contractions to −e^{†}. We also de�ne a _{m}-action on U(λ, 2) × Σ by setting

k · (u, x) = (λ(k^{−1} )uλ(k), ρ(k)(x))

for all k ∈ _{m}. It is clear that lim_{k→0} k · (u, x) = (1, −e^{†}) for all (u, x) ∈ U(λ, 2) × Σ so this is also a contraction and the action map α is _{m}-equivariant with respect to these actions.

Consider the tangent space T ≅ 𝔲(λ, 2) × 𝔰 of U(λ, 2) × Σ at the point (1, −e^{†}) and the tangent space S ≅ 𝔭(λ) of −e^{†} + 𝔭(λ) at −e^{†} = α(1, −e^{†}) (note these are canonical vector space isomorphisms). We claim that the di�erential d_{(1,−e^{†})} α : T → S is an isomorphism. Firstly, note that the di�erential is given by d_{(1,−e^{†})} α(u, x) = [u, −e^{†}] + x for all (u, x) ∈ T so to prove the map is surjective it su�ces to show that

𝔭(λ) = [𝔲(λ, 2), −e^{†}] ⊕ 𝔰.

Now the map ad_{−e^{†}} : 𝔲(λ, 2) → 𝔭(λ) is injective because 𝔠_{𝔲(λ,2)}(−e^{†}) = {0} and [𝔲(λ, 2), −e^{†}] ∩ 𝔰 = {0} by Corollary 3.17 (note that [𝔤, −e^{†}] = T_{−e^{†}} (Ad G(−e^{†}))) hence we have [𝔲(λ, 2), −e^{†}] ⊕ 𝔰 ⊆ 𝔭(λ). However, as dim 𝔰 = dim 𝔠_{𝔤}(−e^{†}) = dim 𝔠_{𝔤}(e) we have

dim([𝔲(λ, 2), −e^{†}] ⊕ 𝔰) = dim 𝔲(λ, 2) + dim 𝔠_{𝔤}(e) = dim 𝔭(λ)

so we must have an equality. The same dimension counting argument also proves the map d_{(1,−e^{†})} α is injective. The result now follows from Lemma 6.7. ■

From now on the subspace 𝔰 ⊆ 𝔤 considered in Proposition 6.9 will be �xed and Σ will denote the transverse slice −e^{†} + 𝔰.

6.10. With this in hand we now obtain [Lus92, Proposition 2.5] in exactly the same way. We give the argument here for smoothness of the exposition. From the de�nitions and Lemma 5.15 we have for any y ∈ 𝔤^{F} that

F(Γ_{u})(y) = q^{dim 𝔤(λ,1)/2−dim 𝔲(λ)} ∑_{\begin{matrix}x∈𝔤^{F},g∈G \\ Ad g(x)∈𝔲(λ)^{F} \end{matrix}} χ_{q}(κ(y, x) + κ(e^{†}, Ad g(x)))

= q^{dim 𝔤(λ,1)/2−dim 𝔲(λ)} ∑_{\begin{matrix}g∈G \\ x∈𝔲(λ)^{F} \end{matrix}} ζ_{y,g}(x),

where ζ_{y,g} is the linear character x ↦ χ_{q}(κ(e^{†} + Ad g(y), x)) of 𝔲(λ)^{F} (as an abelian group). As ζ_{y,g} is a character of a �nite abelian group we have the sum of its values is either 0 or q^{dim 𝔲(λ)} (which happens if and only if ζ_{y,g} is the trivial character). By 5.8 and the non-degeneracy of κ we have the character ζ_{y,g} is trivial precisely when e^{†} + Ad g(y) ∈ 𝔲(λ) so we have

F(Γ_{u})(y) = q^{dim 𝔤(λ,1)/2} |{(g, x) ∈ G × 𝔲(λ)^{F} | Ad g(y) = −e^{†} + x}|.

Applying Proposition 6.9 we see that the element −e^{†} + x can be written uniquely as Ad h(−e^{†} + z) for some h ∈ U(λ, 2) and z ∈ 𝔰^{F} (which are �xed by F through the uniqueness). In particular, we may rewrite the above as

F(Γ_{u})(y) = q^{dim 𝔤(λ,1)/2} |{(g, h, z) ∈ G × U(λ, 2) × 𝔰^{F} | Ad h^{−1} g(y) = −e^{†} + z}|.

Thus, changing the variable h^{−1} g → g we obtain the following.

Proposition 6.11 (Lusztig, [Lus92, Proposition 2.5]). For any y ∈ 𝔤^{F} we have

F(Γ_{u})(y) = q^{r_{u}}|{g ∈ G | Ad g(y) ∈ Σ}|

where

r_{u} = dim 𝔤(λ, 1)/2 + dim 𝔲(λ, 2) = (dim G − dim C_{G}(u))/2.

7. Poincar�e Duality

Notation

7.1. For any algebraic variety X over we will denote by 𝒟X := 𝒟_{c}^{b}X the bounded derived category of \overline{}_{ℓ}-constructible sheaves on X. Assume now that the Frobenius endomorphism induces a morphism of varieties on X then we say A ∈ 𝒟X is F-stable if there exists an isomorphism ϕ : F^{∗}A → A in 𝒟X. With such a choice of isomorphism we will denote by χ_{A,ϕ} : X^{F} → \overline{}_{ℓ} the corresponding characteristic function de�ned by

χ_{A,ϕ}(x) = ∑_{i∈} (−1)^{i} Tr(ϕ, ℋ_{x}^{i}(A))

for any x ∈ X^{F}. We will denote by supp(A) the set {x ∈ X | ℋ_{x}^{i}(A) ≠ 0 for some i ∈ } which we call the support of A.

7.2. The full subcategory of 𝒟X consisting of the perverse sheaves on X will be denoted by ℳ X. Assume now that X ⊆ Y is a subvariety. We will naturally consider any complex A ∈ 𝒟X as a complex on Y through extension by 0 (we will sometimes do so without explicit mention). Assume now that X is a smooth open dense subset of \overline{X} and that ℒ is a local system on X. We will denote by IC(\overline{X}, ℒ)[dim X] ∈ ℳ \overline{X} the corresponding intersection cohomology complex determined by ℒ.

7.3. For any connected reductive algebraic group H we denote by V_{H} = V_{H}^{nil} the set of all pairs ι = (O_{ι}, ℰ_{ι}) consisting of a nilpotent H-orbit O_{ι} ⊆ Lie(H) and an irreducible L-equivariant local system ℰ_{ι} on O_{ι} (taken up to isomorphism). We will write V_{H}^{0} ⊆ V_{H} for the subset consisting of all pairs ι such that ℰ_{ι}is a cuspidal local system (see [Lus84b, De�nition 2.4]); we call the elements of V_{H}^{0} cuspidal pairs.

Induction Diagram

7.4. Let L ⩽ G be the Levi complement of a parabolic subgroup P ⩽ G and let 𝔩 = Lie(L) ⊆ 𝔭 = Lie(P) be the corresponding Lie algebras. If 𝔷(𝔩) is the centre of the Lie algebra 𝔩 then we de�ne

𝔷(𝔩)_{reg} = {t ∈ 𝔷(𝔩) | 𝔠_{𝔤}(t) = 𝔩},

which is an open subset of 𝔷(𝔩). We will assume ι_{0} = (O_{0}, ℰ_{0}) ∈ V_{L}^{0} is a cuspidal pair. Setting Π = O_{0}+ 𝔷(𝔩) and Π_{reg} = O_{0} + 𝔷(𝔩)_{reg} we have, as in [Lus92, 3.1], the following induction diagram

O_{0}  Ŷ  Ỹ  Y ; \begin{matrix}α \\ β \\ γ \end{matrix}

where

Y = ⋃_{g∈G} Ad g(Π_{reg})
Ŷ = {(x, g) ∈ 𝔤 × G | Ad g^{−1} (x) ∈ Π_{reg}}
Ỹ = {(x, gL) ∈ 𝔤 × (G/L) | Ad g^{−1} (x) ∈ Π_{reg}}
α(x, g) = π_{0}(Ad g^{−1} (x))  β(x, g) = (x, gL)  γ(x, gL) = g

and π_{0} : Π → O_{0} is the natural projection. As discussed in [Lus92, 3.1] and [Let05, pg. 75] from this data one constructs a local system ℒ on Y and a corresponding intersection cohomology complex K_{0} = IC(\overline{Y}, ℒ)[dim Y] (which is denoted by ind_{Σ}^{G}(ℰ_{0}) in [Let05]).

From now on we assume that the parabolic P, Levi subgroup L and the cuspidal pair ι_{0} = (O_{0}, ℰ_{0}) ∈ V_{L}^{0} are �xed.

Lemma 7.5. Let us assume that −e^{†} ∈ \overline{Y} then the following hold:

(i) The �bres of the smooth morphism π : G × Σ → 𝔤 given by π(g, x) = Ad g(x) are of pure dimension equal to dim 𝔠_{𝔤}(e).

(ii) The intersection Y ∩ N is locally closed in 𝔤, irreducible and smooth of dimension dim G/L + dim O_{0}.

(iii) dim(\overline{Y}∩ N ∩ Σ) = − dim L + dim O_{0} + dim 𝔠_{𝔤}(e).

(iv) Let σ : G × (\overline{Y}∩ N ∩ Σ) → \overline{Y}∩ N be the restriction of π to G × (\overline{Y}∩ N ∩ Σ), then σ is a smooth morphism with all �bres of pure dimension equal to dim 𝔠_{𝔤}(e).

Proof. Firstly let us note that 𝔤 and G are irreducible varieties and that any irreducible component of G × Σ has dimension dim 𝔤 + dim Σ. Then applying [Har77, III, §9, 9.6] we have, for any y ∈ 𝔤, that every irreducible component of the �bre π^{−1} (y) has dimension dim Σ = dim 𝔠_{𝔤}(e) which gives us (i).

Note that Y ∩ N = Ad G(O_{0}) is a nilpotent G-orbit so everything except the dimension follows from this (see for instance [Jan04, §2.1]). To obtain the dimension we simply argue as in [Let05, Lemma 5.1.28], which �nishes (ii).

By [Jan04, 2.10, Lemma] we see that \overline{Y}∩ N ∩ Σ is invariant under the contracting _{m}-action ρ on Σ de�ned in the proof of Proposition 6.9. We easily see that the proof given in Corollary 3.17 also shows that σ is smooth (so we will not repeat it). As \overline{Y}∩ N is irreducible we can conclude that every irreducible component of G × (\overline{Y}∩ N ∩ Σ) has the same dimension because σ is smooth. Furthermore, as Y ∩ N = Ad G(O_{0}) and Σ intersects all Ad G-orbits transversally we have

dim(\overline{Y}∩ N ∩ Σ) = dim Y ∩ N + dim Σ − dim G = − dim L + dim O_{0} + dim 𝔠_{𝔤}(e),

which gives (iii). Using the argument from the proof of (i) this completes (iv). ■

7.6. According to [Lus84b, 6.6.1] we have the restriction K_{0}[dim(\overline{Y}∩ N)]|_{\overline{Y}∩N} is a perverse sheaf. Let us take on the assumption that −e^{†} ∈ \overline{Y} and let σ be as in Lemma 7.5 then according to [BBD82, 4.2.4] we have

σ^{∗}(K_{0}[dim G + d]|_{\overline{Y}∩N} ) ∈ 𝒟(\overline{Y}∩ N ∩ Σ)  (7.7)

is also a perverse sheaf. Now G acts on Y (via Ad), on Ŷ and Ỹ (via Ad on the �rst coordinate and left translation on the second) and �nally Σ (trivially). As discussed in [Let05, pg. 75] we see that, with respect to these actions, ℒ is a G-equivariant local system and K_{0} is a G-equivariant perverse sheaf (where ℒ and K_{0} are as in 7.4). According to [Lus92, 3.3] if K̃_{0} ∈ 𝒟(\overline{Y}∩ N ∩ Σ) is the restriction of K_{0} to \{1\} × (\overline{Y}∩ N ∩ Σ) then K̃_{0}[d_{0}] is a perverse sheaf, where d_{0} = dim(\overline{Y}∩ N ∩ Σ). Furthermore, if K̃_{0}^{∨} is the restriction of K_{0}^{∨} = IC(\overline{Y}, ℒ^{∨})[dim Y], where ℒ^{∨} is the local system dual to ℒ, then we have K̃_{0}^{∨}[d_{0}] is also a perverse sheaf which is naturally the Verdier dual of K̃_{0}[d_{0}]. We then have a canonical non-singular pairing (Poincar� e duality)

H^{j}(\overline{Y}∩ N ∩ Σ, K̃_{0}) ⊗ H_{c}^{2d−j} (\overline{Y}∩ N ∩ Σ, K̃_{0}^{∨}) → \overline{}_{ℓ}(−d_{0}).  (7.8)

between the corresponding hypercohomology groups.

7.9. If ρ is the _{m}-action on 𝔤 de�ned in the proof of Proposition 6.9 then we let _{m} act on Y (via ρ(k)), on Ŷ and Ỹ (via ρ(k) on the �rst coordinate and Ad λ(k^{−1} ) on the second) and on O_{0} (via k · x = k^{−2} x). As in [Lus92, 3.4] we may use the contracting action to deduce that the canonical homomorphism

H^{j}(\overline{Y}∩ N ∩ Σ, K̃_{0}) → ℋ_{−e^{†}}^{j} (K̃_{0})  (7.10)

is an isomorphism. In particular, the pairing of (7.8) becomes a pairing

ℋ_{−e^{†}}^{j} (K̃_{0}) ⊗ H_{c}^{2d−j} (\overline{Y}∩ N ∩ Σ, K̃_{0}^{∨}) → \overline{}_{ℓ}(−d_{0}).  (7.11)

8. Blocks

8.1. Let us denote by A the endomorphism algebra End_{𝒟\overline{Y}} (K_{0}) of the complex K_{0} (note this is naturally isomorphic to the endomorphism algebra End_{𝒟O_{0}} (ℰ_{0})). The following is a combination of [Lus84b, Theorem 9.2(a)] and [Lus86, 2.4] (see also [Let05, Proposition 5.3.6]).

Proposition 8.2. There exists a set of basis elements \{θ_{w} | w ∈ W_{G}(L)\} for A such that the \overline{}_{ℓ}-linear extension of the map w ↦ θ_{w} de�nes a \overline{}_{ℓ}-algebra isomorphism \overline{}_{ℓ}W_{G}(L) ≅ A.

8.3. Combining Proposition 8.2 with [Lus92, 4.2(b)] (see also [Tay14, 4.3]) we have a canonical isomorphism

K_{0} ≅ ⨁_{E∈Irr(W_{G}(L))}(E ⊗ K_{E}),  (8.4)

where K_{E} = Hom_{A}(E, K_{0}) ∈ ℳ \overline{Y} and we identify Irr(A) and Irr(W_{G}(L)) through the isomorphism in Proposition 8.2. Again from [Lus92, 4.2] (see also [Lus84b, Theorem 6.5]) there exists a unique pair ι ∈ V_{G} such that

K_{E}|_{\overline{Y}∩N} ≅ IC(\overline{O}_{ι}, ℰ_{ι})[dim O_{ι}− dim(\overline{Y}∩ N)].  (8.5)

The map E → K_{E} → ι then gives us an injective map Irr(W_{G}(L)) → V_{G}. We will denote the perverse sheaf K_{E} by K_{ι}.

8.6. Let us denote by W̃_{G} the set of all pairs (M, υ) consisting of a Levi complement M ⩽ G of a parabolic subgroup of G and a cuspidal pair υ ∈ V_{M}^{0}. We have G acts naturally on W̃_{G} by conjugation and we denote by [M, υ] the orbit containing (M, υ). We also denote by W_{G} the set of all such orbits. Using the map in 8.3 we can associated to every pair (M, υ) ∈ W̃_{G} a subset ℐ [M, υ] ⊆ V_{G} which depends only on the orbit [M, υ]. We then have a disjoint union

V_{G} = ⨆_{[M,υ]∈M_{G}} ℐ [M, υ]

and we call ℐ [M, υ] a block of V_{G}. If υ ∈ V_{M}^{0} is cuspidal then the relative Weyl group W_{G}(M) = N_{G}(M)/M is a Coxeter group and 8.3 yields a bijection

ℐ [M, υ] ⟷ Irr(W_{G}(M))  (8.7)

for all [M, υ] ∈ W_{G}, which we denote by ι ↦ E_{ι}.

8.8. The Frobenius endomorphism F acts naturally on the set of pairs V_{G} via

ι ↦ F^{−1} (ι) = (F^{−1} (O_{ι}), F^{∗}ℰ_{ι}).

We say ι is F-stable if F(O_{ι}) = O_{ι}and there exists an isomorphism F^{∗}ℰ_{ι}→ ℰ_{ι}and we denote by V_{G}^{F} ⊆ V_{G} the subset of all F-stable pairs. The Frobenius acts also on the set W̃_{G}, hence also on W_{G}, via (M, υ) ↦ (F^{−1} (M), F^{−1} (υ)). We say pair (M, υ) (resp. the orbit [M, υ]) is F-stable if F^{−1} (M) = M and F^{−1} (υ) = υ (resp. [M, υ] = [F^{−1} (M), F^{−1} (υ)]) and we denote by W̃_{G}^{F} (resp. W_{G}^{F}) the subset of all F-stable pairs (resp. orbits). Note that for any orbit [M, υ] ∈ W_{G}^{F} we have [M, υ] ∩ W̃_{G}^{F} ≠ ∅ and the map V_{G} → W_{G} is compatible with the corresponding actions of F so that we have an induced map V_{G}^{F} → W_{G}^{F}. Furthermore if (M, υ) ∈ W_{G}^{F} then we have the bijection in (8.7) is also compatible with F, namely it restricts to a bijection

ℐ [M, υ]^{F} ⟷ Irr(W_{G}(M))^{F}.  (8.9)

9. Deligne�Fourier Transform

9.1. We de�ne the Deligne�Fourier transform of a perverse sheaf as follows (see [Lus92]). Let _{a} act transitively on the a�ne line ^{1} = via a · t = t + a − a^{p}. Now the character χ_{p} �xed in 5.10 gives rise to a _{a}-equivariant local system of rank 1 on ^{1} which we denote by ℒ_{χ_{p}}. Its inverse image under the non-degenerate form κ is then a local system of rank 1 on 𝔤 × 𝔤. With this we de�ne the Fourier transform F(A) of any complex A ∈ 𝒟(𝔤) by setting

F(A) = (pr_{2})_{!}(pr_{1}^{∗}(A) ⊗ κ^{∗}ℒ_{χ_{p}})  (9.2)

where pr_{i} : 𝔤 × 𝔤 → 𝔤 is the projection onto the ith factor. Note that the Deligne�Fourier transform of complexes and the Fourier transform of functions are related in the following way (see [Let05, 5.2.3]).

Lemma 9.3. Assume A ∈ 𝒟(𝔤) is an F-stable complex and ϕ : F^{∗}A → A is an isomorphism then we have

χ_{F(A),F(ϕ)} = F(χ_{A,ϕ}),

where F(ϕ) : F(F^{∗}A) → F(A) is the induced isomorphism.

Remark 9.4. We will frequently cite results from the work of Letellier [Let05] throughout this article. We point out here that our de�nitions of the Deligne�Fourier transform and Fourier transform do not coincide with those of [Let05] but that they di�er only up to shifts and a scalar. The correction is easily seen by comparing Lemma 9.3 and [Let05, 5.2.3].

9.5. Let A_{0} ∈ 𝒟(𝔤) be the complex obtained as the extension by 0 of the complex K_{0}|_{\overline{Y}∩N} . According to the proof of [Let05, Proposition 6.2.9] (see also [Lus87, §8]) we have an isomorphism

F(K_{0}) → A_{0}.

Comparing (8.4) and (8.5) we see that for each ι ∈ ℐ [L, ι_{0}] there exists a unique pair ι̂∈ ℐ [L, ι_{0}] such that F(K_{ι}) ≅ IC(\overline{Oι̂}, ℰι̂) up to some shift. In particular the map ˆ : ℐ [L, ι_{0}] → ℐ [L, ι_{0}] given by ι ↦ ι̂is a bijection. We will need the following property of this bijection (and its corollary).

Lemma 9.6. Assume ℐ [L, ι_{0}] ⊆ V_{G} is a block containing precisely two elements (i.e. L is the Levi complement of a maximal parabolic subgroup of G) then the bijection ˆ is not the identity.

Corollary 9.7 (Lusztig, [Lus92, 5.5]). Under the bijection in (8.7) we have the map ι ↦ ι̂corresponds to the natural bijection

E ↦ E ⊗ sgn

where sgn ∈ Irr(W_{G}(M)) is the sign representation.

9.8. Lemma 9.6 can be proved as a consequence of the validity of Theorem 11.10 for a split Frobenius endomorphism F just as in [Lus92, 7.7]. Note that the statement of Lemma 9.6 does not depend upon the _{q}-structure, so we may use any Frobenius to prove it. We will not use Lemma 9.6 in any step towards the proof of Theorem 11.10 for split Frobenius endomorphisms. However, we will use the validity of Lemma 9.6 (more speci�cally Corollary 9.7) to prove Theorem 11.10 for twisted Frobenius endomorphisms. This is a �ne needle to thread but the reader is free to check that the logic is sound.

9.9. Reading carefully the rest of [Lus92, §5] one sees that the entire discussion holds unchanged. One need only note that the results cited in [Lus87] are proved in [Let05] under our weaker assumption that p is an acceptable prime for G (see in particular [Let05, 5.2.9, 5.2.10]).

10. _{q}-Structures

From now on we assume that the orbit [L, ι_{0}] of (L, ι_{0}) is F-stable. Furthermore, we assume that (L, ι_{0}) ∈ W̃_{G}^{F} is F-stable and that the parabolic subgroup P is F-stable. Finally we assume chosen an involutive automorphism : \overline{}_{ℓ} → \overline{}_{ℓ} which maps any root of unity to its inverse.

Isomorphisms

10.1. We will now follow the path of [Lus92, §6]. However, unlike [Lus92] we will not assume that F is a split Frobenius endomorphism.

We will denote by φ_{0} : F^{∗}ℰ_{0} → ℰ_{0} a �xed isomorphism chosen such that the induced isomorphism (ℰ_{0})_{x} → (ℰ_{0})_{x} at the stalk of any element x ∈ O_{0}^{F} has �nite order. Recalling the notation from 7.4 we have the choice of φ_{0} naturally determines an isomorphism ϕ_{0} : F^{∗}K_{0} → K_{0} and hence a corresponding automorphism ϕ_{0} : ℋ_{x}^{i}(K_{0}) → ℋ_{x}^{i}(K_{0}) for any x ∈ \overline{Y}^{F} and i ∈ (see [Tay14, 6.12]). Taking the contragradient of φ_{0} we obtain an isomorphism φ_{0}^{∨} : F^{∗}ℰ_{0}^{∨} → ℰ_{0}^{∨} which in turn induces isomorphisms ϕ_{0}^{∨} : F^{∗}K_{0}^{∨} → K_{0}^{∨} and ϕ̃_{0}^{∨} : F^{∗}K̃_{0}^{∨} → K̃_{0}^{∨}. These in turn induce automorphisms ℋ_{x}^{i}(K_{0}^{∨}) and H_{c}^{i}(\overline{Y}∩ N ∩ Σ, K̃_{0}^{∨}) for any x ∈ \overline{Y}^{F} and i ∈ .

10.2. The choice of parabolic subgroup P naturally determines a set of Coxeter generators of the relative Weyl group W_{G}(L) (see for instance [Bon04, Proposition 1.12]). Furthermore, as both L and P are chosen to be F-stable we have F induces an automorphism W_{G}(L) → W_{G}(L) which stabilises . For any irreducible W_{G}(L)-module E we denote by E_{F} the \overline{}_{ℓ}W_{G}(L)-module obtained from E by twisting with F^{−1} . Now the module E is F-stable if and only if there exists an isomorphism E → E_{F} of \overline{}_{ℓ}W_{G}(L)- modules. Furthermore, any choice of such an isomorphism naturally determines a \overline{}_{ℓ}W̃_{G}(L)-module structure on E (where W̃_{G}(L) = W_{G}(L) ⋊ ⟨F⟩).

Let ι ∈ ℐ [L, ι_{0}]^{F} be an F-stable pair in the block then we wish to choose an isomorphism ϕ_{ι}: F^{∗}K_{ι}→ K_{ι}. Recall that E_{ι} ∈ Irr(W_{G}(L))^{F} denotes the corresponding simple module under (8.9). Following the construction in [Tay14, 6.13] we see that choosing an isomorphism ϕ_{ι} is equivalent to choosing an isomorphism ψ_{ι} : E → E_{F} of \overline{}_{ℓ}W_{G}(L)-modules. Thus we de�ne ϕ_{ι} by requiring that the action of F^{−1} , through ψ_{ι}, makes E_{ι}Lusztig’s preferred extension of E_{ι}as a \overline{}_{ℓ}W̃_{G}(L)-module (see [Lus85, 17.2]).

10.3. From the isomorphism (8.5) we recover the local system ℰ_{ι}via the isomorphism

ℋ ^{− dim O_{ι}+dim(\overline{Y}∩N)} (K_{ι})|_{O_{ι}} ≅ ℰ_{ι}.

Through this we see that ϕ_{ι}determines an isomorphism F^{∗}ℰ_{ι}→ ℰ_{ι}which is of the form q^{b_{ι}/2}φ_{ι}where

b_{ι}= − dim O_{ι}+ dim(\overline{Y}∩ N) = (dim G − dim O_{ι}) − (dim L − dim O_{0})  (10.4)

and φ_{ι}: F^{∗}ℰ_{ι} → ℰ_{ι} is an isomorphism which induces an automorphism φ_{ι}: (ℰ_{ι})_{x} → (ℰ_{ι})_{x} of �nite order for any x ∈ O_{ι}^{F} (see [Lus84b, 24.2.4]). For convenience we will also de�ne the value

a_{ι}= − dim O_{ι}− dim Z^{∘}(L)  (10.5)

for any ι ∈ ℐ [L, ι_{0}].

10.6. For w ∈ W_{G}(L) we consider an F-stable Levi subgroup L_{w} = gLg^{−1} , where g ∈ G is such that g^{−1} F(g) = ẇ^{−1} ∈ N_{G}(L) is a representative of w^{−1} ∈ W_{G}(L). Conjugating the cuspidal pair ι_{0} we also obtain a corresponding F-stable cuspidal pair (O_{w}, ℰ_{w}) ∈ V_{L_{w}}^{0} where

O_{w} = ^{g}O_{0} ℰ_{w} = (Ad g^{−1} )^{∗}ℰ_{0}.

We now obtain complexes K_{w}, K̃_{w} and K_{w}^{∨} from (L_{w}, O_{w}, ℰ_{w}) just as K_{0}, K̃_{0} and K_{0}^{∨} were obtained from (L, O_{0}, ℰ_{0}). As in [Tay14, 6.9] using Lusztig’s basis element θ_{w} of End(ℰ_{0}) (c.f. Proposition 8.2) we see that the �xed isomorphism φ_{0} : F^{∗}ℰ_{0} → ℰ_{0} determines an isomorphism φ_{w} : F^{∗}ℰ_{w} → ℰ_{w}. In turn, this induces isomorphisms

ϕ_{w} : F^{∗}K_{w} → K_{w}  ϕ_{w}^{∨} : F^{∗}K_{w}^{∨} → K_{w}^{∨}  ϕ̃_{w} : F^{∗}K̃_{w} → K̃_{w},

where ϕ_{w}^{∨} is the contragradient of ϕ_{w} (see [Tay14, 6.12]).

Remark 10.7. In light of Corollary 9.7 and 10.2 we would like to recall the following property of Lusztig’s preferred extension as observed by Digne�Lehrer�Michel in [DLM03, Remark 3.6]. For any pair ι ∈ ℐ [L, ι_{0}]^{F} there exists a unique sign ε_{ι}∈ {±1} such that

Tr(wF, Eι̂) = ε_{ι}sgn(w) Tr(wF, E_{ι})

for all w ∈ W_{G}(L). Note that here we are using the characterisation of the map ι ↦ ι̂given in Corollary 9.7.

Lusztig’s Algorithm

10.8. We now de�ne functions Y_{ι}, X_{ι}, X̃_{ι}: 𝔤^{F} → \overline{}_{ℓ} by setting

Y_{ι}(y)= Tr(φ_{ι}, (ℰ_{ι})_{y})
X_{ι}(x) = ∑_{i∈} (−1)^{i} Tr(ϕ_{ι}, ℋ_{x}^{i}(K_{ι}))
X̃_{ι}(x) = ∑_{i∈} (−1)^{i} Tr(ϕ_{ι}^{−1} , ℋ_{x}^{i}(K_{ι}))

if y ∈ O_{ι}^{F}, x ∈ \overline{O_{ι}}^{F} and 0 otherwise. For any x ∈ 𝔤 we denote by A_{G}(x) the component group C_{G}(x)/C_{G}^{∘}(x) of the centraliser (under the Ad G action). With this notation we have the following lemma.

Lemma 10.9. Let y = y_{1}, …, y_{m} be a set of representatives for the orbits of G acting on O_{ι}^{F} then the following orthogonality relations hold:

\begin{aligned}∑_{i=1}^{m} [A_{G}(y) : A_{G}(y_{i})^{F}]Y_{ι}(y_{i})\overline{Y_{ι′}(y_{i})} & = |A_{G}(y)|δ_{ι′,ι} \\ ∑_{\begin{matrix}ι′∈V_{G} \\ O_{ι′}=O_{ι} \end{matrix}} Y_{ι′}(y_{i})\overline{Y_{ι′}(y_{j})} & = |A_{G}(y_{i})^{F}|δ_{i,j}. \end{aligned}

Here δ_{ι′,ι}and δ_{i,j} denote the Kronecker delta.

Proof. Let y ∈ O_{ι}^{F} be a �xed class representative and denote by H^{1}(F, A_{G}(y)) the F-conjugacy classes of the component group. Then we may realise the set of representatives {y_{1}, …, y_{m}} as the set of y_{a} = gyg^{−1} for every a ∈ H^{1}(F, A_{G}(y)), where g^{−1} F(g) ∈ C_{G}(y) is a representative of a ∈ A_{G}(y). From the de�nition we see that A_{G}(y_{a})^{F} is naturally isomorphic to the F-centraliser C_{A_{G}(y),F}(a) = {b ∈ A_{G}(y) | b^{−1} aF(b) = a}. Let us denote by χ_{ι}∈ Irr(A_{G}(u)) the irreducible character corresponding to the local system ℰ_{ι}. Then χ_{ι}is F-stable (as ℰ_{ι}is F-stable) and we can choose an extension χ̃_{ι} to the semidirect product A_{G}(y) ⋊ ⟨F⟩. According to [Sho06, 1.3] there exists a scalar ξ_{ι}∈ \overline{}_{ℓ}^{×} such that

Y_{ι}(y_{a}) = ξ_{ι}χ̃_{ι}(aF)

for all a ∈ A_{G}(y). In fact, as both Y_{ι}(y_{a}) and χ̃_{ι}(aF) have �nite order we must have ξ_{ι} is a root of unity so \overline{ξ_{ι}} = ξ_{ι}^{−1} . The result now follows from the orthogonality relations of cosets (see [DM85, II, Corollaire 2.10]). ■

10.10. The set of functions {Y_{ι} | ι ∈ V_{G}^{F}} forms a basis for the space Cent_{nil}(𝔤^{F}) of Ad G-invariant functions 𝔤^{F} → \overline{}_{ℓ} which are supported on N^{F} (see [Lus85, 24.2.7]). In particular, for any two pairs ι, ι′∈ V_{G}^{F} there exists a scalar P_{ι′,ι}∈ \overline{}_{ℓ} such that

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

By the de�nition of the functions X_{ι}and Y_{ι′}we see that

P_{ι′,ι}= \{ \begin{matrix}1 & if ι′= ι, \\ 0 & if O_{ι′}⊈ \overline{O_{ι}} or if O_{ι′}= O_{ι}and ι ≠ ι′. \end{matrix}  (10.11)

for any ι′, ι ∈ V_{G}^{F} (see also [Lus84b, 24.2.10, 24.2.11]). We now also de�ne scalars λ_{ι′,ι}∈ \overline{}_{ℓ} by setting

λ_{ι′,ι}= λ_{ι,ι′}= ∑_{y∈N^{F}} Y_{ι}(y)\overline{Y_{ι}(y)}.

It is clear from the de�nition that we have

λ_{ι′,ι}= 0 if O_{ι′}≠ O_{ι}.  (10.12)

10.13. For any w ∈ W_{G}(L) let L_{w} be as in 10.6 then for any ι′, ι ∈ ℐ [L, ι_{0}]^{F} we set

ω_{ι,ι′}= ω_{ι′,ι}= q^{− dim G−(a_{ι}+a_{ι′})/2} \frac{1}{|W_{G}(L)|} ∑_{w∈W_{G}(L)} Tr(wF, E_{ι}) Tr(wF, E_{ι′})\frac{|G^{F}|}{|Z^{∘}(L_{w})^{F}|} .

Here Z^{∘}(L_{w}) denotes the connected centre of L_{w} and F is acting on E_{ι}via the isomorphism ψ_{ι}^{−1} , similarly for E_{ι′}(c.f. 10.2). If either ι or ι′are not contained in the block ℐ [L, ι_{0}] then we set ω_{ι,ι′}= 0. Our de�nition of the term ω_{ι,ι′}is slightly di�erent to that given in [Lus85, 24.3.4]. However, one sees that these de�nitions are equivalent by [Lus85, 24.2.1]. With this we have the following theorem of Lusztig.

Theorem 10.14 (Lusztig, [Lus85, Theorem 24.4]). Let us denote by P, Λ and Ω the matrices (P_{ι′,ι}), (λ_{ι′,ι}) and (ω_{ι′,ι}) respectively. Then the entries of the matrices P and Λ are the unique solution to the system of equations given by PΛP^{T} = Ω, (10.11) and (10.12). Furthermore, we have:

(i) P, Λ and Ω are non-singular integer valued matrices,

(ii) P_{ι′,ι}= λ_{ι′,ι}= 0 if ι′, ι ∈ V_{G}^{F} lie in di�erent blocks.

10.15. Raising the arithmetic Frobenius to the power n ∈ (see [Gec03, 4.1.2]) we obtain a corresponding _{q^{n}}-structure on G which is the �xed points under some Frobenius endomorphism F′ on G. Replacing q by q^{n} in the entries for P we see that we obtain the corresponding matrix for G^{F′}. In this way we may view the entries of P as polynomials in a single variable, say q. We now denote by P^{⋆} = (P_{ι′,ι}^{⋆}) the rational valued matrix obtained from P by evaluating q at q^{−1} . This matrix is such that

X̃_{ι}(y) = ∑_{ι′∈V_{G}^{F}} P_{ι′,ι}^{⋆}\overline{Y_{ι′}(y)}  (10.16)

for all ι ∈ V_{G}^{F} and y ∈ N^{F}.

10.17. Let us denote by Q = (Q_{ι′,ι}), Λ̃ = (λ̃_{ι′,ι}) and Ω̃ = (ω̃_{ι′,ι}) the inverse matrices to P, Λ and Ω respectively (c.f. Theorem 10.14). It is clear that we have

QΛ̃ Q^{T} = Ω̃  (10.18)
Y_{ι}= ∑_{ι′∈V_{G}^{F}} Q_{ι′,ι}X_{ι′}  (10.19)

for all ι ∈ V_{G}^{F}. Using the coset orthogonality relations for �nite groups we deduce that

ω̃_{ι,ι′} = q^{dim G+(a_{ι}+a_{ι′})/2} \frac{1}{|W_{G}(L)|} ∑_{w∈W_{G}(L)} Tr(wF, E_{ι}) Tr(wF, E_{ι′})\frac{|Z^{∘}(L_{w})^{F}|}{|G^{F}|} ,  (10.20)

if ι, ι′ ∈ ℐ [L, ι_{0}]^{F} (see also [DLM03, Lemma 5.1]). Clearly ω̃_{ι,ι′} = 0 if ι, ι′ ∈ V_{G}^{F} are not contained in the same block.

From this point forward we assume that −e^{†} ∈ \overline{Y}.

Formula for the Fourier Transform of the GGGR

10.21. Let L_{w} for some w ∈ W_{G}(L) be as in 10.6 then by the argument in [Lus85, 10.6, 25.6.3] (see also [Tay14, 6.15] and [Lus92, 6.9]) we have the following equalities

χ_{K_{w},ϕ_{w}}(y) = χ_{K_{0},θ_{w}∘ϕ_{0}}(y) = ∑_{ι∈ℐ [L,ι_{0}]^{F}}q^{b_{ι}}Tr(wF, E_{ι})X_{ι}(y),  (10.22)
χ_{K_{w},ϕ_{w}^{−1}} (y) = χ_{K_{0},θ_{w}∘ϕ_{0}^{−1}} (y) = ∑_{ι∈ℐ [L,ι_{0}]^{F}}q^{−b_{ι}} Tr(wF, E_{ι})X̃_{ι}(y),  (10.23)
χ_{K_{w},ϕ_{w}^{∨}}(y) = χ_{K_{0},θ_{w}∘ϕ_{0}^{∨}} (y) = ∑_{ι∈ℐ [L,ι_{0}]^{F}}q^{b_{ι}}Tr(wF, E_{ι})\overline{X_{ι}(y)},  (10.24)

for any y ∈ (\overline{Y}∩ N)^{F}.

10.25. By Grothendieck’s trace formula we have

∑_{i∈} (−1)^{i} Tr(ϕ̃_{w}^{∨}, H_{c}^{i}(\overline{Y}∩ N ∩ Σ, K̃_{w}^{∨})) = ∑_{x∈(\overline{Y}∩N ∩Σ)^{F}} ∑_{i∈} (−1)^{i} Tr(ϕ_{w}^{∨}, ℋ_{x}^{i}(K_{w}^{∨})).  (10.26)

Using (7.11) we get

Tr(ϕ̃_{w}^{∨}, H_{c}^{2d_{ι_{0}}−i} (\overline{Y}∩ N ∩ Σ, K̃_{w}^{∨})) = q^{d_{ι_{0}}}Tr(ϕ_{w}^{−1} , ℋ_{−e^{†}}^{i} (K_{w}))  (10.27)

where d_{ι_{0}} is the integer in (iii) of Lemma 7.5. Now, combining (10.26) and (10.27) we obtain

q^{d_{ι_{0}}} ∑_{i∈} (−1)^{i} Tr(ϕ_{w}^{−1} , ℋ_{−e^{†}}^{i} (K_{w})) = ∑_{x∈(\overline{Y}∩N ∩Σ)^{F}} ∑_{i∈} (−1)^{i} Tr(ϕ_{w}^{∨}, ℋ_{x}^{i}(K_{w}^{∨}))

and applying (10.23) and (10.24) to this equality we obtain

∑_{ι∈ℐ [L,ι_{0}]^{F}}q^{d_{ι_{0}}−b_{ι}/2} X̃_{ι}(−e^{†})f_{ι}= ∑_{x∈(\overline{Y}∩N ∩Σ)^{F}} ∑_{ι∈ℐ [L,ι_{0}]^{F}}q^{b_{ι}/2}\overline{X_{ι}(x)}f_{ι},

where f_{ι}(w) = Tr(wF, E_{ι}) for all w ∈ W_{G}(L). Note that the set of functions {f_{ι}| ι ∈ ℐ [L, ι_{0}]^{F}} is linearly independent hence we get

q^{d_{ι_{0}}−b_{ι}/2} X̃_{ι}(−e^{†}) = ∑_{x∈(\overline{Y}∩N ∩Σ)^{F}} q^{b_{ι}/2}\overline{X_{ι}(x)}.

Rewriting the X_{ι}’s in terms of the Y_{ι}’s and conjugating by we have

q^{d_{ι_{0}}−b_{ι}} ∑_{ι′∈V_{G}^{F}} P_{ι′,ι}^{⋆}Y_{ι′}(−e^{†}) = ∑_{x∈(\overline{Y}∩N ∩Σ)^{F}} ∑_{ι′∈V_{G}^{F}} P_{ι′,ι}Y_{ι′}(x).  (10.28)

(note that P_{ι′,ι}^{⋆}and P_{ι′,ι}are both rational and neither sum on the right hand side depends on ι).

Finally, as Q is the inverse to P we have ∑ _{ι∈ℐ [L,ι_{0}]^{F}} P_{ι′,ι}Q_{ι,ι′′}= δ_{ι′,ι′′}(the Kronecker delta) by (10.11) for any ι′∈ V_{G}^{F} and ι′′∈ ℐ [L, ι_{0}]^{F}. Thus, multiplying both sides of (10.28) by ∑ _{ι∈ℐ [L,ι_{0}]^{F}} Q_{ι,ι′′}we obtain

∑_{x∈(\overline{Y}∩N ∩Σ)^{F}} Y_{ι′′}(x) = ∑_{ι,ι′∈ℐ [L,ι_{0}]^{F}}q^{d_{ι_{0}}−b_{ι}}P_{ι′,ι}^{⋆}Q_{ι,ι′′}Y_{ι′}(−e^{†}),  (10.29)

which holds for any ι′′∈ ℐ [L, ι_{0}]^{F}. As is pointed out in [Lus92, 6.9] we use here the assumption that −e^{†} ∈ \overline{Y} but in fact this equality holds regardless as both sides are zero when −e^{†} ∉ \overline{Y}.

10.30. By 10.10 we see there exist unique scalars α_{ι_{1}}, β_{ι_{1}} ∈ \overline{}_{ℓ}, for ι_{1} ∈ V_{G}^{F}, such that

F(Γ_{u})|_{N^{F}} = ∑_{ι_{1}∈V_{G}^{F}} α_{ι_{1}}Y_{ι_{1}} = ∑_{ι_{1}∈V_{G}^{F}} β_{ι_{1}}X_{ι_{1}}.

Our goal is to now try and determine these scalars.

Multiplying F(Γ_{u})|_{N^{F}} by the complex conjugate of Y_{ι_{2}} and summing over N^{F} (then inverting the matrix Λ) we get

α_{ι_{1}} = ∑_{ι_{2}∈ℐ [L,ι_{0}]^{F}} λ̃_{ι_{2},ι_{1}} ∑_{x∈N^{F}} Γ̂_{u}(x)\overline{Y_{ι_{2}}(x)}

for any ι_{1} ∈ V_{G}^{F} (c.f. 10.10). Applying Proposition 6.11 we may rewrite this as

α_{ι_{1}} = ∑_{ι_{2}∈ℐ [L,ι_{0}]^{F}} λ̃_{ι_{2},ι_{1}} ∑_{x∈N^{F}} q^{r_{u}}|\{g ∈ G | Ad g(x) ∈ Σ\}|\overline{Y_{ι_{2}}(x)}

= q^{r_{u}}|G| ∑_{ι_{2}∈ℐ [L,ι_{0}]^{F}} λ̃_{ι_{2},ι_{1}} ∑_{x∈N^{F}∩Σ} \overline{Y_{ι_{2}}(x)}

where r_{u} is as in Proposition 6.11. Now if λ̃_{ι_{2},ι_{1}} ≠ 0 then we must have ι_{1} and ι_{2} are in the same block but if this is the case then Y_{ι_{2}}(x) ≠ 0 implies x ∈ \overline{Y}. So the right most sum can be taken over (\overline{Y}∩ N ∩ Σ)^{F}. Thus, using (10.29) we get

α_{ι_{1}} = |G| ∑_{ι_{2},ι,ι′∈ℐ [L,ι_{0}]^{F}}q^{d_{ι_{0}}−b_{ι}+r_{u}} λ̃_{ι_{2},ι_{1}}P_{ι′,ι}^{⋆}Q_{ι,ι_{2}}\overline{Y_{ι′}(−e^{†})}

Note that the scalars α_{∗} and β_{∗} are related via the equation β_{ι_{1}} = ∑ _{ι′_{1}∈ℐ [L,ι_{0}]^{F}} Q_{ι_{1},ι′_{1}}α_{ι′_{1}} so applying this to the above expression for α_{ι′_{1}} we obtain

β_{ι_{1}} = |G| ∑_{ι,ι′∈ℐ [L,ι_{0}]^{F}}q^{d_{ι_{0}}−b_{ι}+r_{u}} P_{ι′,ι}^{⋆}\overline{Y_{ι′}(−e^{†})} ∑_{ι′_{1},ι_{2}∈ℐ [L,ι_{0}]^{F}}Q_{ι,ι_{2}}λ̃_{ι_{2},ι′_{1}}Q_{ι_{1},ι′_{1}}

= |G| ∑_{ι,ι′∈ℐ [L,ι_{0}]^{F}}q^{d_{ι_{0}}−b_{ι}+r_{u}} P_{ι′,ι}^{⋆}ω̃_{ι,ι_{1}}\overline{Y_{ι′}(−e^{†})}

where in the second equality we have used (10.18). Finally, using (10.20) we have the following.

Proposition 10.31. We de�ne Γ̂_{u,ℐ [L,ι_{0}]} : N^{F} → \overline{}_{ℓ} to be the function

∑_{ι,ι′,ι_{1}∈ℐ [L,ι_{0}]^{F}}q^{f(ι,ι_{1})}\frac{1}{|W_{G}(L)|} ∑_{w∈W_{G}(L)} Tr(wF, E_{ι}) Tr(wF, E_{ι_{1}})|Z^{∘}(L_{w})^{F}|P_{ι′,ι}^{⋆}\overline{Y_{ι′}(−e^{†})} X_{ι_{1}}

where

f(ι, ι_{1})= d_{ι_{0}}− b_{ι}+ r_{u} + dim G + (a_{ι}+ a_{ι_{1}})/2
= dim G − dim Z^{∘}(L) + (dim O_{ι}− dim O_{ι_{1}}− dim O_{u})/2.

Then we have

F(Γ_{u})|_{N^{F}} = ∑_{[L,ι_{0}]∈W_{G}^{F}} Γ̂_{u,ℐ [L,ι_{0}]},

where the sum is taken over all F-stable blocks.

11. A Decomposition of Γ_{u}

11.1. By the de�nition of the bijection ˆ : ℐ [L, ι_{0}] → ℐ [L, ι_{0}] and [Let05, 5.2.3] we see that for any ι ∈ ℐ [L, ι_{0}] there exists a scalar c_{ι}∈ \overline{}_{ℓ}^{×} such that

F(X_{ι})|_{N^{F}} = c_{ι}Xι̂|_{N^{F}}.  (11.2)

For any function f : 𝔤^{F} → \overline{}_{ℓ} we will denote by f^{∗} : 𝔤^{F} → \overline{}_{ℓ} the function obtained as the extension by 0 of f|_{N^{F}}. With this in hand we have the following lemma.

Lemma 11.3. There exists a sign ν ∈ {±1} such that Ψ(χ_{K_{0},ϕ_{0}}) = νχ_{K_{0},ϕ_{0}}, where Ψ is as in Lemma 6.5and

F(χ_{K_{0},ϕ_{0}}) = γq^{(dim G+dim Z^{∘}(L))/2} χ_{K_{0},ϕ_{0}}^{∗}

where γ ∈ \overline{}_{ℓ}^{×} is such that γ^{2} = ν (hence γ^{4} = 1). The constant γ does not depend upon the choice of isomorphism ϕ_{0} and we have γ = ν = 1 if L is a torus. Furthermore, assume Lemma 9.6holds then we have

F(χ_{K_{0},θ_{w}∘ϕ_{0}}) = γ sgn(w)q^{(dim G+dim Z^{∘}(L))/2} χ_{K_{0},θ_{w}∘ϕ_{0}}^{∗}  (11.4)

for any w ∈ W_{G}(L).

Proof. All parts follow from the proof of [Let05, 6.2.9] together with [Let05, 4.4.6, 5.2.3, 5.2.8] and [Let05, 6.2.8, 6.2.12, 6.2.15]. One only has to take into consideration that our de�nitions of the Fourier and Deligne�Fourier transforms are slightly di�erent to those used in [Let05]. ■

Proposition 11.5. Assume either that F is split or that the conclusion of Lemma 9.6holds then there exists a fourth root of unity ζ ∈ \overline{}_{ℓ}^{×} such that

c_{ι}= ε_{ι}ζq^{(dim G−dim Z^{∘}(L)−dim Oι̂+dim O_{ι})/2} ,

where ε_{ι}is as in Remark 10.7. In particular, we have ζ_{ℐ} = ζ depends only on the block ℐ = ℐ [L, ι_{0}] and not on ι itself. Furthermore, if L is a torus then ζ_{ℐ} = 1.

Proof. We are going to use Lemma 11.3 to determine the scalar c_{ι}. Applying F^{−1} ∘ Ψ (c.f. Lemma 6.5) to the equality (11.4) we have

F(χ_{K_{0},θ_{w}∘ϕ_{0}}^{∗}) = νγ^{−1} sgn(w)q^{(dim G−dim Z^{∘}(L))/2} χ_{K_{0},θ_{w}∘ϕ_{0}}.  (11.6)

Now let us consider the restriction of the equality (11.6) to N^{F} then applying (10.22) we obtain

∑_{ι∈ℐ [L,ι_{0}]^{F}}q^{b_{ι}/2} Tr(wF, E_{ι})F(X_{ι})|_{N^{F}} = νγ^{−1} q^{(dim G−dim Z^{∘}(L))/2} ∑_{ι∈ℐ [L,ι_{0}]^{F}}q^{b_{ι}/2} sgn(w) Tr(wF, E_{ι})X_{ι}|_{N^{F}}, (11.7)

for any w ∈ W_{G}(L). Using (11.2) and the change of variable ι ↦ ι̂this becomes

∑_{ι∈ℐ [L,ι_{0}]^{F}}q^{b_{ι}/2} Tr(wF, E_{ι})c_{ι}Xι̂|_{N^{F}} = νγ^{−1} q^{(dim G−dim Z^{∘}(L))/2} ∑_{ι∈ℐ [L,ι_{0}]^{F}}q^{bι̂/2} sgn(w) Tr(wF, Eι̂)Xι̂|_{N^{F}}. (11.8)

The set of functions {Xι̂|_{N^{F}} | ι ∈ ℐ [L, ι_{0}]^{F}} is linearly independent. Thus, using Remark 10.7 we get that

q^{b_{ι}/2} Tr(wF, E_{ι})c_{ι}= ε_{ι}νγ^{−1} q^{(bι̂+dim G−dim Z^{∘}(L))/2} Tr(wF, E_{ι})

Both sides of this equation may be 0 as Tr(wF, E_{ι}) could be 0. However, choosing a w ∈ W_{G}(L) such that Tr(wF, E_{ι}) ≠ 0 we deduce the result with ζ = νγ^{−1} . Note that it is immediately clear from Lemma 11.3 that γ = 1 if L is a torus.

Now assume F is split then F acts trivially on W_{G}(L) and so it acts as the identity on the representation E_{ι}. In this case we may take w = 1 in the above argument as we will have Tr(wF, E_{ι}) = dim(E_{ι}) ≠ 0. When w = 1 we have the results of Lemma 11.3 hold without assuming Lemma 9.6 holds, so the statement follows. ■

11.9. We point out that Digne�Lehrer�Michel both stated and indicated how to prove Proposition 11.5 in the proof of [DLM03, Proposition 6.1]. We now come to the following main result of this article, which is due to Lusztig when p is large enough so that log and exp de�ne inverse bijections between N and U.

Theorem 11.10. Let Γ_{u,ℐ [L,ι_{0}]} : 𝔤^{F} → \overline{}_{ℓ} denote the function

ζ_{ℐ [L,ι_{0}]}^{−1} ∑_{ι,ι′,ι_{1}∈ℐ [L,ι_{0}]^{F}}q^{f′(ι,ι_{1})}\frac{1}{|W_{G}(L)|} ∑_{w∈W_{G}(L)} Tr(wF, E_{ι}) Tr(wF, Eι̂_{1})|Z^{∘}(L_{w})^{F}|P_{ι′,ι}^{⋆}\overline{Y_{ι′}(−e^{†})} ε_{ι_{1}}X_{ι_{1}}

where

f′(ι, ι_{1})= f(ι, ι_{1}) − (dim G − dim Z^{∘}(L) − dim O_{ι_{1}}+ dim O_{ι_{1}})/2

= (dim G − dim Z^{∘}(L) + dim O_{ι}− dim O_{ι_{1}}− dim O_{u})/2.

Then we have

Γ_{u} = ∑_{[L,ι_{0}]∈W_{G}^{F}}Γ_{u,ℐ [L,ι_{0}]},  (11.11)

where the sum is taken over all F-stable blocks.

Proof. If F is split then this is proved in exactly the same way as [Lus92, Theorem 7.3]. Once the theorem is deduced when F is split we may prove Lemma 9.6 as in [Lus92, Theorem 7.7]. Finally, now that Lemma 9.6 is proved we have Proposition 11.5 holds so we may again apply the proof in [Lus92, Theorem 7.3] to deduce the theorem when F is twisted. ■

11.12. We now wish to transfer Theorem 11.10 to a statement about the GGGR Γ_{u}. Let V_{G}^{uni} denote the set of pairs (O, ℰ) consisting of a unipotent conjugacy class of G and an irreducible local system on O. It is easy to see that the Springer isomorphism induces a bijection V_{G}^{nil} → V_{G}^{uni} given by (O, ℰ) ↦ (ϕ_{spr}^{−1}(O), ϕ_{spr}^{∗}ℰ) (this is independent of the choice of ϕ_{spr} by Proposition 3.4). Assume now that A ∈ 𝒟N is an F-stable complex and let γ : F^{∗}A → A be an isomorphism. We then have ϕ_{spr}^{∗}A ∈ 𝒟U is a complex and ϕ_{spr}^{∗}γ de�nes an isomorphism

F^{∗}ϕ_{spr}^{∗}A = ϕ_{spr}^{∗}F^{∗}A → ϕ_{spr}^{∗}A

because F and ϕ_{spr} commute. This has the following a�ect at the level of characteristic functions

χ_{ϕ_{spr}^{∗}A,ϕ_{spr}^{∗}γ} = χ_{A,γ} ∘ ϕ_{spr}.

Applying this to the functions X_{ι} and Y_{ι} we may easily translate the statement in Theorem 11.10 to the following statement about Γ_{u}. For ease, we simply write X_{ι} for the unipotently supported class function corresponding to the nilpotently supported function.

Theorem 11.13. Recall that G is any connected reductive algebraic group with Frobenius endomorphism F : G → G and p is an acceptable prime for G. Let Γ_{u,ℐ} : G → \overline{}_{ℓ} denote the function

ζ_{ℐ}^{−1} ∑_{ι,ι′,ι_{1}∈ℐ [L,ι_{0}]^{F}}q^{f′(ι,ι_{1})}\frac{1}{|W_{G}(L)|} ∑_{w∈W_{G}(L)} Tr(wF, E_{ι}) Tr(wF, Eι̂_{1})|Z^{∘}(L_{w})^{F}|P_{ι′,ι}^{⋆}\overline{Y_{ι′}(u^{∗})}ε_{ι_{1}}X_{ι_{1}}

where f′(ι, ι_{1}) is as in Theorem 11.10and u^{∗} = ϕ_{spr}(−e^{†}). Then

Γ_{u} = ∑_{ℐ} Γ_{u,ℐ} ,

where the sum is taken over all F-stable blocks.

11.14. In [Lus92] Lusztig slightly modi�ed the GGGRs to obtain a new basis for the space Cent_{uni}(G^{F}) of unipotently supported class functions, which tends to be more convenient than the GGGRs themselves. This is done as follows. Let O ⊆ U be an F-stable unipotent conjugacy class of G then we denote by {u_{1}, …, u_{r}} ⊆ O^{F} a set of representatives for the G^{F}-classes contained in O^{F}. We then de�ne

Γ_{ι^{∗}} = ∑_{i=1}^{r} [A_{G}(u_{i}) : A_{G}(u_{i})^{F}]Y_{ι^{⋆}}(u_{i})Γ_{u_{i}},  (11.15)

for any ι^{∗} ∈ V_{G}^{uni} with O_{ι^{∗}} = O. We now get the following expression for Γ_{ι^{∗}}.

Lemma 11.16. Recall the assumptions of Theorem 11.13. Let ℐ ⊆ V_{G}^{uni} be the block containing ι^{∗} then we have

Γ_{ι^{∗}} = ζ_{ℐ}^{−1} ∑_{ι,ι_{1}∈ℐ^{F}} q^{f′(ι,ι_{1})}\frac{|A_{G}(u)|}{|W_{G}(L)|} ∑_{w∈W_{G}(L)} Tr(wF, E_{ι}) Tr(wF, Eι̂_{1})|Z^{∘}(L_{w})^{F}|P_{ι^{∗},ι}^{⋆}ε_{ι_{1}}X_{ι_{1}}

where f′(ι, ι_{1}) is as in Theorem 11.10and u ∈ O_{ι^{∗}}.

Proof. Using Theorem 11.13 and the de�nition of Γ_{ι^{∗}} we see this follows from Proposition 5.3, the de�nition of the functions Y_{ι}and the orthogonality relations given in Lemma 10.9. ■

12. Weyl Groups and Unipotent Classes

From now until the end of this article we will only assume that p is a good prime for G and not an acceptable prime for G.

12.1. We will denote by S(T_{0}) the set of (isomorphism classes of) tame local systems on T_{0}. Assume now that ℒ ∈ S(T_{0}) is such a local system then we denote by W_{G}(ℒ) the quotient group N_{G}(T_{0}, ℒ)/T_{0} where

N_{G}(T_{0}, ℒ) = {x ∈ N_{G}(T_{0}) | Innx^{∗}(ℒ) ≅ ℒ }.

If we wish to emphasise the torus then we will write W_{G}(T_{0}, ℒ) instead of W_{G}(ℒ). For any root α ∈ Φ we denote by s_{α} ∈ W_{G}(T_{0}) the corresponding re�ection then following [Lus85, 2.3] we set

Φ_{ℒ} = {α ∈ Φ | s_{α} ∈ W_{G}(ℒ)}.

This is a root system and W_{G}^{∘}(ℒ) = ⟨s_{α} | α ∈ Φ_{ℒ} ⟩ is the corresponding re�ection group. In fact, this is a large normal subgroup of W_{G}(ℒ) (which can of course be the whole of W_{G}(ℒ)). Setting Φ_{ℒ}^{+} = Φ_{ℒ} ∩ Φ^{+} we get a system of positive roots for Φ_{ℒ} which determines a corresponding set of Coxeter generators _{ℒ} for W_{G}^{∘}(ℒ). Denoting by A_{G}(ℒ) the group {w ∈ W_{G}(ℒ) | w _{ℒ} w^{−1} = _{ℒ} } we obtain a semidirect product decomposition W_{G}(ℒ) = W_{G}^{∘}(ℒ) ⋊ A_{G}(ℒ). Let us now set W = W_{G}(T_{0}), H = W_{G}(ℒ), H^{∘} = W_{G}^{∘}(ℒ) and Ω = A_{G}(ℒ) then following [Lus85, 16.3] we make the following de�nition.

De�nition 12.2. A subset ℭ ⊆ H is called a two-sided cell if there exists a two-sided cell ℭ^{∘} ⊆ H^{∘} (de�ned as in [Lus84a, §5.1]) such that ℭ = Ωℭ^{∘}Ω.

12.3. The two-sided cells of H are naturally in bijection with the Ω-orbits of the two-sided cells of H^{∘}. In particular, we have H is a disjoint union of its two-sided cells. Using the leading coe�cients of representations of the corresponding extended Hecke algebra Lusztig associates to each two-sided cell ℭ ⊆ H a subset of irreducible representations Irr(H | ℭ) of H (see [Lus92, 10.4]). We will call these subsets families.

In each family he identi�es a unique irreducible representation E_{ℭ} ∈ Irr(H | ℭ), called a special representation, which has the following property. Let {ℭ_{1}^{∘}, …, ℭ_{r}^{∘}} be the Ω-orbit of two-sided cells of H^{∘} corresponding to ℭ then

Res_{H^{∘}}^{H}(E_{ℭ}) = E_{ℭ_{1}^{∘}} + ⋯+ E_{ℭ_{r}^{∘}},  (12.4)

where E_{ℭ_{i}^{∘}} ∈ Irr(H^{∘}) is the unique special representation corresponding to ℭ_{i}^{∘} (in the sense of [Lus84a, (4.1.9)]).

We denote by sgn the sign representation of W and also its restriction to any subgroup of W. Now, given any two-sided cell ℭ of H we denote by ℭ^{∗} the unique two-sided cell of H satisfying the condition:

• For any representation E ∈ Irr(H | ℭ) we have E ⊗ sgn ∈ Irr(H | ℭ^{∗}). This map ℭ → ℭ^{∗} de�nes a permutation on the set of two-sided cells of H.

12.5. Let V be the natural module for H then for each representation E ∈ Irr(H^{∘}) we denote by b_{E} the minimal i ∈ such that E occurs in the ith symmetric power S^{i}V of V. Note that taking H^{∘} = H = W in this de�nition we again obtain an integer b_{E} for every E ∈ Irr(W). With this in hand we have the Lusztig� Macdonald�Spaltenstein induction map j_{H^{∘}}^{W} : Irr(H^{∘}) → Cent(W), which is de�ned in the following way. Let E ∈ Irr(H^{∘}) be an irreducible representation then we set

j_{H^{∘}}^{W}(E) = ∑_{\begin{matrix}E′∈Irr(W) \\ b_{E}=b_{E′} \end{matrix}}⟨Ind_{H^{∘}}^{W}(E), E′⟩E′.

12.6. According to [Lus09, 1.3] we have for each two sided cell ℭ^{∘} ⊆ H^{∘} that the j-induced representation j_{H^{∘}}^{W}(E_{ℭ^{∘}}) is irreducible. Furthermore, by [Lus09, 1.5(a)] we see that j_{H^{∘}}^{W}(E_{ℭ^{∘}}) corresponds under the Springer correspondence to some E_{ι}∈ Irr(W) with ι = (O, \overline{}_{ℓ}) ∈ ℐ [T_{0}, \overline{}_{ℓ}] ⊆ V_{G}^{uni}. We denote by O_{H^{∘},ℭ^{∘}} the unipotent class O. Now assume ℭ ⊆ H is a two-sided cell of H and ℭ^{∘} ⊆ H^{∘} is a two-sided cell such that ℭ = Ωℭ^{∘}Ω then we denote by O_{ℒ,ℭ} the class O_{H^{∘},ℭ^{∘}} (recall that H = W_{G}(ℒ)). As in [Lus92] we have de�ned a map

{two-sided cells of H} → {unipotent conjugacy classes of G}
ℭ ↦ O_{ℒ,ℭ} .

(12.7)

Note that, by (12.4), we see that this map is well de�ned as O_{ℒ,ℭ} does not depend upon the choice of two-sided cell ℭ^{∘} ⊆ H^{∘} satisfying ℭ = Ωℭ^{∘}Ω.

13. Unipotent Supports for Character Sheaves

13.1. For any local system ℒ ∈ S(T_{0}) Lusztig has de�ned in [Lus85, 2.10] a set of (isomorphism classes of) perverse sheaves Ĝ_{ℒ} on G. The de�nition of this set depends only on the W_{G}(T_{0})-orbit of ℒ. With this we have

Ĝ = ⨆_{ℒ ∈S(T_{0})/W_{G}(T_{0})} Ĝ_{ℒ}

is the set of character sheaves of G, where the union is over the W_{G}(T_{0})-orbits of tame local systems on T_{0}. The set Ĝ_{ℒ} is then further partitioned into families

Ĝ_{ℒ} = ⨆_{ℭ⊆W_{G}(ℒ)} Ĝ_{ℒ,ℭ}

where the union runs over all the two-sided cells of W_{G}(ℒ) (see [Lus85, Corollary 16.7]).

De�nition 13.2. For any conjugacy class C of G let C_{uni} be the set of all unipotent elements occuring in the Jordan decomposition of some element in C (it is a unipotent conjugacy class of G). If A ∈ Ĝ_{ℒ,ℭ} is a character sheaf then we say O ∈ U/G is a unipotent support of A if following properties hold:

(i) For any conjugacy class C of G we have A|_{C} ≠ 0 implies that either dim C_{uni} < dim O or C_{uni} = O.

(ii) There exists a conjugacy class C of G and a character sheaf A′ ∈ Ĝ_{ℒ,ℭ} such that C_{uni} = O and A′|_{C} ≠ 0.

13.3. It is clear that every character sheaf has a unipotent support. What we would like to now show is that every character sheaf has a unique unipotent support in good characteristic. Before doing so we recall the following result of Shoji.

Theorem 13.4 (Shoji, [Sho96, Theorem 4.2]). Assume that p is good for G and that Z(G) is connected then the results of [Lus90] are true without restriction on q.

13.5. We will denote by q_{0}(G) ⩾ 1 a constant (as in [Lus90]) such that if q > q_{0}(G) the results of [Lus90] are true. By Theorem 13.4 we may set q_{0}(G) = 1 if Z(G) is connected. Now, combining Theorem 11.10 with a careful reading of the remaining parts of [Lus92] we see that the following is true.

Corollary 13.6. Assume that p is an acceptable prime for G and q > q_{0}(G) (recall that this means no assumption on q if Z(G) is connected). Then after replacing log and exp by ϕ_{spr} and ϕ_{spr}^{−1} we see that all the results of [Lus92, §8 � §11] are true in our more general situation.

13.7. With this in hand we may prove the following theorem, which is due to Lusztig assuming that p is su�ciently large (c.f. 11.9) and Aubert in certain special cases when p is good (see [Lus92] and [Aub03]).

Theorem 13.8. Assume p is a good prime for G then every character sheaf A ∈ Ĝ has a unique unipotent support denoted by O_{A}. Furthermore, if A is contained in the family Ĝ_{ℒ,ℭ} then O_{A} = O_{ℒ,ℭ} .

13.9. Our strategy for proving Theorem 13.8 will be to reduce the problem to the case where G is simple and simply connected. If G is not of type A_{n} then any good prime is an acceptable prime for G. In particular, by Corollary 13.6 we have Theorem 13.8 follows from [Lus92, Theorem 10.7] in this case. However, if G is of type A_{n} then we require special arguments to obtain Theorem 13.8. Our idea for this case is to characterise the restriction of character sheaves to conjugacy classes in terms of combinatorial data which is (in a suitable sense) independent of p. We then use the validity of the result in the case where p is large to deduce the case of small p.

Proposition 13.10. Assume G is simple and simply connected of type A_{n} then Theorem 13.8holds for G.

Proof. For any element z ∈ Z(G) let t_{z} : G → G be the morphism given by t_{z}(g) = zg. Now given any character sheaf A ∈ Ĝ_{ℒ,ℭ} there exists an element z ∈ Z(G) and a character sheaf A′ ∈ Ĝ_{ℒ,ℭ} such that A = t_{z}^{∗}A′ and supp(A′) ⊆ U (see [Lus85, §17.17], the proof of [Lus85, Proposition 18.5] and [Lus86, 2.9]). In particular, we have

A|_{C} = A′|_{zC}

for any conjugacy class C ⊆ G. Clearly if A|_{C} ≠ 0 then we must have C = z^{−1} C_{uni}. From this we see that a unipotent conjugacy class is a unipotent support of A′ if and only if it is a unipotent support of A. Thus we may and will assume that A is such that supp(A) ⊆ U.

Now, there exists a parabolic subgroup Q ⩽ G and Levi complement M ⩽ Q such that M supports a cuspidal character sheaf A_{0} ∈ M̂ and A is a summand of the induced complex ind_{M⊆Q}^{G} (A_{0}) (see [Lus85, Theorem 4.4(a)]). Replacing A by an isomorphic character sheaf we may assume that M ⩽ Q are standard in the sense that T_{0} ⩽ B_{0} ⩽ Q and T_{0} ⩽ M. Furthermore, we must have that A_{0} ∈ M̂_{ℒ} and supp(A_{0}) ⊆ U ∩ M (see [Lus85, Proposition 4.8(b)] and [Lus86, 2.9]). In fact, we have

A_{0} = IC(\overline{O_{0}}Z^{∘}(M), ℰ_{0} ℱ)[dim O_{0} + dim Z^{∘}(M)]  (13.11)

where ι_{0} = (O_{0}, ℰ_{0}) ∈ V_{M}^{uni} is a cuspidal pair and ℱ ∈ S(Z^{∘}(M)) is a tame local system.

Replacing ℱ by \overline{}_{ℓ} in (13.11) we obtain a new cuspidal unipotently supported character sheaf which we denote by A_{1}. According to [Lus86, 2.4] there is a canonical parameterisation of the simple summands of ind_{M⊆Q}^{G} (A_{0}) (resp. ind_{M⊆Q}^{G} (A_{1})) by the set of simple modules of \overline{}_{ℓ}W_{G}(M, ℱ) (resp. \overline{}_{ℓ}W_{G}(M)). Note these groups are de�ned using the obvious generalisation of the construction in 12.1. Let us assume A = A_{E} is parameterised by E ∈ Irr(W_{G}(M, ℱ)) then according to [Lus86, 2.4] we have

A_{E}|_{U} ≅ ⨁_{ι∈ℐ [M,ι_{0}]}(K_{ι}|_{U})^{⊕m_{E,ι}}.  (13.12)

Here E_{ι} ∈ Irr(W_{G}(M)) is the simple module corresponding to ι under the generalised Springer correspondence, K_{ι} is the summand of ind_{M⊆Q}^{G} (A_{1}) parameterised by E_{ι} and m_{E,ι}= ⟨Ind_{W_{G}(M,ℱ)}^{W_{G}(M)} (E), E_{ι}⟩. The K_{ι}described here are the group analogues of the perverse sheaves described in 8.3.

Let us denote by G′a simple simply connected algebraic group again of type A_{n} (the same n) de�ned over an algebraic closure of the �nite �eld _{p′} where p′ > n and p′ ≠ ℓ (i.e. p′ is a very good prime for G′). We �x a maximal torus and Borel subgroup T′_{0} ⩽ B′_{0} ⩽ G′ then we assume M′ ⩽ Q′ ⩽ G′ are the standard parabolic subgroup and Levi complement naturally in correspondence with M ⩽ Q (i.e. they are determined by the same set of simple roots). Note that by Lemma 6.3 and Corollary 13.6, [Lus92, Theorem 10.7] we have Theorem 13.8 holds for G′.

Recall from [LS85, §5] that the component group Z(M) = Z(M)/Z^{∘}(M) of the centre of M acts on the local system ℰ_{0} by a faithful irreducible character φ ∈ Irr(Z(M)). By this we mean that Z(M) acts on each stalk of the local system by multiplication with φ. This irreducible character characterises the cuspidal pair on M. The component group Z(M′) is isomorphic to Z(M) and �xing an isomorphism we may identify the sets of irreducible characters Irr(Z(M)) and Irr(Z(M′)). We denote by φ′∈ Irr(Z(M′)) the faithful irreducible character corresponding to φ in this way.

As for M we see that M′ then admits a cuspidal pair ι′_{0} = (O′_{0}, ℰ′_{0}) ∈ V_{M′}^{uni} such that Z(M′) acts on ℰ′_{0} by the irreducible character φ′. Let m ∈ _{⩾1} be the minimum integer such that ℱ^{⊗m} ≅ \overline{}_{ℓ} then choosing p′ > m (as well as maintaining our previous assumptions) we can �nd a tame local system ℱ′ ∈ S(Z^{∘}(M′)) so that we have W_{G}(M, ℱ) ≅ W_{G}(M, ℱ′). Using this data in (13.11) we thus obtain a cuspidal character sheaf A′_{0} ∈ M̂′. The relative Weyl groups W_{G}(M) and W_{G}(M′) are isomorphic so choosing an isomorphism we may identify the sets of irreducible characters. Assume E′ ∈ Irr(W_{G}(M′)) corresponds to E ∈ Irr(W_{G}(M)) in this way then we have a corresponding summand A_{E′} of the induced complex ind_{M′⊆Q′}^{G′} (A′_{0}). Note that, as above we may assume A′_{0} ∈ Ĝ_{ℒ′,ℭ′} where ℒ′ ∈ S(T′_{0}) is a tame local system such that W_{G}(T_{0}, ℒ) ≅ W_{G′}(T′_{0}, ℒ′) and the two-sided cell ℭ′is identi�ed with ℭ.

We are now in a position to prove the proposition. We will identify the unipotent classes of G and G′ in the obvious way. Let Õ_{A} be a unipotent conjugacy class such that A_{E}|Õ_{A} ≠ 0 and Õ_{A} has maximal dimension amongst all classes with this property. By (13.12), we see that any such class is obtained as a class O_{ι}where ι ∈ ℐ [M, ι_{0}] satis�es the property

m_{E,ι′}≠ 0 ⇒ dim O_{ι′}⩽ dim O_{ι}.

In particular, we see that the restriction of this character sheaf to Õ_{A} is characterised in terms of data which is (in a suitable sense) independent of p. Now applying this argument in G′ we see that A_{E′}|Õ_{A} ≠ 0. As Theorem 13.8 holds in G′we can deduce that either dim Õ_{A} < dim O_{ℒ,ℭ} or Õ_{A} = O_{ℒ,ℭ} .

To �nish the argument it remains to �nd a character sheaf in Ĝ_{ℒ,ℭ} whose restriction to O_{ℒ,ℭ} is nonzero. In the proof of [Lus92, Theorem 10.7] Lusztig constructs such a character sheaf for G′ which is obtained as above with M′= T′_{0} and ι′_{0} = (\{1\}, \overline{}_{ℓ}). Thus, applying the same style of argument as before we see that such a character sheaf exists in Ĝ_{ℒ,ℭ} . ■

Proof (of Theorem 13.8). Let π : G → \overline{G} = G/Z^{∘}(G) and σ : G → G/G_{der} be the canonical quotient maps. We will denote by \overline{T}_{0} the image of T_{0} by π. Any local system ℒ ∈ S(T_{0}) is of the form ℱ ⊗ ℰ where ℱ is the inverse image under π of a local system ℱ′ ∈ S(\overline{T}_{0}) and ℰ is the inverse image under σ of a local system ℰ_{0} ∈ S(G/G_{der}). Note that here we consider ℰ as a local system on T_{0} by restriction.

Assume A ∈ Ĝ is contained in the series Ĝ_{ℒ} then A may be written as π^{∗}(\overline{A}) ⊗ ℰ with \overline{A} ∈ ˆ \overline{G}_{ℱ′} . Note that π induces a natural isomorphism W_{G}(T_{0}, ℒ) → W_{G_{ad}}(\overline{T}_{0}, ℱ′), in particular we may identify the two-sided cells of these groups. In this way we see that A ∈ Ĝ_{ℒ,ℭ} if and only if \overline{A} ∈ ˆ \overline{G}_{ℱ′,ℭ} . From this description it is clear that we have

A|_{C} ≠ 0 ⇔ \overline{A}|_{π(C)} ≠ 0.

for any conjugacy class C of G. In particular, we may clearly assume that G is semisimple.

If G is semisimple then there exists a simply connected cover π : G_{sc} → G and we have the following reduction arguments (see [Lus85, §17.16]). Let T_{sc} ⩽ G_{sc} be the unique maximal torus satisfying π(T_{sc}) = T_{0}. Assume A ∈ Ĝ_{ℒ} then we set ℒ′ ∈ S(T_{sc}) to be the inverse image of ℒ under π. Note that, identifying W_{G}(T_{0}) and W_{G_{sc}}(T_{sc}) under the natural isomorphism we have W_{G}^{∘}(ℒ) = W_{G_{sc}}^{∘}(ℒ′) and W_{G}(ℒ) ⊆ W_{G_{sc}}(ℒ′). Furthermore, for any character sheaf A ∈ Ĝ_{ℒ,ℭ} we have π^{∗}A = A_{1} ⊕ ⋯⊕ A_{r} is a direct sum of character sheaves with A_{i} ∈ Ĝ_{sc,ℒ′,ℭ′} where ℭ′ ⊆ W_{G_{sc}}(ℒ′) is the unique two-sided cell containing ℭ. From this it is clear that we may assume G is simply connected.

Finally assume G is simply connected then by [Lus85, §17.11] we may assume that G is simple and simply connected. The result now follows from Proposition 13.10 and the remarks in 13.9. ■

14. Wave Front Sets for Irreducible Characters

Families of Irreducible Characters

14.1. Recall that a family of character sheaves Ĝ_{ℒ,ℭ} contains an F-stable character sheaf if and only if the W_{G}(T_{0})-orbit of (ℒ, ℭ) is F-stable. Assume this is the case then as in [Lus92, 11.1] we de�ne a corresponding set of irreducible characters

E(G, ℒ, ℭ) = {ρ ∈ Irr(G) | ⟨χ_{A,ϕ}, ρ⟩ ≠ 0 for some A ∈ Ĝ_{ℒ,ℭ}^{F} with supp(A) = G}.

Clearly this de�nition is independent of the choice of isomorphism F^{∗}A → A. We now wish to consider the relationship between this set and the usual notion of a Lusztig series.

14.2. Assume (T, ℒ) is a pair consisting of an F-stable maximal torus and a tame F-stable local system ℒ ∈ S(T)^{F}. To this pair we have a corresponding F-stable complex K_{T}^{ℒ} ∈ 𝒟G de�ned as in [Sho95, I, 1.7] (this is simply the complex obtained by inducing ℒ to G). There is a unique isomorphism φ : F^{∗}ℒ → ℒ such that the induced isomorphism over the stalk of the identity is the identity. Let χ_{ℒ} be the resulting characteristic function then we have χ_{ℒ} (1) is a positive integer. With this we have a bijection

\begin{matrix}S(T)^{F} & → Irr(T^{F}) \\ ℒ & ↦ χ_{ℒ} . \end{matrix}  (14.3)

The isomorphism φ chosen above naturally induces an isomorphism ϕ : F^{∗}K_{T}^{ℒ} → K_{T}^{ℒ} . We will denote by χ_{K_{T}^{ℒ}} the resulting characteristic function determined by ϕ. If θ corresponds to ℒ under (14.3) then [Sho95, I, Corollary 2.3] states that

χ_{K_{T}^{ℒ}} = (−1)^{dim T}R_{T}^{G}(θ),  (14.4)

where R_{T}^{G}(θ) is the corresponding Deligne�Lusztig virtual character (as de�ned in [DL76]).

14.5. If A ∈ Ĝ_{ℒ,ℭ}^{F} is such that supp(A) = G then we must have A is a constituent of a complex K_{T}^{ℒ} where both T and ℒ are F-stable (see [Lus86, 2.9] and [Lus85, 10.5]). The endomorphism algebra End(K_{T}^{ℒ} ) is isomorphic to the group algebra \overline{}_{ℓ}W_{G}(T, ℒ) and so A is indexed by an irreducible representation E ∈ Irr(W_{G}(T, ℒ))^{F}. If we choose an extension Ẽ of E to the semidirect product W_{G}(T, ℒ) ⋊ ⟨F⟩ then this determines an isomorphism ϕ_{A} : F^{∗}A → A.

For each w ∈ W_{G}(T, ℒ) we denote by T_{w} a conjugate gTg^{−1} such that g^{−1} F(g) = ẇ^{−1} where ẇ ∈ N_{G}(T) is a representative of w ∈ W_{G}(T). We also denote by ℒ_{w} the F-stable local system (Inng^{−1} )^{∗}ℒ on T_{w}. In [Lus85, 10.6] Lusztig has de�ned an isomorphism φ_{w} : F^{∗}ℒ_{w} → ℒ_{w} from the choice of isomorphism φ : F^{∗}ℒ → ℒ. By [Bon04, Corollary 6.9] we see that φ_{w} is the canonical isomorphism considered in 14.2. Thus, [Lus85, 10.4.5, 10.6.1] together with (14.4) and the previous remark shows that

χ_{A,ϕ_{A}} = (−1)^{dim T} \frac{1}{|W_{G}(T, ℒ)|} ∑_{w∈W_{G}(T,ℒ)} Tr((wF)^{−1} , Ẽ)R_{T_{w}}^{G}(θ_{w}),  (14.6)

where θ_{w} is the character corresponding to ℒ_{w} under (14.3).

14.7. Now, let us �x a triple (G^{⋆}, T_{0}^{⋆}, F^{⋆}) dual to (G, T_{0}, F) then as in [Bon06, 18.A] we have an isomorphism T_{0}^{⋆} ≅ S(T_{0}) which respects the actions of W_{G}(T_{0}), W_{G^{⋆}}(T_{0}^{⋆}), F and F^{⋆}. Assume now that the W_{G}(T_{0})-orbit of the pair (ℒ, ℭ) is F-stable, as in 14.1, then if s ∈ T_{0}^{⋆} corresponds to ℒ ∈ S(T_{0}) under the previous isomorphism we have the G^{⋆}-conjugacy class of s is F^{⋆}-stable. In particular, the corresponding geometric Lusztig series E(G, s) is de�ned (see [Bon06, 11.A]). From the de�nitions and (14.6) we see that

E(G, ℒ, ℭ) ⊆ E(G, s).

We now claim (as in [Lus92, 11.1]) that we have a partition

E(G, s) = ⨆_{ℭ⊆W_{G}(ℒ)} E(G, ℒ, ℭ),

where the union is over all F-stable two-sided cells. If Z(G) is connected then by Shoji’s result [Sho95, II, Theorems 3.2 and 4.1] this follows from the disjointness theorem of Lusztig [Lus84a, 6.17]. Note that, in this case, the sets E(G, ℒ, ℭ) are precisely the families of characters considered in [Lus84a] (see [Lus84a, Theorem 5.25]). In general this follows from the results of [Lus88] (see for instance [DM90, Proposition 5.2]).

Wave Front Sets

De�nition 14.8. Assume ρ ∈ Irr(G) is an irreducible character of G and O is an F-stable unipotent conjugacy class of G. We say O is a wave front set for ρ if ⟨Γ_{u}, ρ⟩ ≠ 0 for some u ∈ O^{F} and O has maximal dimension amongst all unipotent classes with this property.

14.9. The following result was conjectured to hold by Kawanaka in [Kaw85, Conjecture 3.3.3]. In [Lus92, Theorem 11.2] Lusztig proved Kawanaka’s conjecture under the assumption that p and q are su�ciently large. Here we give the general case where p is a good prime, thus completing the proof of Kawanaka’s conjecture. Note that our proof uses in an essential way the ideas and techniques used by Lusztig in [Lus92, Theorem 11.2].

Theorem 14.10. Assume p is a good prime for G then every irreducible character ρ ∈ Irr(G) has a unique wave front set denoted by O_{ρ}^{∗}. Furthermore, if ρ is contained in the series E(G, ℒ, ℭ) then O_{ρ}^{∗} = O_{ℒ,ℭ^{∗}} (c.f. 12.3).

14.11. Before proving the theorem we will consider the following two reduction steps (which are similar to those used in [Gec96]). Note that the second reduction is only required because Theorem 11.13 does not necessarily hold in good characteristic when Z(G) is connected. In the following lemmas it will be implicitly assumed that p is a good prime.

Lemma 14.12. Assume G ↪ G̃ is a regular embedding into a group with connected centre (c.f. [Lus88, §7]) then Theorem 14.10holds for G if and only if it holds for G̃.

Proof. Let u ∈ G be a unipotent element then by the de�nition of the GGGR it is clear to see that we have

ΓG̃ ; u = IndG̃ ; G(Γ_{u}^{G})

and all GGGRs of G̃ are obtained in this way. Assume ρ ∈ Irr(G̃) is an irreducible character then according to [Lus88, Proposition 5.1] we have ResG̃ ; G(ρ) = ρ_{1} + ⋯+ ρ_{r} for some irreducible characters ρ_{i} ∈ Irr(G). In particular, by Frobenius reciprocity we have

⟨ΓG̃ ; u, ρ⟩G̃ = ∑_{i=1}^{r} ⟨Γ_{u}^{G}, ρ_{i}⟩_{G}.

As the right hand side is a sum of non-negative integers we have ⟨ΓG̃ ; u, ρ⟩G̃ ≠ 0 if and only if ⟨Γ_{u}^{G}, ρ_{i}⟩_{G} ≠ 0 for some i ∈ {1, …, r}. This proves the �rst part of Theorem 14.10 holds in G if and only if it holds in G̃. In fact, this argument shows that we have O_{ρ_{i}}^{∗} = O∗ ; ρ̃ ∩ G if such a class exists. The second part of Theorem 14.10 follows from the e�ect of restriction on Deligne�Lusztig characters (see [Bon06, Proposition 11.5]) and the de�nition of the class O_{ℒ,ℭ^{∗}} . ■

Lemma 14.13. Assume G has a connected centre and let π : G̃ → G be a surjective homomorphism de�ned over q such that: Ker(π) ⩽ Z(G̃) is connected, Z(G̃) is connected and the derived subgroup of G̃ is simply connected. Then Theorem 14.10holds for G if it holds for G̃.

Proof. Note that the map π descends to a surjective map π : G̃ → G of F-�xed points as the kernel of π is connected. We denote by InfG̃ ; G : Cent(G) → Cent(G̃) the in�ation map given by InfG̃ ; G(f) = f ∘ π. As this map induces an isometry onto its image we have

⟨Γ_{u}^{G}, χ⟩_{G} = ⟨InfG̃ ; G(Γ_{u}^{G}), InfG̃ ; G(χ)⟩G̃

for all unipotent elements u ∈ G and irreducible characters χ ∈ Irr(G). We want to show that the in�ation InfG̃ ; G(Γ_{u}^{G}) is a summand of the corresponding GGGR in G̃.

Note that π induces a bijection between the unipotent elements of G̃ and G so we may consider u as an element of G̃. Let us recall the notation of De�nition 5.14 then, by the previous remark, we may identify the subgroup U(λ, 1) ⩽ G with the corresponding subgroup of G̃. Denoting by Ũ(λ, 1) ⩽ G̃ the subgroup U(λ, 1) Ker(π)^{F} (a direct product) we have the GGGR corresponding to u in G̃ is given by

ΓG̃ ; u = IndG̃ ; U(λ,1)(φ̃_{u}) = (IndG̃ ; Ũ(λ,1) ∘ Ind_{U(λ,1)}^{Ũ(λ,1)})(φ̃_{u}).

There is a unique irreducible constituent of Ind_{U(λ,1)}^{Ũ(λ,1)}(φ̃_{u}) whose kernel contains Ker(π)^{F} and inducing this to G̃ we obtain the in�ation InfG̃ ; G(Γ_{u}^{G}) as a summand of ΓG̃ ; u. In particular, this shows that we have

⟨Γ_{u}^{G}, χ⟩_{G} = ⟨ΓG̃ ; u, InfG̃ ; G(χ)⟩G̃

as InfG̃ ; G(Γ_{u}^{G}) contains all irreducible constituents of ΓG̃ ; ũ with Ker(π)^{F} in their kernel. From this the result follows immediately. ■

Proof (of Theorem 14.10). Let us assume that G is GL_{n}( ) or that Z(G) is connected and the derived subgroup G_{der} has no simple factors of type A. In this situation both Theorem 11.13 and the results of

[Lus90] are available to us (assuming that p is a good prime) and the theorem can be proved in exactly the same way as [Lus92, Theorem 11.2]. We will not repeat the argument here.

Assume now that G is simple and simply connected then we may choose a regular embedding G ↪ G̃ such that G̃ is GL_{n}( ) if G is of type A_{n−1} . By the previous case and Lemma 14.12 we see the theorem holds for G.

Now assume G is semisimple and simply connected then we may write G as a direct product G^{(1)} × ⋯× G^{(r)} where each G^{(i)} is a direct product of simple groups permuted transitively by F. Clearly if the result holds for each G^{(i)} then it holds for G so we may assume that G = G^{(1)} = G_{1} × ⋯× G_{r} is a direct product of simple groups. However, in this situation we have G^{F} ≅ G_{1}^{F^{r}} so the result follows from the previous case. Thus the theorem holds for semisimple simply connected groups.

Assume now that G has a connected centre and simply connected derived subgroup G_{der}. Applying Lemma 14.12 and the previous case to the natural regular embedding G_{der} ↪ G we get that the theorem holds for G.

Finally, assume G is any group with a connected centre then we may �nd a surjective morphism G̃ → G as in Lemma 14.13 (see [Lus84a, §8.8]). In particular, the theorem holds for G by Lemma 14.13 and the previous case. Finally, assume G is arbitrary then choosing a regular embedding G ↪ G̃ we deduce the theorem from Lemma 14.12 and the previous case. ■

Remark 14.14. In [Lus92, Theorem 11.2(ii)] Lusztig proves that the multiplicity ⟨Γ_{u}, ρ⟩, for any u ∈ O_{ρ}^{∗}^{F} is small in the sense that it is bounded independently of q. In fact, chasing through the proof of Theorem 14.10 one sees that this can also be deduced, in our situation, from [Lus92, Theorem 11.2(ii)]. However, with our approach one obtains a bound which is slightly larger than that given by Lusztig. For instance, let \overline{A}_{G}(u) be the quotient A_{G}(u)/Z_{G}(u) where Z_{G}(u) is the image of the natural map Z(G) → A_{G}(u) and let π_{1}(Φ) be the fundamental group of the root system (i.e. the quotient of the weight lattice by the root lattice) then we have |π_{1}(Φ)| · |\overline{A}_{G}(u)| · |W_{G}(T_{0})| is a su�cient bound.

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Email: taylor@mathematik.uni-kl.de