GENERIC BLOCKS OF FINITE REDUCTIVE GROUPS

Michel Broué, Gunter Malle and Jean Michel

Ecole Normale Supérieure and Universität Heidelberg

To Charlie Curtis

Contents

0. Introduction

1. Notation, prerequisites and complements A. Generic finite reductive groups

• Usual invariants of a generic finite reductive group

• Class functions on 𝔾

• d–split Levi subgroups B. Generic characters

• Some consequences of Lusztig’s results

• Generic unipotent characters, uniform functions

• The generic Deligne–Lusztig induction and restriction 2. d–cuspidality and the uniform theory

A. d–cuspidality B. Regular unipotent characters C. Regular characters

• Introduction: the regular character of E_{π}(G^{F} , 1)

• The generic formalism D. First application to actual finite reductive groups

3. Generalized Harish–Chandra theory A. The fundamental theorem B. d–Harish–Chandra theory for unipotent characters

Typeset by AMS-TEX Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20, 20G.

We thank the M.S.R.I. in Berkeley, and the second author thanks the Ecole Normale Supérieure, for their hospitality during the elaboration of part of this work S.M.F.

Astérisque 212* (1993)

M. BROUÉ, G. MALLE, J. MICHEL

4. Generic Φ_{d}-blocks

A. More on regular unipotent characters and Mackey for-

mula B. Φ_{d}–defect groups C. Regular characters of generic d–blocks

5. π–Blocks and Isotypies A. π–blocks, Brauer morphisms B. Isotypies C. ℓ–Blocks Appendix : Tables.

0. Introduction

This work has been motivated by two problems, which interacted with one another: attempts to solve one helped to solve the other one.

The first one originated in [Br1], where is presented a conjecture about the structure of any block with abelian defect of any finite group (see [Br1], 6.1), which states that such a block should have the same type as the corresponding block of the normalizer of its defect group. One of our goals was to check this conjecture in the case of the finite reductive groups G^{F} (we denote by G a connected reductive algebraic group over an algebraic closure of a finite field 𝔽_{q}, by F: G → G a Frobenius endomorphism defining a rational structure on this finite field, and by G^{F} the group of rational points). This is achieved in the present work (see §5.C, 5.24), at least for “large” prime numbers (i.e., in the split case, prime numbers which do not divide the order of the Weyl group of G).

By general properties of isotypies (see [Br1]), our results imply in particular that, for large prime numbers, conjectures as Alperin’s weight conjecture or the Alperin–McKay height conjecture are verified for unipotent blocks (and so for “almost” all blocks) of all finite reductive groups.

The second question which motivated this work was to understand better the “generic” aspects of block theory of finite reductive groups.

Throughout the intensive work which has been done recently by many authors about blocks of finite reductive groups (among whom Fong–Srinivasan, Schewe, Cabanes–Enguehard, Hiß, Geck, and the authors), it had become gradually clear that, for large prime numbers ℓ, properties of ℓ–blocks of G^{F} do not really depend on the prime number ℓ, but rather depend on the cyclotomic factor of the “polynomial order” of G^{F} which is divisible by ℓ (see [BrMa] and §1.A below).

This remark had to be put in the perspective opened by the work of C.W. Curtis, who defined in particular the “generic degrees” of characters of G^{F}, and subsequently of his student R. Boyce (see [Boy]) who defined characters “with Φ_{d}–defect zero”.

A few experiments convinced us that it was indeed very often possible to replace ℓ by the corresponding Φ_{d} in many of the congruence relations involved in modular character theory.

One of the aims of the present work is to present further evidence for the existence of a “generic representation theory of generic groups”. Using extensively [BrMa] (where are defined what we call here “generic finite reductive groups”, as well as their Φ_{d}–subgroups and their centralizers), we introduce here a formalism which allows us in particular to group the unipotent characters into Φ_{d}–blocks and to define a suitable notion of “Φ_{d}–defect group” of a unipotent character (see definition 4.7 below). From many points of view, this Φ_{d}–defect group (a rational torus in G) behaves like an ordinary defect group. Moreover, if ℓ is a large prime which divides Φ_{d}(q), then the actual ℓ–defect group is just the Sylow ℓ–subgroup of the group of rational points of the Φ_{d}–defect group.

One of the by–products of this approach is indeed the fact that we can treat the case of π–blocks (π a set of large prime numbers which divide the same cyclotomic factor) the same way we would treat ℓ–blocks, and this is what we present here.

With these two questions in mind, we came upon the main theorem of this work (see 3.2). This theorem is certainly the main tool to study our blocks, but it is also interesting in itself. It shows in particular that the usual so–called Harish–Chandra theory of characters, as well as many aspects of the Howlett– Lehrer–Lusztig theory about induction of cuspidal characters, must be viewed as a particular case (the case associated with the first cyclotomic polynomial Φ_{1}) of a general set of results which hold for every cyclotomic polynomial, provided one replaces the family of “Harish–Chandra” Levi subgroups by the family of centralizers of Φ_{d}–subgroups (called here d–split subgroups), and the cuspidal characters by the “d–cuspidal characters”.

In particular, theorem 3.2 shows that, provided each irreducible unipotent character is equipped with an appropriate sign, Deligne–Lusztig induction from d–split subgroups is nothing but the ordinary induction in Weyl–type groups which are independent of q.

We thank Paul Fong for a careful and useful reading of the first version of this manuscript.

1. Notation, prerequisites and complements

A. Generic finite reductive groups.

In §1.A, we recall notation, definitions and some results from [BrMa] concerning generic finite reductive groups.

We call “generic finite reductive group” what is called “donnée radicielle complète” in [BrMa]. Let us recall briefly the definition and set our notation.

A root datum of rank r is a quadruple (X, R, Y, R^{∨}) such that

(rd.1) X and Y are free ℤ–modules of rank r, endowed with a duality X ×

Y → ℤ denoted by (x, y) ↦ <x, y>. (rd.2) R and R^{∨} are finite subsets of X and Y respectively, endowed with a

bijection R → R^{∨} denoted by α ↦ α^{∨}. (rd.3) For α ∈ R we have <α, α^{∨}> = 2. Let s_{α} be the involutive auto-

morphism of X defined by s_{α}(x) = x −<x, α^{∨}>α, and let s_{α}^{∨} be its adjoint, an automorphism of Y (one has s_{α}^{∨}(y) = y −<a, y>α^{∨}). Then

s_{α}(R) = R and s_{α}^{∨}(R^{∨}) = R^{∨}.

The Weyl group of the root datum (X, R, Y, R^{∨}) is the subgroup of the group of automorphisms of Y generated by the s_{α}^{∨} for α ∈ R.

The following definition is valid in the general case; for the particular twisted cases ^{2}B_{2}, ^{2}F_{4}, ^{2}G_{2}, the reader may refer to [BrMa].

Definition.

• An automorphism of the root datum (X, R, Y, R^{∨}) is an automorphism ϕ of Y which stabilizes R^{∨} and such that ϕ^{∨} stabilizes R.

• A generic finite reductive group is a pair 𝔾 = (Γ_{𝔾}, W_{𝔾}ϕ), where Γ_{𝔾} is a root datum, W_{𝔾} is its Weyl group, and ϕ is an automorphism of finite order of Γ_{𝔾}.

Let p be a prime number and let 𝔽̄_{p} be a chosen algebraic closure of 𝔽_{p}. To a root datum Γ_{𝔾} is then associated a pair (G, T) where

• G is a connected reductive algebraic group over 𝔽̄_{p},

• T is a maximal torus of G,

and such a pair is uniquely determined up to inner automorphisms of G induced by the elements of T.

The isogeny theorem (cf. for example [Sp2], 11.4.9) implies the following:

(1.1) To a linear map of Y of the form qϕ, where q is a power of p and ϕ an automorphism of Γ_{𝔾}, is associated an isogeny F_{qϕ} : G → G of the algebraic group G, uniquely determined up to inner automorphisms of G induced by elements of T.

If q = 1, the isogeny F_{ϕ} is in fact an algebraic automorphism of G; otherwise, the isogeny F_{qϕ} is the Frobenius endomorphism associated to some 𝔽_{q}-structure on G. Note that, since T is commutative, if ϕ and ϕ′ commute then F_{qϕ} commutes with F_{q′ϕ′} for any other power q′ of p.

Thus, given a generic group 𝔾, the choice of a power q > 1 of p and of an element ϕ in the coset W ϕ determines a triple (G, T, F), where F = F_{qϕ} is a Frobenius endomorphism (such a construction is also possible in the twisted cases, where q has to be replaced by an odd power of \sqrt{2} or \sqrt{3}, and F is a special isogeny of the corresponding group of type B_{2}, F_{4} or G_{2} cf. [BrMa], §2). Let us call such a triple a “(q, ϕ)–triple associated to 𝔾”. As in [BrMa], §2 we will write 𝔾(q) for G^{F}.

Usual invariants of a generic finite reductive group (cf. [BrMa]).

Preliminary remark. What follows is written in the “general case”. In order to apply it to the twisted cases (^{2}B_{2}, ^{2}F_{4}, ^{2}G_{2}), one has to perform a few modifications, such as to replace the ring ℤ and the field ℚ by, respectively, ℤ[\sqrt{2}^{−1}] and ℚ(\sqrt{2}) (for ^{2}B_{2} and ^{2}F_{4}), or by ℤ[\sqrt{3}^{−1}] and ℚ(\sqrt{3}) (for ^{2}G_{2}). We leave this work to the reader.

Let 𝔾 = ((X, R, Y, R^{∨}), W_{𝔾}ϕ) be a generic finite reductive group.

We set V = ℚ⊗Y and W = W_{𝔾}. We denote by SV the symmetric algebra of V, by (SV)^{W} the graded subalgebra of W –invariant elements of SV, and by 𝔄_{𝔾} the ideal of (SV)^{W} consisting of elements without degree zero terms.

• The vector space 𝔄_{𝔾}/𝔄_{𝔾}^{2} has dimension r, and is endowed with an action of the image ϕ̄ of ϕ modulo W. We set

ε_{𝔾} := (−1)^{r} det ϕ̄ .

• We set R𝔾 := SV/(SV 𝔄_{𝔾}) . Then R𝔾 is a finite dimensional graded algebra which, as a ℚW –module, is isomorphic to the regular representation of W. We denote by R^{n}𝔾 the subspace of elements of degree n of R𝔾, and by 2N(𝔾) the number of elements of the root system R. Then we have

R𝔾 = ∑_{n=0}^{N(𝔾)} R^{n}𝔾 .

• The polynomial order of 𝔾 (denoted by O_{𝔾}(x) in [BrMa]) is the polynomial denoted here by |𝔾| and defined by the formula

(1.2)  |𝔾| := \frac{ε_{𝔾}x^{N(𝔾)}}{\frac{1}{|W |} ∑_{w∈W} \frac{1}{det_{V} (1 − xwϕ)}} .

If (q, ϕ) (whence (G, T, F)) is chosen, one has |G^{F} | = |𝔾|(q) .

We recall now how various definitions are translated in the setting of generic groups.

Tori and Levi subgroups.

• A generic torus is a generic group such that R = R^{∨} = ∅ (so W ϕ reduces to a single element). Generic subtori of 𝔾 (called “sous–données toriques” in [BrMa]) are generic tori of the form (X′, Y′, wϕ|_{Y′}) where w ∈ W and Y′ is a wϕ-stable direct factor of Y and X′ its dual (a quotient of X).

• Generic Levi subgroups of 𝔾 (called in [BrMa] “sous–données de Levi”, and often abbreviated herein “Levi subgroups”) are generic groups of the form ((X, R′, Y, R^{′∨}), W_{R′}wϕ), where w ∈ W, where R^{′∨} is a parabolic subsystem of R^{∨} which is wϕ-stable and where W_{R′} is the Weyl group of R^{′∨}.

Lifting scalars.

Let Γ be a root datum, and let 𝔾 = (Γ, W ϕ) be an associated generic finite group. Let a ∈ ℕ.

(1.3) We define the generic finite group 𝔾^{(a)} = (Γ^{(a)}, ϕ^{(a)}) by the following rules: (ls.1) Γ^{(a)} := Γ × ⋯ × Γ (a times), (ls.2) ϕ^{(a)} is the product of ϕ (acting diagonally on Γ × ⋯ × Γ) and the

a–cycle which permutes the various factors Γ of Γ^{(a)}.

We have |𝔾^{(a)}|(x) = |𝔾|(x^{a}). Let (G, T, F_{a}) be a (q^{a}, ϕ)–triple associated with 𝔾, and let (G^{(a)}, T^{(a)}, F^{(a)}) be a (q, ϕ^{(a)})–triple associated with 𝔾^{(a)}. Then

(1.4)  G^{(a)F^{(a)}} ≃ G^{F_{a}} .

In other words, we have 𝔾^{(a)}(q) ≃ 𝔾(q^{a}) .

Changing q into −q.

Let 𝔾 = (Γ_{𝔾}, W_{𝔾}ϕ) be a generic finite group. We recall (see [BrMa]) that we define the generic finite group 𝔾^{−} by

(1.5)  𝔾^{−} := (Γ_{𝔾}, W_{𝔾}(−ϕ)) .

Note that if −Id ∈ W_{𝔾}, then 𝔾^{−} = 𝔾.

We have |𝔾^{−}|(x) = (−1)^{r}|𝔾|(−x).

The radical, the “semi–simple quotient”, the adjoint group, the dual group. If 𝔾 = ((X, R, Y, R^{∨}), W ϕ) is a generic finite reductive group, we denote

by Q(R) the ℤ–submodule of X generated by R.

• The radical of 𝔾 is by definition the generic torus defined by

Rad(𝔾) := ((X/Q(R)^{⊥⊥}, Q(R)^{⊥}), ϕ_{|_{Q(R)^{⊥}}}) .

For a triple (G, T, F) associated with 𝔾, the algebraic group associated with Rad(𝔾) is Z^{o}(G) (the connected component of 1 in the center Z(G) of G).

• The generic “semi–simple quotient” of 𝔾 is “the quotient of 𝔾 by its radical Rad(𝔾)”, namely

𝔾_{ss} := ((Q(R)^{⊥⊥}, R, Y/Q(R)^{⊥}, R^{∨}), W ϕ)

(here “ϕ” stands for the automorphism of Y/Q(R)^{⊥} induced by ϕ, and R^{∨} stands for the image of R^{∨} in Y/Q(R)^{⊥}).

For a triple (G, T, F) associated with 𝔾, the algebraic group associated with 𝔾_{ss} is G/Z^{o}(G). We have |𝔾_{ss}| = |𝔾|/|Rad(𝔾)| .

• The dual generic finite group is defined by

𝔾^{∗} := ((Y, R^{∨}, X, R), W ϕ^{∨}^{−1}) ,

where the automorphism ϕ^{∨} of X is the adjoint of ϕ, and where W is identified with its contragredient action on X, i.e., W is the group generated by the s_{α} for α ∈ R (note that 𝔾^{∗} is covariant in 𝔾).

• We denote by P (R) the dual of Q(R^{∨}) in ℚ ⊗ Q(R), i.e., the set of all v ∈ ℚ ⊗ Q(R) such that <v, α^{∨}> ∈ ℤ for all α^{∨} ∈ R^{∨}.

The adjoint generic finite group of 𝔾 is defined by

𝔾_{ad} := ((Q(R), R, P (R^{∨}), R^{∨}), W ϕ) .

For a triple (G, T, ϕ) associated with 𝔾, the algebraic group associated with 𝔾_{ad} is G_{ad}.

• We define the derived generic finite group D(𝔾) by (with obvious abuse of notation)

D(𝔾) := ((X/Q(R^{∨})^{⊥}, R, Q(R^{∨})^{⊥⊥}, R^{∨}), W ϕ) .

Thus we have a generic finite torus

𝔾/D(𝔾) = ((Q(R^{∨})^{⊥}, Y/Q(R^{∨})^{⊥⊥}), ϕ_{|_{Y /Q(R^{∨})^{⊥⊥}}}) .

We have

(1.6)  Rad(𝔾^{∗}) = (𝔾/D(𝔾))^{∗} .

Let (G, T, F) be a (q, ϕ)–triple associated with 𝔾. Let [G, G] be the derived group of G. Then the algebraic group associated with D(𝔾) (resp. 𝔾/D(𝔾)) is [G, G] (resp. G/[G, G]).

Good primes.

The following definition can be found in [SpSt], 4.1. It depends only on the root system of 𝔾.

(1.7) We say that a prime ℓ is good for 𝔾 if there is no closed root subsystem R_{1} of R such that ℓ divides the order of the torsion subgroup of Q(R)/Q(R_{1}). A prime which is not good is called bad.

It results from [SpSt], 4.4, that if ℓ is good for 𝔾, then there is no closed root subsystem R_{1} of R such that ℓ divides the order of the torsion subgroup of Q(R^{∨})/Q(R_{1}^{∨}). Hence if ℓ is good for 𝔾 it is also good for the dual generic group 𝔾^{∗}.

Intersections of Levi subgroups.

Let 𝕃 = ((X, R_{𝕃}, Y, R_{𝕃}^{∨}), W_{𝕃}vϕ) and 𝕄 = ((X, R_{𝕄}, Y, R_{𝕄}^{∨}), W_{𝕄}wϕ) be two Levi subgroups of 𝔾.

(1.8) We say that 𝕃 ∩ 𝕄 is defined if

W_{𝕃}vϕ ∩ W_{𝕄}wϕ ≠ ∅ .

In that case, choosing an element u ∈ W_{𝕃}v ∩ W_{𝕄}w, we define

𝕃 ∩ 𝕄 := ((X, R_{𝕃} ∩ R_{𝕄}, Y, R_{𝕃}^{∨} ∩ R_{𝕄}^{∨}), (W_{𝕃} ∩ W_{𝕄})uϕ) ,

Note that W_{𝕃}vϕ = W_{𝕃}uϕ and W_{𝕄}wϕ = W_{𝕄}uϕ, whence W_{𝕃}vϕ ∩ W_{𝕄}wϕ = (W_{𝕃} ∩ W_{𝕄})uϕ. It is easy to see that 𝕃 ∩ 𝕄 is a well–defined (i.e., independent of the choice of u) Levi subgroup of both 𝕃 and 𝕄.

By the preceding definition, it is clear that (1.9) 𝕃 ∩ 𝕄 is defined if and only if there exists a maximal generic torus 𝕋 of 𝔾 such that 𝕋 is contained in both 𝕃 and 𝕄. In this case, 𝕋 is contained in 𝕃 ∩ 𝕄.

1.10. Definition. For 𝕃 and 𝕄 two Levi subgroups of 𝔾, we denote by S_{W_{𝔾}}(𝕃, 𝕄) the set of all w ∈ W_{𝔾} such that ^{w}𝕃 ∩ 𝕄 is defined.

If (G, T, F) is a (q, ϕ)–triple associated to 𝔾, we recall (cf. [BrMa], 2.1) that there is a well–defined bijection between W_{𝔾}–classes of generic Levi subgroups of 𝔾 and G^{F} –conjugacy classes of F –stable Levi subgroups of G. Then S_{W_{𝔾}}(𝕃, 𝕄) is non empty if and only if there exist L and M in the corresponding classes of Levi subgroups in G such that L ∩ M contains a maximal torus; in that case, L ∩ M is a Levi subgroup of G, whose rational conjugacy class corresponds to the W_{𝔾}–conjugacy class of 𝕃 ∩ 𝕄.

Class functions on 𝔾.

Let CF(𝔾) be the space of all W –invariant functions on the coset W ϕ with values in ℚ[x], called class functions on 𝔾.

We shall see later on that this space can be “specialized” onto the space of unipotent uniform functions for a chosen (q, ϕ)–triple associated to 𝔾.

If α and α′ ∈ CF(𝔾), we set <α, α′>_{𝔾} := \frac{1}{|W |} ∑_{w∈W} α(wϕ)α′(wϕ) .

Induction and restriction.

Let 𝕃 = ((X, R_{𝕃}, Y, R_{𝕃}^{∨}), W_{𝕃}wϕ) be a Levi subgroup of 𝔾, and let α ∈ CF(𝔾) and β ∈ CF(𝕃).

We denote

• by Res_{𝕃}^{𝔾} α the restriction of α to the coset W_{𝕃}wϕ,

• by Ind_{𝕃}^{𝔾}β the class function on 𝔾 defined by

(1.11)  Ind_{𝕃}^{𝔾}β(uϕ) = \frac{1}{|W_{𝕃}|} ∑_{v∈W} β̃(vuϕv^{−1}) for uϕ ∈ W_{𝔾}ϕ,

where β̃(xϕ) = β(xϕ) if x ∈ W_{𝕃}w, and β̃(xϕ) = 0 if x ∉ W_{𝕃}w . In other words, we have

(1.12)  Ind_{𝕃}^{𝔾}β(uϕ) = ∑_{v∈W_{𝔾}/W_{𝕃},^{v}(uϕ)∈W_{𝕃}wϕ} β(^{v}(uϕ)) .

We have the Frobenius reciprocity:

(1.13)  <α, Ind_{𝕃}^{𝔾}β>_{𝔾} = <Res_{𝕃}^{𝔾} α, β>_{𝕃} .

The Mackey formula.

It is clear that W_{𝕃} acts on S_{W_{𝔾}}(𝕃, 𝕄) from the right, while W_{𝕄} acts on S_{W_{𝔾}}(𝕃, 𝕄) from the left. If we let w run over a chosen double coset W_{𝕄}vW_{𝕃} for some v ∈ S_{W_{𝔾}}(𝕃, 𝕄), we see that ^{w}𝕃∩𝕄 is defined up to W_{^{w}𝕃}–conjugation as a subgroup of ^{w}𝕃 and up to W_{𝕄}–conjugation as a subgroup of 𝕄, which proves that the operations Ind_{𝕄∩^{w}𝕃}^{𝕄} and Res_{𝕄∩^{w}𝕃}^{^{w}𝕃} depend only on the double coset of w. This gives sense to the following formula

(1.14)  Res_{𝕄}^{𝔾} ·Ind_{𝕃}^{𝔾} = ∑_{w∈W_{𝕄}\S_{W_{𝔾}} (𝕃,𝕄)/W_{𝕃}} Ind_{𝕄∩^{w}𝕃}^{𝕄} · Res_{𝕄∩^{w}𝕃}^{^{w}𝕃} ·ad(w) ,

whose proof goes like the proof of the Mackey formula for ordinary induction and restriction (note that S_{W_{𝔾}}(𝕃, 𝕄) may be empty).

Some important class functions.

• We denote by tr_{R𝔾} the class function on 𝔾 defined by the character of the “graded regular representation” R𝔾 (see above). Thus the value of the function tr_{R𝔾} on wϕ is tr_{R𝔾}(wϕ) := ∑_{n=0}^{N(𝔾)} tr(wϕ; R^{n}𝔾)x^{n} .

• For α ∈ CF(𝔾), we shall consider the following polynomial (which will be identified later on with the “fake degree” Deg(Φ_{α}^{𝔾})) :

(1.15)  D(α) := <α, tr_{R𝔾}>_{𝔾} = ∑_{n=0}^{N(𝔾)}(\frac{1}{|W |} ∑_{w∈W} α(wϕ)tr(wϕ; R^{n}𝔾))x^{n} .

We have (cf. [BrMa], prop. 1.6′)

(1.16)  \frac{1}{|W |} ∑_{w∈W} \frac{α(wϕ)}{det_{V} (1 − xwϕ)} = D(α)\frac{1}{|W |} ∑_{w∈W} \frac{1}{det_{V} (1 − xwϕ)} ,

or, in other words

<α, tr_{SV} >_{𝔾} = <α, tr_{R𝔾}>_{𝔾}<1^{𝔾}, tr_{SV} >_{𝔾} ,

where 1^{𝔾} is the constant function 1 on W_{𝔾}ϕ.

• For a Levi subgroup 𝕃 of 𝔾, by the Frobenius reciprocity (1.13), we have

D(Ind_{𝕃}^{𝔾}1^{𝕃}) = <1^{𝕃}, Res_{𝕃}^{𝔾} tr_{R𝔾}>_{𝕃} = ∑_{n=0}^{N(𝔾)} tr(wϕ; (R^{n}𝔾)^{W_{𝕃}})x^{n} .

where W_{𝕃}wϕ is the coset associated to 𝕃 and (R^{n}𝔾)^{W_{𝕃}} are the W_{𝕃}–invariants in R^{n}𝔾.

It follows from 1.2, 1.13 and 1.16 (see also [BrMa], prop. 1.7) that

(1.17)  |𝔾|/|𝕃| = ε_{𝔾}ε_{𝕃}x^{N(𝔾)−N(𝕃)}D(Ind_{𝕃}^{𝔾}1^{𝕃}) ;

D(Ind_{𝕃}^{𝔾}1^{𝕃}) will be identified later on with the generic degree Deg(R_{𝕃}^{𝔾}(1)).

• Every element wϕ ∈ W ϕ defines a maximal generic subtorus (or, equivalently, a minimal Levi subgroup) 𝕋_{wϕ} := ((X, Y), wϕ) . We have

tr_{R𝔾}(wϕ) := D(Ind_{𝕋_{wϕ}}^{𝔾}1^{𝕋_{wϕ}}) .

So

(1.18)  |𝔾|/|𝕋_{wϕ}| = ε_{𝔾}ε_{𝕋_{wϕ}}x^{N(𝔾)}tr_{R𝔾}(wϕ) ,

and also tr_{R𝔾}(wϕ) = D(κ_{wϕ}), where κ_{wϕ} is |C_{W} (wϕ)| times the characteristic function of the W –conjugacy class of wϕ.

It follows from 1.17 and 1.18 that

(1.19)  Res_{𝕃}^{𝔾} tr_{R𝔾} = D(Ind_{𝕃}^{𝔾}1^{𝕃})tr_{R𝕃} .

(1.20) For β ∈ CF(𝕃), we have D(Ind_{𝕃}^{𝔾}β) = D(Ind_{𝕃}^{𝔾}1^{𝕃})<β, tr_{R𝕃}>_{𝕃} .

Indeed, D(Ind_{𝕃}^{𝔾}β) = <β, Res_{𝕃}^{𝔾} tr_{R𝔾}>_{𝕃} = D(Ind_{𝕃}^{𝔾}1^{𝕃})<β, tr_{R𝕃}>_{𝕃} .

Class functions on 𝔾^{−}.

The map σ^{𝔾}: CF(𝔾) → CF(𝔾^{−}), given by σ^{𝔾}α(w(−ϕ)) = α(wϕ), is an isometry. The map 𝕃 ↦ 𝕃^{−} is a W_{𝔾}–equivariant bijection from the set of all generic Levi subgroups of 𝔾 onto the set of all generic Levi subgroups of 𝔾^{−}. It is clear that

(1.21)

σ^{𝔾} · Ind_{𝕃}^{𝔾} = Ind_{𝕃^{−}}^{𝔾^{−}} · σ^{𝕃} , σ^{𝕃} · Res_{𝕃}^{𝔾} = Res_{𝕃^{−}}^{𝔾^{−}} ·σ^{𝔾} ,
D(Ind_{𝕃^{−}}^{𝔾^{−}}1^{𝕃^{−}})(x) = D(Ind_{𝕃}^{𝔾}1^{𝕃})(−x) .

Class functions on 𝔾^{(a)}.

Let a ∈ ℕ. We have (see 1.3) W_{𝔾^{(a)}} = (W_{𝔾})^{a}, and it is clear that the map (w_{1}, w_{2}, …, w_{a})ϕ^{(a)} ↦ w_{1}w_{2} ⋯ w_{a}ϕ defines a bijection between the set of classes of W_{𝔾^{(a)}}ϕ^{(a)} under W_{𝔾^{(a)}}–conjugacy and the set of classes of W_{𝔾}ϕ under W_{𝔾}–conjugacy. Thus it induces an isometry

σ_{𝔾}^{(a)} : CF(𝔾^{(a)}) ⟶^{∼} CF(𝔾) .

1.22. Proposition. We have

σ_{𝔾}^{(a)} · Ind_{𝕃^{(a)}}^{𝔾^{(a)}} = Ind_{𝕃}^{𝔾} · σ_{𝕃}^{(a)}
Res_{𝕃}^{𝔾} ·σ_{𝔾}^{(a)} = σ_{𝕃}^{(a)} · Res_{𝕃^{(a)}}^{𝔾^{(a)}} .

d–split Levi subgroups.

Let 𝕋 = ((X, Y), ϕ) be a generic torus. We recall that in this case the polynomial order |𝕋| is just the characteristic polynomial of ϕ on Y. A generic Φ_{d}–group (cf. [BrMa], §3, abbreviated here by “Φ_{d}–group” or even by “dgroup”) is a generic torus whose polynomial order is a power of Φ_{d}. It is proved in [BrMa], 3.3, that a Φ_{d}–group is completely specified up to isomorphism by its polynomial order.

Remark. If (T, F) is an associated algebraic torus, the fact that 𝕋 is a Φ_{d}– group translates to the fact that T splits “exactly” over 𝔽_{q^{d}} (“exactly” in the sense that no subtorus splits over a smaller field).

It is proved in [BrMa] that the maximal Φ_{d}–subgroups of 𝔾 (called “Sylow Φ_{d}–subgroups of 𝔾) are all conjugate under W_{𝔾}.

We call d–split Levi subgroups the centralizers in 𝔾 of the Φ_{d}–subgroups of 𝔾 (as in [BrMa], 1.3, we define the centralizer C_{𝔾}(𝕋) of a subtorus 𝕋 of 𝔾 as the Levi subgroup whose root system consists of the roots orthogonal to 𝕋 and such that its associated coset of automorphisms contains the automorphism ϕ associated to 𝕋). In particular, the centralizers of the Sylow Φ_{d}–subgroups are the minimal d–split Levi subgroups.

We shall use the following technical remark.

By [BrMa], prop. 3.8, we know that whenever 𝕃 is a minimal d–split Levi subgroup of 𝔾, we have ε_{𝔾}ε_{𝕃}x^{N(𝔾)−N(𝕃)} ≡ 1 mod Φ_{d} . It then follows that the preceding congruence holds for every d–split Levi subgroup of 𝔾, from which we deduce by 1.17:

(1.23) For any d–split Levi subgroup 𝕃 of 𝔾, we have

|𝔾|/|𝕃| ≡ D(Ind_{𝕃}^{𝔾}1^{𝕃}) mod Φ_{d} .

The Mackey formula becomes particularly simple when we restrict ourselves to pairs of d–split Levi subgroups.

1.24. Proposition. Let 𝕄_{1} and 𝕄_{2} be two d–split Levi subgroups of 𝔾.

(1) 𝕄_{1} ∩𝕄_{2} is defined if and only if 𝕄_{1} and 𝕄_{2} contain a common Sylow Φ_{d}–subgroup of 𝔾. In particular S_{W_{𝔾}}(𝕄_{1}, 𝕄_{2}) ≠ ∅.

(2) If 𝕄_{1} ∩ 𝕄_{2} is defined, then it is a d–split Levi subgroup of 𝔾.

(3) Let 𝕄_{1} and 𝕄_{2} be two d–split Levi subgroups containing the minimal d–split Levi subgroup 𝕃. Then we have

S_{W_{𝔾}}(𝕄_{1}, 𝕄_{2}) = W_{𝕄_{2}}N_{W_{𝔾}}(𝕃)W_{𝕄_{1}} .

In particular, we have

Res_{𝕄_{2}}^{𝔾} ·Ind_{𝕄_{1}}^{𝔾} = ∑_{w∈W_{𝕄_{2}} (𝕃)\W_{𝔾}(𝕃)/W_{𝕄_{1}} (𝕃)} Ind_{𝕄_{2}∩^{w}𝕄_{1}}^{𝕄_{2}} · Res_{𝕄_{2}∩^{w}𝕄_{1}}^{^{w}𝕄_{1}} ·ad(w) ,

where, following the notation of [BrMa], §1.B, for a Levi subgroup 𝕃 of 𝔾, we put W_{𝔾}(𝕃) = N_{W_{𝔾}}(𝕃)/W_{𝕃}.

Proof. For a generic Levi subgroup 𝕄 of 𝔾, let us denote by Z_{d}(𝕄) the Sylow Φ_{d}–subgroup of its center. Then 𝕄 is d–split if and only if 𝕄 = C_{𝔾}(Z_{d}(𝕄)).

(1) Suppose first that 𝕄_{1} ∩ 𝕄_{2} is defined. By 1.9, there exists a generic maximal torus 𝕋 of 𝔾 contained in both 𝕄_{1} and 𝕄_{2}. Then the generic groups Z_{d}(𝕄_{1}) and Z_{d}(𝕄_{2}) are both contained in the Sylow Φ_{d}–subgroup 𝕋_{d} of 𝕋. Now if 𝕊 is a Sylow Φ_{d}–subgroup of 𝔾 containing 𝕋_{d}, we see that 𝕄_{1} and 𝕄_{2} both contain C_{𝔾}(𝕊) hence 𝕊. Conversely, if 𝕄_{1} and 𝕄_{2} both contain a minimal d–split Levi subgroup, a fortiori they both contain a maximal torus 𝕋, thus 𝕄_{1} ∩ 𝕄_{2} is defined by 1.9.

The last assertion in (1) follows from the Sylow theorems in [BrMa].

(2) Let 𝕋 be a common maximal torus for both 𝕄_{1} and 𝕄_{2}. Then the generic subgroup Z_{d}(𝕄_{1}).Z_{d}(𝕄_{2}) of 𝕋 is defined in an obvious way (cf. [BrMa], §3.E), and it is then clear that 𝕄_{1} ∩ 𝕄_{2} = C_{𝔾}(Z_{d}(𝕄_{1}).Z_{d}(𝕄_{2})).

(3) Set 𝕊 = Z_{d}(𝕃) (thus 𝕊 is a Sylow Φ_{d}–subgroup of 𝔾). Assume that w ∈ S_{W_{𝔾}}(𝕄_{1}, 𝕄_{2}). Then by what precedes, there exists a Sylow Φ_{d}–subgroup 𝕊′ of 𝔾 such that 𝕊′ is contained in both ^{w}𝕄_{1} and 𝕄_{2}. By Sylow theorems (cf. [BrMa], th. 3.4), there exist w_{1} ∈ W_{𝕄_{1}} and w_{2} ∈ W_{𝕄_{2}} such that ^{w^{−1}}𝕊′ = w_{1}𝕊 and 𝕊′ = ^{w_{2}}𝕊, from which it follows that w_{2}^{−1}ww_{1} ∈ N_{W_{𝔾}}(𝕊). The assertion now follows from the fact that N_{W_{𝔾}}(𝕊) = N_{W_{𝔾}}(𝕃) (cf. [BrMa], th. 3.4). □

Given a Levi subgroup ((X, R_{𝕃}, Y, R_{𝕃}^{∨}), W_{𝕃}wϕ), we define its image in 𝔾_{ss} (resp. its image in 𝔾_{ad}) to be ((Q(R)^{⊥⊥}, R_{𝕃}, Y/Q(R)^{⊥}, R_{𝕃}^{∨}), W_{𝕃}wϕ) (resp. ((Q(R), R_{𝕃}, P (R^{∨}), R_{𝕃}^{∨}), W_{𝕃}wϕ)).

1.25. Remark. A Levi subgroup is d-split if and only if its image in 𝔾_{ss} (resp. in 𝔾_{ad}) is d–split.

B. Generic characters.

In §1.B, we introduce the formalism which is necessary to treat unipotent characters as generic objects, and to prove that related constructions such as Deligne–Lusztig induction are indeed defined at the “generic level”.

Some consequences of Lusztig’s results.

For chosen q and ϕ, whence a triple (G, T, F) (cf. §1.A), one has a well– defined bijection between the W –conjugacy classes of the coset W ϕ and the G^{F} –conjugacy classes of F –stable maximal tori of G (cf. [BrMa], 2.1). Thus in particular, for wϕ ∈ W ϕ, the virtual Deligne-Lusztig character R_{T_{wϕ}}^{G}(1),

denoted here R_{wϕ}^{G^{F}}, is well defined.

The following result is a particular case of a more general theorem which will be proved later (see 1.33).

1.26. Theorem. (Lusztig) There exists a finite set, denoted by Uch(𝔾), and a map Uch(𝔾) → CF(𝔾), denoted by γ ↦ m_{γ}, with the following property: whenever (G, T, F) is a (q, ϕ)–triple associated to 𝔾, there is a bijection γ ↦ ρ_{γ}^{G^{F}} from Uch(𝔾) onto the set E(G^{F} , 1) of unipotent characters of G^{F} such that for every wϕ ∈ W ϕ, we have

m_{γ}(wϕ) = (ρ_{γ}^{G^{F}} , R_{wϕ}^{G^{F}})_{G^{F}} .

Thus in particular

(1) for all γ ∈ Uch(𝔾), m_{γ} takes only integral values,

(2) by the known orthogonality relations on the virtual characters R_{wϕ}^{G^{F}}, we have

∑_{γ∈Uch(𝔾)} m_{γ}(wϕ)m_{γ}(w′ϕ) = \{ \begin{matrix}|C_{W} (wϕ)| & if wϕ and w′ϕ are W –conjugate , \\ 0 & if not . \end{matrix}

Sketch of proof of theorem 1.26. Let Fam(W) be the set of families of ordinary irreducible representations of W. One knows (see [Lu1], chap. 4, and also [DiMi1], §4) how to associate to each element F ∈ Fam(W) a well–defined finite group G(F).

1. Assume first that ϕ is chosen in W ϕ. Let Fam(W)^{ϕ} be the set of all ϕ–invariant families. The choice of ϕ defines, for each F ∈ Fam(W)^{ϕ}, an automorphism ϕ_{F} of G(F) (see [DiMi1], 4.1). We denote by Uch(𝔾, ϕ) the union (for all F ∈ Fam(W)^{ϕ}) of the sets of G(F)–conjugacy classes of pairs (gϕ_{F}, χ), where g ∈ G(F) and χ ∈ Irr(C_{G(F)}(gϕ_{F})).

It is one of Lusztig’s main theorems in [Lu1] cf. th. 4.23) that there is a map Uch(𝔾, ϕ) → CF(𝔾), say γ ↦ m_{γ}, such that for each choice of a suitable q, there is a bijection γ ↦ ρ_{γ}^{G^{F}} : Uch(𝔾, ϕ) → E(G^{F} , 1) with m_{γ}(wϕ) =

(ρ_{γ}^{G^{F}} , R_{wϕ}^{G^{F}} )_{G^{F}} .

2. If ϕ′ is another element of W ϕ (defining a triple (G, T, F′)) there is a well–defined bijection σ_{ϕ,ϕ′} from Uch(𝔾, ϕ) onto Uch(𝔾, ϕ′), such that

(ρ_{γ}^{G^{F}} , R_{wϕ}^{G^{F}} )_{G^{F}} = (ρ_{σ_{ϕ,ϕ′} (γ)}^{G^{F′}}, R_{wϕ}^{G^{F′}} )_{G^{F′}} .

The system (Uch(𝔾, ϕ), σ_{ϕ,ϕ′})_{ϕ,ϕ′∈W ϕ} is projective. We define Uch(𝔾) as the limit of this system. □

Remark. It results from [Lu1] chap. 4, that the set Uch(𝔾), as well as the “multiplicities” m_{γ}, depend only on the action of ϕ on W_{𝔾} and on the order of ϕ on Q(R). In particular, to compute the multiplicities we may reduce to the case where X = Q(R) and R is irreducible. Also,

(1.27) any automorphism ϕ′ of the generic group 𝔾 which commutes with ϕ defines a permutation of the set Uch(𝔾), which depends only on the action of ϕ′ on W_{𝔾} and on the order of ϕ′.

Remark. We refer to [DiMi1], 6.4, for what follows. The map γ ↦ ρ_{γ}^{G^{F}} is uniquely determined by the condition (ρ_{γ}^{G^{F}} , R_{wϕ}^{G^{F}})_{G^{F}} = m_{γ}(wϕ) for all rational unipotent characters ρ_{γ}^{G^{F}}. This includes all unipotent characters in the case of classical groups. To ensure unicity in the case of exceptional groups for non-rational characters, the following supplementary conditions have to be added:

• To γ ∈ Uch(𝔾) there is associated a root of unity λ_{γ} (cf. [DiMi1], before 6.4 for the definition — we just mention that when ϕ_{F} acts trivially on G(F) and γ is given by (gϕ_{F}, χ) then λ_{γ} = χ(g)/χ(1)). We ask that, if F^{δ} is the smallest split power of F, the eigenvalues of F^{δ} associated to ρ_{γ}^{G^{F}} in any Deligne-Lusztig variety of G be equal to λ_{γ} up to some power of q^{δ/2}.

• To deal with the principal series unipotent characters corresponding to characters χ_{q} of degree 512 of the Hecke algebra of a group of type E_{7} or of degree 4096 of the Hecke algebra of a group of type E_{8}, we must add another condition: Lusztig ([Lu1], chap. 4) identifies F to a subset of the couples (gϕ_{F}, χ); the γ we are looking at is an element of F via this identification, which gives us a character χ_{γ} of W. We ask that χ_{γ} be the specialization for q = 1 of the character χ_{q} associated to ρ_{γ}^{G^{F}}.

In what follows we will assume that these conditions are satisfied. This has the following consequence:

1.28. Proposition. Let ϕ′ be an automorphism of Γ_{𝔾} which commutes with ϕ, and let U(ϕ′) be the corresponding permutation of Uch(𝔾) (cf. 1.27). Then F_{q′ϕ′}(ρ_{γ}^{G^{F}}) = ρ_{U(ϕ′)(γ)}^{G^{F}} for any power q′ of p, and for γ ∈ Uch(𝔾).

In particular we get that the action of F_{q′ϕ′} on E(G^{F} , 1) does not depend on q′.

Generic unipotent functions, uniform functions.

Generic generalized unipotent functions.

We denote by ℚ[x] Uch(𝔾) the free ℚ[x]–module on Uch(𝔾), endowed with the quadratic form for which the canonical basis {γ}_{γ∈Uch(𝔾)} is orthonormal. The scalar product of two elements ψ and ψ′ of ℚ[x] Uch(𝔾) is denoted by (ψ, ψ′)_{𝔾}. The elements of ℚ[x] Uch(𝔾) are called the generic generalized unipotent functions.

Generic uniform functions.

For wϕ ∈ W ϕ, we set R_{wϕ}^{𝔾} := ∑_{γ∈Uch(𝔾)} m_{γ}(wϕ)γ . Thus R_{wϕ}^{𝔾} depends only on the W_{𝔾}–conjugacy class of wϕ, and the system {R_{wϕ}^{𝔾}}_{wϕ}, where wϕ runs over a set of representatives for the W_{𝔾}–conjugacy classes of W_{𝔾}ϕ, is an orthogonal system in ℚ[x] Uch(𝔾) (by the remark (2) following theorem 1.26).

For all γ ∈ Uch(𝔾), we have m_{γ}(wϕ) = (γ, R_{wϕ}^{𝔾})_{𝔾} and more generally, for ψ ∈ ℚ[x] Uch(𝔾), we set m_{ψ}(wϕ) = (ψ, R_{wϕ}^{𝔾})_{𝔾} .

We call generic uniform functions on 𝔾 the elements of ℚ[x] Uch(𝔾) which are ℚ[x]–linear combinations of the R_{wϕ}^{𝔾}.

For α ∈ CF(𝔾), we set

(1.29)  Φ_{α}^{𝔾} := \frac{1}{|W |} ∑_{w∈W} α(wϕ)R_{wϕ}^{𝔾} ,

so that α = m_{Φ_{α}^{𝔾}}. The linear map Φ^{𝔾} : CF(𝔾) → ℚ[x] Uch(𝔾), α ↦ Φ_{α}^{𝔾}, is an isometric embedding whose image is the subspace of generic uniform functions.

The orthogonal projection from ℚ[x] Uch(𝔾) onto the subspace of uniform functions is denoted π_{u}^{𝔾}. Thus for all ψ ∈ ℚ[x] Uch(𝔾), we have

(1.30)  π_{u}^{𝔾}(ψ) = \frac{1}{|W |} ∑_{w∈W}(ψ, R_{wϕ}^{𝔾})_{𝔾} · R_{wϕ}^{𝔾} ,

and π_{u}^{𝔾}(ψ) = Φ_{m_{ψ}}^{𝔾}.

Generic degrees.

For all w ∈ W_{𝔾}, we set (see §1.A) Deg(R_{wϕ}^{𝔾}) := tr_{R𝔾}(wϕ) . We extend linearly the map Deg to all uniform functions, by setting Deg(Φ_{α}^{𝔾}) := D(α) = <α, tr_{R𝔾}>_{𝔾} , and then to all of ℚ[x] Uch(𝔾) by composition with π_{u}^{𝔾} (this corresponds to the fact that, for the actual finite reductive groups, the degree of a character equals the degree of its uniform projection). Thus we have

(1.31) Deg(ψ) := Degπ_{u}^{𝔾}(ψ) = \frac{1}{|W |} ∑_{w∈W} m_{ψ}(wϕ)Deg(R_{wϕ}^{𝔾}) = Deg(Φ_{m_{ψ}}^{𝔾}) ,

and we see in particular that for γ ∈ Uch(𝔾), we have Deg(γ) ∈ \frac{1}{|W |}ℤ[x].

We may notice the following more precise statement due to Lusztig:

1.32. Theorem. There exists an integer c divisible only by bad primes for

𝔾 (cf. 1.7) such that Deg(γ) ∈ \frac{1}{c}ℤ[x].

Proof. Let {Ẽ}_{E∈Irr(W)^{ϕ}} be an orthonormal basis formed by the restrictions to W ϕ of one extension to W ⟨ϕ⟩ of each ϕ-invariant irreducible character of W. Then Deg(Φ𝔾 ; Ẽ) = <Ẽ, tr_{R𝔾}> is in ℤ[x], and Degγ = ∑_{E}(Φ𝔾 ; Ẽ, γ)_{𝔾}Deg(Φ𝔾 ; Ẽ); the theorem then follows from the formula given in [Lu1], 4.26, which implies that (Φ𝔾 ; Ẽ, γ)_{𝔾} are rational numbers whose denominators are divisible only by bad primes for 𝔾. □

Remark. Whenever (q, ϕ) is chosen, we have Deg(γ)(q) = ρ_{γ}^{G^{F}} (1). Indeed, it follows from 1.31 that Deg(γ)(q) is equal to the scalar product

of ρ_{γ}^{G^{F}} with the character \frac{1}{|W |} ∑_{w∈W} Deg(R_{wϕ}^{G^{F}} (1))R_{wϕ}^{G^{F}} (1) , which is the

unipotent projection of the regular character of G^{F}.

The polynomial Deg(γ) is called the generic degree of γ.

The generic Deligne-Lusztig induction and restriction.

If 𝕃 = ((X, R_{𝕃}, Y, R_{𝕃}^{∨}), W_{𝕃}wϕ) is a Levi subgroup of 𝔾, a choice of ϕ′ ∈ W_{𝔾}ϕ is said to be 𝕃–adapted if ϕ′ ∈ W_{𝕃}wϕ. The choice of such a ϕ′ in the coset W_{𝔾}ϕ and any choice of q determines, as explained below 1.1, a quadruple (G, L, T, F) where G is a connected reductive algebraic group over 𝔽̄_{p}, F: G → G is a surjective endomorphism, L is an F –stable Levi complement of some parabolic subgroup of G and T is an F –stable maximal torus of L.

The following theorem translates into our language the known fact that the Deligne-Lusztig induction is “generic”.

1.33. Theorem. For each Levi subgroup 𝕃 of 𝔾 there exists a linear map

R_{𝕃}^{𝔾}: ℚ[x] Uch(𝕃) → ℚ[x] Uch(𝔾)

with the following properties:

(1) Whenever λ ∈ Uch(𝕃) and R_{𝕃}^{𝔾}(λ) = ∑_{γ∈Uch(𝔾)} n_{γ}γ , then n_{γ} ∈ ℤ,

and for any 𝕃–adapted choice of ϕ, R_{L}^{G}(ρ_{λ}^{L^{F}}) = ∑_{γ∈Uch(𝔾)} n_{γ}ρ_{γ}^{G^{F}} .

(2) R_{𝕃}^{𝔾} sends the generic uniform functions on 𝕃 into the generic uniform functions on 𝔾 and induces Ind_{𝕃}^{𝔾} on the level of class functions, i.e., we have R_{𝕃}^{𝔾} · π_{u}^{𝕃} = π_{u}^{𝔾} · R_{𝕃}^{𝔾} and R_{𝕃}^{𝔾} · Φ^{𝕃} = Φ^{𝔾} · Ind_{𝕃}^{𝔾}.

(3) If 𝕄 is a Levi subgroup of 𝔾 and if 𝕃 is a Levi subgroup of 𝕄, then R_{𝕃}^{𝔾} = R_{𝕄}^{𝔾} · R_{𝕃}^{𝕄} .

Sketch of proof of 1.33.

Notice that once (1) is proved, (2) and (3) are just translations of known properties of the map R_{L}^{G}. We prove (1).

Let (G, F) be associated with 𝔾 as above.

There exists an embedding π : G → G̃ where G̃ has a connected center and π(G) contains the derived group of G̃. Let L̃ = π(L), and let π_{L} be the restriction of π to L. Since for any γ ∈ Uch(𝔾) we have ρ_{γ}^{G^{F}} = ρG̃^{F} ; γ · π (cf. [DiMi2], 13.20), and RG̃ L̃(ρL̃^{F} ; γ) · π = R_{L}^{G}(ρL̃^{F} ; γ · π_{L}) (cf. [DiMi2], 13.22), it is enough to prove the theorem for G̃, i.e., for a group with connected center. Applying the same arguments to the quotient map G̃ → G̃_{ad} (where G̃_{ad} is the adjoint group) which has a connected kernel (so [DiMi2], 13.22, still applies), we may assume G adjoint.

When G is adjoint, (G, F) is a direct product of the form (G_{i}^{(a_{i})}, F^{(a_{i})}) (restriction of scalars from 𝔽_{q^{a_{i}}} to 𝔽_{q} of (G_{i}, F)) where G_{i} are simple groups. Since all our constructions behave nicely with respect to products, we may assume that the group is of the form (G^{(a)}, F^{(a)}) with G simple.

Now the isomorphisms G^{(a)F^{(a)}} ≃ G^{F_{a}} and L^{(a)F^{(a)}} ≃ L^{F_{a}} (see 1.4) “commute” with R_{L}^{G}, and map isomorphically E(G^{(a)F^{(a)}}, 1) to E(G^{F_{a}}, 1). So we can reduce our problem to the case of (G, F_{a}), i.e., to the case where G is simple. We will use results of Shoji [Sho1], [Sho2] in this case.

Let m be sufficiently divisible so that (G, F^{m}) is split, i.e., the generic group 𝔾_{0} associated to it is such that ϕ = 1. It follows from 1.28 that there exists a map U(ϕ) on Uch(𝔾_{0}) such that F (ρ_{γ}^{G^{F^{m}}}) = ρ_{U(ϕ)(γ)}^{G^{F^{m}}} . Moreover, the set of fixed points Uch(𝔾_{0})^{U(ϕ)} has the same cardinality as Uch(𝔾). In this context, for any γ ∈ Uch(𝔾_{0})^{U(ϕ)}, Shoji ([Sho2], 3.1.2 and 3.1.3) defines an element of ℚ Uch(𝔾) (that he calls R_{x} or Rx̂ depending on the case), which we shall denote here by Sh^{0}γ (here Sh stands for “Shintani” — see [DiMi1] for the notation used here, such as Sh_{F^{m}/F}), such that:

• γ ↦ Sh^{0}γ is an isometry, and the set {Sh^{0}γ; (γ ∈ Uch(𝔾_{0})^{U(ϕ)})} spans ℚ Uch(𝔾),

• for m divisible enough, there is a well-defined G^{F^{m}}–class function ρ̃_{γ} on G^{F^{m}}.F (whose values are the product of the value of a suitable extension of ρ_{γ}^{G^{F^{m}}} to the group G^{F^{m}}·⟨F ⟩ by a root of unity) such that Sh_{F^{m}/F} (ρ̃_{γ}) = ρ_{Sh^{0}γ}^{G^{F}} (where this last function is defined by linearly

extending the notation ρ_{γ}^{G^{F}}).

We shall give a formula for R_{L}^{G} in the basis ρ_{Sh^{0}γ}^{G^{F}} which will make clear it is generic.

Let m be divisible enough so that (L, F^{m}) is split and let 𝕃_{0} be the corresponding generic group. Let λ ∈ Uch(𝕃_{0})^{U(ϕ)}. By Harish-Chandra theory, there exists a Levi subgroup M of L, such that (M, F^{m}) is split, and an element μ ∈ Uch(𝕄_{0}) (where 𝕄_{0} is the generic group corresponding to (M, F^{m})) such that ρ_{μ}^{M^{F^{m}}} is cuspidal and ρ_{λ}^{L^{F^{m}}} is a constituent of R_{M}^{L}(ρ_{μ}^{M^{F^{m}}}). Fur-

thermore, since ρ_{λ}^{L^{F^{m}}} is F -stable, we may choose (M, ρ_{μ}^{M^{F^{m}}}) to be F -stable. Let 𝕄 be a generic group corresponding to (M, F) and let W_{𝕄}vϕ be the corresponding coset. Set W_{𝔾_{0}}(𝕄_{0}, μ) := N_{W_{𝔾_{0}}} (𝕄_{0}, μ)/W_{𝕄_{0}}. Then the F -stability of μ implies a natural action of vϕ on W_{𝔾_{0}}(𝕄_{0}, μ) and on W_{𝕃_{0}}(𝕄_{0}, μ), and the extension ρ̃_{λ} corresponds naturally to a function χ_{λ} on W_{𝕃_{0}}(𝕄_{0}, μ)vϕ (an extension of the vϕ-invariant irreducible character of W_{𝕃_{0}}(𝕄_{0}, μ) corresponding to ρ_{λ}^{L^{F^{m}}} by Howlett–Lehrer–Lusztig theory). Conversely any W_{𝕃_{0}}(𝕄_{0}, μ)- invariant function on W_{𝕃_{0}}(𝕄_{0}, μ)vϕ corresponds to a linear combination of extensions ρ̃_{λ}.

It is proved in [DiMi3], 9.7, that (assuming Shoji’s results)

R_{L}^{G}(ρ_{Sh^{0}λ}^{L^{F}}) = ρ_{Sh^{0}λ′}^{G^{F}}

where λ′ is the linear combination of extensions ρ̃_{γ} corresponding to the function Ind_{W_{𝕃_{0}} (𝕄_{0},μ)vϕ}^{W_{𝔾_{0}} (𝕄_{0},μ)vϕ}χ_{λ}. □

Note that

• Uch(𝕋_{wϕ}) consists of only one element denoted by 1, and R_{𝕋_{wϕ}}^{𝔾}(1) = R_{wϕ}^{𝔾}, hence (by 1.33, (3)), for all w′ϕ ∈ W_{𝕃}wϕ, we have R_{𝕃}^{𝔾}(R_{w′ϕ}^{𝕃}) = R_{w′ϕ}^{𝔾},

• similarly, the image of 1^{𝕃} by the isometric embedding Φ^{𝔾} of 1.29 is denoted by 1^{𝕃} or just 1; we have Deg(R_{𝕃}^{𝔾}(1)) = D(Ind_{𝕃}^{𝔾}1^{𝕃}) (as promised after 1.17).

1.34. Proposition. For all ζ ∈ ℚ[x] Uch(𝕃) we have

DegR_{𝕃}^{𝔾}(ζ) = Deg(R_{𝕃}^{𝔾}(1))Deg(ζ) .

Proof. This is an immediate consequence of 1.20 and of the formula R_{𝕃}^{𝔾}(Φ_{β}^{𝕃}) = Φ_{Ind_{𝕃}^{𝔾}β}^{𝔾} , for all β ∈ CF(𝕃). □

The adjoint map of R_{𝕃}^{𝔾} is a map ^{∗}R_{𝕃}^{𝔾}: ℚ[x] Uch(𝔾) → ℚ[x] Uch(𝕃) which is in particular such that

• for all α ∈ CF(𝔾), we have ^{∗}R_{𝕃}^{𝔾}(Φ_{α}^{𝔾}) = Φ_{Res_{𝕃}^{𝔾} α}^{𝕃} ,

• for all wϕ ∈ W_{𝔾}ϕ, we have ^{∗}R_{𝕋_{wϕ}}^{𝔾}(γ) = m_{γ}(wϕ)1 .

The following theorem, which extends the Mackey formula (see 1.14) to the preceding context, is a consequence of an unpublished result of Deligne (see [DiMi2], 11.6) and of 1.33, (1) and 1.24, (3).

1.35. Theorem.

(1) Let 𝕃 and 𝕄 be two Levi subgroups of 𝔾. Then

∗R_{𝕄}^{𝔾} · R_{𝕃}^{𝔾} = ∑_{w∈W_{𝕄}\S_{W_{𝔾}} (𝕃,𝕄)/W_{𝕃}} R_{𝕄∩^{w}𝕃}^{𝕄} · ^{∗}R_{𝕄∩^{w}𝕃}^{^{w}𝕃} · ad(w) .

(2) Suppose that 𝕄_{1} and 𝕄_{2} are two d–split Levi subgroups containing the minimal d–split Levi subgroup 𝕃. Then

∗R_{𝕄_{2}}^{𝔾} · R_{𝕄_{1}}^{𝔾} = ∑_{w∈W_{𝕄_{2}} (𝕃)\W_{𝔾}(𝕃)/W_{𝕄_{1}} (𝕃)} R_{𝕄_{2}∩^{w}𝕄_{1}}^{𝕄_{2}} · ^{∗}R_{𝕄_{2}∩^{w}𝕄_{1}}^{^{w}𝕄_{1}} · ad(w) .

(Indeed, Deligne proved the Mackey formula for actual finite reductive groups under the assumption that q is large. Thus 1.35 results from the genericity of R_{𝕃}^{𝔾}.)

The arguments at the beginning of the proof of 1.33, together with 1.28, prove the following two propositions that we will use later to reduce some questions to the case of simple groups.

Going to the adjoint group.

The reader may refer to the remark before 1.27 to prove the following statement.

1.36. Proposition. Let 𝔾 = ((X, R, Y, R^{∨}), W ϕ) be a generic group, and let 𝕃 = ((X, R′, Y, R^{′∨}), W_{R′}wϕ) be a generic Levi subgroup of 𝔾. The sets Uch(𝔾) and Uch(𝔾_{ad}) may be identified, and similarly the sets Uch(𝕃) and Uch(\overline{𝕃}), where \overline{𝕃} is the generic Levi subgroup of 𝔾_{ad} defined by

\overline{𝕃} = ((Q(R), R′, P (R^{∨}), R^{′∨}), W_{R′}wϕ).

Then the following diagram is commutative:

ℤ Uch(𝔾) ⟶^{∼} ℤ Uch(𝔾_{ad})
R_{𝕃}^{𝔾}↑  ↑^{R𝔾_{ad} ; \overline{𝕃}}
ℤ Uch(𝕃) ⟶^{∼} ℤ Uch(\overline{𝕃})

Remark. If G is the algebraic group associated to 𝔾, and π : G → G_{ad} is the natural isogeny, and if L is associated to 𝕃, then π(L) is associated to \overline{𝕃}.

Lifting scalars.

1.37. Proposition. Let 𝔾 be a generic finite reductive group and let a ∈ ℕ. For all 𝔾 there is a natural identification σ_{𝔾}^{(a)} between Uch(𝔾) and Uch(𝔾^{(a)}) such that the following diagram is commutative:

ℤ Uch(𝔾^{(a)}) ⟶^{σ_{𝔾}^{(a)}} ℤ Uch(𝔾)
R_{𝕃^{(a)}}^{𝔾^{(a)}} ↑  ↑^{R_{𝕃}^{𝔾}}
ℤ Uch(𝕃^{(a)}) ⟶^{σ_{𝕃}^{(a)}} ℤ Uch(𝕃)

We will use the following result, whose proof is based on an argument given by Geck [Ge] in the case of a torus.

1.38. Theorem. Let γ ∈ Uch(𝔾) and 𝕃 be a Levi subgroup of 𝔾. Then the

polynomial Deg(γ) divides \frac{|𝔾|}{|𝕃|}Deg(^{∗}R_{𝕃}^{𝔾}(γ)) (in ℚ[x]).

Proof. Let (G, L, T, F) be a quadruple associated to 𝔾 and 𝕃 for a choice of q. Whenever q is large enough, there exists some regular element s ∈ Z^{o}(L)^{F}, i.e., an element such that C_{G}(s) = L. Then by the “Curtis–type formula” (see, e.g. [DiMi2], 12.5) we have (^{∗}R_{L}^{G}(ρ_{γ}^{G^{F}}))(s) = (ρ_{γ}^{G^{F}})(s). On the other hand, since ^{∗}R_{L}^{G}(ρ_{γ}^{G^{F}}) is a sum of unipotent characters and s ∈ Z(L), we have (^{∗}R_{L}^{G}(ρ_{γ}^{G^{F}}))(s) = (^{∗}R_{L}^{G}(ρ_{γ}^{G^{F}}))(1). Thus, using that for any

s the expression \frac{|G^{F} /C_{G^{F}} (s)|(ρ_{γ}^{G^{F}})(s)}{(ρ_{γ}^{G^{F}})(1)} is an algebraic integer, we get that

\frac{|G^{F} /L^{F} |(^{∗}R_{L}^{G}(ρ_{γ}^{G^{F}}))(1)}{(ρ_{γ}^{G^{F}})(1)} is an integer. Since this is true for an infinity of

values of q, we must have divisibility of the corresponding polynomials in ℚ[x], whence the theorem. □

When 𝕃 is a maximal torus 𝕋_{wϕ}, we get

1.39. Whenever γ ∈ Uch(𝔾) and wϕ ∈ W_{𝔾}ϕ are such that m_{γ}(wϕ) ≠ 0, then the polynomial Deg(γ) divides x^{N(𝔾)}Deg(R_{wϕ}^{𝔾}) (in ℚ[x]).

This was first proved by Boyce [Boy].

2. d–cuspidality and the uniform theory

A. d–cuspidality.

In §2.A, we define the notion of d–cuspidality of unipotent class functions as the generalization of the usual notion of cuspidality (which must be viewed as the 1-cuspidality).

Let ρ_{γ}^{G^{F}} be a unipotent character of G^{F}. We say that ρ_{γ}^{G^{F}} is d–cuspidal if, whenever T is a maximal F –stable torus of G such that ^{∗}R_{T}^{G}(ρ_{γ}^{G^{F}}) ≠ 0, then the Sylow Φ_{d}–subgroup of T is contained in Z(G). The translation in terms of generic groups (independent of q) is achieved by the following definition.

2.1. Definition.

(1) A class function α on 𝔾 is d–cuspidal if whenever wϕ ∈ W ϕ is such that α(wϕ) ≠ 0, then ker(Φ_{d}(wϕ)) is orthogonal to all the roots of 𝔾.

(2) ψ ∈ ℚ[x] Uch(𝔾) is said to be d–cuspidal if the corresponding class function m_{ψ} is d–cuspidal, i.e., whenever 𝕋 is a maximal torus of 𝔾 such that ^{∗}R_{𝕋}^{𝔾}(ψ) ≠ 0, then the Sylow Φ_{d}–subgroup of 𝕋 is contained in Rad(𝔾).

Remarks.

1. Note that for d = 1 we get the usual notion of cuspidality for the elements of Uch(𝔾) (i.e., γ ∈ Uch(𝔾) is 1-cuspidal if and only if ρ_{γ}^{G^{F}} is a cuspidal character for any associated group G^{F}). Indeed, condition (2) above can be reformulated as follows: “ ^{∗}R_{𝕋}^{𝔾}(ψ) = 0 whenever 𝕋 contains some non-central Φ_{d}-subtorus 𝕊”. If 𝕃 is the centralizer of 𝕊 in 𝔾, condition (2) becomes: “whenever 𝕃 is a proper d-split Levi subgroup of 𝔾, the uniform projection of the function ^{∗}R_{𝕃}^{𝔾}(ψ) is zero” (in particular, being d-cuspidal is a property of π_{u}(ψ)). Now, for d = 1 and γ ∈ Uch(𝔾) the “positivity” of ordinary Harish-Chandra restriction shows that if ^{∗}R_{𝕃}^{𝔾}(γ) ≠ 0, then its degree is non zero, and so its uniform projection cannot vanish; thus we see that γ is 1–cuspidal if and only if ^{∗}R_{𝕃}^{𝔾}(γ) = 0 for any proper 1-split Levi subgroup, which is the usual notion of cuspidality.

2. In the case where 𝔾 = 𝔾𝕃_{n}, then W = 𝔖_{n}, ϕ = 1, and a d–cuspidal class function on 𝔖_{n} is a function which vanishes on all elements whose cycle decomposition contains a cycle of length a multiple of d.

Moreover, Uch(𝔾) may be viewed as the set of partitions of n. Thus it results from the Murnaghan–Nakayama formula (see for example [JaKe]) that γ is d–cuspidal if and only if the partition of n corresponding to γ has no d– hook (i.e., the d–cuspidal partitions are the d–cores of length n). Thus this paragraph may be considered as the generalization to all Weyl groups of the “yoga” of hooks for the symmetric groups.

2.2. Definition. Let 𝕃 be a d–split Levi subgroup of 𝔾. We denote by CF_{d}(𝔾, 𝕃) the set of all class functions α on 𝔾 with the following property: if wϕ ∈ W_{𝔾}ϕ is such that α(wϕ) ≠ 0, then there exists w′ ∈ W_{𝔾} such that ker Φ_{d}(wϕ)^{w′} is orthogonal to all roots of 𝕃.

Remarks.

• Let ψ ∈ ℚ[x] Uch(𝔾). Then m_{ψ} ∈ CF_{d}(𝔾, 𝕃) is equivalent to the following: whenever 𝕋 is a maximal torus of 𝔾 such that ^{∗}R_{𝕋}^{𝔾}(ψ) ≠ 0, then the maximal Φ_{d}–subgroup of 𝕋 is contained in a W_{𝔾}–conjugate of Rad(𝕃).

• CF_{d}(𝔾, 𝔾) is the set of all d–cuspidal functions.

• If 𝕃 is the centralizer of a Sylow Φ_{d}–subgroup of 𝔾, then CF_{d}(𝔾, 𝕃) = CF(𝔾).

2.3. Proposition. Let 𝕃 be a d–split Levi subgroup of 𝔾.

(1) Let α ∈ CF_{d}(𝔾, 𝕃). Then Res_{𝕃}^{𝔾} α is d–cuspidal.

(2) Let β be a d–cuspidal class function on 𝕃. Then Ind_{𝕃}^{𝔾}β ∈ CF_{d}(𝔾, 𝕃),

and

Res_{𝕃}^{𝔾} Ind_{𝕃}^{𝔾}β = ∑_{w∈W_{𝔾}(𝕃)} ^{w}β .

Proof. The first assertion is obvious. The second assertion is an application of formula 1.12. □

Generic degrees and d–cuspidality.

For a polynomial P ∈ ℚ[x], we denote by P_{d} the largest power of the cyclotomic polynomial Φ_{d} which divides P (i.e., the Φ_{d}–part of P).

2.4. Proposition. If ψ ∈ ℚ[x] Uch(𝔾) is d–cuspidal, then |𝔾_{ss}|_{d} divides Deg(ψ).

Proof. We have by 1.31

Deg(ψ) = \frac{1}{|W |} ∑_{w∈W} m_{ψ}(wϕ)Deg(R_{wϕ}^{𝔾}) ,

where m_{ψ} is d-cuspidal. It follows from definition 2.1 that the preceding sum need only be taken over those w’s such that the Sylow Φ_{d}–subgroup of 𝕋_{wϕ} is contained in Rad(𝔾), which implies that |𝔾_{ss}|_{d} divides |𝔾|/|𝕋_{wϕ}|. Since (cf. 1.18 and the definition of R_{wϕ}^{𝔾}) |𝔾|/|𝕋_{wϕ}| = ±x^{N(𝔾)}Deg(R_{wϕ}^{𝔾}), we deduce that |𝔾_{ss}|_{d} divides Deg(ψ). □

We may notice the following lemma, which is immediate by 1.30.

2.5. Lemma. For ψ ∈ ℚ[x] Uch(𝔾) we have

π_{u}^{𝔾}(ψ) = ∑_{[𝕋]_{W_{𝔾}}} \frac{1}{|W_{𝔾}(𝕋)|}R_{𝕋}^{𝔾}(^{∗}R_{𝕋}^{𝔾}(ψ)) .

We introduce a new definition which will be technically useful. Let us call “d–anisotropic maximal tori” of 𝔾 the maximal tori 𝕋 such that the Sylow Φ_{d}–subgroup of 𝕋 is contained in Rad(𝔾). We denote by T_{d}(𝔾) the set of all d–anisotropic maximal tori. For ψ ∈ ℚ[x] Uch(𝔾), we define the “d-cuspidal projection” of ψ as

(2.6)  c_{d}(ψ) := ∑_{[𝕋∈T_{d}(𝔾)]_{W_{𝔾}}} \frac{1}{|W_{𝔾}(𝕋)|}R_{𝕋}^{𝔾}(^{∗}R_{𝕋}^{𝔾}(ψ)) .

So we have by 1.34

(2.7)  Deg(c_{d}(ψ)) := ∑_{[𝕋∈T_{d}(𝔾)]_{W_{𝔾}}} \frac{1}{|W_{𝔾}(𝕋)|}Deg(R_{𝕋}^{𝔾}(1))Deg(^{∗}R_{𝕋}^{𝔾}(ψ)) ,

which, since Deg(^{∗}R_{𝕋}^{𝔾}(ψ)) = (^{∗}R_{𝕋}^{𝔾}(ψ), 1)_{𝕋} = (ψ, R_{𝕋_{wϕ}}^{𝔾}(1)), can be written:

Deg(c_{d}(ψ)) := \frac{1}{|W |} ∑ m_{ψ}(wϕ)Deg(R_{wϕ}^{𝔾})

where the sum is taken over those w’s such that the Sylow Φ_{d}–subgroup of 𝕋_{wϕ} is contained in Rad(𝔾). It is clear from the above proof of 2.4 that

(2.8) |𝔾_{ss}|_{d} divides Deg(c_{d}(ψ)), and if ψ is d–cuspidal, then c_{d}(ψ) = ψ.

The following result is specific to the elements of Uch(𝔾).

2.9. Proposition. If γ ∈ Uch(𝔾) and if |𝔾_{ss}|_{d} divides Deg(γ), then γ is d–cuspidal. In particular, for γ ∈ Uch(𝔾), the following statements are equivalent:

(i) γ is d–cuspidal (ii) Deg(γ)_{d} = |𝔾_{ss}|_{d}, (iii) Deg(γ) = Deg(c_{d}(γ)) .

Proof of 2.9. Assume that γ ∈ Uch(𝔾) is such that |𝔾_{ss}|_{d} divides Deg(γ). Whenever wϕ ∈ W ϕ is such that m_{γ}(wϕ) ≠ 0, it follows from remark 1.39 that (Deg(γ))_{d} divides Deg(R_{wϕ}^{𝔾}). Thus we see that whenever m_{γ}(wϕ) ≠ 0, |𝔾_{ss}|_{d} divides |𝔾|/|𝕋_{wϕ}|, which proves that m_{γ}, whence γ, is d–cuspidal. □

B. Regular unipotent characters.

In §2.B, we give various formulae for the computation of the unipotent part of the regular character — viewed here again in the generic context — and called “regular unipotent character of 𝔾”. These formulae will be used later on to prove some divisibility properties for characters of the actual finite groups.

The unipotent regular character of 𝔾 is the element of ℚ[x] Uch(𝔾) defined by the formula

(2.10)  UReg^{𝔾} := \frac{1}{|W |} ∑_{w∈W} Deg(R_{wϕ}^{𝔾}).R_{wϕ}^{𝔾} .

Since by 1.31, Deg(ψ) = (ψ, UReg^{𝔾})_{𝔾}, we see that

UReg^{𝔾} = ∑_{γ∈Uch(𝔾)} Deg(γ).γ .

We change the notation in formula 2.10 : instead of summing over w ∈ W, we sum over a set [𝕋]_{W_{𝔾}} of representatives for the W_{𝔾}-classes of maximal tori 𝕋_{wϕ}. We recall that W_{𝔾}(𝕋_{wϕ}) = C_{W} (wϕ) and we get :

(2.11)  UReg^{𝔾} = ∑_{[𝕋]_{W_{𝔾}}} \frac{Deg(R_{𝕋}^{𝔾}(1))}{|W_{𝔾}(𝕋)|}R_{𝕋}^{𝔾}(1) .

Similarly, choosing a natural integer d,

c_{d}(UReg^{𝔾}) = \frac{1}{|W |} ∑ Deg(R_{wϕ}^{𝔾}).R_{wϕ}^{𝔾} ,

where the sum is taken over those w’s such that the Sylow Φ_{d}–subgroup of 𝕋_{wϕ} is contained in Rad(𝔾). We have

c_{d}(UReg^{𝔾}) = ∑_{γ∈Uch(𝔾)} Deg(c_{d}(γ)).γ .

Performing a similar change of notation as in 2.10, we get:

(2.12)  c_{d}(UReg^{𝔾}) = ∑_{[𝕋∈T_{d}(𝔾)]_{W_{𝔾}}} \frac{Deg(R_{𝕋}^{𝔾}(1))}{|W_{𝔾}(𝕋)|}R_{𝕋}^{𝔾}(1) .

More generally, we have the following decomposition of uniform functions:

2.13. Proposition. Let ψ ∈ ℚ[x] Uch(𝔾). We have:

π_{u}^{𝔾}(ψ) = ∑_{[𝕄 d–split]_{W_{𝔾}}} \frac{1}{W_{𝔾}(𝕄)}R_{𝕄}^{𝔾}(c_{d}(^{∗}R_{𝕄}^{𝔾}(ψ))).

where the sum is over a set of representatives for the W_{𝔾}-classes of d-split Levi subgroups of 𝔾.

Proof. To prove this, we use the 2.14. Lemma. Let φ be a W_{𝔾}–stable function on the set of all maximal tori of 𝔾. Let 𝕄 be a d–split Levi subgroup of 𝔾. Then

∑_{[𝕋; 𝕋_{d}=_{W_{𝔾}} Rad(𝕄)_{d}]_{W_{𝔾}}} \frac{φ(𝕋)}{|W_{𝔾}(𝕋)|} = \frac{1}{|W_{𝔾}(𝕄)|} ∑_{[𝕋∈T_{d}(𝕄)]_{W_{𝕄}}} \frac{φ(𝕋)}{|W_{𝕄}(𝕋)|} .

Proof of 2.14. Since 𝕄 = C_{𝔾}(Rad(𝕄)_{d}), we see that 𝕋_{d} = Rad(𝕄)_{d} if and only if 𝕋 is a d–anisotropic maximal torus of 𝕄. Moreover, since N_{W_{𝔾}}(𝕄) = N_{W_{𝔾}}(Rad(𝕄)_{d}), two such tori are W_{𝔾}–conjugate if and only if they are N_{W_{𝔾}}(𝕄)–conjugate. It then follows that

∑_{[𝕋; 𝕋_{d}=_{W_{𝔾}} Rad(𝕄)_{d}]_{W_{𝔾}}} \frac{φ(𝕋)}{|W_{𝔾}(𝕋)|} = ∑_{[𝕋; 𝕋_{d}=Rad(𝕄)_{d}]_{N_{W_{𝔾}} (𝕄)}} \frac{φ(𝕋)}{|W_{𝔾}(𝕋)|}

= ∑_{[𝕋; 𝕋_{d}=Rad(𝕄)_{d}]_{W_{𝕄}}} \frac{|W_{𝕄} : N_{W_{𝕄}}(𝕋)|}{|N_{W_{𝔾}}(𝕄) : N_{W_{𝔾}}(𝕋)|} \frac{φ(𝕋)}{|W_{𝔾}(𝕋)|}

= \frac{1}{|W_{𝔾}(𝕄)|} ∑_{[𝕋∈T_{d}(𝕄)]_{W_{𝕄}}} \frac{φ(𝕋)}{|W_{𝕄}(𝕋)|} .

□

Applying this lemma to φ(𝕋) = R_{𝕋}^{𝔾}(^{∗}R_{𝕋}^{𝔾}ψ), we get from 2.5

π_{u}^{𝔾}(ψ) = ∑_{[𝕄 d–split]_{W_{𝔾}}} \frac{1}{|W_{𝔾}(𝕄)|} ∑_{[𝕋∈T_{d}(𝕄)]_{W_{𝕄}}} \frac{R_{𝕋}^{𝔾}(^{∗}R_{𝕋}^{𝔾}(ψ))}{|W_{𝕄}(𝕋)|}

which gives the proposition, using the definition of c_{d} and the equality R_{𝕋}^{𝔾} · ∗R_{𝕋}^{𝔾}(ψ) = R_{𝕄}^{𝔾} · R_{𝕋}^{𝕄} · ^{∗}R_{𝕋}^{𝕄}^{∗}R_{𝕄}^{𝔾}(ψ). □

2.15. Proposition. Let 𝕃 be a generic Levi subgroup of 𝔾. Then

∗R_{𝕃}^{𝔾}(UReg^{𝔾}) = Deg(R_{𝕃}^{𝔾}(1))UReg^{𝕃} .

Proof. For ν ∈ Uch(𝕃), we have

(^{∗}R_{𝕃}^{𝔾}(UReg^{𝔾}), ν)_{𝕃} = (UReg^{𝔾}, R_{𝕃}^{𝔾}(ν))_{𝔾}
= Deg(R_{𝕃}^{𝔾}(ν)) = Deg(R_{𝕃}^{𝔾}(1)).Deg(ν)
= Deg(R_{𝕃}^{𝔾}(1)) (UReg^{𝕃}, ν)_{𝕃} .

□ 2.16. Proposition. We have

UReg^{𝔾} = ∑_{[𝕄 d–split]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(1))}{|W_{𝔾}(𝕄)|}R_{𝕄}^{𝔾}(c_{d}(UReg^{𝕄})) .

Proof. It is immediate from 2.11, 2.13, 2.15 and the multiplicativity of the degrees Deg(R_{𝕋}^{𝔾}(1)) = Deg(R_{𝕄}^{𝔾}(1))Deg(R_{𝕋}^{𝕄}(1)). □

The following statement generalizes proposition 2.4.

2.17. Proposition. Let ψ ∈ ℚ[x] Uch(𝔾) be such that m_{ψ} ∈ CF_{d}(𝔾, 𝕃). (1) |𝕃_{ss}|_{d} divides both Deg(ψ) and Deg(^{∗}R_{𝕃}^{𝔾}(ψ)).

(2) We have

\frac{Deg(ψ)}{|𝕃_{ss}|_{d}} ≡ \frac{Deg(R_{𝕃}^{𝔾}(1))}{|W_{𝔾}(𝕃)|} \frac{Deg(^{∗}R_{𝕃}^{𝔾}(ψ))}{|𝕃_{ss}|_{d}} mod Φ_{d} .

Proof. We know (cf. 2.3) that ^{∗}R_{𝕃}^{𝔾}(ψ) is d–cuspidal, and it follows from 2.4 that |𝕃_{ss}|_{d} divides Deg(^{∗}R_{𝕃}^{𝔾}(ψ)).

Moreover, we have

Deg(ψ) = (ψ, UReg^{𝔾})_{𝔾} = ∑_{[𝕄]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(1))}{|W_{𝔾}(𝕄)|}(ψ, R_{𝕄}^{𝔾}(c_{d}(UReg^{𝕄})))_{𝔾} .

Since (by 2.12)

(ψ, R_{𝕄}^{𝔾}(c_{d}(UReg^{𝕄})))_{𝔾} = ∑_{[𝕋; 𝕋∈T_{d}(𝕄)]_{W_{𝕄}}} \frac{Deg(R_{𝕋}^{𝕄}(1))}{|W_{𝕄}(𝕋)|}(ψ, R_{𝕋}^{𝔾}(1))_{𝔾} ,

we see that this scalar product is zero unless 𝕋_{d} ⊂_{W_{𝔾}} Rad(𝕃), whence that

Φ_{d} divides \frac{(ψ, R_{𝕄}^{𝔾}(c_{d}(UReg^{𝕄})))_{𝔾}}{|𝕃_{ss}|_{d}} unless 𝕋_{d} =_{W_{𝔾}} Rad(𝕃). This proves that

\frac{(ψ, R_{𝕄}^{𝔾}(c_{d}(UReg^{𝕄})))_{𝔾}}{|𝕃_{ss}|_{d}} ≡ 0 mod Φ_{d}

unless 𝕄 =_{W_{𝔾}} 𝕃, from which 2.17 is immediate. □

C. Regular characters.

In §2.C, we define a suitable generic version of some projections of the regular character which occur in block theory (see [BrMi]), and we establish some technical formulae.

Introduction: the regular character of E_{π}(G^{F} , 1).

In what follows we shall give a “generic version” of the series E_{π}(G^{F} , 1) introduced in [BrMi], and we start by computing the regular character of this series in a particular case, in order to justify the notation which will be introduced in the generic case.

Notation and prerequisites.

Let (G, T, F) be a (q, ϕ)–triple associated with 𝔾.

We recall that there is a natural indexing of the set of minimal closed non trivial unipotent subgroups of G normalized by T by the root system R of 𝔾. For α ∈ R, we denote by U_{α} the corresponding unipotent “root subgroup”.

Notice that, since [G, G] is connected, we have

(G/[G, G])^{F} = G^{F} /[G, G]^{F} .

2.18. Definition. Let 𝕂 be a characteristic zero field, containing enough roots of unity for G^{F}.

(1) We denote by AbIrr(G^{F}) the group of 𝕂–characters of the abelian group G^{F} /[G, G]^{F}, considered as characters of G^{F}. We denote by AbReg^{G^{F}} = ∑_{θ∈AbIrr(G^{F})} θ the corresponding regular character.

(2) For a set of prime numbers π, we denote by Ab_{π} Irr(G^{F}) the subgroup of AbIrr(G^{F}) consisting of characters whose order is a π–number. We denote by Ab_{π} Reg^{G^{F}} = ∑_{θ∈Ab_{π} Irr(G^{F})} θ the corresponding regular character.

Notice that the notation AbIrr(G^{F}) is slightly abusive, since G^{F} /[G, G]^{F} depends on G and F, and not only on G^{F}.

Whenever M is a Levi subgroup of G, we have M = Z^{o}(M)[M, M]. In particular for a Levi subgroup L of M we have M = L[M, M], and M/[M, M] = L/L∩[M, M]. Thus M^{F} /[M, M]^{F} is a quotient of L^{F} /[L, L]^{F}, and the restriction of characters induces inclusions

AbIrr(G^{F}) ⊆ AbIrr(M^{F}) ⊆ AbIrr(L^{F}) .

From now on, we assume that (2.19) π is a set of good prime numbers for 𝔾.

Centralizers of π–elements.

Since T = 𝔽̄_{p}^{×} ⊗ Y, the group 𝔽̄_{p}^{×} ⊗ X is identified with Hom(T, 𝔽̄_{p}^{×}). For s ∈ T and χ ∈ X, we denote by χ(s) the corresponding element of 𝔽̄_{p}^{×}.

Let s ∈ T. We set

(2.20)  R_{s} := {α ; (α ∈ R)(α(s) = 1)} ,

and we denote by G(s) the connected centralizer C_{G}^{o}(s) of s in G.

2.21. Proposition. Let s be a π–element of T.

(1) R_{s} is a parabolic subsystem of R.

(2) G(s) is the Levi subgroup of G generated by T and by the U_{α} for α ∈ R_{s}.

Proof. For (1), cf. [GeHi], 2.1. For (2), see for example [BoTi], 3.4, and also [DiMi2], 2.3. □

Centralizers of π–characters.

Let θ be a character of T^{F}, whose order is divisible only by primes in π : we then say that θ is a π–character of T^{F}.

Suppose chosen once for all an isomorphism 𝔽̄_{p}^{×}⟶^{∼}(ℚ/ℤ)_{p′} . Then it follows from [DeLu], (5.2.3), that we have an exact sequence

(2.22)  {0} ⟶ Y ⟶^{F −1} Y ⟶ T^{F} ⟶ {1} ,

which allows us to view T^{F} as a quotient of Y, and thus to define the image θ(y) of an element y ∈ Y through a character θ of T^{F}. For such a character, we then define the closed subsystem

(2.23)  R_{θ} := {α ; (α ∈ R)(θ(α^{∨}) = 1)} ,

and the subgroup G(T, θ) of G by

G(T, θ) := T.⟨U_{α}⟩_{α∈R_{θ}} .

2.24. Proposition. Let θ be a π–character of T^{F}.

(1) The subsystem R_{θ} is parabolic in R.

(2) The group G(T, θ) is the largest F –stable Levi subgroup M of G such

that • T ⊂ M,

• θ belongs to the image of AbIrr(M^{F}) in AbIrr(T^{F}).

Proof of 2.24.

(1) We may assume that T is part of a (q, ϕ)–triple associated to 𝔾 and let (G^{∗}, T^{∗}, F^{∗}) be a (q, ϕ^{∨}^{−1})–triple associated with 𝔾^{∗}. We choose an isomorphism

(*)  T^{∗}^{F^{∗}} ⟶^{∼}Irr(T^{F}) .

The reciprocal image of θ through the isomorphism (*) is a π–element s ∈ T^{∗}^{F^{∗}}. The system R_{θ} is then the dual of R_{s}^{∨} (see 2.20), hence is parabolic by 2.21, (1).

(2) The second assertion of 2.24 results from [DeLu], proposition 5.11, (i), and from the fact that a closed connected subgroup H of G containing T is generated by T and by {U_{α} ; (α ∈ R)(U_{α} ⊂ H)} (see [BoTi], 3.4). □

We set G^{F} (T, θ) := G(T, θ)^{F} .

Following Lusztig’s notation, we denote by E(G^{F} , 1) the set of unipotent characters of G^{F}. For any ρ ∈ E(G^{F} (T, θ), 1) we then have (see [Lu2]) a well defined irreducible character of G^{F}, defined by the formula

χ_{(G(T,θ),θ,ρ)}^{G^{F}} := ε_{G}ε_{G(T,θ)}R_{G(T,θ)}^{G}(θρ)

(this is a particular case of the “Jordan decomposition of characters”).

Two characters χ_{(G(T,θ),θ,ρ)}^{G^{F}} and χ_{(G(T′,θ′),θ′,ρ′)}^{G^{F}} are equal if and only if the triples (G(T, θ), θ, ρ) and (G(T′, θ′), θ′, ρ) are G^{F} –conjugate (see [Lu2], and also [DeLu], 5.20).

A particular case.

We assume now that:

(2.25) Every π–element of G^{F} is G^{F} –conjugate to an element of T^{F}.

We shall see in §4 that the preceding hypothesis is satisfied if π = {ℓ} for a “large” prime number ℓ (“d-adapted” in the sense of §5) and T contains a Sylow Φ_{d}-subgroup of G.

It is then easy to see that the regular character of the series E_{π}(G^{F} , 1) (see [BrMi], above 2.1) is given by the following formula:

(2.26)  Reg_{π}^{G^{F}} := ∑_{[(G(T,θ),θ,ρ)]_{G^{F}}} χ_{(G(T,θ),θ,ρ)}^{G^{F}}(1).χ_{(G(T,θ),θ,ρ)}^{G^{F}} .

For an F –stable Levi subgroup M of G containing T, we define

(2.27)  Θ_{G,π}^{M^{F}} := ∑_{{θ; (θ∈Ab_{π} Irr(M^{F}))(M=G(T,θ))}} θ .

2.28. Lemma. Under the preceding hypothesis, we have

Reg_{π}^{G^{F}} = ∑_{[M]_{G^{F}}} \frac{Deg(R_{M}^{G}(1))}{|W_{G^{F}} (M)|}R_{M}^{G}(Θ_{G,π}^{M^{F}}. UReg^{M^{F}})

= ∑_{[(M,μ)]_{G^{F}}} \frac{Deg(R_{M}^{G}(μ))}{|W_{G^{F}} (M, μ)|}R_{M}^{G}(Θ_{G,π}^{M^{F}}.μ) ,

where M runs over representatives of G^{F} -conjugacy classes of F -stable Levi subgroups of G, where μ runs over the set E(M^{F} , 1) of irreducible unipotent characters of M^{F}, where

UReg^{M^{F}} := ∑_{μ∈E(M^{F} ,1)} μ(1)μ

and

W_{G^{F}} (M) := N_{G^{F}} (M)/M^{F} , W_{G^{F}} (M, μ) := N_{G^{F}} (M, μ)/M^{F} .

The proof is left to the reader (it is immediate once the notation is understood).

We note that the sum in the above lemma is over the F –stable Levi subgroups M of G for which there exists an element θ ∈ Ab_{π} Irr(T^{F}) such that M is G^{F} –conjugate to G(T, θ). These will be generically the “π-split” Levi subgroups of G.

The generic formalism.

Lemma 2.28 justifies the introduction of the following formalism, meant to define a “generic” version of the character Reg_{π}^{G^{F}} .

• The group W_{𝔾} acts naturally on the set of pairs (𝕄, μ) (extending remark 1.27 to isomorphisms), where 𝕄 is a d–split generic Levi subgroup of 𝔾 and where μ ∈ Uch(𝕄); we set W_{𝔾}(𝕄, μ) = N_{W_{𝔾}}(𝕄, μ)/W_{𝕄}. We denote by R_{d}(𝔾) the free ℚ[x]–module on the set of W_{𝔾}-conjugacy classes of such pairs, with basis denoted by {χ_{(𝕄,μ)}^{𝔾}}.

We define (cf. 2.28)

(2.29)  Reg_{d}^{𝔾} := ∑_{[(𝕄,μ)]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(μ))}{|W_{𝔾}(𝕄, μ)|}.χ_{(𝕄,μ)}^{𝔾} .

• For any generic Levi subgroup ℕ of 𝔾, we denote by A_{d,𝔾}(ℕ) the free ℚ[x]–module on the set of pairs (𝕄, ν) such that 𝕄 is a d–split Levi subgroup of 𝔾 containing ℕ and ν ∈ Uch(ℕ), with basis denoted by {Θ_{𝔾,𝕄}^{ℕ} ⊗ ν} .

We set Θ_{𝔾,𝕄}^{ℕ} := Θ_{𝔾,𝕄}^{ℕ} ⊗ 1 . For a d–split generic Levi subgroup 𝕄 of 𝔾, we set Θ_{𝔾}^{𝕄} := Θ_{𝔾,𝕄}^{𝕄} .

[Θ_{𝔾,𝕄}^{ℕ} ⊗ ν should be viewed as the formal version of Res_{N^{F}}^{M^{F}} (Θ_{G,π}^{M^{F}})ρ_{ν}^{N^{F}}.]

• We define R_{ℕ}^{𝔾} : A_{d,𝔾}(ℕ) → R_{d}(𝔾) by the formulae (2.30)

\{ \begin{matrix}R_{𝕄}^{𝔾}(Θ_{𝔾}^{𝕄} ⊗ μ) & = χ_{(𝕄,μ)}^{𝔾} for ℕ = 𝕄, a d–split Levi subgroup of 𝔾, \\ R_{ℕ}^{𝔾}(Θ_{𝔾,𝕄}^{ℕ} ⊗ ν) & = R_{𝕄}^{𝔾}(Θ_{𝔾}^{𝕄} ⊗ R_{ℕ}^{𝕄}(ν)) for ℕ ⊂ 𝕄, 𝕄 d–split. \end{matrix}

We shall see that the map R_{𝕃}^{𝔾} of 1.33 identifies to a special case of this one.

• We set

(2.31)  Ab_{d} Reg^{ℕ} := ∑_{{𝕄; (ℕ⊆𝕄)(𝕄 d–split)}} Θ_{𝔾,𝕄}^{ℕ} .

[Note that Ab_{d} Reg^{ℕ} should be viewed as the formal version of Ab_{π} Reg^{N^{F}} := ∑_{θ∈Ab_{π} Irr(N^{F})} θ.]

With this notation, it follows from definition 2.29 that

(2.32)

Reg_{d}^{𝔾} = ∑_{[(𝕄,μ)]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(1))}{|W_{𝔾}(𝕄, μ)|}R_{𝕄}^{𝔾}(Θ_{𝔾}^{𝕄} ⊗ Deg(μ).μ)

= ∑_{[𝕄]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(1))}{|W_{𝔾}(𝕄)|}R_{𝕄}^{𝔾}(Θ_{𝔾}^{𝕄} ⊗ UReg^{𝕄}) .

The following theorem is the analogue of 2.16.

2.33. Theorem.

(1)

Reg_{d}^{𝔾} = ∑_{[𝕋]_{W_{𝔾}}} \frac{Deg(R_{𝕋}^{𝔾}(1))}{|W_{𝔾}(𝕋)|}R_{𝕋}^{𝔾}(Ab_{d} Reg^{𝕋}) ,

where 𝕋 runs over the set of maximal tori of 𝔾. (2)

Reg_{d}^{𝔾} = ∑_{[(𝕄,μ)]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(1))}{|W_{𝔾}(𝕄, μ)|}R_{𝕄}^{𝔾}(Ab_{d} Reg^{𝕄} ⊗ Deg(c_{d}(μ)).μ)

= ∑_{[𝕄]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(1))}{|W_{𝔾}(𝕄)|}R_{𝕄}^{𝔾}(Ab_{d} Reg^{𝕄} ⊗ c_{d}(UReg^{𝕄})) ,

where 𝕄 runs over the set of d–split Levi subgroups of 𝔾.

Proof. (1) By 2.32, we see that

Reg_{d}^{𝔾} = \frac{1}{|W_{𝔾}|} ∑_{𝕄} |W_{𝕄}|Deg(R_{𝕄}^{𝔾}(1))R_{𝕄}^{𝔾}(Θ_{𝔾}^{𝕄} ⊗ UReg^{𝕄}) .

Also, by 2.11, we see that

UReg^{𝕄} = \frac{1}{|W_{𝕄}|} ∑_{𝕋⊆𝕄} Deg(R_{𝕋}^{𝕄}(1))R_{𝕋}^{𝕄}(1) .

Thus we get, using 2.30,

Reg_{d}^{𝔾} = \frac{1}{|W_{𝔾}|} ∑_{𝕋⊆𝔾} Deg(R_{𝕋}^{𝔾}(1))( ∑_{\{𝕄; 𝕋⊆𝕄\}} R_{𝕋}^{𝔾}(Θ_{𝔾,𝕄}^{𝕋} ⊗ 1))

and the result follows from 2.31.

(2) We extend c_{d} of 2.7 to Reg_{d}^{𝔾} by setting

c_{d}(Reg_{d}^{𝔾}) := ∑_{[𝕋∈T_{d}(𝔾)]_{W_{𝔾}}} \frac{Deg(R_{𝕋}^{𝔾}(1))}{|W_{𝔾}(𝕋)|}R_{𝕋}^{𝔾}(Ab_{d} Reg^{𝕋}) .

By lemma 2.14 applied with φ(𝕋) = Deg(R_{𝕋}^{𝔾}(1))R_{𝕋}^{𝔾}(Ab_{d} Reg^{𝕋}), we see that

Reg_{d}^{𝔾} = ∑_{[𝕄; 𝕄 d–split]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(1))}{|W_{𝔾}(𝕄)|}R_{𝕄}^{𝔾}(c_{d}(Reg_{d}^{𝕄})) .

Thus it suffices to prove the following lemma.

2.34. Lemma. We have c_{d}(Reg_{d}^{𝔾}) = Ab_{d} Reg^{𝔾} ⊗ c_{d}(UReg^{𝔾}) .

Proof of 2.34. We have

Ab_{d} Reg^{𝔾} ⊗ c_{d}(UReg^{𝔾}) = ∑_{[𝕋∈T_{d}(𝔾)]_{W_{𝔾}}} \frac{Deg(R_{𝕋}^{𝔾}(1))}{|W_{𝔾}(𝕋)|}Ab_{d} Reg^{𝔾} ⊗ R_{𝕋}^{𝔾}(1) ,

and it suffices to check that, for 𝕋 ∈ T_{d}(𝔾), we have

Ab_{d} Reg^{𝔾} ⊗ R_{𝕋}^{𝔾}(1) = R_{𝕋}^{𝔾}(Ab_{d} Reg^{𝕋}) ,

which is immediate by 2.30 and 2.31, since 𝕋 is contained in no proper d–split Levi subgroup of 𝔾. □

□

In order to define a generic version of the adjoint ^{∗}R_{ℕ}^{𝔾} of R_{ℕ}^{𝔾} which may be “specialized” to the ordinary ^{∗}R_{N}^{G}, we must introduce more notation.

Let R_{d,𝔾}(ℕ) be the free ℚ[x]–module on the set of W_{ℕ}–conjugacy classes of pairs (𝕄, δ), where

• 𝕄 is a d–split Levi subgroup of 𝔾 such that 𝕄 ∩ ℕ is defined,

• δ ∈ Uch(𝕄 ∩ ℕ) , with basis denoted by {χ_{(𝕄,δ)}^{ℕ}} .

Remark. We must see χ_{(𝕄,δ)}^{ℕ} as R_{𝕄∩ℕ}^{ℕ}(Θ_{𝔾,𝕄}^{𝕄∩ℕ} ⊗ δ) .

We define R_{ℕ}^{𝔾}: R_{d,𝔾}(ℕ) → R_{d}(𝔾) by the formula

(2.35)  R_{ℕ}^{𝔾}(χ_{(𝕄,δ)}^{ℕ}) := R_{𝕄}^{𝔾}(Θ_{𝔾}^{𝕄} ⊗ δ) ,

and then we define ^{∗}R_{ℕ}^{𝔾}: R_{d}(𝔾) → R_{d,𝔾}(ℕ) in such a way as to fit with the Mackey formula, i.e., for any d–split pair (𝕄, μ) of 𝔾, we set

(2.36) ^{∗}R_{ℕ}^{𝔾}(χ_{(𝕄,μ)}^{ℕ}) := ∑_{W_{ℕ}\S_{W_{𝔾}} (𝕄,ℕ)/W_{𝕄}} R_{ℕ∩^{w}𝕄}^{ℕ} (Θ_{𝔾,^{w}𝕄}^{^{w}𝕄∩ℕ} ⊗ ^{∗}R_{^{w}𝕄∩ℕ}^{^{w}𝕄}(^{w}μ)) .

2.37. Theorem. Let ℕ be a generic Levi subgroup of 𝔾. Then

∗R_{ℕ}^{𝔾}(Reg_{d}^{𝔾}) = Deg(R_{ℕ}^{𝔾}(1))Reg_{d}^{ℕ} .

Sketch of proof. Since, by 2.33,

Reg_{d}^{𝔾} = ∑_{[𝕋]_{W_{𝔾}}} \frac{Deg(R_{𝕋}^{𝔾}(1))}{|W_{𝔾}(𝕋)|}R_{𝕋}^{𝔾}(Ab_{d} Reg^{𝕋}) ,

and since Ab_{d} Reg^{𝕋} = Reg_{d}^{𝕋}, it follows that

∗R_{𝕋}^{𝔾}(Reg_{d}^{𝔾}) = Deg(R_{𝕋}^{𝔾}(1))Reg_{d}^{𝕋}

and also that

∗R_{ℕ}^{𝔾}(Reg_{d}^{𝔾}) = ∑_{[{𝕋; 𝕋⊆ℕ}]_{W_{ℕ}}} R_{𝕋}^{ℕ}(^{∗}R_{𝕋}^{ℕ}(^{∗}R_{ℕ}^{𝔾}(Reg_{d}^{𝔾}))) .

Thus the result follows from the transitivity of both R_{ℕ}^{𝔾} and ^{∗}R_{ℕ}^{𝔾}. □

D. First applications to actual finite reductive groups.

In §2.D, as a first application, we prove that the specialization of our previous regular characters behaves as the regular character of a “Φ_{d}–idempotent”. The actual application to true ℓ–idempotents will be given in §5.

Let (G, T, F) be a (q, ϕ)–triple associated with 𝔾. Following the notation introduced at the beginning of §2.C, let (G/[G, G])_{d′} be the product of all the Sylow Φ_{e}–subgroups (e ≠ d) of the torus G/[G, G]. We then denote by Ab_{d} Irr(G^{F}) the group of all characters of G^{F} /[G, G]^{F} (viewed as characters of G^{F}) whose kernel contains the group of rational points of (G/[G, G])_{d′}. We denote by Ab_{d} Reg^{G^{F}} := ∑_{θ∈Ab_{d} Irr(G^{F})} θ the corresponding regular character.

2.38. Definition–Proposition.

(1) Let N be an F –stable Levi subgroup of G. The class function Θ_{G}^{N^{F}} on N^{F} is defined inductively by the formula

∑_{{M ; (M d–split)(N⊆M)}} Res_{N^{F}}^{M^{F}} (Θ_{G}^{M^{F}}) = Ab_{d} Reg^{N^{F}} .

(2) (Equivalent definition) For an F –stable Levi subgroup N of G, we define

Irr(Θ_{G}^{N^{F}}) := Ab_{d} Irr(N^{F}) − ⋃_{{M ; (M d–split)(N⊂M)}} Ab_{d} Irr(M^{F}) .

Then we have Θ_{G}^{N^{F}} = ∑_{θ∈Irr(Θ_{G}^{N^{F}} )} θ .

For a unipotent character μ of M^{F}, we set

W_{G^{F}} (M, μ) := N_{G^{F}} (M, μ)/M^{F}

and (with obvious notation)

Reg_{d}^{G^{F}} := ∑_{[(M,μ)]_{G^{F}}} \frac{DegR_{M}^{G}(μ)}{|W_{G^{F}} (M, μ)|}R_{M}^{G}(Θ_{G}^{M^{F}} .μ) .

The following result is a kind of “generic version” of [BrMi], thm. 2.2.

2.39. Theorem. All the values of |W_{G}| Reg_{d}^{G^{F}} are divisible by |𝔾|_{d}(q).

Proof of 2.39. This depends on the following general property of the Deligne– Lusztig induction.

2.40. Proposition. Let L be an F –stable Levi subgroup of G. If ζ is a class function on L^{F} which takes integral values, then the function |G^{F} |_{p}R_{L}^{G}(ζ) takes integral values.

Proof of 2.40. Let P be a parabolic subgroup of G such that L is a Levi complement of P, and let U be the unipotent radical of P. We denote by V_{U} the associated Deligne–Lusztig variety, on which the finite group G^{F} acts by left multiplication, while the finite group L^{F} acts by right multiplication. We denote by Λ_{U} the class function on G^{F} × L^{F} defined by the formula

Λ_{U}(g, l) := ∑_{i} tr((g, l^{−1}) ; H^{i}(V_{U}, ℚ_{ℓ})) .

It is known (cf. [DeLu], prop. 3.3) that the function Λ_{U} takes integral values, which are independent of ℓ (ℓ ≠ p). Moreover, the map R_{L}^{G} is defined by Λ_{U} as follows (see for example [Br1], §1). For a class function ζ on L^{F}, the value of R_{L}^{G}(ζ) at g ∈ G^{F} is given by the following formula:

R_{L}^{G}(ζ)(g) = (Λ_{U}(g, ·), ζ)_{L^{F}}

where we denote by Λ(g, ·) the class function on L^{F} whose value at l ∈ L^{F} is Λ(g, l).

By [Br1], 4.1, it suffices then to prove that Λ_{U} is “ℓ–perfect” for every ℓ ≠ p. This follows from [Br1], prop. 2.1 (see also [DeLu], prop. 3.5). □

We can now prove Theorem 2.39.

By 2.33 and using 1.28, which shows that we may identify W_{𝔾}(𝕄, μ) and W_{G}(M, ρ_{μ}^{M^{F}}), we see (note that our notation has been chosen so that everything specializes nicely) that (2.41)

Reg_{d}^{G^{F}} = ∑_{[(M,μ)]_{G^{F}}} \frac{DegR_{M}^{G}(1)}{|W_{G^{F}} (M, μ)|}R_{M}^{G}(Ab_{d} Reg^{M^{F}} .Deg(c_{d}(μ))(q).μ) .

It suffices to prove the following lemma.

2.42. Lemma. Whenever M is a d–split Levi subgroup of G and μ is a unipotent character of M^{F},

|W_{M}|\frac{DegR_{M}^{G}(1)}{|W_{G^{F}} (M, μ)|}R_{M}^{G}(Ab_{d} Reg^{M^{F}}.Deg(c_{d}(μ))(q).μ)

takes values which are divisible by |𝔾|_{d}(q).

Proof of 2.42. On one hand, since W_{G^{F}} (M, μ) = N_{G^{F}} (M, μ)/M^{F}, we have

\frac{DegR_{M}^{G}(1)}{|W_{G^{F}} (M, μ)|} = \frac{ε_{G}ε_{M}}{|W_{G^{F}} (M, μ)|_{p}} |G^{F} : N_{G^{F}} (M, μ)|_{p′} .

On the other hand, by 2.8, we see that |W_{M}|Deg(c_{d}(μ))(q) is divisible by |𝕄_{ss}|_{d}(q) (since |W_{M}|Deg(c_{d}(μ)) and |𝕄_{ss}|_{d} are both polynomials in ℤ[x], the second one monic), and thus |W_{M}|Ab_{d} Reg^{M^{F}} .Deg(c_{d}(μ))(q).μ takes values divisible by |𝕄|_{d}(q).

Since |𝕄|_{d}(q) = |𝔾|_{d}(q), 2.42 is proved. □

□

3. Generalized Harish–Chandra theory

A. The fundamental theorem.

In §3.A, in order to decompose our regular characters, we define partitions of the set of unipotent characters in a way which generalizes Harish–Chandra and Howlett–Lehrer–Lusztig theories.

We call d–split pairs of 𝔾 the pairs (𝕄, μ) where 𝕄 is the centralizer in 𝔾 of a Φ_{d}–torus of 𝔾, and where μ ∈ Uch(𝕄). We say that such a pair is d–cuspidal if μ is d–cuspidal.

3.1. Definition. Let (𝕄_{1}, μ_{1}) and (𝕄_{2}, μ_{2}) be two d–split pairs. We write

(𝕄_{1}, μ_{1}) ≼ (𝕄_{2}, μ_{2})

if (1) 𝕄_{1} is a Levi subgroup of 𝕄_{2},

(2) (R_{𝕄_{1}}^{𝕄_{2}}(μ_{1}), μ_{2})_{𝔾} ≠ 0.

The Weyl group W_{𝔾} of 𝔾 acts on the set of d–split pairs of 𝔾 and it stabilizes the relation ≼ . If (𝕄_{1}, μ_{1}) and (𝕄_{2}, μ_{2}) are two d–split pairs, we set (𝕄_{1}, μ_{1}) ≼_{W_{𝔾}} (𝕄_{2}, μ_{2}) if and only if there exists w ∈ W_{𝔾} such that (𝕄_{1}, μ_{1})^{w} ≼ (𝕄_{2}, μ_{2}).

For a d–cuspidal pair (𝕃, λ) of 𝔾, we denote by Uch(𝔾, (𝕃, λ)) the set of γ ∈ Uch(𝔾) such that (𝕃, λ) ≼ (𝔾, γ). Note that Uch(𝔾, (𝕃, λ)) depends only on the W_{𝔾}–conjugacy class of (𝕃, λ).

The following statement is fundamental. It expresses that Deligne–Lusztig induction can be viewed as ordinary induction through generalized Weyl groups, provided each unipotent character is equipped with the appropriate sign, and if we work only with d–split Levi subgroups of 𝔾.

3.2. Fundamental Theorem.

(1) For each d, the sets Uch(𝔾, (𝕃, λ)) (where (𝕃, λ) runs over a complete set of representatives of the W_{𝔾}–conjugacy classes of d–cuspidal pairs of 𝔾) partition Uch(𝔾).

(2) There exists a collection of isometries

I_{(𝕃,λ)}^{𝕄}: ℤIrr(W_{𝕄}(𝕃, λ)) → ℤ Uch(𝕄, (𝕃, λ)) ,

where • 𝕄 runs over the set of all d–split generic Levi subgroups of 𝔾,

• (𝕃, λ) runs over the set of d–cuspidal pairs of 𝕄, such that

(a) for all 𝕄 and all (𝕃, λ), we have

R_{𝕄}^{𝔾} · I_{(𝕃,λ)}^{𝕄} = I_{(𝕃,λ)}^{𝔾} · Ind_{W_{𝕄}(𝕃,λ)}^{W_{𝔾}(𝕃,λ)} .

(b) The collection (I_{(𝕃,λ)}^{𝕄})_{𝕄,(𝕃,λ)} is stable under the conjugation

action by W_{𝔾}.

(c) I_{(𝕃,λ)}^{𝕃} maps the trivial character of the trivial group W_{𝕃}(𝕃, λ) to λ.

We believe that 3.2 will have a general and nice proof, involving in particular a generalization of the notion of Hecke algebras (cf. [Br1] last paragraph, and [BrMa2]) and a suitable generalization of Howlett–Lehrer theory (which must be understood as adapted to the case d = 1 — see below) to a “d– Howlett–Lehrer theory”. Unfortunately such a theory seems out of reach for us at the moment.

Indeed, the case d = 1 is known: the first assertion is the so–called Harish– Chandra theory, and the second assertion can be deduced from theorem 5.9 in [HoLe].

We provide a case by case proof of the fundamental theorem. The case of type A will be treated first; for the classical groups, we give a rough sketch (we refer to [En2] for a more detailed proof), and a case-by-case analysis for the exceptional groups using tables given in the appendix.

It will be convenient to prove 3.2 together with the following result, stating a general “Ennola”–duality via σ^{𝔾} for Deligne–Lusztig induction:

3.3. Theorem (“Ennola”–duality for R_{𝕃}^{𝔾}). There exists a natural bijective isometry σ^{𝔾} : Uch(𝔾) → Uch(𝔾^{−}) such that whenever 𝕃 is d–split for some d, the following diagram is commutative :

ℤ Uch(𝔾) ⟶^{σ^{𝔾}} ℤ Uch(𝔾^{−})
R_{𝕃}^{𝔾}↑  ↑_{R_{𝕃^{−}}^{𝔾^{−}}}
ℤ Uch(𝕃) ⟶^{σ^{𝕃}} ℤ Uch(𝕃^{−})

Remark. It is very likely that the preceding theorem holds without the hypothesis about the Levi subgroup 𝕃. This is actually the case if 𝔾 is of classical type, as shown below. It is still to be checked for the case where 𝔾 is of exceptional type.

We define a sign ε_{γ} and γ^{−} by σ^{𝔾}(γ) = ε_{γ}γ^{−} where γ ∈ Uch(𝔾) and γ^{−} ∈ Uch(𝔾^{−}). Note that the σ^{𝔾} defined above extends the one of 1.21 via the isometry of CF(𝔾) with the subspace of uniform functions of ℚ[x] Uch(𝔾).

Proof of 3.2 and 3.3. First note that it suffices to prove both statements in the case where 𝔾 is simple. Namely, by 1.36 and remark 1.25, we may reduce to the case of adjoint generic groups, which are direct products of simple generic groups 𝔾^{(a)}. Since R_{𝕃}^{𝔾} is compatible with products, we may hence assume that we have just one factor 𝔾^{(a)}, and finally, by 1.37, that a = 1.

The case of the type A.

Since the property we want to prove concerns only the unipotent characters, we may as well assume that we are dealing with the general linear or unitary group.

Let us denote by 𝔾𝕃_{n} the generic finite group corresponding to the general linear groups of rank n. Thus 𝔾𝕃_{n} = ((ℤ^{n}, R, ℤ^{n}, R^{∨}), 𝔖_{n}) with R = R^{∨} = {(e_{i} − e_{j}) ; (1 ≤ i ≠ j ≤ n)}, where {e_{1}, …, e_{n}} is the canonical basis of ℤ^{n}.

Then the generic finite group corresponding to the unitary groups of rank n is 𝔾𝕌_{n} := 𝔾𝕃_{n}^{−}, where 𝔾𝕃_{n}^{−} = ((ℤ^{n}, R, ℤ^{n}, R^{∨}), 𝔖_{n}.(−Id)) (cf. [BrMa], §2).

If we have shown 3.3, then it suffices to prove 3.2 for 𝔾𝕃_{n}. But 3.3 is well known in the case of type A. Namely, by [LuSr] (see for example [DiMi2], 15.4) if we take as reference torus the “diagonal” maximal torus of GL_{n}(𝔽̄_{q}) (which is stable by both the standard and the “twisted” Frobenius endomorphism), the generic irreducible unipotent characters of 𝔾𝕃_{n} are the functions

Φ_{χ}^{𝔾𝕃_{n}} := \frac{1}{n!} ∑_{w∈𝔖_{n}} χ(w)R_{w}^{𝔾𝕃_{n}} for χ ∈ Irr(𝔖_{n}) ,

while the generic irreducible unipotent characters of 𝔾𝕌_{n} are the ε_{χ}Φ_{χ}^{𝔾𝕌_{n}} where ε_{χ} = ±1 and

Φ_{χ}^{𝔾𝕌_{n}} := \frac{1}{n!} ∑_{w∈𝔖_{n}} χ(w)R_{w}^{𝔾𝕌_{n}} for χ ∈ Irr(𝔖_{n}) .

Identifying Uch(𝔾) and Uch(𝔾^{−}) with Irr(𝔖_{n}), we then see that the map χ ↦ ε_{χ}χ is the map σ_{𝔾}, which shows 3.3.

The natural bijection 𝕄 ↦ 𝕄^{−} between the set of generic Levi subgroups of 𝔾𝕃_{n} and the set of generic Levi subgroups of 𝔾𝕌_{n} is such that the Φ_{d}(x)– subgroups of 𝔾𝕃_{n} correspond to the Φ_{d}(−x)–subgroups of 𝔾𝕌_{n} (recall that, for d > 2, Φ_{d}(−x) = Φ_{2d}(x), Φ_{d/2}(x) , or Φ_{d}(x) if d is respectively odd, congruent to 2 modulo 4, or divisible by 4). So to prove 3.2 for type A, we may as well assume that 𝔾 = 𝔾𝕃_{n}, which we do from now on.

We identify CF(𝔾) and the subspace of uniform functions of ℚ[x] Uch(𝔾) via the map α ↦ Φ_{α}^{𝔾} (see 1.29). Thus the map R_{𝕃}^{𝔾} is identified to the induction Ind_{𝕃}^{𝔾} on class functions (see 1.33, (2)) and ^{∗}R_{𝕃}^{𝔾} to Res_{𝕃}^{𝔾}.

Let γ be an irreducible character of 𝔖_{n} and let w ∈ 𝔖_{n}. Then ^{∗}R_{𝕋_{w}}^{𝔾}(γ) = γ(w)1 and (see remark 2 after 2.1) γ is d–cuspidal if and only if the partition of n corresponding to γ is a d-core.

Moreover, a d–split Levi subgroup of 𝔾 is of type 𝔾𝕃_{n_{1}}^{(d)} × 𝔾𝕃_{n_{2}}^{(d)} × ⋯ × 𝔾𝕃_{n_{a}}^{(d)} × 𝔾𝕃_{s} (cf. 1.3). It is then easy to see that a d–cuspidal pair of 𝔾 is a pair (𝕃, λ) such that 𝕃 is the direct product 𝕋 × 𝔾𝕃_{r} where

• 𝕋 is a torus such that |𝕋| = (x^{d} − 1)^{a},

• λ (with the appropriate identifications) is a character of 𝔖_{r} corresponding to a d–core of cardinal r.

Thus n = ad + r, and 𝕃 = (Γ_{𝕃}, W_{𝕃}v), where

• Γ_{𝕃} is a suitable root datum,

• W_{𝕃} ≃ 𝔖_{r},

• v is an element of cycle type d^{a} of 𝔖_{ad}

(here we identify 𝔖_{ad} with the obvious subgroup of the Young subgroup 𝔖_{ad}× 𝔖_{r} in 𝔖_{n}).

We have γ ∈ Uch(𝔾, (𝕃, λ)) if and only if λ is the d–core of γ. This proves in particular the first assertion of 3.2.

A d–split Levi subgroup 𝕄 containing 𝕃 is of type 𝔾𝕃_{n_{1}}^{(d)}×⋯×𝔾𝕃_{n_{a}}^{(d)}×𝔾𝕃_{s} , where s ≥ r. We can compute R_{𝕄}^{𝔾} by computing successively R_{𝕄}^{𝕄_{1}}, then R_{𝕄_{1}}^{𝕄_{2}}, etc., where 𝕄_{1} is of type 𝔾𝕃_{n_{1}}^{(d)} × ⋯ × 𝔾𝕃_{n_{a−1}}^{(d)} × 𝔾𝕃_{dn_{a}+s} , 𝕄_{2} is of type 𝔾𝕃_{n_{1}}^{(d)} × ⋯ × 𝔾𝕃_{n_{a−2}}^{(d)} × 𝔾𝕃_{dn_{a−1}+dn_{a}+s} , and so we see that it is enough to assume (since R_{𝕃}^{𝔾} behaves nicely with respect to products)

𝔾 = 𝔾𝕃_{n} and 𝕄 = 𝔾𝕃_{b}^{(d)} × 𝔾𝕃_{s}

with 𝕃 ⊆ 𝕄 , which we do from now on. Thus n = ad+r = bd+s with r ≤ s, 𝔾 = (Γ_{𝔾}, W_{𝔾}) with W_{𝔾} ≃ 𝔖_{n}, and 𝕄 = (Γ_{𝕄}, W_{𝕄}w) with

• W_{𝕄} ≃ (𝔖_{b})^{d} × 𝔖_{s}, a Young subgroup of 𝔖_{n}

• w is an element of cycle type d^{b} in 𝔖_{bd} (note that W_{𝕄}w = W_{𝕄}v).

The class functions on 𝔾 are the usual class functions on 𝔖_{n}, while the set CF(𝕄) may be identified with the set of usual class functions on 𝔖_{b} × 𝔖_{s}. Indeed, every β ∈ CF(𝕄) (i.e., β is a W_{𝕄}–stable function on W_{𝕄}w) defines a class function β_{b} on 𝔖_{b} as follows: for (σ_{1}, σ_{2}, …, σ_{d}) ∈ (𝔖_{b})^{d}, we set β_{b}(σ_{1}σ_{2} ⋯ σ_{d}) := β((σ_{1}, σ_{2}, …, σ_{d}).w) .

We have

W_{𝔾}(𝕃, λ) = C_{𝔖_{ad+r}}(v)/𝔖_{r} ≃ ((ℤ/dℤ) ≀ 𝔖_{a}) ,

and

W_{𝕄}(𝕃, λ) ≃ 𝔖_{b} × ((ℤ/dℤ) ≀ 𝔖_{a−b}) .

With the preceding identifications, the description of I_{(𝕃,λ)}^{𝔾} and I_{(𝕃,λ)}^{𝕄} amounts to the description of corresponding maps I_{d,λ}^{m} (m = n, s), where

I_{d,λ}^{n}: ℤIrr((ℤ/dℤ) ≀ 𝔖_{a}) → ℤIrr(𝔖_{n}, (d, λ)) ,
I_{d,λ}^{s}: ℤIrr((ℤ/dℤ) ≀ 𝔖_{a−b}) → ℤIrr(𝔖_{s}, (d, λ)) ,

where Irr(𝔖_{m}, (d, λ)) denotes the set of irreducible characters of 𝔖_{m} (partitions of m) with d–core λ. We want these maps to be such that the following diagram is commutative :

(3.4)

ℤIrr((ℤ/dℤ) ≀ 𝔖_{a}))  ⟶^{I_{d,λ}^{n}}  ℤIrr(𝔖_{n})
↓  ↓
ℤIrr(𝔖_{b} × (ℤ/dℤ) ≀ 𝔖_{a−b}) ⟶^{Id_{𝔖_{b}} ⊗I_{d,λ}^{s}} ℤIrr(𝔖_{b} × 𝔖_{s})

where the vertical arrows are the restriction maps.

With appropriate choices (see [Os]), an irreducible character of (ℤ/dℤ)≀𝔖_{a} (resp. of (ℤ/dℤ) ≀ 𝔖_{a−b}) is defined by a family of partitions (π_{i} ⊢ a_{i})_{i=1,...,d} where ∑_{i=1}^{d} a_{i} = a (resp. ∑_{i=1}^{d} a_{i} = a − b). Such a collection may be interpreted as the “d–quotient” of a partition π of n (resp. of s) with d– core λ , and defines a sign ε_{π} (see for example [En1], §3) in a way which is “compatible” with the Murnaghan–Nakayama formula, and this defines the maps I_{d,λ}^{m}.

Assume first that a = b. In this case 3.4 is equivalent to the following property :

For all ζ ∈ Irr((ℤ/dℤ) ≀ 𝔖_{a}), for all (σ_{1}, σ_{2}, …, σ_{d}) ∈ (𝔖_{a})^{d} and for all σ ∈ 𝔖_{r}, we have

I_{d,λ}^{n}(ζ)((σ_{1}, σ_{2}, …, σ_{d}).w.σ) = ζ(σ_{1}σ_{2} ⋯ σ_{d})λ(σ) .

This equality results from the Murnaghan–Nakayama formula for wreath products (see [Os] or also [Rou]) and from the fact that the cycle type of ((σ_{1}, σ_{2}, …, σ_{d}).w) ∈ 𝔖_{ad} is d times the cycle type of σ_{1}σ_{2} ⋯ σ_{d} ∈ 𝔖_{a}.

The general case a > b is analogous, and is translated as follows.

We let w be as above. Then for ζ ∈ Irr((ℤ/dℤ) ≀ 𝔖_{a}) the value of I_{d,λ}^{n}(ζ) on ((𝔖_{b})^{d} × (𝔖_{a−b})^{d} × 𝔖_{r}).w is given as follows:

For (σ_{1}, σ_{2}, …, σ_{d}, σ′_{1}, σ′_{2}, …, σ′_{d}) ∈ (𝔖_{b})^{d} × (𝔖_{a−b})^{d} and σ ∈ 𝔖_{r} we have

I_{d,λ}^{n}(ζ)((σ_{1}, σ_{2}, …, σ_{d}, σ′_{1}, σ′_{2}, …, σ′_{d}).w.σ) =

ζ((σ_{1}σ_{2} ⋯ σ_{d})(σ′_{1}σ′_{2} ⋯ σ′_{d}))λ(σ).

Let now w′ be an element of cycle type (d)^{a−b} in 𝔖_{(a−b)d}, and let ζ_{b} denote the restriction of ζ to 𝔖_{b} × (ℤ/dℤ) ≀ 𝔖_{a−b}. Then the value of (Id_{𝔖_{b}} ⊗ I_{d,λ}^{s})(ζ_{b}) on (𝔖_{b} × (𝔖_{a−b})^{d} × 𝔖_{r}).w′ is given as follows:

For (σ_{1}, σ_{2}, …, σ_{d}, σ′_{1}, σ′_{2}, …, σ′_{d}) ∈ (𝔖_{b})^{d} × (𝔖_{a−b})^{d} and σ ∈ 𝔖_{r} we have

(Id_{𝔖_{b}} ⊗ I_{d,λ}^{s})(ζ_{b})((σ_{1}σ_{2} ⋯ σ_{d})(σ′_{1}, σ′_{2}, …, σ′_{d}).w′.σ) =
ζ((σ_{1}σ_{2} ⋯ σ_{d})(σ′_{1}σ′_{2} ⋯ σ′_{d}))λ(σ) .

It is clear that the preceding formulae prove the commutativity of 3.4.

Note also that statement (b) of the theorem holds trivially since all automorphisms of 𝔾𝕃_{n} fix all unipotent characters.

The case of classical groups.

We use Asai’s results on the decomposition of the Lusztig functor R_{𝕃}^{𝔾} as cited in the paper [FoSr] of Fong and Srinivasan.

Let 𝔾 be a generic group of type B_{l}, C_{l}, D_{l} or ^{2}D_{l}. Lusztig has shown that the unipotent characters Uch(𝔾) of these groups may be parametrized by means of so called symbols. A symbol is an unordered set {S, T } of two strictly increasing sequences S = x_{1} < ⋯ < x_{a}, T = y_{1} < ⋯ < y_{b} of nonnegative integers, usually denoted as

Λ = ( \begin{matrix}x_{1} & … & x_{a} \\ y_{1} & … & y_{b} \end{matrix} ) .

Two symbols are considered to be equivalent if they can be transformed into each other by a sequence of steps

( \begin{matrix}x_{1} & … & x_{a} \\ y_{1} & … & y_{b} \end{matrix} ) ∼ ( \begin{matrix}0 & x_{1} + 1 & … & x_{a} + 1 \\ 0 & y_{1} + 1 & … & y_{b} + 1 \end{matrix} ) .

The rank of a symbol Λ is

rank(Λ) := ∑ x_{i} + ∑ y_{i} − [(\frac{a + b − 1}{2})^{2}] ,

and |a−b| is called the defect of Λ. The unipotent characters of groups of type B_{l} and C_{l} are in bijection with the equivalence classes of symbols of rank l and odd defect. The unipotent characters of groups of type D_{l} are parametrized by (classes of) symbols of rank l and defect divisible by 4, except that if the two rows of Λ are equal, two unipotent characters correspond to the same symbol; such symbols are called degenerate. Finally the unipotent characters of groups of type ^{2}D_{l} are parametrized by symbols Λ of rank l and defect congruent 2 (mod 4) (cf. [Ca], 13.8).

Let us first assume that the integer d of the theorem is odd. We will need the decomposition of R_{𝕃}^{𝔾} in the particular case of a Levi subgroup 𝕃 = 𝕋 × ℍ, where ℍ is a generic classical group of the same type as 𝔾 (or possibly 𝔾^{−} in the case of even orthogonal groups) and 𝕋 is a torus such that |𝕋| = x^{f} − 1 (we will always have d|f). In this particular situation a unipotent character of ℍ and hence of 𝕃 is indexed by a symbol Λ = (S, T). The decomposition of R_{𝕃}^{𝔾}(γ_{Λ}) is explicitly described by a simple formula of Asai given in (3.1) of [FoSr]. Namely R_{𝕃}^{𝔾} equals

(3.5) ∑_{\begin{matrix}x∈S \\ x+f∉S \end{matrix}}(−1)^{M_{S,x}}γ_{S−\{x\}∪\{x+f\},T} + ∑_{\begin{matrix}y∈T \\ y+f∉T \end{matrix}}(−1)^{M_{T,y}}γ_{S,T −\{y\}∪\{y+f\}},

where M_{S,x} := |{s ∈ S | x < s < x + f}|. Also by convention if a degenerate symbol occurs on the right hand side of the formula, then it is understood that both unipotent characters corresponding to it appear in the sum. The formula above has a convenient interpretation in terms of f-hooks: The symbols occurring as indices of unipotent characters on the right hand side are precisely those which are obtained from Λ by adding an f-hook. (The concept of hooks, cohooks, cores and cocores of symbols is explained in detail in [Ol] and [FoSr], for example.)

The proof will be in several steps. We first describe the d-cuspidal pairs. Each symbol Λ has a uniquely determined d-core. We claim that the dcuspidal unipotent characters γ_{Λ} of 𝔾 are precisely those where Λ itself is a d-core. Indeed it is easily seen that the generic d-split subgroups 𝕃 of 𝔾 all have type

𝔾𝕃_{n_{1}}^{(d)} × ⋯ × 𝔾𝕃_{n_{a}}^{(d)} × ℍ,

where ℍ has the same type as 𝔾 (or possibly 𝔾^{−}). Now by 1.37 the d-cuspidal unipotent characters of 𝔾𝕃_{n}^{(d)} are in natural bijection with the 1-cuspidal characters of 𝔾𝕃_{n}, and as we already saw above such characters exist only if n = 1. Hence d-split subgroups 𝕃 can have d-cuspidal unipotent characters only if 𝕃 is of type

(3.6)  𝕋 × ℍ,  where |𝕋| = (x^{d} − 1)^{a},

and where ℍ has the type of 𝔾^{±}. In turn, if a > 0 by induction these dcuspidal characters have the form λ = 1 ⊗ ⋯ ⊗ 1 ⊗ γ_{Λ}, where Λ is a d-core.

Now R_{𝕃}^{𝔾} behaves well with respect to direct products. So the constituents of R_{𝕃}^{𝔾}(λ) for (𝕃, λ) as above may be computed by repeated application of Asai’s formula 3.5 in the particular case f = d, using transitivity. Clearly all constituents of R_{𝕃}^{𝔾}(λ) have the same d-core Λ, so we obtain a partition of the unipotent characters of 𝔾 into series, such that two characters lie in the same series if and only if their associated symbols have the same d-core. This completes the description of the d-cuspidal pairs and of the d-series and proves 3.2(1).

We next show that the series are disjoint even with respect to induction from intermediate d-split Levi subgroups. Obviously by induction it suffices to do this for maximal subgroups 𝕄 (see the reasoning for type A). Up to 𝔾-conjugacy any maximal d-split generic subgroup of 𝔾 containing 𝕃 has the form

(3.7)  𝕄 = 𝔾𝕃_{b}^{(d)} × ℍ′

where b ≤ a and ℍ′ has the same type as 𝔾^{±}. Although the formula of Asai doesn’t explicitly yield the decomposition of R_{𝕄}^{𝔾}(μ) for μ a unipotent character of 𝕄, we can still deduce it. Namely, the unipotent characters of 𝔾𝕃_{b}^{(d)} are indexed by the irreducible characters χ of 𝔖_{b} = W (𝔾𝕃_{b}). The unipotent character of 𝔾𝕃_{b} corresponding to χ is

(3.8)  γ_{χ} = \frac{1}{|𝔖_{b}|} ∑_{w∈𝔖_{b}} χ(w)R_{𝕋_{w}}^{𝔾𝕃_{b}}(1),

where |𝕋_{w}| = ∏_{i}(x^{db_{i}} − 1) when w has cycle type ∏_{i}(b_{i}). It is now clear by repeated application of Asai’s formula for f = db_{i} that the constituents of R_{𝕄}^{𝔾}(μ) are indexed by symbols obtained from Λ by adding db_{i}-hooks, and since the addition of a db_{i}-hook may be mimicked by the addition of b_{i} dhooks, R_{𝕄}^{𝔾} preserves the series defined above, which proves disjointness.

Thus it remains to establish the existence of the collection of maps I_{(𝕃,λ)}^{𝕄}. As in the case of type A above it is sufficient to do this for 𝕄 maximal. Let 𝕃 be a fixed d-split subgroup of 𝔾 as in 3.6 with ℍ of rank r, so that l = ad + r, and λ = γ_{Λ} a d-cuspidal character of ℍ. Then 𝕄 has the form 3.7. Let us assume for the moment that 𝔾 has type B_{l} or C_{l}. Then W_{𝔾}(𝕃, λ) ≅ (ℤ/2dℤ) ≀ 𝔖_{a} (where implicitly we have already made all the identifications as in the proof for type A). The characters of this wreath product are in bijection with 2d-tuples of partitions (ω_{i} ⊢ a_{i}) with ∑ a_{i} = a.

As we have seen, the unipotent characters in Uch(𝔾, (𝕃, λ)) are parametrized by symbols Ψ with given d-core Λ and hence weight a. Since a symbol Λ = {S, T } is uniquely determined (up to equivalence) by its defect and by the two partitions represented by the β-numbers S, T, we get a bijection from the set of symbols with a given d-core onto the set of pairs of partitions of a_{1}, a_{2} with prescribed d-cores and a_{1} + a_{2} = a (we get ordered pairs since the defect of Λ is odd). These in turn are in bijection with the set of pairs of d-quotients (see [JaKe]), hence with 2d-tuples of partitions (ω_{i} ⊢ a_{i}) of a = ∑ a_{i}. This defines a bijection between Uch(𝔾, (𝕃, λ)) and Irr(W_{𝔾}, (𝕃, λ)).

For 𝕄 as above we also have W_{𝕄}(𝕃, λ) ≅ 𝔖_{b} × ((ℤ/2dℤ) ≀ 𝔖_{a−b}). The set Uch(𝕄, (𝕃, λ)) is parametrized by pairs (ω, Γ) consisting of a partition ω ⊢ b and a symbol Γ for ℍ′ with d-core Λ. We have an obvious bijection from these to Irr(W_{𝕄}(𝕃, λ)).

We now have to compare the decomposition of R_{𝕄}^{𝔾}(μ), for μ a constituent of R_{𝕃}^{𝕄}(λ), with the corresponding decomposition of induced characters in the Weyl group. First note that by (3B) of [FoSr] there is associated a well defined sign ϵ to each unipotent character of 𝔾 so that no cancelling takes place in the induction.

We make again use of the decomposition 3.8 for the unipotent characters of 𝔾𝕃_{b}. The different tori 𝕋_{w} appearing in the sum correspond to the one element subsets {w}. Induction to W_{𝔾}(𝕃, λ) just yields the characteristic function on the corresponding class, so that the multiplicity of any irreducible character χ of W_{𝔾}(𝕃, λ) is just equal to the value χ(w). This value may be computed by the generalized Murnaghan-Nakayama formula of Osima [Os] for wreath products, by successively removing hooks of length n_{i} from w of cycle type ∏_{i} n_{i}. On the other hand, the multiplicity of a unipotent character of 𝔾 in R_{𝕋_{w}}^{𝔾}(1) is seen by 3.5 to be obtained (up to sign) by the same procedure of adding hooks of length dn_{i}. Adding a hook of length dn_{i} is equivalent to adding a hook of length n_{i} to one of the partitions of the d-quotient. Hence the multiplicities on both sides agree and the assertion of 3.2 follows for types B and C.

In types D and ^{2}D, two complications arise. Firstly, degenerate symbols may occur in 3.5, and secondly, sometimes the normalizer W_{𝔾}(𝕃, λ) is not the full wreath product. It will become clear presently that both problems are intimately linked. But if both do not occur, the above proof carries through precisely as for types B and C.

As the d-core of a degenerate symbol is easily seen to be also degenerate, the first complication occurs when the cuspidal character λ is indexed by a degenerate symbol. But also, the normalizer W_{𝔾}(𝕃, λ) is not the full wreath product precisely when Λ is degenerate. So it remains to check the existence of I_{(𝕃,λ)}^{𝕄} in this case. Then W_{𝔾}(𝕃, λ) is a subgroup of index two in (ℤ/2dℤ)≀𝔖_{a}, generated by the elements of 𝔖_{a} and the even elements in (ℤ/2dℤ)^{a}. It hence suffices to see that precisely those characters χ of (ℤ/2dℤ) ≀ 𝔖_{a} indexed by 2d-tuples of partitions (ω_{i} ⊢ a_{i}) of a = ∑ a_{i} with a_{i} = a_{i+d} and ω_{i} = ω_{i+d}, 1 ≤ i ≤ d, split when restricted to the subgroup described above. This in turn is equivalent to saying that precisely those χ vanish on all odd elements of (ℤ/2dℤ) ≀ 𝔖_{a}. This final fact follows from the Murnaghan-Nakayama formula in [Os], and the proof for odd d is completed.

It remains to treat the case when d is even. This resembles very much the previous case, so we do not go into any details. (Indeed, the particular case where d is twice an odd number will follow immediately from 3.3, which is proved below.) Let 𝕃 be a generic Levi subgroup of 𝔾 of type 𝕋 × ℍ, where ℍ is a generic classical group of the same type as 𝔾 (or possibly 𝔾^{−}) and 𝕋 is a torus such that |𝕋| = x^{f} + 1. Again, the decomposition of R_{𝕃}^{𝔾}(γ_{Λ}), where γ_{Λ} denotes the unipotent character of 𝕃 indexed by the symbol Λ, was explicitly given in [FoSr], (3.2). Namely, R_{𝕃}^{𝔾}(γ_{Λ}) equals

(3.9)  \begin{aligned}∑_{\begin{matrix}x∈S \\ x+f∉T \end{matrix}}(−1)^{N_{S,x}+N_{T,x+f}} & γ_{S−\{x\},T ∪\{x+f\}} \\ & + ∑_{\begin{matrix}y∈T \\ y+f∉S \end{matrix}}(−1)^{N_{S,y+f} +N_{T,y}}γ_{S∪\{y+f\},T −\{y\}} \end{aligned}

Here N_{S,x} := |{s ∈ S | s < x}|, and the same convention concerning degenerate symbols as in 3.5 applies. The symbols occurring as indices of unipotent characters on the right hand side are precisely those which are obtained from Λ by adding an f-cohook. Each symbol has a well defined f-cocore, and arguments as in the previous part now show that the d-cuspidal pairs of 𝔾 are the (𝕃, λ) where λ is a unipotent character of the form 1 ⊗ ⋯ ⊗ 1 ⊗ γ_{Λ} of

𝕃 = 𝕋 × ℍ,  where |𝕋| = (x^{e} + 1)^{a},  e := d/2,

and where Λ is an e-cocore. As above this is an easy consequence of 3.9. In types B_{l} and C_{l} we have W_{𝔾}(𝕃, λ) ≅ Z_{2e} ≀ 𝔖_{a}. The same holds for D_{l} and 2D_{l}, except that when Λ is degenerate, we only obtain the subgroup of even elements as previously. It is well known that for symbols the mechanism of d–hooks when d is odd is completely parallel to that of d/2–cohooks when d is even. Since this same parallelity also holds for Asai’s formula 3.9, the previous arguments may be applied with only minor changes. A more detailed account is contained in [En2].

We next prove the general Ennola–duality for R_{𝕃}^{𝔾} in the classical groups. Let Λ = (S, T) be a symbol. Define Λ^{−} = (S^{−}, T^{−}) to be the symbol obtained from Λ by moving all odd entries from S to T and vice versa. In fact, as (S, T) = (T, S) by definition, we might as well exchange the even entries of S and T to obtain Λ^{−}. We claim σ^{𝔾} is the mapping

γ_{Λ} ↦ ϵ_{Λ}γ_{Λ^{−}},

where ϵ_{Λ} is the parity of the degree in x of Deg(γ_{Λ}) (note that ϵ_{Λ} is the sign by which one has to adjust Deg(γ_{Λ})(−q) to obtain a positive value). To prove 3.3, observe that any Levi subgroup of a classical group contains at most one factor of classical type, i.e., not of type A. Since all unipotent characters of type-A groups are uniform, the formulae of Asai 3.5, 3.9, enable us to compute the decomposition of R_{𝕃}^{𝔾} for any Levi subgroup 𝕃 of any classical group. For type A the desired commutativity has already been proved, and the definition of the sign ϵ_{Λ} agrees with the one given there. So by induction it therefor suffices to prove the commutativity of 3.3 in the case that 𝕃 has the particular form assumed in Asai’s formula.

The operation Λ ↦ Λ^{−} interchanges odd hooks and odd cohooks, and leaves even hooks and even cohooks alone. It follows that the symbols for constituents of R_{𝕃}^{𝔾}(γ_{Λ^{−}}) are just the images under ^{−} of the constituents of R_{𝕃}^{𝔾}(γ_{Λ}). Hence we are reduced to checking the correctness of the signs. Assume that 𝕃 = 𝕋 × ℍ as in 3.5, and Ψ is obtained by adding an f-hook at position x of Λ. Shifting the symbol if necessary, we may assume that x is even. The equality to be proved after evaluating 3.5 and 3.9 is

(−1)^{M_{S,x}} = ϵ_{Λ}ϵ_{Ψ}(−1)^{N_{S^{−},x}+N_{T^{−},x+f}} .

By the definition of S^{−}, T^{−}, M and N, this may be written as

ϵ_{Λ}ϵ_{Ψ} = (−1)^{1+N_{S,x}+N_{T,x}+N_{1}},

where N_{1} := |{s ∈ S | x ≤ s < x+f, s even}|+|{t ∈ T | x ≤ t < x+f, t even}|. But by the degree formula for γ_{Λ} given in [Ca], 13.8, the right hand side is the change in parity of the degree of Deg(γ_{Λ}) when Λ is replaced by Ψ, whence the desired result follows for odd length hooks. (Note that if 𝕃 is of type D, exactly three of the four groups 𝕃, 𝕃^{−}, 𝔾 and 𝔾^{−} are of the same orthogonal type. This accounts for a factor −1 in the above equation.) If f is even, one obtains similarly

ϵ_{Λ}ϵ_{Ψ} = (−1)^{1+N_{2}},

where N_{2} := |{s ∈ S | x ≤ s < x+f, s odd}|+|{t ∈ T | x ≤ t < x+f, t odd}|. Again this identity is verified using the degree formulae. This completes the proof of 3.3 for classical groups.

The case of exceptional groups.

We now prove theorem 3.2 for exceptional groups of Lie type, together with ”Ennola”–duality 3.3 in the case of d–split Levi subgroups of such groups.

We assume that 𝔾 has a connected Dynkin diagram of exceptional type, so that the corresponding groups of Lie type are simple of exceptional type. In this case we do not have such a nice result on the decomposition of R_{𝕃}^{𝔾}(λ) as Asai’s formula for the classical groups (cf. 3.5, 3.9). But since we are dealing with groups of bounded rank, it can be replaced by the Mackey formula for twisted induction 1.35 (which had been already proved by Deligne if q is large enough, and which follows in general from Shoji’s results, cf. §1.B). Using the Mackey formula the decomposition may uniquely be determined except for some ambiguities stemming from algebraic conjugate eigenvalues of Frobenius. These we can not resolve, but the attainable information is sufficient to prove the fundamental theorem 3.2.

Our approach will consist in a case by case analysis, but some frequently used arguments are collected at the beginning. Clearly if we know the decomposition of R_{𝕃}^{𝔾}(λ) for each d–split generic Levi subgroup 𝕃 of 𝔾 and each unipotent character λ ∈ Uch(𝕃), then the assertion of the theorem may be checked. Namely the disjointness and the positivity will follow from the lists, and the existence of the collection of maps I_{(𝕃,λ)}^{𝕄} just by comparing these finitely many decompositions with tables showing the decompositions of induced characters in the corresponding Weyl groups. This trivial step of verification will not be made explicit in the proof. We shall just sketch how to find the decomposition of R_{𝕃}^{𝔾}, and list the relevant centralizers W_{𝔾}(𝕃, λ) in the Weyl group.

If our d–split Levi subgroup 𝕃 happens to be a maximal torus 𝕋, then R_{𝕃}^{𝔾} is just the usual Deligne–Lusztig induction R_{𝕋}^{𝔾}. The decomposition of R_{𝕋}^{𝔾} is known in all cases by the work of Lusztig. This already accounts for about half the cases in the exceptional groups. When 𝕃 is contained in a Harish–Chandra generic subgroup 𝕄, i.e., centralizes a 1–subgroup (see [BoTi], Sect. 4), then R_{𝕃}^{𝕄} is known by induction and R_{𝕄}^{𝔾} coincides with ordinary Harish–Chandra induction, so the decomposition follows from the result of Howlett and Lehrer [HoLe] (the case d = 1). Finally, if λ lies in the space of uniform functions of 𝕃 then it may be written as a linear combination of Deligne–Lusztig characters R_{𝕋}^{𝕃}(1). By transitivity of Lusztig–induction we can thus also express R_{𝕃}^{𝔾}(λ) as a linear combination of R_{𝕋}^{𝔾}(1)’s. Hence the determination of the decomposition is reduced to computations with Weyl group characters. So far we don’t even have to assume the Mackey formula to obtain the decompositions.

Unfortunately, not all cases (𝔾, d, 𝕃, λ) are of one of the above types. We now first collect a list of all quadruples (𝔾, d, 𝕃, λ), where 𝔾 is simple exceptional, d an integer with Φ_{d}(x) dividing |𝔾|, 𝕃 a d–split generic Levi subgroup and λ ∈ Uch(𝕃) is d–cuspidal (this last information is either already known if 𝕃 is classical, or by induction for exceptional 𝕃). This list appears as Table 1. The notation for the unipotent characters of the exceptional groups follows [Ca], 13.9. For groups of type A_{l}, the unipotent characters are indexed by partitions, for types D_{l} and ^{2}D_{l} by symbols (see the proof for classical groups).

However we have omitted the cases with d = 1, as these are known by the classical Howlett–Lehrer theory, and the cases where 𝕃 = 𝕋, (hence) λ = 1, (then d is regular for 𝔾 in the sense of Springer [Sp1]), because here the decomposition is also clear due to 𝕃 = 𝕋. The cases are numbered, and a ^{∗} indicates that in this case the decomposition of R_{𝕃}^{𝔾}(λ) follows from the above considerations, i.e., without having to apply to the results of Deligne and Shoji.

Note that the groups W_{𝔾}(𝕃, λ) occurring in case d = 2 are all well known Coxeter groups, as they should be by the ”Ennola”–duality, since we have G(2, 1, 2) ≅ W (B_{2}), G(1, 1, 3) ≅ W (A_{2}), G(2, 1, 3) ≅ W (B_{3}), G_{28} ≅ W (F_{4}) and G(6, 6, 2) ≅ W (G_{2}).

The table already shows that for the types ^{2}B_{2}, ^{2}G_{2}, G_{2}, ^{3}D_{4} and ^{2}F_{4} the decomposition of R_{𝕃}^{𝔾}(λ) for all cuspidal pairs (𝕃, λ) can be determined from the knowledge of R_{𝕋}^{𝔾}. But this is not the case for the decomposition of all R_{𝕄}^{𝔾}(λ), where 𝕄 < 𝔾 is maximal d–split but λ not cuspidal. In particular we will have to return to ^{2}F_{4} later on.

3.10. Theorem. The decomposition of the R_{𝕃}^{𝔾}(λ) for the (𝔾, d, 𝕃, λ) listed in Table 1 is given by Table 2.

From this, the fundamental theorem 3.2 follows for exceptional groups. At the same time, the ”Ennola”–duality 3.3 follows for all Levi subgroups 𝕃 which are d–split for some d.

Remark. In the cases marked by a ^{∗} in the last column, the decomposition can be proved without using the general Mackey formula.

Remark. The decompositions for 𝔾 not of type E_{8}, i.e., cases 1–38, were already obtained by Schewe [Sch] assuming a conjecture on Shintani descent, but note that some of his tables contain inaccuracies.

Outline of the proof of 3.10. In all cases marked with a ^{∗} in the next to last column of Table 1, the d–cuspidal character λ of 𝕃 lies in the space of uniform functions of 𝕃. So by our above remarks on transitivity of Deligne–Lusztig induction, R_{𝕃}^{𝔾} can be computed purely mechanical from information inside the Weyl group. All these computations with characters of Weyl groups were done with the computer algebra system CAS [NPP].

This leaves 28 cases (𝔾, d, 𝕃, λ) to be treated. These can not be solved with the methods discussed so far. We shall now assume in addition the validity of the Mackey formula for R_{𝕃}^{𝔾} in full generality, as allowed by 1.35 and 1.33.

It enables us to calculate the norm of R_{𝕃}^{𝔾}(λ) from the knowledge of W_{𝔾}(𝕃). For λ ∈ Uch(𝕃), denote by π_{u}^{𝕃}(λ) its projection to the space of uniform class functions on 𝕃. By definition, this projection may be written as a linear combination of Deligne–Lusztig characters R_{𝕋}^{𝔾}(1), so as above we may calculate R_{𝕃}^{𝔾}(π_{u}^{𝕃}(λ)) from the character tables of W_{𝕃} and W_{𝔾}. Since we have R_{𝕃}^{𝔾}(π_{u}^{𝕃}(λ)) = π_{u}^{𝔾}(R_{𝕃}^{𝔾}(λ)) by 1.33 (2), the uniform part of R_{𝕃}^{𝔾}(λ) is known in all cases, as is the norm by the Mackey formula. These two informations suffice to determine the decomposition except for some small ambiguities resulting from irrationalities. Namely, it will turn out that there exists an essentially unique element γ ∈ ℤ Uch(𝔾) with π_{u}^{𝔾}(γ) = π_{u}^{𝔾}(R_{𝕃}^{𝔾}(λ)) of minimal norm, and that this norm coincides with the norm of R_{𝕃}^{𝔾}(λ) calculated from the Mackey formula.

By transitivity of Deligne–Lusztig induction we may recover the decomposition of R_{𝕃}^{𝔾} if for some maximal Levi subgroup 𝕄 < 𝔾 containing 𝕃 we know the decomposition of R_{𝕄}^{𝔾} (and the one of R_{𝕃}^{𝕄} by induction). We will therefore from now on assume that 𝕃 is already maximal in 𝔾.

If 𝕃 is a direct product of two groups 𝕃 = 𝕃_{1} × 𝕃_{2}, and all unipotent characters of 𝕃_{1} happen to be uniform, then R_{𝕃}^{𝔾} may be written as a sum of R_{𝕋×𝕃_{2}}^{𝔾}, for some maximal subtori 𝕋 of 𝕃_{1}. If moreover 𝕋 × 𝕃_{2} is contained in another maximal Levi subgroup 𝕃′ for which we know R_{𝕃′}^{𝔾}, then by induction we can also compute R_{𝕃}^{𝔾}. This is for example useful if R_{𝕃′}^{𝔾} tends to contain only few constituents (and hence is easily determined), while R_{𝕃}^{𝔾} has large norm. As an example we cite the case d = 2 in 𝔾 of type E_{8}, where 𝕃 has type D_{5} + A_{2} and 𝕃′ has type E_{7}.

We now consider the restriction of the uniform part π_{u}^{𝔾}(R_{𝕃}^{𝔾}(λ)) to a family F of unipotent characters as defined by Lusztig (see [Ca], chap. 13). The characters in a family, as well as their uniform projections, span pairwise orthogonal spaces of class functions. If a family consists of just one unipotent character, then this character coincides with its uniform projection and hence its multiplicity in R_{𝕃}^{𝔾} may be recovered from the uniform projection.

Next assume that the family F is a four member family ρ_{1}, …, ρ_{4}, i.e., is associated to the group 𝔖_{2}. Then except for one case in E_{7} and two cases in E_{8} the orthogonal complement of uniform functions has a one dimensional intersection with F and is spanned (after a suitable renumbering of the ρ_{i}) by ρ_{1} − ρ_{2} − ρ_{3} + ρ_{4}. It turns out that in all cases of interest to us the uniform projections of the R_{𝕃}^{𝔾}(λ), restricted to a four element family, coincide with the uniform projection of exactly one ρ_{i}. It is now clear that the minimal norm is achieved if indeed R_{𝕃}^{𝔾}(λ)|_{F} = ρ_{i}. In case that adding up the minimal norms one just arrives at the norm of R_{𝕃}^{𝔾}(λ), the decomposition is uniquely determined. This accounts for the cases (17), (27), (62), (63), (64), (65). For cases (42) and (43) the decomposition can not be deduced unambiguously, since there two unipotent characters in the four–element family have the same uniform projection, but at least the sum of the two decompositions follows from the norm condition. Since the pairs of characters concerned do only occur with equal multiplicity in all other R_{𝕃}^{𝔾}, the ambiguity here does not harm the proof of the theorem. Whichever of the possibilities holds, the collection of maps I_{(𝕃,λ)}^{𝕄} exists.

Next we consider the eight member family associated to the group 𝔖_{3}. Here the orthogonal complement to the space of uniform functions has dimension three, except for type G_{2}, but there all unipotent characters of every proper Levi subgroup are uniform. In the other cases, the vector of shortest length in the sublattice spanned by generalized characters is ρ_{7} − ρ_{8} of length two; all other vectors have length at least four. Checking the cases one sees that (5), (9), (11) and (14) follow from this observation.

In the case of the 21–member family in F_{4} associated to 𝔖_{4}, the shortest vectors in the lattice of generalized characters orthogonal to all uniform functions are F_{4}[θ] − F_{4}[θ^{2}] and F_{4}[i] − F_{4}[−i] of norm two, and all other vectors have length at least four. This determines the cases (1), (2) and (3).

Finally in the 39–member family F in E_{8} associated to 𝔖_{5} one again finds that all vectors of length two come from pairs of unipotent characters with algebraically conjugate eigenvalues of Frobenius. Moreover, all other vectors have length at least four. This information is not yet sufficient do obtain all missing decompositions. So for example in case (47) we have 𝔾(q) = E_{8}(q), 𝕄(q) = Φ_{3}E_{6}(q), and it turns out that the norm of R_{𝕄}^{𝔾}(λ) restricted to the family F may be as big as seven. But at least we know how to decompose R_{𝕄}^{𝔾}(γ) for those γ lying in the principal series of E_{6}(q). Also, by the Mackey formula, the scalar products of these with the still unknown R_{𝕄}^{𝔾}(λ) are known. This, combined with the information on nonuniform characters in F, allows to compute the decomposition completely.

The same type of reasoning takes care of the 13–member family of characters in groups of type ^{2}F_{4} for Φ′_{8}–split subgroups 𝕄 with structure 𝕄(q) = Φ′_{8} ^{2}B_{2}(q). Although some ambiguities with respect to characters with algebraically conjugate eigenvalues of Frobenius remain, these are not relevant to the proof of the theorem, since it holds whichever of the different possible cases occurs.

After the determination of all R_{𝕄}^{𝔾} it remains to compare the decompositions for 𝕄 maximal with those of the corresponding characters induced from W_{𝕄}(𝕃, λ) to W_{𝔾}(𝕃, λ). The groups W_{𝔾}(𝕃, λ) turn out to be irreducible complex reflection groups in all cases. We have indicated these groups, using the standard notation given for example in [Be] or [Co], in the last column of Table 1 for the cases listed there, and collected them in all other cases where d ≠ 1, 2 and W_{𝔾}(𝕃, λ) is noncyclic in Table 3. (For d = 1, 2 they are the usual Weyl groups of 𝔾 or 𝔾^{−}.)

In the two cases (22) and (23), W_{𝔾}(𝕃) is the reflection group G_{8} but the stabilizer of the d–cuspidal character λ of 𝕃 is a Sylow 2–subgroup of G_{8} of index three, isomorphic to the imprimitive reflection group G(4, 1, 2).

With the help of the reflection representations for the relevant reflection groups given in [Be] it is now possible to find the fusion of W_{𝕄}(𝕃, λ) into W_{𝔾}(𝕃, λ) and then compute the decomposition of all induced characters, using the computer algebra system CAS [NPP]. Then a straightforward verification proves the existence of the maps I_{(𝕃,λ)}^{𝕄}. The details are omitted. This completes the proof of 3.10. □

□ It might be noted that the bijections I_{(𝕃,λ)}^{𝕄} are not at all unique if W_{𝔾}(𝕃, λ)

is cyclic; indeed on this level any bijective map is admissible. But once we have chosen the bijections on the level of cyclic groups, it is ”almost” determined for all bigger groups W_{𝔾}(𝕃, λ). To be more precise, the characters in the (𝕃, λ)– series are determined by their multiplicities in the R_{𝕄}^{𝔾}(γ) for (𝕃, λ) ≼ (𝕄, γ) (provided that some numbering is chosen on the 1–dimensional level), up to a small number of cases which basically result from algebraically conjugate eigenvalues of Frobenius, making characters indistinguishable by induction. This is again a remarkable analogy to the case d = 1, where Benson and Curtis have proved a similar result (here the ”cyclic” case is just the trivial group), also up to one ambiguity in E_{7} and two in E_{8}.

B. d–Harish–Chandra theory for unipotent characters.

In §3.B, we draw some straightforward consequences of the preceding fundamental theorem about the structure of the poset of d–split pairs.

The following statements generalize Harish–Chandra theory, which corresponds here to the case d = 1. They were recently conjectured or partially proved by many authors (see for example [Sch] or [FoSr]), and are immediate consequences of theorem 3.2.

3.11. Theorem. (transitivity) Let (𝕃, λ) be a d–cuspidal pair, and let (𝕄, μ) and (𝔾, γ) be d–split pairs such that (𝕃, λ) ≼ (𝕄, μ) and (𝕄, μ) ≼ (𝔾, γ). Then (𝕃, λ) ≼ (𝔾, γ).

This theorem, together with the disjointness asserted in 3.2(1), shows the existence of a “d–Harish–Chandra theory”. We draw some straightforward consequences of the existence of such a theory.

3.12. Proposition. Let (𝕃, λ) be a d–cuspidal pair, and let (𝕄, μ) and (𝔾, γ) be d–split pairs such that (𝕃, λ) ≼ (𝔾, γ) and (𝕄, μ) ≼ (𝔾, γ). Then (𝕃, λ) ≼_{W_{𝔾}} (𝕄, μ).

Proof of 3.12. By 3.2, (1), there exists a d–cuspidal pair (𝕃′, λ′) such that (𝕃′, λ′) ≼ (𝕄, μ). Then by 3.11 we have (𝕃′, λ′) ≼ (𝔾, γ). By 3.2, (1) again, we then see that (𝕃′, λ′) is W_{𝔾}–conjugate to (𝕃, λ). □

3.13. Corollary. The d–cuspidal pairs are the minimal d–split pairs for the order relation ≼ .

Proof of 3.13. Apply the preceding proposition with (𝕃, λ) = (𝔾, γ). □

The following consequence of d-Harish-Chandra theory is a result of “control of fusion”.

3.14. Proposition. Let (𝕃, λ) and (𝕄, μ) be two d–split pairs. We assume that (𝕃, λ) is d–cuspidal, and that (𝕃, λ) ≼ (𝕄, μ).

(1) Assume that w ∈ W_{𝔾} is such that also (𝕃, λ) ≼ (𝕄, μ)^{w}. Then there exists v ∈ W_{𝕄} and n ∈ N_{W_{𝔾}}(𝕃, λ) such that w = vn.

(2) The natural map N_{W_{𝔾}}((𝕃, λ), (𝕄, μ))/N_{W_{𝕄}}(𝕃, λ) → N_{W_{𝔾}}(𝕄, μ)/W_{𝕄} is an isomorphism. In particular (with obvious notation) we have

N_{W_{𝔾}(𝕃,λ)}(𝕄, μ)/W_{𝕄}(𝕃, λ) = W_{𝔾}(𝕄, μ) .

Proof of 3.14. If (𝕃, λ) is contained in both (𝕄, μ) and (𝕄, μ)^{w}, then (𝕃, λ) and ^{w}(𝕃, λ) are d–cuspidal pairs contained in (𝕄, μ), and hence (see 3.13) are conjugate under W_{𝕄}. This proves (1). We get (2) by applying (1) to the case where w ∈ N_{W_{𝔾}}(𝕄, μ). □

The next result is also an immediate consequence of 3.12.

3.15. Proposition. Suppose that (𝕃, λ) is a d–cuspidal pair such that (𝕃, λ) ≼ (𝔾, γ). Then

∗R_{𝕃}^{𝔾}(γ) = (γ, R_{𝕃}^{𝔾}(λ))_{𝔾} ∑_{w∈W_{𝔾}(𝕃)/W_{𝔾}(𝕃,λ)}^{w}λ .

In particular, we have

Deg(^{∗}R_{𝕃}^{𝔾}(γ)) = (γ, R_{𝕃}^{𝔾}(λ))_{𝔾}|W_{𝔾}(𝕃) : W_{𝔾}(𝕃, λ)|Deg(λ) ,

and Deg(^{∗}R_{𝕃}^{𝔾}(γ)) ≠ 0.

4. Generic Φ_{d}–blocks.

A. More on regular unipotent characters and Mackey formula.

In §4.A, using §3, we split unipotent regular characters into a sum of characters, for which we refine results of §2.B. We also refine the Mackey formula.

Let (𝕃, λ) be a d–cuspidal pair. We denote by pr_{𝕃,λ}^{𝔾} the orthogonal projection from ℚ[x] Uch(𝔾) onto the submodule with basis Uch(𝔾, (𝕃, λ)).

We define

(4.1)  UReg_{𝕃,λ}^{𝔾} := pr_{𝕃,λ}^{𝔾}(UReg^{𝔾}) = ∑_{{γ ; (𝕃,λ)≼(𝔾,γ)}} Deg(γ).γ ,

and we have (see formula one line above 2.12)

(4.2) c_{d}(UReg_{𝕃,λ}^{𝔾}) = pr_{𝕃,λ}^{𝔾}(c_{d}(UReg^{𝔾})) = ∑_{{γ ; (𝕃,λ)≼(𝔾,γ)}} Deg(c_{d}(γ)).γ .

We have the following version of 2.13:

4.3. Proposition. Let γ ∈ Uch(𝔾, (𝕃, λ)), and let ψ ∈ ℚ[x] Uch(𝔾) be a uniform function. Then

(γ, ψ)_{𝔾} = ∑_{[(𝕄,μ) ; (𝕃,λ)≼_{W_{𝔾}} (𝕄,μ)]_{W_{𝔾}}} \frac{(γ, R_{𝕄}^{𝔾}(μ))_{𝔾}}{|W_{𝔾}(𝕄, μ)|} (μ, c_{d}(^{∗}R_{𝕄}^{𝔾}(ψ)))_{𝕄}

Proof. From 2.13, we have

(γ, ψ)_{𝔾} = ∑_{[𝕄 d–split]_{W_{𝔾}}} \frac{1}{|W_{𝔾}(𝕄)|}(^{∗}R_{𝕄}^{𝔾}(γ), c_{d}(^{∗}R_{𝕄}^{𝔾}(ψ)))_{𝕄}

= ∑_{[𝕄 d–split]_{W_{𝔾}}} \frac{1}{|W_{𝔾}(𝕄)|} ∑_{μ∈Uch(𝕄)}(γ, R_{𝕄}^{𝔾}(μ))_{𝔾}(μ, c_{d}(^{∗}R_{𝕄}^{𝔾}(ψ)))_{𝕄}

whence the result, since the pairs (𝕄, μ) in the summation have to be above (𝕃, λ). □

From 4.3 we deduce the following refinement of 2.16.

4.4. Proposition. We have

UReg_{𝕃,λ}^{𝔾} = ∑_{[(𝕄,μ) ; (𝕃,λ)≼_{W_{𝔾}} (𝕄,μ)]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(1))}{|W_{𝔾}(𝕄, μ)|}R_{𝕄}^{𝔾}(Deg(c_{d}(μ)).μ) .

Finally, the preceding machinery allows us to prove a refined version of the Mackey formula (see 1.35). The proof of the following statement follows from 1.35 and from 3.14 and is left to the reader.

4.5. Proposition. If (𝕄_{1}, μ_{1}) and (𝕄_{2}, μ_{2}) are two d–split pairs containing the d–cuspidal pair (𝕃, λ), we have

(R_{𝕄_{1}}^{𝔾}(μ_{1}), R_{𝕄_{2}}^{𝔾}(μ_{2}))_{𝔾} =

∑_{w∈W_{𝕄_{2}} (𝕃,λ)\W_{𝔾}(𝕃,λ)/W_{𝕄_{1}} (𝕃,λ)}(^{∗}R_{𝕄_{2}∩^{w}𝕄_{1}}^{^{w}𝕄_{1}}(^{w}μ_{1}),^{∗}R_{𝕄_{2}∩^{w}𝕄_{1}}^{𝕄_{2}}(μ_{2}))_{𝕄_{2}∩^{w}𝕄_{1}} .

The preceding statement can be reformulated in terms entirely analogous to the previous “Mackey formulae”, provided we consider maps “over (𝕃, λ)”.

For a d–split generic Levi subgroup 𝕄 containing 𝕃, let us denote by

R_{𝕄}^{𝔾}(𝕃, λ): ℤ Uch(𝕄, (𝕃, λ)) → ℤ Uch(𝔾, (𝕃, λ))

the restriction of R_{𝕄}^{𝔾}, and by

∗R_{𝕄}^{𝔾}(𝕃, λ): ℤ Uch(𝔾, (𝕃, λ)) → ℤ Uch(𝕄, (𝕃, λ))

its adjoint, i.e., the restriction to ℤ Uch(𝔾, (𝕃, λ)) of pr_{𝕃,λ}^{𝕄} · ^{∗}R_{𝕄}^{𝔾} .

4.6. Proposition. Let (𝕃, λ) be a d–cuspidal pair, and let 𝕄_{1} and 𝕄_{2} be two d–split Levi subgroups of 𝔾 containing 𝕃. Then

∗R_{𝕄_{2}}^{𝔾}(𝕃, λ) · R_{𝕄_{1}}^{𝔾}(𝕃, λ) =

∑_{w∈W_{𝕄_{2}} (𝕃,λ)\W_{𝔾}(𝕃,λ)/W_{𝕄_{1}} (𝕃,λ)} R_{𝕄_{2}∩^{w}𝕄_{1}}^{𝕄_{2}}(𝕃, λ) · ^{∗}R_{𝕄_{2}∩^{w}𝕄_{1}}^{^{w}𝕄_{1}}(𝕃, λ) · ad(w) .

B. Φ_{d}–defect groups.

In §4.B, we reformulate some of the previous results in order to define a suitable notion of generic Φ_{d}–defect group of a unipotent character.

The d–Harish–Chandra theory allows us to define the notion of “Φ_{d}–defect groups” or “defect (generic) tori” of an element γ ∈ Uch(𝔾), as follows.

4.7. Definition. Let γ ∈ Uch(𝔾).

(1) For each d let S_{d}(γ) be the set of all Φ_{d}–tori 𝕊 of 𝔾 for which there exists a maximal torus 𝕋 containing 𝕊 such that ^{∗}R_{𝕋}^{𝔾}(γ) ≠ 0.

(2) The maximal elements of S_{d}(γ) are called the Φ_{d}–defect tori of γ.

4.8. Theorem. Let γ ∈ Uch(𝔾), and let (𝕃, λ) be a d–cuspidal pair (unique up to W_{𝔾}–conjugation) such that γ ∈ Uch(𝔾, (𝕃, λ)). Then 𝕊 ∈ S_{d}(γ) if and only if 𝕊 is W_{𝔾}–conjugate to a Φ_{d}–subgroup of Rad(𝕃). In particular the defect tori of γ are the W_{𝔾}–conjugates of Rad(𝕃)_{d}.

Remark. An element γ ∈ Uch(𝔾) is d–cuspidal if and only if the Sylow Φ_{d}– subtorus of Rad(𝔾) is its defect torus.

Proof of 4.8. We set 𝕄 := C_{𝔾}(𝕊).

Assume first that 𝕊 ∈ S_{d}(γ). Let 𝕋 be a maximal torus of 𝔾 such that 𝕋 ⊃ 𝕊 and ^{∗}R_{𝕋}^{𝔾}(γ) ≠ 0. Since 𝕋 ⊆ 𝕄, we have ^{∗}R_{𝕄}^{𝔾}(γ) ≠ 0 and there exists μ ∈ Uch(𝕄) such that (𝕄, μ) ≼ (𝔾, γ). By 3.2(i) there exists w ∈ W_{𝔾} such that 𝕃 is a subgroup of 𝕄^{w}, from which it follows that 𝕊^{w} is contained in the maximal Φ_{d}–subtorus of Rad(𝕃).

Assume now that 𝕊 is contained in the maximal Φ_{d}–subgroup of Rad(𝕃). Then 𝕃 ⊆ 𝕄. Since (cf. 3.15) Deg(^{∗}R_{𝕃}^{𝔾}(γ)) ≠ 0, we see that there exists a maximal torus 𝕋 of 𝕃, whence of 𝕄, such that ^{∗}R_{𝕋}^{𝔾}(γ) ≠ 0, which shows that 𝕊 ∈ S_{d}(γ). □

For the notation used below, the reader may refer to §2.A.

4.9. Theorem. Let γ ∈ Uch(𝔾) with Φ_{d}–defect group 𝕊. Let 𝕃 := C_{𝔾}(𝕊). Then

(1) 𝕊 = Rad(𝕃)_{d} and m_{γ} ∈ CF_{d}(𝔾, 𝕃),

(2) Deg(γ)_{d} = |𝔾|_{d}/|𝕊|,

(3) ^{∗}R_{𝕄}^{𝔾}(γ) = 0 whenever 𝕄 is a d–split Levi subgroup of 𝔾 such that Z_{d}(𝕄) is not W_{𝔾}–conjugate to a subgroup of 𝕊.

Proof. (1) results from the definition of a defect group.

(2) Since |𝔾|_{d}/|𝕊| = |𝕃_{ss}|_{d}, we see by 2.17 that

\frac{Deg(γ)}{|𝕃_{ss}|_{d}} ≡ \frac{Deg(R_{𝕃}^{𝔾}(1))}{|W_{𝔾}(𝕃)|} \frac{Deg(^{∗}R_{𝕃}^{𝔾}(γ))}{|𝕃_{ss}|_{d}} mod Φ_{d} ,

and it suffices to prove that \frac{Deg(^{∗}R_{𝕃}^{𝔾}(γ))}{|𝕃_{ss}|_{d}} is not divisible by Φ_{d}. But this

results from 3.15 and from 2.9.

(3) Z_{d}(𝕄) is not W_{𝔾}–conjugate to a subgroup of 𝕊 if and only if 𝕃 is not conjugate to a subgroup of 𝕄. Thus (3) is an easy consequence of 3.2(i). □

Remark. We shall see later on that a generic defect group does behave, in many respects, as the right “generization” of an actual defect group. Moreover, for suitable choices of q and ℓ (see §4), the Sylow ℓ– subgroups of the rational points of a generic defect group are the defect groups of a corresponding actual ℓ–block.

Assertions (2) and (3) of 4.9 may be stated in the following way, which generalizes the properties of d–cuspidal elements of Uch(𝔾).

4.10. Proposition. Let (𝕃, λ) be a d-cuspidal pair, and let γ be an element of Uch(𝔾, (𝕃, λ)). Then

(1) Deg(γ)_{d} = |𝕃_{ss}|_{d} ,

(2) for a d–split generic Levi subgroup 𝕄 of 𝔾, we have ^{∗}R_{𝕄}^{𝔾}(γ) = 0 unless 𝕃 ⊆_{W_{𝔾}} 𝕄 .

Finally, we have the following version of 2.13:

4.11. Proposition. Let γ ∈ Uch(𝔾) and let 𝕊 ∈ S_{d}(γ); let ψ be a uniform function in ℚ[x] Uch(𝔾). Then

(γ, ψ)_{𝔾} = ∑_{[𝕄 d–split;Rad(𝕄)_{d}≤𝕊]_{W_{𝔾}}} \frac{1}{|W_{𝔾}(𝕄)|}(γ, R_{𝕄}^{𝔾}(c_{d}(^{∗}R_{𝕄}^{𝔾}(ψ))))_{𝔾}

Proof. This is immediate from 2.13 if we notice that c_{d}(^{∗}R_{𝕄}^{𝔾}(ψ)) is a sum of R_{𝕋}^{𝕄}’s where 𝕋 ∈ T_{d}(𝕄): this shows that, for the scalar product to be non zero, those 𝕋’s must have 𝕋_{d} = Rad(𝕄)_{d} ≤ 𝕊. □

C. Regular characters of generic d–blocks.

In §4.C, results of §2.C are refined the same way results of §2.B were refined in §4.A. In particular we define what must be considered as the regular characters of the generic Φ_{d}–blocks.

Following the notation introduced in §2.C, we have a decomposition

R_{d}(𝔾) = ⨁_{[𝕃,λ]_{W_{𝔾}}} R_{𝕃,λ}(𝔾)

where (𝕃, λ) runs over the set of d–cuspidal pairs of 𝔾, and where we denote by R_{𝕃,λ}(𝔾) the free ℚ[x]–submodule of R_{d}(𝔾) generated by all the χ_{(𝕄,μ)}^{𝔾} for (𝕃, λ) ≼_{W_{𝔾}} (𝕄, μ).

We “extend” the definition of the orthogonal projection pr_{𝕃,λ}^{𝔾} (see §3.A above) by defining

pr_{𝕃,λ}^{𝔾}(χ_{(𝕄,μ)}^{𝔾}) := \{ \begin{matrix}χ_{(𝕄,μ)}^{𝔾} & if (𝕃, λ) ≼_{W_{𝔾}} (𝕄, μ) \\ 0 & if not . \end{matrix}

The following property of pr_{𝕃,λ}^{𝔾} is a translation of the consequences of 3.2 listed in §3.A above.

4.12. Lemma. Whenever ℕ and 𝕄 are two d–split generic Levi subgroups of 𝔾 such that ℕ ⊆ 𝕄, and ν ∈ Uch(ℕ), we have

pr_{𝕃,λ}^{𝔾}(R_{ℕ}^{𝔾}(Θ_{𝔾,𝕄}^{ℕ} ⊗ ν)) = \{ \begin{matrix}R_{ℕ}^{𝔾}(Θ_{𝔾,𝕄}^{ℕ} ⊗ ν) & if (𝕃, λ) ≼_{W_{𝔾}} (ℕ, ν) \\ 0 & if not . \end{matrix}

Proof. By 2.30, we know that

R_{ℕ}^{𝔾}(Θ_{𝔾,𝕄}^{ℕ} ⊗ ν) = R_{𝕄}^{𝔾}(Θ_{𝔾}^{𝕄} ⊗ R_{ℕ}^{𝕄}(ν)) ,

or, in other words,

R_{ℕ}^{𝔾}(Θ_{𝔾,𝕄}^{ℕ} ⊗ ν) = ∑_{{μ; (ℕ,ν)≼(𝕄,μ)}}(R_{ℕ}^{𝕄}(ν), μ)_{𝕄}R_{𝕄}^{𝔾}(Θ_{𝔾}^{𝕄} ⊗ R_{ℕ}^{𝕄}(ν)) .

The result then follows from the following property:

Assume (ℕ, ν) ≼ (𝕄, μ). Then

(𝕃, λ) ≼_{W_{𝔾}} (𝕄, μ) if and only if (𝕃, λ) ≼_{W_{𝔾}} (ℕ, ν) ,

which is immediate by 3.13. □

We define (cf. 2.28)

Reg_{(𝕃,λ)}^{𝔾} := pr_{𝕃,λ}^{𝔾}(Reg_{d}^{𝔾}) .

Thus by definition,

Reg_{(𝕃,λ)}^{𝔾} = ∑_{[(𝕄,μ) ; (𝕃,λ)≼_{𝔾}(𝕄,μ)]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(μ))}{|W_{𝔾}(𝕄, μ)|}.χ_{(𝕄,μ)}^{𝔾}

= ∑_{[(𝕄,μ) ; (𝕃,λ)≼_{𝔾}(𝕄,μ)]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(1))}{|W_{𝔾}(𝕄, μ)|}R_{𝕄}^{𝔾}(Θ_{𝔾}^{𝕄} ⊗ Deg(μ).μ) , (4.13)

and also

(4.14)  Reg_{d}^{𝔾} = ∑_{[(𝕃,λ)]_{W_{𝔾}}} Reg_{(𝕃,λ)}^{𝔾} ,

where (𝕃, λ) runs over the set of d–cuspidal pairs of 𝔾.

4.15. Theorem. We have

Reg_{(𝕃,λ)}^{𝔾} = ∑_{[(𝕄,μ) ; (𝕃,λ)≼_{𝔾}(𝕄,μ)]_{W_{𝔾}}} \frac{Deg(R_{𝕄}^{𝔾}(1))}{|W_{𝔾}(𝕄, μ)|}R_{𝕄}^{𝔾}(Ab_{d} Reg^{𝕄}⊗Deg(c_{d}(μ)).μ) .

Proof. It suffices to apply pr_{𝕃,λ}^{𝔾} to both sides of the corresponding formula for Reg_{d}^{𝔾} (see 2.33), and to use lemma 4.12 above. □

The preceding results allows us to improve 2.39, by decomposing the character Reg_{d}^{G^{F}} (see §2.D) into a sum of characters with similar divisibility properties.

The reader may refer to §2.D above for the notation used here.

Let (G, T, F) be a (q, ϕ)–triple associated with 𝔾, chosen in such a way that T contains a Sylow Φ_{d}–subgroup of G. Let (L, λ) be a d–cuspidal pair of G. We set

Reg_{L,λ}^{G^{F}} := ∑_{[(M,μ) ; (L,λ)≼_{G^{F}} (M,μ)]_{G^{F}}} \frac{DegR_{M}^{G}(μ)}{|W_{G^{F}} (M, μ)|}R_{M}^{G}(Θ_{G}^{M^{F}} .μ) ,

4.16. Theorem.

(1) We have Reg_{d}^{G^{F}} = ∑_{[(L,λ)]_{G^{F}}} Reg_{L,λ}^{G^{F}} .

(2) All the values of |W_{G}| Reg_{L,λ}^{G^{F}} are divisible by |𝔾|_{d}(q).

Proof. The first assertion is immediate. As in the proof of 2.39, the second assertion is a consequence of 2.42. □

5. π–blocks and isotypies

In §5, we apply what precedes to the study of ordinary blocks of actual finite reductive groups.

The preceding “generic” approach shows that the group G^{F} (at least as far as unipotent characters are concerned) hardly differentiates between two prime numbers dividing the same factor Φ_{d}(q). Thus, many usual objects and properties of the classical ℓ–modular representation theory can be replaced, without harm, by corresponding objects or properties when the single prime ℓ is replaced by a set of primes dividing the same factor Φ_{d}(q). In this section, we illustrate this with an application of the generic theory, assuming that we are dealing with “large” primes (which in practice means primes not dividing |W_{𝔾}⟨ϕ⟩|).

Since many notions (Brauer morphisms, generalized decomposition maps, isotypies, etc.) are often known only for the case of a single prime, we give (or recall) briefly, when needed, their definitions for a set of primes. The reader may refer to [Rob] and [RoSt] for more details — or read what follows assuming that π consists of a single prime.

A. π–blocks, Brauer morphisms.

In §5.A, for suitable sets of primes π, we describe π–idempotents which turn out to be the π–blocks of G^{F}. We compute their images under the usual Brauer morphisms.

Large primes.

Let 𝔾 be a generic finite reductive group, let p be a prime number, and let (G, T, F) be a (q, ϕ)–triple associated to 𝔾 such that q is a power of p. Let ℓ be a prime number different from p and dividing |G^{F} |.

5.1. Definition. We say that ℓ is large (for 𝔾) if ℓ ∤ |W_{𝔾}⟨ϕ⟩|.

5.2. Proposition. If ℓ is large, then

(1) there exists a unique integer d such that Φ_{d} divides |𝔾| and ℓ divides Φ_{d}(q) (in other words, we have |G^{F} |_{ℓ} = |𝔾|_{d}(q)_{ℓ}),

(2) ℓ is good for G,

(3) ℓ divides neither |(Z(G)/Z^{o}(G))^{F} | nor |(Z(G^{∗})/Z^{o}(G^{∗}))^{F^{∗}}|.

Proof. For (1), see [Ge].

(2) follows from the fact that bad primes divide |W |.

(3) The group of characters of Z(G)/Z^{o}(G) is isomorphic to the p′-part of the torsion subgroup of X/Q(R). Since the torsion subgroup of X/Q(R), namely the group Q(R)^{⊥⊥}/Q(R), is a subgroup of P (R)/Q(R), it is enough to check that |P (R)/Q(R)| and |P (R^{∨})/Q(R^{∨})| (this last to get the result for G^{∗}) divides |W |. But by definition |P (R)/Q(R)| is the connection index f_{R} of R, and f_{R} = f_{R^{∨}} divides |W | by [Bou], ch. VI, §2, prop.7. □

5.3. Definition. Let π be a set of prime numbers such that p ∉ π. We say that π is (G, F)–adapted if there exists an integer d such that |G^{F} |_{π} = |𝔾|_{d}(q)_{π}. We then say that π is (G, F, d)–adapted.

Remarks.

1. If π is (G, F, d)–adapted and L is an F –stable Levi subgroup of G, then π is (L, F, d)–adapted.

2. If π = {ℓ} where ℓ is a large prime, then π is (G, F)–adapted by 5.2.

5.4. Proposition. Let π be a set of large prime numbers which is (G, F, d)– adapted. Let L be an F -stable Levi subgroup of G.

(1) Let S = Z^{o}L. Let S be a π-subgroup of S^{F}. Then C_{G}^{o}(S) is a d–split Levi subgroup of G.

(2) Let T be an F -stable maximal torus of L, and let θ be a π-character of (L/[L, L])^{F}. Then G(T, θ) is a d–split Levi subgroup of G.

Proof. (1) Since every element of π is good for G, the group C_{G}^{o}(S) is a Levi subgroup of G. Moreover, by 5.2, (3), we have S ⊂ Z^{o}(C_{G}^{o}(S)). Now C_{G}^{o}(S) contains C_{G}(S), a Levi subgroup whose center by assumption is S, so we have Z^{o}(C_{G}^{o}(S)) ⊂ S, and thus π is d–adapted for Z^{o}(C_{G}^{o}(S)). It follows that S is contained in the Sylow Φ_{d}-subgroup Z^{o}(C_{G}^{o}(S))_{d} of Z^{o}(C_{G}^{o}(S)), since the quotient of the corresponding groups of rational points is a π′-group. Then we have C_{G}^{o}(S) = C_{G}^{o}(Z^{o}(C_{G}^{o}(S))_{d}), which shows that C_{G}^{o}(S) is a d–split Levi subgroup of G.

(2) The proof of the second assertion follows from the first one applied in the group G^{∗}. □ π–blocks. Let π be a (G, F, d)–adapted set of large primes, and let Reg_{π,(L,λ)}^{G^{F}} be

the projection of Reg_{L,λ}^{G^{F}} (see §3.C above) onto the subspace generated by characters in

E_{π}(G^{F} , 1) := ⋃_{s∈(G^{∗}^{F^{∗}})_{π}} E(G^{F} , (s)) ,

(cf. [BrMi]). We will prove Reg_{π,(L,λ)}^{G^{F}} is the regular character of a central “π–idempotent” of G^{F} (which will turn out to be a block if π = {ℓ}).

From now on, we assume that (G, T, F) is chosen so that T contains a Sylow Φ_{d}–subgroup of G. So T is contained up to G^{F} –conjugation in all d–split Levi subgroups of G.

5.5. Theorem. Let π be a set of large primes which is (G, F, d)–adapted.

(1) The projection of Reg_{d}^{G^{F}} onto E_{π}(G^{F} , 1) is the character Reg_{π}^{G^{F}}.

(2) Let (L, λ) be a d–cuspidal pair of G such that T ⊆ L. Then

Reg_{π,(L,λ)}^{G^{F}} = ∑_{[(M,θ,μ)]_{G^{F}}} χ_{(M,θ,μ)}^{G^{F}}(1).χ_{(M,θ,μ)}^{G^{F}}

where • θ runs over Ab_{π} Irr(L^{F}),

• M = G(T, θ) (cf. 2.24).

• (M, μ) is a d–split pair such that (L, λ) ≼_{G^{F}} (M, μ),

The character Reg_{π,(L,λ)}^{G^{F}} is the character associated to a central π–idempotent e_{π,(L,λ)}^{G^{F}}. In particular, its values are divisible by |G^{F} |_{π}.

(3) We have

Reg_{π}^{G^{F}} = ∑_{[(L,λ)]_{G^{F}}} Reg_{π,(L,λ)}^{G^{F}} ,

(in other words, e_{π}^{G^{F}} = ∑_{[(L,λ)]_{G^{F}}} e_{π,(L,λ)}^{G^{F}} ), where (L, λ) runs over the set of all d–cuspidal pairs of G.

Proof. (1) For a d–split Levi subgroup M of G, let us denote by Irr_{π}(Θ_{G}^{M^{F}}) the set of π–elements of Irr(Θ_{G}^{M^{F}}). Then, if pr_{π}^{G^{F}} denotes the projection onto the subspace generated by characters in E_{π}(G^{F} , 1), we have

pr_{π}^{G^{F}} (R_{M}^{G}(Θ_{G}^{M^{F}} .μ)) = ∑_{θ∈Irr_{π}(Θ_{G}^{M^{F}} )} R_{M}^{G}(θ.μ) .

We need the following lemma, which follows from 5.2 applied to a dual group L^{∗} of L. The reader may refer to 2.38 for the notation.

5.6. Lemma. Let L be an F –stable Levi subgroup of G containing T. Let π be a (Z^{o}(L), d)–adapted set of prime numbers which are large for 𝔾. Let M be a d–split Levi subgroup of G containing L. Let θ be a π–element of Irr(Θ_{G}^{M^{F}}). Then M = G(T, θ).

It then results from 5.6 that

pr_{π}^{G^{F}} (R_{M}^{G}(Θ_{G}^{M^{F}} .μ)) = ∑_{{θ; (θ π–element)(G(T,θ)=M)}} R_{M}^{G}(θ.μ) .

(1), as well as the first part of (2), are now immediate. Let us now prove that the values of Reg_{π,(L,λ)}^{G^{F}} are divisible by |G^{F} |_{π}. Let e_{π}^{G^{F}} denote the central idempotent corresponding to E_{π}(G^{F} , 1). We know (see [BrMi], th. 2.2) that e_{π}^{G^{F}} is a π–idempotent, i.e., e_{π}^{G^{F}} ∈ ℤ̄[1/r]_{r∈π′}. Since π is d–adapted for G and no element of π divides |W_{𝔾}|, we know by 4.16 that all values of Reg_{L,λ}^{G^{F}} are

divisible by |G^{F} |_{π}. For all g ∈ G^{F}, we have Reg_{π,(L,λ)}^{G^{F}}(g) = Reg_{L,λ}^{G^{F}}(g.e_{π}^{G^{F}}) and this proves now the second assertion of 5.5.

Finally the third assertion of 5.5 is an immediate consequence of 4.14. □

Brauer morphisms.

Let π be a set of prime numbers, and let s be a π–element of G^{F}. Whenever e is a central π–idempotent of ℚ̄G^{F}, we denote by Br_{s}(e) the image of e through the corresponding Brauer morphism (cf. [Rob] or [RoSt], and also [BrMi], §3). For the convenience of the reader, we recall the definition of Br_{s}(e).

• Let ϖ_{s}^{G^{F}} be the class function on G^{F} with value |C_{G^{F}} (s)| on the G^{F} – conjugacy class of s and 0 elsewhere.

• For a class function ψ on G^{F}, let dec_{π}^{s,G^{F}} (ψ) denote the class function on C_{G^{F}} (s) such that

dec_{π}^{s,G^{F}} (ψ)(t) = \{ \begin{matrix}ψ(st) if t is a π′–element \\ 0 if not. \end{matrix}

• For a class function ψ on G^{F}, let e.ψ be the class function on G^{F} defined by e.ψ(g) := ψ(eg). Then the following formula defines the idempotent Br_{s}(e) :

(5.7)  Br_{s}(e). Reg^{C_{G^{F}} (s)} = dec_{π}^{s,G^{F}} (e.ϖ_{s}^{G^{F}}) .

Thus Br_{s}(e) is a central π–idempotent of ℚ̄C_{G^{F}} (s).

Iteration of 5.7 allows us to define (see for example [RoSt] or [Rob]) an idempotent Br_{S}(e) for all abelian π–subgroups S of G^{F}. In the case where π = {ℓ}, this definition coincides with the original definition of Brauer.

5.8. Theorem. Let (L, λ) be a d–cuspidal pair of G. Let π be a set of large prime numbers which is (G, F, d)–adapted for L. Let S be a subgroup of Z^{o}(L)_{π}^{F} and set M := C_{G}^{o}(S). Then

Br_{S}(e_{π,(L,λ)}^{G^{F}}) = ∑_{(L′,λ′)} e_{π,(L′,λ′)}^{M^{F}}

where (L′, λ′) runs over a complete set of representatives for the M^{F} –conjugacy classes of d–cuspidal pairs of M which are G^{F} –conjugate to (L, λ).

Proof. We may assume by induction that S is cyclic. Let s be a generator of S. By “Curtis–type formula” (see [Br2], 4.3), we know that

dec_{π}^{s,G^{F}} = dec_{π}^{s,M^{F}} · ^{∗}R_{M}^{G} .

So by 5.7 we must prove that

(dec_{π}^{s,M^{F}} · ^{∗}R_{M}^{G})(e_{π,(L,λ)}^{G^{F}}.ϖ_{s}^{G^{F}}) = ( ∑_{(L′,λ′)} e_{π,(L′,λ′)}^{M^{F}}) . Reg^{M^{F}} .

Let us denote by E_{π}(G^{F} , (L, λ)) the subspace of the space of all class functions ψ on G^{F} such that e_{π,(L,λ)}^{G^{F}}.ψ = ψ.

For each d–cuspidal pair (L, λ), ^{∗}R_{M}^{G} sends the ℤ–span of E_{π}(G^{F} , (L, λ)) into the ℤ–span of ⋃_{(L′,λ′)} E_{π}(M^{F} , (L′, λ′)). Thus we see that

(L) ( ∑_{(L′,λ′)} e_{π,(L′,λ′)}^{M^{F}}) .^{∗}R_{M}^{G}(e_{π,(L,λ)}^{G^{F}}.ϖ_{s}^{G^{F}}) = ^{∗}R_{M}^{G}(e_{π,(L,λ)}^{G^{F}}.ϖ_{s}^{G^{F}}) ,

and also that, for (N, ν) any d–cuspidal pair not G^{F} –conjugate to (L, λ),

(N)  ( ∑_{(L′,λ′)} e_{π,(L′,λ′)}^{M^{F}}) .^{∗}R_{M}^{G}(e_{π,(N,ν)}^{G^{F}}.ϖ_{s}^{G^{F}}) = 0 .

It follows from (L) and (N) that

( ∑_{(L′,λ′)} e_{π,(L′,λ′)}^{M^{F}}) .^{∗}R_{M}^{G}(e_{π,(L,λ)}^{G^{F}}.ϖ_{s}^{G^{F}}) = ^{∗}R_{M}^{G}(ϖ_{s}^{G^{F}}) ,

and so

(dec_{π}^{s,M^{F}} · ^{∗}R_{M}^{G})(e_{π,(L,λ)}^{G^{F}}.ϖ_{s}^{G^{F}}) = ( ∑_{(L′,λ′)} e_{π,(L′,λ′)}^{M^{F}}) .(dec_{π}^{s,M^{F}} · ^{∗}R_{M}^{G})(ϖ_{s}^{G^{F}})

= ( ∑_{(L′,λ′)} e_{π,(L′,λ′)}^{M^{F}}) .dec_{π}^{s,G^{F}} (ϖ_{s}^{G^{F}})

= ( ∑_{(L′,λ′)} e_{π,(L′,λ′)}^{M^{F}}) . Reg^{M^{F}} .

□

B. Isotypies.

In §5.B, we check that the isotypy conjectures of [Br1] hold for our π– idempotents.

Preliminaries.

The following notation and hypothesis are in force throughout this section. Let π be a set of large prime numbers for 𝔾, which is (G, F, d)–adapted.

Let (L, λ) be a d–cuspidal pair of G. Let T be an F –stable maximal torus of G which contains Z^{o}(L).

In order to describe the irreducible characters of Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ) we have to find the inertia group of θ ∈ Irr(Z^{o}(L)_{π}^{F}) in W_{G^{F}} (L, λ). This is the purpose of the next proposition.

5.9. Proposition.

(1) The restriction map from L^{F} down to Z^{o}(L)^{F} induces an isomorphism

Ab_{π} Irr(L^{F})⟶^{∼}Irr(Z^{o}(L)_{π}^{F}) .

(2) Let θ be a character of Z^{o}(L)_{π}^{F}, and let θ denote as well the linear character of T^{F} corresponding to it by (1). Then the inertial subgroup of θ in W_{G^{F}} (L, λ) is W_{G^{F} (T,θ)}(L, λ).

Proof. (1) We know that L = [L, L]Z^{o}(L). Moreover, Z^{o}(L) ∩ [L, L] is a subgroup of the finite group Z([L, L]). Since π consists of large prime numbers for 𝔾, we see by 5.2 that Z^{o}(L) ∩ [L, L] is a π′–group. Thus the projection Z^{o}(L)_{π}^{F} → L^{F} /[L, L]^{F} is injective. Since |Z^{o}(L)_{π}^{F} | = |L^{F} /[L, L]^{F} |_{π}, the image of Z^{o}(L)_{π}^{F} in L^{F} /[L, L]^{F} is a supplement of (L^{F} /[L, L]^{F})_{π′}, which proves (1).

(2) Let θ̃ be the linear character of L^{F} corresponding to θ by (1). We have to show that N_{G^{F}} (L, λ, θ̃) = N_{G^{F} (T,θ)}(L, λ). This will result from the more general equality N_{G^{F}} (L, θ̃) = N_{G^{F} (T,θ)}(L). Let T′ be a maximally split F -stable maximal torus of L, and let θ′ = θ̃|_{T′}. By 2.24 we have G^{F} (T, θ) = G^{F} (T′, θ′). Since two maximally split F -stable maximal tori of L are L^{F} – conjugate, we can find representatives of N_{G^{F}} (L, θ̃)/L^{F} in N_{G^{F}} (L, T′, θ′). It is thus enough to show that N_{G^{F}} (T′, θ′) ⊂ G^{F} (T′, θ′). This results from the fact (applied in G^{∗}) that centralizers of π-elements are connected and are Levi subgroups of G. □

In order to simplify the notation, we set

G(θ) := G(T, θ) .

Let θ̃ be the character of the group Z^{o}(L)_{π}^{F} ⋊ W_{G^{F} (θ)}(L, λ) defined by θ̃(zw) := θ(z) where z ∈ Z^{o}(L)_{π}^{F}, w ∈ W_{G^{F} (θ)}(L, λ). Then, whenever τ is an irreducible character of W_{G^{F} (θ)}(L, λ), the character

Ind_{Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (θ)}(L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(θ̃.τ)

is an irreducible character of the group Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ). We have (with the above notation)

Irr(Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ)) = {Ind_{Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (θ)}(L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(θ̃.τ)}_{[(θ,τ)]_{W_{G^{F}} (L,λ)}} .

We denote by e_{π,(L,λ)}^{G^{F}} the central π–idempotent of G^{F} associated with π and (L, λ) (cf. 5.5). For x ∈ G^{F} we set

G(x) := C_{G}^{o}(x) and G^{F} (x) := C_{G}^{o}(x)^{F} .

The centralizer in Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ) of an element x ∈ Z^{o}(L)_{π}^{F} is Z^{o}(L)_{π}^{F} ⋊ W_{G^{F} (x)}(L, λ).

The next proposition is a result about control of fusion.

5.10. Proposition. The map x ↦ (x, e_{π,(L,λ)}^{G^{F} (x)}) induces a bijection between the conjugacy classes of π–elements of the group Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ) and the G^{F} –conjugacy classes of pairs (x, e_{π,(L,λ)}^{G^{F} (x)}) (for a π–element x of Z^{o}(L)_{π}^{F}).

Proof. Since e_{π,(L,λ)}^{G^{F} (x)} = e_{π,(L′,λ′)}^{G^{F} (x)} if and only if (L, λ) and (L′, λ′) are conjugate under G^{F} (x), we must check the following property: the elements x and x′ of Z^{o}(L)_{π}^{F} are conjugate under Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ) if and only if there exists g ∈ G^{F} such that

(a) x′ = x^{g},

(b) (L, λ)^{g} is G^{F} (x′)–conjugate to (L, λ). This is obvious since W_{G^{F}} (L, λ) = N_{G^{F}} (L, λ)/L^{F}. □

π–perfect isometries.

We shall use the notion of “perfect isometry” as defined in [Br1]. Since this was only defined for ℓ–idempotents and not for π–idempotents, we give here briefly a whole account of the necessary definitions and elementary properties of π–perfect isometries.

Let H be a finite group and let π be a set of prime numbers. Let K be a field of characteristic zero which is a splitting field for the group H. Let O denote the integral closure in K of the ring ℤ[{1/ℓ}_{ℓ∉π}].

Let e be an idempotent of the center ZOH (a “central π–idempotent” for H).

We denote by CF(H, e; K) the K–vector space consisting of class functions α on H with values in K and such that α(eh) = α(h) for all h ∈ H. Let CF_{π′}(H, e; K) be the subspace of functions vanishing outside of the set H_{π′} of π′–elements of H. We define the O–modules CF(H, e; O) and CF_{π′}(H, e; O) in an analogous way. We denote by Irr(H, e) the set of irreducible characters of H on K which belong to CF(H, e; K), and so to CF(H, e; O).

5.11. Definition. (see [Br1], 4.1) Let H and H′ be finite groups, and let e and e′ be two central π–idempotents for H and H′ respectively. Let

I: ℤIrr(H, e)⟶^{∼}ℤIrr(H′, e′)

be an isometry. I is π–perfect if the following two conditions are fulfilled: (P1) I extends linearly to an isometric isomorphism

CF(H, e; O)⟶^{∼}CF(H′, e′; O) .

(P2) The restriction of I to CF_{π′}(H, e; O) induces an isometric isomor-

phism

CF_{π′}(H, e; O)⟶^{∼}CF_{π′}(H′, e′; O) .

The proof of the following property of π–perfect isometries is the same as the proof of the corresponding result given in [Br1], 1.5, for the case π = {ℓ}.

5.12. Proposition. Assume that

I: ℤIrr(H, e)⟶^{∼}ℤIrr(H′, e′)

is a π–perfect isometry. Then the algebra isomorphism from ZKHe onto ZKH′e′ defined by I induces an algebra isomorphism from ZOHe onto ZOH′e′. In particular e is primitive if and only if e′ is primitive.

Generalized decomposition maps.

For the convenience of the reader, we also recall the definition of generalized decomposition maps in the particular settings we are working in (see for example [Br1], 4A).

5.13. Definition.

(1) Given a class function ψ of G^{F}, let dec_{π,(L,λ)}^{x,G^{F}}(ψ) be the class function on G^{F} (x) defined as follows (see for example [Br1], 4.3):

• dec_{π,(L,λ)}^{x,G^{F}}(ψ) vanishes outside of the set of π′–elements of G^{F} (x),

• for x′ ∈ G^{F} (x)_{π′},

dec_{π,(L,λ)}^{x,G^{F}}(ψ)(x′) := ψ(xx′e_{π,(L,λ)}^{G^{F} (x)}) .

(2) Similarly, given a class function φ of Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ), let

dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(φ) be the class function on the group

Z^{o}(L)_{π}^{F} ⋊ W_{G^{F} (x)}(L, λ)

defined as follows:

• dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(φ) vanishes outside of the set of π′–elements of

the group Z^{o}(L)_{π}^{F} ⋊ W_{G^{F} (x)}(L, λ),
• for a π′–element x′ of Z^{o}(L)_{π}^{F} ⋊ W_{G^{F} (x)}(L, λ),
dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(φ)(x′) := φ(xx′) .

Let ^{t}dec_{π,(L,λ)}^{x,G^{F}} denote the adjoint of dec_{π,(L,λ)}^{x,G^{F}}. For a class function φ of

G^{F} (x) with support on G^{F} (x)_{π′}, ^{t}dec_{π,(L,λ)}^{x,G^{F}}(φ) is the class function of G^{F} with support on the π–section of x in G^{F} given by

tdec_{π,(L,λ)}^{x,G^{F}}(φ)(xx′) = φ(x′)

for all x′ ∈ G^{F} (x)_{π′}. Similarly, we let ^{t}dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} denote the

adjoint of dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}.

The following result is a classical consequence of Brauer’s second main theorem (see for example [Br1], 4A, last part, for the case where π consists of a single prime).

5.14. Lemma.

(1) If ψ is a linear combination of elements of E_{π}(G^{F} , (L, λ)), then

ψ = ∑_{[x∈Z^{o}(L)_{π}^{F}]_{W_{G^{F}} (L,λ)}}(^{t}dec_{π,(L,λ)}^{x,G^{F}} · dec_{π,(L,λ)}^{x,G^{F}})(ψ) .

(2) If ξ is a class function on Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ), then

ξ = ∑_{[x∈Z^{o}(L)_{π}^{F}]_{W_{G^{F}} (L,λ)}}(^{t}dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} · dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)})(ξ) .

The main theorem.

Notation.

Let (L, λ) be a d–cuspidal pair of G. We denote by Uch(G^{F} , (L, λ)) the set of unipotent characters γ of G^{F} such that <R_{L}^{G}(λ), γ> ≠ 0 . It follows from 3.2 that there exists a collection of isometries

I_{(L,λ)}^{M} : ℤIrr(W_{M^{F}} (L, λ)) → ℤ Uch(M^{F} , (L, λ)) ,

where M runs over the set of all d–split Levi subgroups of G, such that

R_{M}^{G} · I_{(L,λ)}^{M} = I_{(L,λ)}^{G} · Ind_{W_{M^{F}} (L,λ)}^{W_{G^{F}} (L,λ)} .

5.15. Theorem.

(1) The map

I_{π,(L,λ)}^{G^{F}}: ℤIrr (Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ)) ⟶ ℤIrr(G^{F} , e_{π,(L,λ)}^{G^{F}})

such that

I_{π,(L,λ)}^{G^{F}}: Ind_{Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (θ)}(L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(θ̃.τ) ↦ R_{G(θ)}^{G}(θ.I_{(L,λ)}^{G(θ)}(τ))

commutes with the generalized decomposition maps in the following sense: whenever x ∈ Z^{o}(L)_{π}^{F}, then

(a)  dec_{π,(L,λ)}^{x,G^{F}} · I_{π,(L,λ)}^{G^{F}} = I_{π,(L,λ)}^{G^{F} (x)} · dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

(2) I_{π,(L,λ)}^{G^{F}} is a π–perfect isometry between (Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ), 1) and

(G^{F} , e_{π,(L,λ)}^{G^{F}}).

Remark. Had we given the general definition of a “π-isotypy” (see [Br1], 4.6, for the definition of an ℓ–isotypy), assertions (1) and (2) above would mean that (G^{F} , e_{π,(L,λ)}^{G^{F}}) and (Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ), 1) are π–isotypic.

Proof of 5.15, (1).

For x ∈ Z^{o}(L)_{π}^{F}, we have the “Curtis type formula” (cf. [Br2], 4.3)

(5.16)  dec_{π,(L,λ)}^{x,G^{F}} = dec_{π,(L,λ)}^{x,G^{F} (x)} · ^{∗}R_{G(x)}^{G}(L, λ) ,

and similarly it is easy to see that

dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} = dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (x)}(L,λ)} · Res_{Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (x)}(L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

Thus we see that (a) in the statement is equivalent to

dec_{π,(L,λ)}^{x,G^{F} (x)}·^{∗}R_{G(x)}^{G}(L, λ) · I_{π,(L,λ)}^{G^{F}} =

(b) I_{π,(L,λ)}^{G^{F} (x)} · dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (x)}(L,λ)} · Res_{Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (x)}(L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

The proof of 5.15 is in several steps.

Step 1. The first step allows us to reduce the proof of (b) to the case where x ∈ Z^{o}(G)_{π}^{F}.

5.17. Proposition. Whenever N is a d–split Levi subgroup of G,

∗R_{N}^{G}(L, λ) · I_{π,(L,λ)}^{G^{F}} = I_{π,(L,λ)}^{N^{F}} · Res_{Z^{o}(L)_{π}^{F} ⋊W_{N^{F}} (L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

Proof of 5.17. It suffices to prove the equivalent adjoint statement:

R_{N}^{G}(L, λ) · I_{π,(L,λ)}^{N^{F}} = I_{π,(L,λ)}^{G^{F}} · Ind_{Z^{o}(L)_{π}^{F} ⋊W_{N^{F}} (L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

Thus we must check that for all d–split Levi subgroups M of N such that L ⊆ M, for θ ∈ Ab_{π} Irr(Z^{o}(L)^{F}) such that N(θ) = M, and for all irreducible characters τ of W_{M}(L, λ),

R_{N}^{G}(L, λ) · I_{π,(L,λ)}^{N^{F}} (Ind_{Z^{o}(L)_{π}^{F} ⋊W_{M^{F}} (L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{N^{F}} (L,λ)}(θ̃τ)) =
I_{π,(L,λ)}^{G^{F}} · Ind_{Z^{o}(L)_{π}^{F} ⋊W_{N^{F}} (L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} (Ind_{Z^{o}(L)_{π}^{F} ⋊W_{M^{F}} (L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{N^{F}} (L,λ)}(θ̃τ)) .

This last equality is equivalent to

R_{G(θ)}^{G} (θR_{M}^{G(θ)}(I_{(L,λ)}^{M}(τ))) = R_{G(θ)}^{G} (θI_{(L,λ)}^{G(θ)} (Ind_{W_{M^{F}} (L,λ)}^{W_{G^{F} (θ)}(L,λ)}(τ))) .

This follows now from the fundamental theorem 3.2 applied in G(θ). □

By 5.17, we see that (b) is equivalent to

dec_{π,(L,λ)}^{x,G^{F} (x)} · I_{π,(L,λ)}^{G^{F} (x)} · Res_{Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (x)}(L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} =

I_{π,(L,λ)}^{G^{F} (x)} · dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (x)}(L,λ)} · Res_{Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (x)}(L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

Thus we see that it is sufficient to prove that, for x ∈ Z^{o}(G)_{π}^{F},

(c)  dec_{π,(L,λ)}^{x,G^{F}} · I_{π,(L,λ)}^{G^{F}} = I_{π,(L,λ)}^{G^{F}} · dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

Step 2. The usual decomposition maps are defined by the formulae

dec_{π,(L,λ)}^{G^{F}} := dec_{π,(L,λ)}^{1,G^{F}} and dec_{π}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} := dec_{π}^{1,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

For x ∈ Z^{o}(G)_{π}^{F}, and for a class function ψ on G^{F}, we denote by t^{x,G^{F}} (ψ) the class function on G^{F} defined by t^{x,G^{F}} (ψ)(g) := ψ(xg) .

Similarly, for x ∈ Z^{o}(G)_{π}^{F}, and a class function φ on Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ), we denote by t^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(φ) the class function defined by

t^{x,G^{F}} (φ)(h) := φ(xh) .

Thus for x ∈ Z^{o}(G)_{π}^{F} we have

dec_{π,(L,λ)}^{x,G^{F}} = dec_{π,(L,λ)}^{G^{F}} · t^{x,G^{F}}

dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} = dec_{π}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} · t^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

Equation (c) becomes then

dec_{π,(L,λ)}^{G^{F}} · t^{x,G^{F}} ·I_{π,(L,λ)}^{G^{F}} =
I_{(L,λ)}^{G^{F}} · dec_{π}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} · t^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

We see that it suffices to prove

(d)  t^{x,G^{F}} · I_{π,(L,λ)}^{G^{F}} = I_{π,(L,λ)}^{G^{F}} · t^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}

and

(e)  dec_{π,(L,λ)}^{G^{F}} · I_{π,(L,λ)}^{G^{F}} = I_{π,(L,λ)}^{G^{F}} · dec_{π}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

Equation (d) is an immediate consequence of the values of Deligne–Lusztig and ordinary induction. Indeed, we see that

(t^{x,G^{F}} ·I_{π,(L,λ)}^{G^{F}})(Ind_{Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (θ)}(L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(θ̃.τ)) =

(I_{π,(L,λ)}^{G^{F}} · t^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}) (Ind_{Z^{o}(L)_{π}^{F} ⋊W_{G^{F} (θ)}(L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(θ̃.τ)) =

θ(x)R_{G(θ)}^{G}(θ.I_{(L,λ)}^{G(θ)}(τ)) .

Step 3 : proof of (e).

The proof is by induction on rk_{ss}(G)−rk_{ss}(L), where we denote by rk_{ss}(G) the semi–simple rank of G.

1. The case G = L.

This is the case where rk_{ss}(G) = rk_{ss}(L). The irreducible characters of L^{F} to be considered are the θ.λ where θ ∈ Ab_{π} Irr(L^{F}). The equality (e) is then proved by the following lemma.

5.18. Lemma.

(1) For a d–cuspidal irreducible character λ of L^{F} and θ ∈ Ab_{π} Irr(L^{F}), we have

dec_{π,(L,λ)}^{L^{F}}(θ.λ) = \frac{1}{|Z^{o}(L)_{π}^{F} |}Ab_{π} Reg^{L^{F}}· λ = \frac{1}{|Z^{o}(L)_{π}^{F} |} ∑_{η∈Irr(Z^{o}(L)_{π}^{F})} η.λ .

(2) For θ ∈ Irr(Z^{o}(L)_{π}^{F}), we have

dec_{π}^{Z^{o}(L)_{π}^{F}} (θ) = \frac{1}{|Z^{o}(L)_{π}^{F} |} Reg^{Z^{o}(L)_{π}^{F}} = \frac{1}{|Z^{o}(L)_{π}^{F} |} ∑_{η∈Irr(Z^{o}(L)_{π}^{F})} η .

Proof of 5.18. The second assertion is obvious. To prove the first one, we notice the following property of d–cuspidal characters.

5.19. Let γ be a d–cuspidal irreducible unipotent character of G^{F}. Let π be a set of large primes which is (G, F, d)–adapted. Let g ∈ G^{F}. Then γ(g) = 0 unless the π–component g_{π} of g belongs to Z^{o}(G)^{F}.

Proof of 5.19. It results from the Curtis type formula (see 5.16) that, for M := C_{G}^{o}(g_{π}), we have

γ(g) = ^{∗}R_{M}^{G}(γ)(g) .

Since M is a d–split Levi subgroup of G (by 5.4), and since γ is d–cuspidal, we see by 4.10, (2), that ^{∗}R_{M}^{G}(γ) = 0 unless M = G, which proves 5.19. □

□

Remark. The preceding result is a particular case of the following property, which also follows from 4.10, (2).

Let γ be a unipotent character of G^{F} such that (γ, R_{L}^{G}(λ))_{G^{F}} ≠ 0 , where L is a d–split Levi subgroup of G, and where λ is a d–cuspidal unipotent character of L^{F}. Let π be a set of large primes which is (G, F, d)–adapted. Let g ∈ G^{F}. Then γ(g) = 0 unless the π–component g_{π} of g belongs to Z^{o}(L)^{F}.

This property can also be viewed as a consequence of Brauer’s second main theorem and theorem 5.8 (see for example formula 5.14, (1)).

2. Applying the induction hypothesis.

By the induction hypothesis, we assume now that (e) holds whenever G is replaced by any of its proper d–split Levi subgroups. It follows that (c)

(whence (b) and (a) as well) holds for all x ∈ Z^{o}(L)_{π}^{F} −Z^{o}(G)_{π}^{F}. The formulae 5.14 applied with ψ = I_{π,(L,λ)}^{G^{F}}(ξ) then show that (5.20)

I_{π,(L,λ)}^{G^{F}}· ( ∑_{z∈Z^{o}(G)_{π}^{F}} ^{t}dec_{π}^{z,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} · dec_{π}^{z,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}) =

( ∑_{z∈Z^{o}(G)_{π}^{F}} ^{t}dec_{π,(L,λ)}^{x,G^{F}} · dec_{π,(L,λ)}^{z,G^{F}}) · I_{π,(L,λ)}^{G^{F}} .

To prove that (e) holds, again by the induction hypothesis and by 5.17, it suffices to prove that, for θ ∈ Irr_{π}(Z^{o}(G)^{F}) and τ ∈ Irr(W_{G^{F}} (L, λ)), we have

(dec_{π,(L,λ)}^{G^{F}} · I_{π,(L,λ)}^{G^{F}})(θ.τ) = (I_{π,(L,λ)}^{G^{F}} · dec_{π}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)})(θ.τ) ,

i.e.,

dec_{π,(L,λ)}^{G^{F}}(θ.I_{(L,λ)}^{G}(τ)) = (I_{π,(L,λ)}^{G^{F}} · dec_{π}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)})(θ.τ) .

It is easy to check that

∑_{θ∈Irr_{π}(Z^{o}(G)^{F})} ∑_{z∈Z^{o}(G)_{π}^{F}} ^{t}dec_{π}^{z,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}· dec_{π}^{z,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(θτ) =

|Z^{o}(G)_{π}^{F} |dec_{π}^{Z^{o}(L)_{π}^{F}} (τ) ,

and also that

∑_{θ∈Irr_{π}(Z^{o}(G)^{F})} ∑_{z∈Z^{o}(G)_{π}^{F}} ^{t}dec_{π,(L,λ)}^{z,G^{F}}· dec_{π,(L,λ)}^{z,G^{F}}(I_{(L,λ)}^{G^{F}}(θτ)) =

|Z^{o}(G)_{π}^{F} |dec_{π,(L,λ)}^{G^{F}}(I_{(L,λ)}^{G^{F}}(τ)) .

Since

dec_{π}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(θ.τ) = dec_{π}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(τ)

and

dec_{π,(L,λ)}^{G^{F}}(θ.I_{(L,λ)}^{G^{F}}(τ)) = dec_{π,(L,λ)}^{G^{F}}(I_{(L,λ)}^{G^{F}}(τ)) ,

we see that (e) follows then from what precedes. This completes step (3) and thus the proof of 5.15, (1). □

Proof of 5.15, (2).

As in step 3 above, the proof is by induction on rk_{ss}(G) − rk_{ss}(L).

1. The case where G = L.

We use notation previously introduced (see 5.11). We shall need the following property of d–cuspidal characters.

5.21. Lemma. Let γ be a d–cuspidal unipotent character of G^{F}. Let π be a set of large prime numbers which is (G, F, d)–adapted for G. Then for all g ∈

G^{F}, |C_{G^{F}} (g) : Z^{o}(G)^{F} |_{π} divides γ(g) (in other words, \frac{γ(g)}{|C_{G^{F}} (g) : Z^{o}(G)^{F} |} ∈

O).

Proof of 5.21. We see by 2.9 (cf. proof of 2.42) that for each ℓ ∈ π, we have γ(1)_{ℓ} = |G^{F} : Z^{o}(G)^{F} |_{ℓ}, and so γ is a character with ℓ–defect zero of the group G^{F} /Z^{o}(G)^{F}. Hence, we know that, for all g ∈ G^{F}, |C_{G^{F}} (g) : Z^{o}(G)^{F} |_{ℓ} divides γ(g), from which it follows that |C_{G^{F}} (g) : Z^{o}(G)^{F} |_{π} divides γ(g). □

We must check that the map

I_{π(L,λ)}^{L^{F}}: ℤIrr(Z^{o}(L)_{π}^{F})⟶^{∼}ℤIrr(L^{F} , e_{π,(L,λ)}^{L^{F}})

such that θ ↦ θλ (here we identify Ab_{π} Irr(L^{F}) and Irr(Z^{o}(L)_{π}^{F}) after proposition 5.9, (1)), is a π–perfect isometry.

• The O–module CF(Z^{o}(L)_{π}^{F}; O) is generated by the set of characteristic functions δ_{x}^{Z^{o}(L)_{π}^{F}} of the elements x of Z^{o}(L)_{π}^{F}, while the submodule CF_{π′}(Z^{o}(L)_{π}^{F}; O) is generated by the characteristic function δ_{1}^{Z^{o}(L)_{π}^{F}}. We have

(a)  δ_{x}^{Z^{o}(L)_{π}^{F}} = \frac{1}{|Z^{o}(L)_{π}^{F} |} ∑_{θ∈Irr(Z^{o}(L)_{π}^{F})} θ(x^{−1})θ .

• The O–module CF(L^{F} , e_{π,(L,λ)}^{L^{F}}; O) is generated by the set of “projections” of the characteristic functions of the conjugacy classes of L^{F} “onto the idempotent e_{π,(L,λ)}^{L^{F}}”. By 5.19, we see that all these characteristic func-

tions project onto zero, except the characteristic functions δ_{(xx′)}^{L^{F}} of elements xx′ ∈ L^{F} whose π–component x belongs to Z^{o}(L)_{π}^{F}. The projection of such a function is then

(b)  e_{π,(L,λ)}^{L^{F}} · δ_{(xx′)}^{L^{F}} := \frac{λ(x′^{−1})}{|C_{L^{F}} (x′)|} ∑_{θ∈Ab_{π} Irr(L^{F})} θ(x^{−1})θλ .

Since

e_{π,(L,λ)}^{L^{F}} · δ_{(xx′)}^{L^{F}} := \frac{λ(x′^{−1})}{|C_{L^{F}} (x′)|}e_{π,(L,λ)}^{L^{F}} · δ_{(x)}^{L^{F}} ,

83 we see by 5.21 that the module CF(L^{F} , e_{π,(L,λ)}^{L^{F}}; O) is actually generated by the functions e_{π,(L,λ)}^{L^{F}} · δ_{(x)}^{L^{F}} for x ∈ Z^{o}(L)_{π}^{F}.

The submodule CF_{π′}(L^{F} , e_{π,(L,λ)}^{L^{F}}; O) is generated by the functions e_{π,(L,λ)}^{L^{F}}· δ_{(x′)}^{L^{F}}. For the same reasons as above, it is actually generated by e_{π,(L,λ)}^{L^{F}} · δ_{(1)}^{L^{F}}.

By (a) and (b), we see that

e_{π,(L,λ)}^{L^{F}} · δ_{(x)}^{L^{F}} = \frac{λ(1)}{|L^{F} : Z^{o}(L)_{π}^{F} |}I_{π,(L,λ)}^{L^{F}}(δ_{x}^{Z^{o}(L)_{π}^{F}}) ,

and this proves indeed that I_{π,(L,λ)}^{L^{F}} induces a bijection from the module CF(L^{F} , e_{π,(L,λ)}^{L^{F}}; O) to CF(Z^{o}(L)_{π}^{F}; O) as well as from CF_{π′}(L^{F} , e_{π,(L,λ)}^{L^{F}}; O) to CF_{π′}(Z^{o}(L)_{π}^{F}; O).

2. Applying the induction hypothesis The following proof is the analogue of the proof of lemma 4.5 in [Br1]. We first need to prove an easy general result about finite groups. Let H be

a finite group, and let e be a central π–idempotent for H. Let us denote by

dec_{π}^{H}: CF(H, e; O) → CF_{π′}(H, e; O)

the “π–decomposition map”, defined as usual (for f ∈ CF(H, e; O), and for h ∈ H_{π′}, we have dec_{π}^{H}(f)(h) = f(h)). 5.22. The O–module CF_{π′}(H, e; O) is generated by dec_{π}^{H}(Irr(H, e)). Proof of 5.22. Since dec_{π}^{H} commutes with the “projection onto e”, we may assume e = 1. Let h be a π′–element of H, and let δ_{π,h}^{H} be the characteristic function of the π′–section of h (namely, we have δ_{π,h}^{H}(g) = 1 if the π′–component of g is conjugate to h, and δ_{π,h}^{H}(g) = 0 otherwise). It suffices to prove that δ_{π,h}^{H} is a linear combination with coefficients in O of elements of Irr(H). This is an easy consequence of Brauer’s characterization of characters.

□ We now assume that for every proper d–split Levi subgroup N of G con-

taining L, the map I_{π,(L,λ)}^{N^{F}} is a π–perfect isometry.

Then by the definition of a perfect isometry (5.11) and by assertion (1) of theorem 5.15, we see that it suffices to prove that I_{π,(L,λ)}^{G^{F}} induces an isomorphism between CF_{π′}(G^{F} , e_{π,(L,λ)}^{G^{F}}; O) and CF_{π′}(Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ); O).

By 5.22, this follows from the fact (5.15, (1)) that

dec_{π(L,λ)}^{G^{F}} · I_{π,(L,λ)}^{G^{F}} = I_{π,(L,λ)}^{G^{F}} · dec_{π,(L,λ)}^{Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)} .

This completes the induction and thus the proof of 5.15, (2). □

5.23. Corollary. The idempotent e_{π,(L,λ)}^{G^{F}} is a π–block of G^{F} (i.e., it is a primitive central π–idempotent).

Proof of 5.23. By 5.15, (2), and by 5.12, it suffices to prove that 1 is a π–block of Z^{o}(L)_{π}^{F} ⋊ W_{G^{F}} (L, λ). By [Rob], thm. 9 (a non trivial generalization to any π of a result which is well known in the case where π consists of a single prime), it suffices to check that W_{G^{F}} (L, λ) acts faithfully on Z^{o}(L)_{π}^{F}. This follows from the fact that L = C_{G}(Z^{o}(L)_{π}^{F}). □

C. ℓ–Blocks.

The following hypothesis are in force in what follows.

(H1) Let G be a connected reductive algebraic group over an algebraic closure of 𝔽_{q}, endowed with a Frobenius endomorphism F: G → G defining a rational structure over 𝔽_{q}.

(H2) Let ℓ be a prime number which does not divide q, which divides |G^{F} |, and which does not divide |W_{G^{F}} ⟨ϕ⟩|.

We denote by 𝔾 a generic finite reductive group corresponding to (G, F), and we set 𝔾 = (Γ, W ϕ).

Following [BrMi], we call unipotent ℓ–blocks of G^{F} the primitive central ℓ–idempotents which are constituents of the “unipotent idempotent” e_{ℓ}^{G^{F}}. We denote by O the extension of the ring of ℓ–adic integers by a root of unity whose order is the l.c.m. of the orders of the elements of G^{F}. Thus e_{ℓ}^{G^{F}} ∈ ZOG^{F}, and the unipotent ℓ–blocks are the primitive idempotents of the algebra ZOG^{F} e_{ℓ}^{G^{F}}.

The following omnibus theorem is an immediate consequence of the preceding sections (specially of theorem 5.15), as well as a consequence of general results about isotypies (see [Br1]). We make free use of the notation which has been introduced previously.

5.24. Theorem. Let d be the unique integer such that Φ_{d}(x) divides |𝔾| and ℓ divides Φ_{d}(q). Let (L, λ) be a d–cuspidal pair of G.

(1) The set of all irreducible constituents of the virtual characters R_{L}^{G}(θλ), where θ ∈ Ab_{ℓ}Irr(L^{F}), is the set of all irreducible characters of an ℓ– block of G^{F}, which we denote by e_{ℓ,(L,λ)}^{G^{F}}.

(2) (Z^{o}(L)_{ℓ}^{F} , e_{ℓ,(L,λ)}^{L^{F}}) is a maximal e_{ℓ,(L,λ)}^{G^{F}}–subpair of G^{F}. In particular,

Z^{o}(L)_{ℓ}^{F} is a defect group of e_{ℓ,(L,λ)}^{G^{F}}.

(3) Whenever S is a subgroup of Z^{o}(L)_{ℓ}^{F}, its centralizer M in G is a d–split Levi subgroup, and the Brauer correspondent Br_{S}(e_{ℓ,(L,λ)}^{G^{F}}) of

e_{ℓ,(L,λ)}^{G^{F}} is given by the following formula:

Br_{S}(e_{ℓ,(L,λ)}^{G^{F}}) = ∑_{(L′,λ′)} e_{ℓ,(L′,λ′)}^{M^{F}}

where (L′, λ′) runs over a complete set of representatives for the M^{F} – conjugacy classes of d–cuspidal pairs of M which are G^{F} –conjugate to (L, λ).

(4) (G^{F} , e_{ℓ,(L,λ)}^{G^{F}}) and (Z^{o}(L)_{ℓ}^{F} ⋊ W_{G^{F}} (L, λ), 1) are isotypic, and so in

particular (see [Br1], thm. 1.5 and §4.D)

(a) the block e_{ℓ,(L,λ)}^{G^{F}} of G^{F} has respectively the same numbers of ordinary and modular characters as the group Z^{o}(L)_{ℓ}^{F} ⋊ W_{G^{F}} (L, λ),

(b) all irreducible characters in e_{ℓ,(L,λ)}^{G^{F}} have height zero,

(c) the Cartan matrix of e_{ℓ,(L,λ)}^{G^{F}} defines the same integral quadratic form as the Cartan matrix of the group Z^{o}(L)_{ℓ}^{F} ⋊ W_{G^{F}} (L, λ),

(d) the generalized decomposition matrix of (G^{F} , e_{ℓ,(L,λ)}^{G^{F}}) defines the same quadratic form (over ℤ[e^{2iπ/ℓ^{a}}], where a is the valuation in ℓ of Φ_{d}(q)) as the generalized decomposition matrix of

(Z^{o}(L)_{ℓ}^{F} ⋊ W_{G^{F}} (L, λ), 1),

(e) the algebras ZOG^{F} e_{ℓ,(L,λ)}^{G^{F}} and ZO[Z^{o}(L)_{ℓ}^{F} ⋊ W_{G^{F}} (L, λ)] are isomorphic.

We have

e_{ℓ}^{G^{F}} = ∑_{[(L,λ)]_{G^{F}}} e_{ℓ,(L,λ)}^{G^{F}} ,

where (L, λ) runs over the set of all d–cuspidal pairs of G^{F} (modulo G^{F} – conjugation). In other words, (L, λ) ↦ e_{ℓ,(L,λ)}^{G^{F}} is a bijection between the set of all unipotent ℓ-blocks of G^{F} and the G^{F} –conjugacy classes of d–cuspidal pairs

Notice that, still assuming ℓ is large (i.e., ℓ ∤ |W ⟨ϕ⟩|), the preceding theorem implies that several of the current conjectures in modular representation theory are true for groups of type G^{F} and unipotent blocks : Alperin conjecture, Alperin–McKay conjecture, Dade’s new weight conjecture, all implied in these cases (blocks with abelian defect groups) by conjecture 6.1 in [Br1].

Notice also that the preceding remark can be strengthened by extending it to non–unipotent blocks of finite reductive groups, using the fact (see [Br1], theorems 2.3 and 5.6) that in many cases the “Jordan decomposition of blocks” is an isotypy.

We hope to extend in a forthcoming paper 5.24 to cover all blocks with abelian defect of reductive groups (i.e., relax the condition on primes being large).

M. BROUÉ, G. MALLE, J. MICHEL

6. Appendix : Tables for exceptional groups

Table 1. Some d-series for exceptional groups.

𝔾 d  𝕃(q)  λ  case #  W_{𝔾}(𝕃, λ)
F_{4} 2  (q + 1)^{2}.B_{2}(q)  ϕ_{1,1}  1  G(2, 1, 2)
4  (q^{2} + 1).B_{2}(q)  ϕ_{11,−}, ϕ_{−,2}  2, 3  Z_{4}
E_{6} 2  (q + 1).A_{5}(q)  ϕ_{321}  4^{∗}  Z_{2}
3  (q^{2} + q + 1).^{3}D_{4}(q)  ^{3}D_{4}[−1]  5  Z_{3}
4  (q^{2} + 1)(q − 1).^{2}A_{3}(q)  ϕ_{22}  6^{∗}  Z_{4}
5  (q^{5} − 1).A_{1}(q)  ϕ_{2}, ϕ_{11}  7^{∗}, 8^{∗}  Z_{5}
2E_{6} 2  (q + 1)^{2}.D_{4}(q)  ϕ_{13,02}  9  G(1, 1, 3)
4  (q^{2} + 1)(q + 1).A_{3}(q)  ϕ_{22}  10^{∗}  Z_{4}
6  (q^{2} − q + 1).^{3}D_{4}(q)  ϕ_{2,1}  11  Z_{3}
10  (q^{5} + 1).A_{1}(q)  ϕ_{2}, ϕ_{11}  12^{∗}, 13^{∗}  Z_{5}
E_{7} 2  (q + 1)^{3}.D_{4}(q)  ϕ_{13,02}  14  G(2, 1, 3)
2  (q + 1).^{2}E_{6}(q)  ^{2}E_{6}[θ], ^{2}E_{6}[θ^{2}]  15, 16  Z_{2}
3  (q^{3} − 1).^{3}D_{4}(q)  ^{3}D_{4}[−1]  17  Z_{6}
3  (q^{2} + q + 1).A_{5}(q)  ϕ_{42}, ϕ_{2211}  18^{∗}, 19^{∗}  Z_{6}
4  (q^{2} + 1)^{2}.A_{1}(q)^{3}  ϕ_{2}^{3}, ϕ_{11}^{3}  20^{∗}, 21^{∗}  G_{8}
ϕ_{2}^{2}ϕ_{11}, ϕ_{2}ϕ_{11}^{2}  22^{∗}, 23^{∗}  G(4, 1, 2)
5  (q^{5} − 1).A_{2}(q)  ϕ_{3}, ϕ_{21}, ϕ_{111}  24^{∗}, 25^{∗}, 26^{∗}  Z_{10}
6  (q^{3} + 1).^{3}D_{4}(q)  ϕ_{2,1}  27  Z_{6}
6  (q^{2} − q + 1).^{2}A_{5}(q)  ϕ_{42}, ϕ_{2211}  28^{∗}, 29^{∗}  Z_{6}
8  (q^{4} + 1).A_{1}(q^{2}).A_{1}(q)  ϕ_{2}^{2}, ϕ_{2}ϕ_{11}, ϕ_{11}ϕ_{2}, ϕ_{11}^{2} 30^{∗}, 31^{∗}, 32^{∗}, 33^{∗} Z_{8}
10  (q^{5} + 1).^{2}A_{2}(q)  ϕ_{3}, ϕ_{21}, ϕ_{111}  34^{∗}, 35^{∗}, 36^{∗}  Z_{10}
12  (q^{4} − q^{2} + 1).A_{1}(q^{3})  ϕ_{2}, ϕ_{11}  37^{∗}, 38^{∗}  Z_{12}
E_{8} 2  (q + 1)^{4}.D_{4}(q)  ϕ_{13,02}  39  G_{28}
2  (q + 1)^{2}.^{2}E_{6}(q)  ^{2}E_{6}[θ], ^{2}E_{6}[θ^{2}]  40, 41  G(6, 6, 2)
2  (q + 1).E_{7}(q)  ϕ_{512,11}, ϕ_{512,12}  42, 43  Z_{2}
3  (q^{2} + q + 1)^{2}.^{3}D_{4}(q)  ^{3}D_{4}[−1]  44  G_{5}
3  (q^{2} + q + 1).E_{6}(q)  ϕ_{81,6}, ϕ_{81,10}, ϕ_{90,8}  45^{∗}, 46^{∗}, 47  Z_{6}
4  (q^{2} + 1)^{2}.D_{4}(q)  ϕ_{3,1}, ϕ_{123,013}, ϕ_{23,01}, ϕ_{12,03} 48^{∗}, 49^{∗}, 50, 51 G_{8}
5 (q^{4} + q^{3} + q^{2} + q + 1).A_{4}(q)  ϕ_{32}, ϕ_{221}  52^{∗}, 53^{∗}  Z_{10}
6  (q^{2} − q + 1)^{2}.^{3}D_{4}(q)  ϕ_{2,1}  54  G_{5}
6  (q^{2} − q + 1).^{2}E_{6}(q)  ϕ′_{9,6}, ϕ′′_{9,6}, ϕ′′_{6,6}  55^{∗}, 56^{∗}, 57  Z_{6}
7  (q^{7} − 1).A_{1}(q)  ϕ_{2}, ϕ_{11}  58^{∗}, 59^{∗}  Z_{14}
8  (q^{4} + 1).^{2}D_{4}(q)  ϕ_{13,−}, ϕ_{0123,13}  60^{∗}, 61^{∗}  Z_{8}
ϕ_{023,1}, ϕ_{123,0}, ϕ_{013,2}, ϕ_{012,3} 62, 63, 64, 65  Z_{8}
9  (q^{6} + q^{3} + 1).A_{2}(q)  ϕ_{3}, ϕ_{21}, ϕ_{111}  66^{∗}, 67^{∗}, 68^{∗}  Z_{18}
10 (q^{4} − q^{3} + q^{2} − q + 1).^{2}A_{4}(q)  ϕ_{32}, ϕ_{221}  69^{∗}, 70^{∗}  Z_{10}
12  (q^{4} − q^{2} + 1).^{3}D_{4}(q)  ϕ′_{1,3}, ϕ′′_{1,3}, ϕ_{2,2}, ^{3}D_{4}[1] 71^{∗}, 72^{∗}, 73, 74 Z_{12}
14  (q^{7} + 1).A_{1}(q)  ϕ_{2}, ϕ_{11}  75^{∗}, 76^{∗}  Z_{14}
18  (q^{6} − q^{3} + 1).^{2}A_{2}(q)  ϕ_{3}, ϕ_{21}, ϕ_{111}  77^{∗}, 78^{∗}, 79^{∗}  Z_{18}

The notation for the unipotent characters in column 3 and in the following table is as in [Ca], the one for the reflection groups in the last column is standard, see [Be].

M. BROUÉ, G. MALLE, J. MICHEL

Table 2. Decomposition of some R_{𝕃}^{𝔾}(λ).

case R_{𝕃}^{𝔾}(λ)
1 ϕ_{4,1} + ϕ_{4,13} − ϕ′′_{4,7} − ϕ′_{4,7} − 2B_{2}, r
2 ϕ′_{2,4} + ϕ′_{2,16} − ϕ′_{4,7} + B_{2}, ϵ′
3 ϕ′′_{2,4} + ϕ′′_{2,16} − ϕ′′_{4,7} + B_{2}, ϵ′′
4 ϕ_{64,4} − ϕ_{64,13}
5 D_{4}, 1 + D_{4}, ϵ − D_{4}, r
6 ϕ_{20,2} − ϕ_{60,5} + ϕ_{60,11} − ϕ_{20,20}
7 ϕ_{1,0} − ϕ_{24,6} + ϕ_{81,10} − ϕ_{64,13} + ϕ_{6,25}
8 ϕ_{6,1} − ϕ_{64,4} + ϕ_{81,6} − ϕ_{24,12} + ϕ_{1,36}
9 ϕ′_{8,3} + ϕ′′_{8,9} + 2ϕ_{16,5}
10 ϕ_{4,1} + ϕ′_{4,7} − ϕ′′_{4,7} − ϕ_{4,13}
11 ϕ′_{8,3} + ϕ′′_{8,9} − ϕ_{16,5}
12 ϕ_{1,0} − ϕ′′_{8,3} + ϕ′′_{9,6} − ^{2}A_{5}, ϵ − ϕ′′_{2,16}
13 ϕ′_{2,4} − ^{2}A_{5}, 1 − ϕ′_{9,6} + ϕ′_{8,9} − ϕ_{1,24}
14 ϕ_{56,3} − ϕ_{120,4} − 3ϕ_{216,9} − 2ϕ_{336,11} + 2ϕ_{336,14} + 3ϕ_{216,16} + ϕ_{120,25} − ϕ_{56,30}

+3ϕ_{280,8} − 3ϕ_{280,17}

15 + 16 E_{6}[θ], 1 − E_{6}[θ], ϵ + E_{6}[θ^{2}], 1 − E_{6}[θ^{2}], ϵ
17 D_{4}, 1 + D_{4}, ϵ_{1} − D_{4}, σ_{2} − D_{4}, σ′_{2} + D_{4}, ϵ_{2} + D_{4}, ϵ
18 ϕ_{27,2} + ϕ_{189,5} − ϕ_{378,9} + ϕ_{189,20} − ϕ_{216,16} + ϕ_{189,17}
19 ϕ_{189,7} − ϕ_{378,14} + ϕ_{189,22} + ϕ_{27,37} − ϕ_{216,9} + ϕ_{189,10}
20 ϕ_{1,0} − 2ϕ_{210,6} − 3ϕ_{105,6} − 3ϕ_{189,22} + ϕ_{21,36} + 2ϕ_{56,3} − 2D_{4}, 1 + 3ϕ_{405,8} + 3ϕ_{189,10}

−4ϕ_{336,11} − 4D_{4}, σ_{2} + 2ϕ_{120,25} − 2D_{4}, ϵ_{2} + ϕ_{315,16} + 2ϕ_{70,18} + ϕ_{35,22}

21 ϕ_{21,3} − 3ϕ_{189,5} − 3ϕ_{105,21} − 2ϕ_{210,21} + ϕ_{1,63} + 2ϕ_{120,4} − 2D_{4}, ϵ_{1} − 4ϕ_{336,14}

−4D_{4}, σ′_{2} + 3ϕ_{405,15} + 3ϕ_{189,17} + 2ϕ_{56,30} − 2D_{4}, ϵ + ϕ_{315,7} + 2ϕ_{70,9} + ϕ_{35,13}

22 ϕ_{7,1} − ϕ_{189,7} + 2ϕ_{378,9} − ϕ_{105,15} + 2ϕ_{210,13} + ϕ_{27,37} − ϕ_{15,7} + ϕ_{105,5} − 2ϕ_{216,16}

−2D_{4}, rϵ_{2} − ϕ_{35,31} + ϕ_{21,33} − 2ϕ_{280,8} − 2D_{4}, r

23 ϕ_{27,2} + 2ϕ_{210,10} − ϕ_{105,12} + 2ϕ_{378,14} − ϕ_{189,20} + ϕ_{7,46} − ϕ_{35,4} + ϕ_{21,6} − 2ϕ_{216,9}

−2D_{4}, rϵ_{1} − ϕ_{15,28} + ϕ_{105,26} − 2ϕ_{280,17} − 2D_{4}, rϵ

24 ϕ_{1,0} + ϕ_{21,3} − ϕ_{189,7} − ϕ_{189,22} − ϕ_{84,12} + ϕ_{336,11} + ϕ_{216,16} − ϕ_{189,17} + ϕ_{56,30} + ϕ_{21,33}
25 ϕ_{7,1} + ϕ_{27,2} − ϕ_{168,6} − ϕ_{378,9} − ϕ_{378,14} − ϕ_{168,21} + ϕ_{27,37} + ϕ_{7,46} + ϕ_{512,11} + ϕ_{512,12}
26 −ϕ_{189,5} − ϕ_{189,20} + ϕ_{21,36} + ϕ_{1,63} + ϕ_{56,3} + ϕ_{21,6} + ϕ_{216,9} − ϕ_{189,10} − ϕ_{84,15} + ϕ_{336,14}
27 ϕ_{56,3} − ϕ_{120,4} + ϕ_{336,11} − ϕ_{336,14} + ϕ_{120,25} − ϕ_{56,30}
28 ϕ_{27,2} − ϕ_{189,5} + ϕ_{378,9} + ϕ_{189,20} − ϕ_{405,15} + D_{4}, rϵ_{2}
29 ϕ_{189,7} + ϕ_{378,14} − ϕ_{189,22} + ϕ_{27,37} − ϕ_{405,8} + D_{4}, rϵ_{1}
30 ϕ_{1,0} − ϕ_{105,6} + ϕ_{189,22} − ϕ_{21,36} + ϕ_{216,9} + D_{4}, rϵ_{1} − ϕ_{280,17} − D_{4}, rϵ
31 ϕ_{7,1} + ϕ_{189,7} − ϕ_{105,15} − ϕ_{27,37} − ϕ_{120,4} + D_{4}, ϵ_{1} + ϕ_{56,30} − D_{4}, ϵ

32 ϕ_{27,2} + ϕ_{105,12} − ϕ_{189,20} − ϕ_{7,46} − ϕ_{56,3} + D_{4}, 1 + ϕ_{120,25} − D_{4}, ϵ_{2}
33 ϕ_{21,3} − ϕ_{189,5} + ϕ_{105,21} − ϕ_{1,63} − ϕ_{216,16} − D_{4}, rϵ_{2} + ϕ_{280,8} + D_{4}, r
34 ϕ_{1,0} − ϕ_{21,3} + ϕ_{189,7} − ϕ_{189,22} − ϕ_{420,10} + D_{4}, σ_{2} + ϕ_{405,15} − D_{4}, rϵ_{2} + ϕ_{35,31} + D_{4}, ϵ
35 −ϕ_{7,1} + ϕ_{27,2} − ϕ_{168,6} + ϕ_{378,9} − ϕ_{378,14} + ϕ_{168,21} − ϕ_{27,37} + ϕ_{7,46} + E_{7}[ξ] + E_{7}[−ξ]
36 −ϕ_{189,5} + ϕ_{189,20} − ϕ_{21,36} + ϕ_{1,63} + ϕ_{35,4} + D_{4}, 1 + ϕ_{405,8} − D_{4}, rϵ_{1} − ϕ_{420,13} + D_{4}, σ′_{2}
37 ϕ_{1,0} + ϕ_{210,6} + ϕ_{21,36} − ϕ_{56,3} + D_{4}, 1 − ϕ_{336,11} − D_{4}, σ_{2} − ϕ_{120,25} + D_{4}, ϵ_{2} + ϕ_{280,18}
+E_{6}[θ], ϵ + E_{6}[θ^{2}], ϵ
38 ϕ_{21,3} + ϕ_{210,21} + ϕ_{1,63} − ϕ_{120,4} + D_{4}, ϵ_{1} − ϕ_{336,14} − D_{4}, σ′_{2} − ϕ_{56,30} + D_{4}, ϵ + ϕ_{280,9}
+E_{6}[θ], 1 + E_{6}[θ^{2}], 1
39 ϕ_{112,3} − 2ϕ_{160,7} − ϕ_{400,7} − 9ϕ_{1296,13} − 4ϕ_{2240,10} − 2ϕ_{3360,13} + 9ϕ_{2800,13} + 8ϕ_{3200,16}

+8ϕ_{3200,22} − 2ϕ_{3360,25} + 9ϕ_{2800,25} − 4ϕ_{2240,28} − 9ϕ_{1296,33} − ϕ_{400,43} − 2ϕ_{160,55} + ϕ_{112,63}

+4ϕ_{1344,8} − 8D_{4}, ϕ′_{8,3} − 8D_{4}, ϕ′′_{8,9} + 4ϕ_{1344,38} − 4ϕ_{1344,19} − 6ϕ_{2016,19} − 6ϕ_{5600,19}

−12ϕ_{448,25} − 16D_{4}, ϕ_{16,5}
40 + 41 E_{6}[θ], ϕ_{1,0} − E_{6}[θ], ϕ′_{1,3} − E_{6}[θ], ϕ′′_{1,3} + E_{6}[θ], ϕ_{1,6} − 2E_{8}[θ] − 2E_{8}[−θ]+

E_{6}[θ^{2}], ϕ_{1,0} − E_{6}[θ^{2}], ϕ′_{1,3} − E_{6}[θ^{2}], ϕ′′_{1,3} + E_{6}[θ^{2}], ϕ_{1,6} − 2E_{8}[θ^{2}] − 2E_{8}[−θ^{2}]

42 + 43 ϕ_{4096,11} − ϕ_{4096,26} + ϕ_{4096,12} − ϕ_{4096,27}
44 D_{4}, ϕ_{1,0} − D_{4}, ϕ′_{2,4} + D_{4}, ϕ′_{1,12} − 2D_{4}, ϕ′_{4,7} − D_{4}, ϕ′′_{2,4} + 2D_{4}, ϕ′′_{8,3} + 2D_{4}, ϕ′_{8,9} − D_{4}, ϕ′_{2,16}

−2D_{4}, ϕ′′_{4,7} + D_{4}, ϕ′′_{1,12} − D_{4}, ϕ′′_{2,16} + D_{4}, ϕ_{1,24} − 2D_{4}, ϕ_{4,1} + 2D_{4}, ϕ′_{8,3} + 2D_{4}, ϕ′′_{8,9}

−2D_{4}, ϕ_{4,13} + D_{4}, ϕ_{4,8} − 2D_{4}, ϕ_{16,5} − 3E_{8}[−θ] − 3E_{8}[−θ^{2}] + 3E_{8}[−1]

45 ϕ_{567,6} + ϕ_{3240,9} − ϕ_{4536,13} − ϕ_{2835,22} + ϕ_{2268,30} + ϕ_{1296,33}
46 −ϕ_{2835,14} − ϕ_{4536,23} + ϕ_{3240,31} + ϕ_{567,46} + ϕ_{2268,10} + ϕ_{1296,13}
47 ϕ_{1008,9} + ϕ_{1575,10} + ϕ_{1575,35} + ϕ_{1008,39} − ϕ_{3150,18} − ϕ_{2016,19}
48 ϕ_{8,1} + 2ϕ_{560,5} + 3ϕ_{4536,13} + ϕ_{4200,21} − 3ϕ_{3240,31} − 3ϕ_{1400,11} + 3ϕ_{840,13} − 4ϕ_{3200,16}

−4D_{4}, ϕ′′_{8,3} + 2ϕ_{2240,28} − 2D_{4}, ϕ′′_{4,7} − 2ϕ_{1344,8} − 2D_{4}, ϕ_{4,1} + ϕ_{1400,37} + 2ϕ_{1008,39} + ϕ_{56,49}

49 −3ϕ_{3240,9} + ϕ_{4200,15} + 3ϕ_{4536,23} + 2ϕ_{560,47} + ϕ_{8,91} + 2ϕ_{2240,10} − 2D_{4}, ϕ′_{4,7} − 4ϕ_{3200,22}

−4D_{4}, ϕ′_{8,9} − 3ϕ_{1400,29} + 3ϕ_{840,31} + ϕ_{1400,7} + 2ϕ_{1008,9} + ϕ_{56,19} − 2ϕ_{1344,38} − 2D_{4}, ϕ_{4,13}

50 ϕ_{28,8} + 2ϕ_{160,7} + ϕ_{300,8} − 3ϕ_{972,12} + 2ϕ_{840,14} − 3ϕ_{700,16} + 2ϕ_{840,26} − 3ϕ_{700,28} − 3ϕ_{972,32}

+ϕ_{300,44} + 2ϕ_{160,55} + ϕ_{28,68} + 4ϕ_{1344,19} − 2D_{4}, ϕ′_{6,6} − 2D_{4}, ϕ′′_{6,6} − 4D_{4}, ϕ_{12,4}

51 ϕ_{84,4} − 2D_{4}, ϕ′_{2,4} − ϕ_{700,6} + 3ϕ_{2268,10} − 2ϕ_{4200,12} − 3ϕ_{2100,16} − 2ϕ_{4200,24} − 3ϕ_{2100,28}

+3ϕ_{2268,30} − ϕ_{700,42} − 2D_{4}, ϕ′′_{2,16} + ϕ_{84,64} + 2ϕ_{2016,19} + 2ϕ_{5600,19} + 4ϕ_{448,25} − 4D_{4}, ϕ_{4,8}

52 ϕ_{35,2} + ϕ_{560,5} − ϕ_{3240,9} + ϕ_{2835,22} − ϕ_{840,14} + ϕ_{3360,13} − ϕ_{2240,28} − ϕ_{840,31} + ϕ_{210,52} + ϕ_{160,55}
53 ϕ_{2835,14} − ϕ_{3240,31} + ϕ_{560,47} + ϕ_{35,74} + ϕ_{210,4} + ϕ_{160,7} − ϕ_{2240,10} − ϕ_{840,13} − ϕ_{840,26} + ϕ_{3360,25}
54 ϕ_{112,3} + ϕ_{160,7} − ϕ_{400,7} + 2ϕ_{2240,10} + ϕ_{3360,13} + 2ϕ_{3200,16} + 2ϕ_{3200,22} + ϕ_{3360,25}

+2ϕ_{2240,28} − ϕ_{400,43} + ϕ_{160,55} + ϕ_{112,63} − 2ϕ_{1344,8} − 2D_{4}, ϕ′_{8,3} − 2D_{4}, ϕ′′_{8,9} − 2ϕ_{1344,38}

−3ϕ_{7168,17} − ϕ_{1344,19} + 2D_{4}, ϕ_{16,5} + 3E_{6}[θ], ϕ_{2,2} + 3E_{6}[θ^{2}], ϕ_{2,2}

55 ϕ_{567,6} − ϕ_{3240,9} + ϕ_{4536,13} − ϕ_{2835,22} + ϕ_{972,32} + D_{4}, ϕ_{9,10}
56 −ϕ_{2835,14} + ϕ_{4536,23} − ϕ_{3240,31} + ϕ_{567,46} + ϕ_{972,12} + D_{4}, ϕ_{9,2}

57 ϕ_{1008,9} − ϕ_{1575,10} − ϕ_{1575,34} + ϕ_{1008,39} + ϕ_{1134,20} + D_{4}, ϕ′_{6,6}
58 ϕ_{1,0} − ϕ_{3240,9} − ϕ_{6075,14} + ϕ_{8,91} + ϕ_{400,7} − ϕ_{300,8} + ϕ_{4096,11} + ϕ_{4096,12} + ϕ_{3200,22}
−ϕ_{2400,23} − ϕ_{972,32} + ϕ_{1296,33} + ϕ_{50,56} − ϕ_{160,55}
59 ϕ_{8,1} − ϕ_{6075,22} − ϕ_{3240,31} + ϕ_{1,120} + ϕ_{50,8} − ϕ_{160,7} − ϕ_{972,12} + ϕ_{1296,13} + ϕ_{3200,16}
−ϕ_{2400,17} + ϕ_{4096,26} + ϕ_{4096,27} + ϕ_{400,43} − ϕ_{300,44}
60 ϕ_{8,1} + ϕ_{4536,13} − ϕ_{4200,21} + ϕ_{3240,31} − ϕ_{2240,10} + D_{4}, ϕ′_{4,7} − ϕ_{1344,38} − D_{4}, ϕ_{4,13}
61 −ϕ_{3240,9} + ϕ_{4200,15} − ϕ_{4536,23} − ϕ_{8,91} + ϕ_{2240,28} − D_{4}, ϕ′′_{4,7} + ϕ_{1344,8} + D_{4}, ϕ_{4,1}
62 ϕ_{84,4} − ϕ_{400,7} + D_{4}, ϕ_{9,2} + ϕ_{700,16} − D_{4}, ϕ′′_{9,6} − ϕ_{972,32} + ϕ_{700,42} − ϕ_{112,63}
63 D_{4}, ϕ_{1,0} + ϕ_{300,8} − ϕ_{2268,10} + ϕ_{2800,13} − ϕ_{2100,28} + ϕ_{1296,33} − D_{4}, ϕ′′_{1,12} − ϕ_{28,68}
64 ϕ_{112,3} − ϕ_{700,6} + ϕ_{972,12} + D_{4}, ϕ′_{9,6} − ϕ_{700,28} − D_{4}, ϕ_{9,10} + ϕ_{400,43} − ϕ_{84,64}
65 ϕ_{28,8} + D_{4}, ϕ′_{1,12} − ϕ_{1296,13} + ϕ_{2100,16} − ϕ_{2800,25} + ϕ_{2268,30} − ϕ_{300,44} − D_{4}, ϕ_{1,24}
66 ϕ_{1,0} − ϕ_{560,47} − ϕ_{50,8} + ϕ_{160,7} + ϕ_{2800,13} + ϕ_{700,16} − ϕ_{5600,21} − ϕ_{3200,22} + ϕ_{4096,26} + ϕ_{4096,27}

+ϕ_{112,63} + ϕ_{28,68} − ϕ_{1008,9} − E_{6}[θ], ϕ_{1,0} − E_{6}[θ^{2}], ϕ_{1,0} − ϕ_{1575,34} − E_{6}[θ], ϕ′′_{1,3} − E_{6}[θ^{2}], ϕ′′_{1,3}

67 ϕ_{8,1} + ϕ_{35,2} + ϕ_{35,74} + ϕ_{8,91} − ϕ_{700,6} − ϕ_{400,7} + ϕ_{2240,10} + ϕ_{1400,11} + ϕ_{2240,28} + ϕ_{1400,29}

−ϕ_{700,42} − ϕ_{400,43} − ϕ_{3150,18} − ϕ_{2016,19} − E_{6}[θ], ϕ_{2,1} − E_{6}[θ^{2}], ϕ_{2,1} − E_{6}[θ], ϕ_{2,2} − E_{6}[θ^{2}], ϕ_{2,2}

68 −ϕ_{560,5} + ϕ_{1,120} + ϕ_{112,3} + ϕ_{28,8} + ϕ_{4096,11} + ϕ_{4096,12} − ϕ_{5600,15} − ϕ_{3200,16} + ϕ_{2800,25} + ϕ_{700,28}

−ϕ_{50,56} + ϕ_{160,55} − ϕ_{1575,10} − E_{6}[θ], ϕ′_{1,3} − E_{6}[θ^{2}], ϕ′_{1,3} − ϕ_{1008,39} − E_{6}[θ], ϕ_{1,6} − E_{6}[θ^{2}], ϕ_{1,6}

69 ϕ_{35,2} − ϕ_{560,5} + ϕ_{3240,9} + ϕ_{2835,22} − ϕ_{4200,12} + D_{4}, ϕ′′_{2,4} − ϕ_{1400,29} − D_{4}, ϕ′′_{4,7} + ϕ_{50,56} + D_{4}, ϕ′′_{2,16}
70 ϕ_{2835,14} + ϕ_{3240,31} − ϕ_{560,47} + ϕ_{35,74} + ϕ_{50,8} + D_{4}, ϕ′_{2,4} − ϕ_{1400,11} − D_{4}, ϕ′_{4,7} − ϕ_{4200,24} + D_{4}, ϕ′_{2,16}
71 ϕ_{8,1} − ϕ_{560,5} + ϕ_{4200,21} − ϕ_{3200,16} − D_{4}, ϕ′′_{8,3} − ϕ_{2240,28} + D_{4}, ϕ′′_{4,7} + ϕ_{1344,8} + D_{4}, ϕ_{4,1}
+ϕ_{448,39} + E_{6}[θ], ϕ_{1,6} + E_{6}[θ^{2}], ϕ_{1,6}
72 ϕ_{4200,15} − ϕ_{560,47} + ϕ_{8,91} − ϕ_{2240,10} + D_{4}, ϕ′_{4,7} − ϕ_{3200,22} − D_{4}, ϕ′_{8,9} + ϕ_{1344,38} + D_{4}, ϕ_{4,13}
+ϕ_{448,9} + E_{6}[θ], ϕ_{1,0} + E_{6}[θ^{2}], ϕ_{1,0}
73 ϕ_{84,4} + D_{4}, ϕ′_{2,4} − ϕ_{700,6} + ϕ_{4200,12} + ϕ_{4200,24} − ϕ_{700,42} + D_{4}, ϕ′′_{2,16} + ϕ_{84,64} − ϕ_{7168,17}
−D_{4}, ϕ_{4,8} + E_{6}[θ], ϕ_{2,2} + E_{6}[θ^{2}], ϕ_{2,2}
74 ϕ_{28,8} − ϕ_{160,7} + ϕ_{300,8} − ϕ_{840,14} − ϕ_{840,26} + ϕ_{300,44} − ϕ_{160,55} + ϕ_{28,68} + ϕ_{1344,19}
−E_{8}[−θ] − E_{8}[−θ^{2}] + E_{8}[−1]
75 ϕ_{1,0} + ϕ_{3240,9} − ϕ_{6075,14} − ϕ_{8,91} − ϕ_{700,6} + D_{4}, ϕ′_{1,12} + E_{7}[ξ], 1 + E_{7}[–ξ], 1 + ϕ_{5600,21}
−D_{4}, ϕ′_{8,9} − ϕ_{2268,30} + D_{4}, ϕ_{9,10} + ϕ_{210,52} − D_{4}, ϕ′′_{2,16}
76 ϕ_{8,1} + ϕ_{6075,22} − ϕ_{3240,31} − ϕ_{1,120} − ϕ_{210,4} + D_{4}, ϕ′_{2,4} + ϕ_{2268,10} − D_{4}, ϕ_{9,2} − ϕ_{5600,15}
+D_{4}, ϕ′′_{8,3} + E_{7}[ξ], ϵ + E_{7}[–ξ], ϵ + ϕ_{700,42} − D_{4}, ϕ′′_{1,12}
77 ϕ_{1,0} + ϕ_{560,47} − ϕ_{210,4} + D_{4}, ϕ′_{2,4} − ϕ_{2100,16} − D_{4}, ϕ′_{9,6} + ϕ_{2400,23} + D_{4}, ϕ′_{8,9} − E_{7}[ξ], ϵ − E_{7}[–ξ], ϵ

−ϕ_{84,64} − D_{4}, ϕ_{1,24} + ϕ_{1008,9} + E_{6}[θ], ϕ_{1,0} + E_{6}[θ^{2}], ϕ_{1,0} − ϕ_{1575,34} − E_{6}[θ], ϕ′′_{1,3} − E_{6}[θ^{2}], ϕ′′_{1,3}

78 −ϕ_{8,1} + ϕ_{35,2} + ϕ_{35,74} − ϕ_{8,91} − ϕ_{300,8} − D_{4}, ϕ′_{1,12} + ϕ_{840,13} + D_{4}, ϕ′_{4,7} + ϕ_{840,31} + D_{4}, ϕ′′_{4,7}

−ϕ_{300,44} − D_{4}, ϕ′′_{1,12} − ϕ_{1134,20} − D_{4}, ϕ′_{6,6} − E_{8}[–θ] − E_{8}[–θ^{2}] − E_{8}[θ] − E_{8}[θ^{2}]

79 ϕ_{560,5} + ϕ_{1,120} − ϕ_{84,4} − D_{4}, ϕ_{1,0} + E_{7}[ξ], 1 + E_{7}[–ξ], 1 + ϕ_{2400,17} + D_{4}, ϕ′′_{8,3} − ϕ_{2100,28} − D_{4}, ϕ′′_{9,6}

−ϕ_{210,52} + D_{4}, ϕ′′_{2,16} − ϕ_{1575,10} − E_{6}[θ], ϕ′_{1,3} − E_{6}[θ^{2}], ϕ′_{1,3} + ϕ_{1008,39} + E_{6}[θ], ϕ_{1,6} + E_{6}[θ^{2}], ϕ_{1,6}

Table 3. W_{𝔾}(𝕋) for d regular and 𝕋 noncyclic.

𝔾  d  |𝕋|  W_{𝔾}(𝕋) 𝔾  d  |𝕋|  W_{𝔾}(𝕋)
3D_{4} 3  Φ_{3}^{2}  G_{4}  ^{2}E_{6} 3  Φ_{3}^{2}Φ_{6}  G_{5}
6  Φ_{6}^{2}  G_{4}  4  Φ_{4}^{2}Φ_{2}^{2}  G_{8}
2F_{4} 4  Φ_{4}^{2}  G_{12}  6  Φ_{6}^{3}  G_{25}
8′  Φ′_{8}^{2}  G_{8}  E_{7}  3  Φ_{3}^{3}Φ_{1}  G_{26}
8′′  Φ′′_{8}^{2}  G_{8}  6  Φ_{6}^{3}Φ_{2}  G_{26}
F_{4}  3  Φ_{3}^{2}  G_{5}  E_{8}  3  Φ_{3}^{4}  G_{32}
4  Φ_{4}^{2}  G_{8}  4  Φ_{4}^{4}  G_{31}
6  Φ_{6}^{2}  G_{5}  5  Φ_{5}^{2}  G_{16}
E_{6}  3  Φ_{3}^{3}  G_{25}  6  Φ_{6}^{4}  G_{32}
4  Φ_{4}^{2}Φ_{1}^{2}  G_{8}  8  Φ_{8}^{2}  G_{9}
6  Φ_{6}^{2}Φ_{3}  G_{5}  10  Φ_{10}^{2}  G_{16}
12  Φ_{12}^{2}  G_{10}

The notation for the reflection groups W_{𝔾}(𝕋) is as in [Be]. Note that the 𝕋 are uniquely determined by their orders, being Sylow Φ_{d}–tori.

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GENERIC BLOCKS OF FINITE REDUCTIVE GROUPS

Index

1  16
Ab_{d} Irr(G^{F})  29
Ab_{d} Reg^{G^{F}}  29
AbIrr(G^{F})  23
AbReg^{G^{F}}  23
Ab_{d} Reg^{ℕ}  26
A_{d,𝔾}  26
adjoint generic finite group  6
𝔄_{𝔾}  4
Br_{S}  52
Br_{s}  51
c_{d}  19
CF_{d}(𝔾, 𝕃)  18
CF(𝔾)  7
CF(H, e; K)  54
CF_{π′}(H, e; K)  54
χ_{(G(T,θ),θ,ρ)}^{G^{F}}  24
χ_{(𝕄,μ)}^{𝔾}  26
χ_{(𝕄,δ)}^{ℕ}  28
class functions on 𝔾  7
d–anisotropic maximal torus 19
d–cuspidal  18
d-cuspidal pair  30
dec_{π,(L,λ)}^{x,G^{F}}  55
tdec_{π,(L,λ)}^{x,G^{F}}  55
dec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}(φ)  55
tdec_{π}^{x,Z^{o}(L)_{π}^{F} ⋊W_{G^{F}} (L,λ)}  55
dec_{π}^{s,G^{F}}  52
Deg  13
derived generic finite group  6
D(𝔾)  6
d-group  9
d–split Levi subgroup  10
d–split pair  30
dual generic finite group  6
ε_{𝔾}  4

E(G^{F} , 1)  11
e_{π,(L,λ)}^{G^{F}}  54
Fam(W)  11
F_{qϕ}  4
|𝔾|  4
𝔾^{∗}  6
𝔾^{−}  5
𝔾^{(a)}  5
𝔾_{ad}  6
(G, F)–adapted  50
Γ^{(a)}  5
Γ_{𝔾}  3
(G, F, d)–adapted  50
generic Levi subgroup  5
generic degree  14
generic finite reductive group 3
generic torus  5
generic uniform functions  13
generic unipotent functions 13
G(F)  11
G^{F} (T, θ)  24
G^{F} (x)  54
good prime  6
𝔾(q)  4
G(s)  23
𝔾_{ss}  5
G(θ)  53
G(T, θ)  24
G(x)  54
I_{(L,λ)}^{M}  56
I_{(𝕃,λ)}^{𝕄}  31
Ind_{𝕃}^{𝔾}  7
I_{π,(L,λ)}^{N^{F}}  56
λ_{γ}  12
m_{γ}  11
N(𝔾)  4
Φ_{α}^{𝔾}  13
Φ_{d}–defect tori  46

Φ_{d}–group  9
ϕ_{F}  12
π–perfect isometry  54
ϖ_{s}^{G^{F}}  52
P (R)  6
pr_{𝕃,λ}^{𝔾}  44
(ψ, ψ′)_{𝔾}  13
π_{u}^{𝔾}  13
Q(R)  5
Rad(𝔾)  5
R_{d}(𝔾)  26
R_{d,𝔾}(ℕ)  28
Reg_{L,λ}^{G^{F}}  48
Reg_{π,(L,λ)}^{G^{F}}  50
Reg_{(𝕃,λ)}^{𝔾}  48
Reg_{π}^{G^{F}}  25
Res_{𝕃}^{𝔾}  7
R𝔾  4
ρ_{γ}^{G^{F}}  11
R_{𝕃}^{𝔾}  14
∗R_{𝕃}^{𝔾}  16
∗R_{𝕄}^{𝔾}(𝕃, λ)  45
R_{𝕄}^{𝔾}(𝕃, λ)  45
R^{n}𝔾  4
R_{s}  23

R_{θ}  24
R_{wϕ}^{𝔾}  13
R_{wϕ}^{G^{F}}  11
S_{d}(γ)  46
σ^{𝔾}  9
SV  4
(SV)^{W}  4
S_{W_{𝔾}}(𝕃, 𝕄)  7
T_{d}(𝔾)  19
θ̃  53
Θ_{G}^{N^{F}}  29
Θ_{𝔾,𝕄}^{ℕ}  26
tr_{R𝔾}  8
𝕋_{wϕ}  9
Uch(𝔾)  11
Uch(𝔾, (𝕃, λ))  31
U(ϕ)  13
UReg^{𝔾}  20
UReg_{𝕃,λ}^{𝔾}  44
UReg^{M^{F}}  25
W_{𝔾}  3
W_{G^{F}} (M)  25
W_{G^{F}} (M, μ)  25
W_{𝔾}(𝕃)  10
W_{𝔾}(𝕄, μ)  26
Z_{d}(𝔾)  10

M. B. and J. M. : D.M.I., 45 rue d’Ulm, F–75005 Paris, France

E-mail address: broue@dmi.ens.fr --- michel@mangue.ens.fr

G. M. : I.W.R., Im Neuenheimer Feld 368, D–6900 Heidelberg, Germany

E-mail address: malle@kalliope.iwr.uni-heidelberg.de