ON BLOCKS OF FINITE REDUCTIVE GROUPS AND TWISTED INDUCTION

Marc Cabanes, Michel Enguehard

UMR 9994, CNRS - Université Paris 7 UFR de Mathématiques, 2 place Jussieu, F-75251 Paris Cedex 05

Introduction

Let G be a connected reductive group over the algebraic closure of a finite field 𝔽_{q} and F a Frobenius endomorphism of G inducing a rational structure over 𝔽_{q}. The group of rational points G^{F} is finite. Assume that ℓ is a prime number not dividing q. Let e be the order of q modulo ℓ.

The main goal of this paper is to describe the ℓ-blocks of G^{F} in terms of G. Lusztig’s twisted induction maps R_{L⊆P}^{G} (our Theorems 2.8 and 4.1). It has been proved in several cases that the ℓ-blocks of G^{F} induce a partition of the set Irr(G^{F}) of irreducible ordinary characters of G^{F} similar to the one of ordinary Harish-Chandra theory. Ordinary induction is replaced with twisted induction, the partition into series is given by generalized Harish-Chandra theory, first introduced by P. Fong and B. Srinivasan for unipotent characters [FS.1]. Standard Levi subgroups are replaced with e-split Levi subgroups as defined by M. Broué, G. Malle and J. Michel [BMM]. One gets a corresponding notion of e-cuspidal characters.

The restrictions on the primes ℓ for which the ℓ-blocks of G^{F} have been studied often refer to the Weyl group or to the fundamental group of G. In [FS.2] P. Fong and B. Srinivasan describe ℓ-blocks for some classical conformal groups and ℓ odd. Thus Z(G), the center of G, is connected and ℓ is prime to the order of the fundamental group. The realizations as unipotent blocks of the Generic Blocks [BMM], studied by M. Broué and J. Michel in [BMi.2], have abelian defect groups, that is a restriction on ℓ in terms of the Weyl group of G. In [CE.1; CE.2], we assumed that ℓ does not divide the number of F -stable connected components of the centers of G and a dual group G^{∗}. M. Geck and G. Hiss have studied basic sets of Brauer characters under the same restriction [G]. Typical examples left aside previously are G^{F} = SL_{n}(εq) for ε = ±1 and ℓ dividing both n and q − ε (SL_{n}(−q) is understood as SU_{n}(q^{2})).

Denote by E(G^{F} , ℓ′) the set of irreducible components of Deligne-Lusztig characters R_{T}^{G}θ for θ of order prime to ℓ.

In the present paper we prove (Theorem 2.5, Theorem 4.1) :

Theorem. Assume ℓ ≥ 7 if some component of G is of type E_{8} , ℓ ≥ 5 otherwise. Let (L, ζ) be a pair consisting of an e-split Levi subgroup L and an e-cuspidal ζ in E(L^{F} , ℓ′). Then, for all parabolic subgroup P of which L is a Levi subgroup, all irreducible components of R_{L⊆P}^{G}ζ belong to the same ℓ-block of G^{F}. This defines a bijection between the G^{F} -conjugacy classes of pairs (L, ζ) as above and the ℓ-blocks of G^{F}.

We also describe the associated Brauer defect ℓ-subgroups of G^{F} (Proposition 5.1).

The proof involves an adaptation of the local methods of [CE.2] along with other general results obtained in 1 and 2 below.

Assume that ℓ is good for G (that is no condition for type A, ℓ ≥ 3 for types B, C, D, ℓ ≥ 7 for type E_{8}, ℓ ≥ 5 for other types). We show that the restrictions of the elements of E(G^{F} , ℓ′) to ℓ-regular elements of G^{F} provide a basis of the space of central functions on this set that are stable under diagonal ℓ-automorphisms of G^{F} (Theorem 1.7). This generalization of [G, Theorem A] applies in particular when ℓ divides the number of connected components of Z(G).

As Z(G) is not assumed to be connected, one has to consider characters in Lusztig series defined by rational elements s of a dual group G^{∗} whose centralizer in G^{∗} is disconnected. Then the Jordan decomposition of characters is not a bijective map between sets of irreducible characters, see [L.3]. Furthermore one has no commutation formula with twisted induction, as in [FS.2] or [CE.2, 2.6], to reduce e-cuspidality of an irreducible character to e-cuspidality of the corresponding unipotent characters. Instead we check one implication (Proposition 1.10.(ii)). Another aspect is the fact that when e ≠ 1, the positivity arguments of ordinary Harish-Chandra theory no longer apply. We have to show at the same time that there exists a generalized Harish-Chandra theory and that it yields the ℓ-blocks. In fact we use block theory to obtain some information about twisted induction, thus completing the characterization of ecuspidal elements in E(G^{F} , ℓ′) (Theorem 4.2) and generalized Harish-Chandra theory (Proposition 5.2). B. Srinivasan has obtained similar results in the case of isolated blocks, that is when the element s above is an isolated element, and G is a special orthogonal or a symplectic group [S].

In 2, we introduce a notion of subpair (see [AB]) associated with connected centralizers in G instead of ordinary centralizers in G^{F}. This is useful in the local study of blocks of G^{F} when ℓ divides the number of connected components of Z(G^{∗}). This also allows us to define a notion of twisted induction R_{L⊆P}^{G} for ℓ-blocks (Theorem 2.5). Just like R. Brauer’s block induction applies to subgroups of a finite group related with centralizers of ℓ-subgroups [B], our twisted induction is defined for Levi subgroups, the E_{q,ℓ}-split Levi subgroups defined in [CE.2], related with centralizers of abelian ℓ-subgroups. For this twisted induction, the map is shown to be independent of the choice of a parabolic subgroup P for L. The fact that, for this class of Levi subgroups, R_{L}^{G} sends a block of L^{F} to a single block also yields a general proof of a phenomenon already observed in many examples — that for a given ℓ-block b of G^{F}, one can give a test for an element χ ∈ Irr(G^{F}) to be in Irr(b) which is expressed only in terms of twisted induction and knowledge of the intersection Irr(b) ∩ E(G^{F} , ℓ′) (Theorem 2.8). Theorem 4.1 may in turn be regarded as an expression of Irr(b) ∩ E(G^{F} , ℓ′) itself in terms of twisted induction.

In 3 are some technicalities about certain subgroups in type A. Section 4 is devoted to the proof of Theorem 4.1 and Theorem 4.2. More results on defect groups, isomorphic blocks and connection with twisted induction are in § 5.

Notations.

The cardinality of a set, or order of a finite group, X is denoted by |X|. If X is an integer (resp. a finite group), then X_{ℓ} is its ℓ-component (resp. the subset of elements of order a power of ℓ) and X_{ℓ′} = X/X_{ℓ} (resp. X_{ℓ′} is the subset of ℓ-regular elements). The normalizer (resp. centralizer) of a subgroup H in a group G is denoted by N_{G}(H) (resp. C_{G}(H)) and Z(G) = C_{G}(G). More generally if G acts on a set X and Y ⊆ X, then N_{G}(Y) is the subgroup of g ∈ G such that g(Y) = Y. If Y = {y} is a singleton in X, then N_{G}(Y) may be denoted by G_{y}. When G is a finite group, one denotes by Irr(G) the set of irreducible characters of the group algebra KG for some field K of characteristic zero containing a primitive |G|-th root of unity. Thus Irr(G) is an orthonormal basis of the space of class functions from G to K, endowed with the usual scalar product denoted by ⟨ , ⟩_{G}.

Algebraic groups are usually denoted by boldface letters. The identity component, i.e. the connected component containing the identity, of an algebraic group C is denoted by C^{∘}, with the conventions that Z^{∘}(C) = Z(C)^{∘} and C_{C}^{∘}(U) = C_{C}(U)^{∘} for any subgroup or element U in C. If T is a maximal torus of a reductive group C, then one denotes by W (C, T) the quotient N_{C}(T)/T and by p_{T} : N_{C}(T) → W (C, T) the canonical surjection.

Let (G, F), q, ℓ, e and E(G^{F} , ℓ′) be as said or defined in the introduction. Let G^{∗} be in duality with G, a duality defined over 𝔽_{q}, with Frobenius F on G^{∗}. The following applies to any F -stable Levi subgroup of G or of G^{∗}. Let \overline{𝔽}_{q} be an algebraic closure of 𝔽_{q} and \overline{𝔽}_{q}^{∗} be its multiplicative group. We shall assume that choices have been made, once and for all, of isomorphisms of \overline{𝔽}_{q}^{∗} with (ℚ/ℤ)_{p′} and of \overline{𝔽}_{q}^{∗} with the group of roots of unity of order prime to p in \overline{ℚ}_{ℓ}. One has the partition Irr(G^{F}) = ∪_{(s)}E(G^{F} , s) into rational series indexed by the G^{∗}^{F} -conjugacy classes of semi-simple elements of G^{∗}^{F} [L.3, 2; DM.1, 14.41]. The elements of E(G^{F} , 1) are the unipotent characters of G^{F}. When s ∈ G^{∗}^{F}, we denote K_{G}(s) = C_{G^{∗}}(s)/C_{G^{∗}}^{∘}(s).

We shall refer quite often to the Jordan decomposition defined by G. Lusztig [L.3], which associates with each element of E(G^{F} , s) a C_{G^{∗}}(s)^{F} -orbit in E(C_{G^{∗}}^{∘}(s)^{F} , 1). When C_{G^{∗}}(s) is connected, this becomes a bijection E(G^{F} , s) → E(C_{G^{∗}}(s)^{F} , 1).

One may find specific notations at the beginning of sections 1.1, 1.2, 1.3, 2.1, 3 and 4.

1. Some remarks on rational series and twisted induction.

1.1. ℓ′-series and linearly independent sets.

The main goal of this section is Theorem 1.7. Assume ℓ is good for G. We generalize [G, Theorem A] to give a basis of Brauer ℓ-modular characters of the group G^{F} extended by its diagonal ℓ-automorphisms. This leads us to consider twisted induction and Jordan decomposition for a mild class of disconnected reductive groups.

Let G ↪ G′ be a regular embedding for a connected reductive group (G, F) defined over 𝔽_{q} (see [DL, 5.18; L.3, 7]), that is Z(G′) is connected and G′ = Z(G′)G. If C is an intermediate group between G^{F} and (G′^{F}), then it is of the form C^{F} where C is an F -stable, possibly disconnected, reductive group standing between G and G′. One may take C = GC, or better C = GA for A = T′ ∩ C where T′ is any F -stable maximal torus of G′, since G^{′F} = G^{F} T^{′F}. According to [G], or in more generality [DM.2], one may consider the Levi subgroups of disconnected reductive groups and analogues of twisted induction. Let P be a parabolic subgroup of G and let P = R_{u}(P)L be a Levi decomposition of P. Let L′ = Z(G′)L. Then L′ is the only Levi subgroup of G′ containing L and P′ = R_{u}(P)L′ is the only parabolic subgroup of G′ containing P. Assume L is F -stable. If A has been chosen in an F -stable maximal torus of L′ (which is always possible as seen above), then PA is a parabolic subgroup of GA and LA is an F -stable Levi complement in it. Note also that LA = C_{GA}(Z^{∘}(L)).

The analogue of Lusztig’s twisted induction is defined as follows. Let S_{LA⊆PA}^{GA} be the variety defined by S_{LA⊆PA}^{GA} = {g ∈ GA ; g^{−1}F (g) ∈ R_{u}(P)} and H_{c}^{i}(S_{LA⊆PA}^{GA}, \overline{ℚ_{ℓ}}) its groups of ℓ-adic cohomology with compact support. Both are acted on by L^{F} A on the right and G^{F} A on the left. The map R_{LA⊆PA}^{GA}: ℤIrr(L^{F} A) → ℤIrr(G^{F} A) is defined by tensor product with the Lefchetz virtual G^{F} A-L^{F} A-module ∑_{i}(−1)^{i}H_{c}^{i}(S_{LA⊆PA}^{GA}, \overline{ℚ_{ℓ}}).

If A ⊆ B ⊆ T′^{F} for T′ an F -stable maximal torus of L′, then Res_{G^{F} A}^{G^{F} B}R_{LB⊆PB}^{GB} =

R_{LA⊆PA}^{GA}Res_{L^{F} A}^{L^{F} B} on Irr(L^{F} B) (see the proof of [G, 1.4], where the case B = T′^{F} is considered).

In what follows, we do not assume that the above maps are independent of the choice of the parabolic subgroup (see Theorem 2.5, Proposition 2.9 below). When we omit the parabolic PA, this means that the choice is fixed and is not changed unless L or G is changed. Nevertheless recall that if T is an F -stable maximal torus of G, then R_{T⊆B}^{G} is independent of the Borel subgroup B [DM.1, 11.15 (ii)].

Let G′^{∗} → G^{∗} be a dual map of the regular embedding and let S be its kernel. This map is surjective and S is connected. The group L^{′∗} maps on a dual L^{∗} of L.

Definition 1.1. Let A be a subgroup of T′^{F}. For any semi-simple element s′ of (L^{′∗})^{F} denote

E(L^{F} A, s′) = {ζ ∈ Irr(L^{F} A) | ⟨ζ, Res_{L^{F} A}^{L^{′F}}ζ′⟩_{L^{F} A} ≠ 0 for some ζ′ ∈ E(L^{′F} , s′)}

Denote E(G^{F} A, ℓ′) = ∪_{s′}E(G^{F} A, s′) where the sum is over the semi-simple s′ ∈ (G′^{∗})^{F} of order prime to ℓ.

The following proposition is standard in the case A = {1}. Note that, when A = {1}, E(G^{F} A, ℓ′) is, by (iii), equal to E(G^{F} , ℓ′) already defined. We abbreviate H_{c}^{i}(S_{LA⊆PA}^{GA}, \overline{ℚ_{ℓ}}) as H^{i}(S_{LA⊆PA}^{GA}).

Proposition 1.2. Let R_{T′}^{G′}θ′ be a Deligne-Lusztig character of G′^{F}, let s′ ∈ (G′^{∗})_{ss}^{F} such that R_{T′}^{G′}θ′ ∈ ℤE(G^{′F} , s′). Let TA = GA ∩ T′, θ_{A} = Res_{T^{F} A}^{T^{′F}}θ′. Then H^{i}(S_{TA⊆BA}^{GA}) ⊗ θ_{A} has all its irreducible components in E(G^{F} A, s′).

(i) If L′ is an F -stable Levi subgroup of G′ and s′ ∈ (L′^{∗})_{ss}^{F}, then R_{LA}^{GA}(E(L^{F} A, s′)) ⊆ ℤE(G^{F} A, s′). (ii) If s′, s′′ ∈ (G′^{∗})_{ss}^{F}, then E(G^{F} A, s′) and E(G^{F} A, s′′) are equal if and only if they are not disjoint. This in turn occurs if and only if there exists z ∈ Z(G′^{∗})^{F} defining by duality a linear character ẑ of G′^{F} with G^{F} A in its kernel and zs′ being (G′^{∗})^{F} - conjugate with s′′. (iii) E(G^{F} A, ℓ′) is the set of irreducible characters occuring in the modules H^{i}(S_{TA⊆BA}^{GA}) ⊗ θ_{A} for θ_{A} ∈ Irr(T^{F} A)_{ℓ′}. So it does not depend on the regular embedding G ↪ G′. (iv) R_{LA}^{GA}(E(L^{F} A, ℓ′)) ⊆ ℤE(G^{F} A, ℓ′) and ^{∗}R_{LA}^{GA}(E(G^{F} A, ℓ′)) ⊆ ℤE(L^{F} A, ℓ′).

(v) Any element of E(G^{F} A, ℓ′) has Z(GA)_{ℓ}^{F} in its kernel.

Proof of the Proposition. By [G, 1.4], H^{i}(S_{TA⊆BA}^{GA}) ⊗ θ_{A} is isomorphic with

Res_{G^{F} A}^{G′^{F}}(H^{i}(S_{T′⊆B′}^{G′}) ⊗ θ′), so all the irreducible components of H^{i}(S_{TA⊆BA}^{GA}) ⊗ θ_{A} are components of the restrictions of irreducible components of H^{i}(S_{T′⊆B′}^{G′}) ⊗ θ′. This, by the definition of E(G^{F} A, s′), reduces the statement to prove to the case GA = G′. The irreducible components of H^{i}(S_{T′⊆B′}^{G′}) ⊗ θ′ are all in the geometric series associated with s′, but G′ has connected center, so this geometric series coincides with E(G^{′F} , s′) (see [DM.1, 13.16, 13.18]).

(i) This follows from the above by a classical application of Künneth formula (see [G, 1.2.(ii)] or the proof of [DM.1, 11.5]).

(ii) The proof is an easy adaptation of the standard one for the case A = 1 (see [DM.1, 13.30] and proof).

(iii) The definition of E(G^{F} A, ℓ′) readily implies that its elements are irreducible components of the modules H^{i}(S_{TA⊆BA}^{GA}) ⊗ θ_{A} for θ_{A} ∈ Irr(T^{F} A)_{ℓ′}. The converse is clear since we know that H^{i}(S_{TA⊆BA}^{GA}) ⊗ θ_{A} has all its irreducible components in

E(G^{F} A, s′) as soon as θ_{A} = Res_{T^{F} A}^{T′^{F}}θ′ and R_{T′}^{G′}θ′ ∈ ℤE(G^{′F} , s′).

(iv) The inclusion R_{LA}^{GA}(E(L^{F} A, ℓ′)) ⊆ ℤE(G^{F} A, ℓ′) follows by (i).

Concerning the second inclusion, note that if s′ ∈ (G′^{∗})_{ss}^{F}, then (ii) implies that E(G^{F} A, s′)∩E(G^{F} A, ℓ′) ≠ ∅ if and only if s′_{ℓ} ∈ Z(G^{′∗}) and ŝ′_{ℓ}(G^{F} A) = 1. Now, when s′ ∈ (L′^{∗})_{ss}^{F}, χ′ ∈ E(L^{F} A, s′) and R_{LA}^{GA}χ′ has an irreducible component in E(G^{F} A, ℓ′), this implies by (i) that E(G^{F} A, s′)∩E(G^{F} A, ℓ′) ≠ ∅. Then s′_{ℓ} ∈ Z(L^{′∗}) and ŝ′_{ℓ}(L^{F} A) = 1, that is χ′ ∈ E(L^{F} A, ℓ′). This yields the second claimed inclusion.

(v) It is clear that Z(G′)_{ℓ}^{F} is a subgroup of (T′)_{ℓ}^{F} acting trivially on any H^{i}(S_{T′⊆B′}^{G′}) ⊗ θ′ for θ′ ∈ Irr(T^{′F})_{ℓ′}. So Z(GA)_{ℓ}^{F} is in the kernel of all the irre-

ducible components of the Res_{G^{F} A}^{G^{′F}}χ′ for χ′ ∈ E(G^{′F} , ℓ′). So it is in the kernel of any element of E(G^{F} A, ℓ′).

We have to review results of Lusztig [L.3] about Jordan decomposition of characters and Clifford theory between G^{F} and G′^{F}. Note that the centralizer of the semi-simple element s′ in G′^{∗} is connected because the center of a dual group G is connected [DM.1, 13.15 (ii)]. Let s ∈ G^{∗}^{F} be the image of s′ by G′^{∗} → G^{∗}. Let K_{G}(s) = C_{G^{∗}}(s)/C_{G^{∗}}^{∘}(s). One has a natural F -isomorphism, defined by commutators, of K_{G}(s) onto a subgroup of S. By duality S^{F} is isomorphic with Irr(G′^{F} /G^{F}), hence there is a monomorphism

m_{G}^{s}: K_{G}(s)^{F} → Irr(G′^{F} /G^{F})

Let χ′ ∈ E(G′^{F} , s′) and let γ ∈ E(C_{G′}^{∗}(s′)^{F} , 1) corresponding with χ′ by Jordan decomposition of characters. As C_{G^{∗}}^{∘}(s) = C_{G′}^{∗}(s′)/S we may identify E(C_{G′}^{∗}(s′)^{F} , 1) and E(C_{G^{∗}}^{∘}(s)^{F} , 1). Let K_{G}(s)_{γ}^{F} be the stabilizer of γ under the natural action of K_{G}(s)^{F} on E(C_{G^{∗}}^{∘}(s)^{F} , 1). By [L.3, 8.1] m_{G}^{s} preserves Jordan decomposition of characters in the following sense : If a ∈ K_{G}(s)^{F}, then a.γ corresponds to m_{G}^{s}(a)χ′ ∈ E(G^{′F} , s′). If λ ∈ Irr(G′^{F} /G^{F}) is not in m_{G}^{s}(K_{G}(s)^{F}), then λχ′ lies in another series. Hence m_{G}^{s} induces an isomorphism from K_{G}(s)_{γ}^{F} onto the stabilizer of χ′ in Irr(G′^{F} /G^{F}).

Now let χ ∈ Irr(G^{F}) be such that ⟨Res_{G^{F}}^{G^{′F}} χ′, χ⟩_{G^{F}} ≠ 0 and let G′_{χ}^{F} be the stabilizer of χ in G′^{F}. According to [L.3, 10] there is no multiplicity in Clifford theory between G^{′F} and G^{F}. Therefore for any λ ∈ Irr(G^{′F} /G^{F}) one has λχ′ = χ′ if, and only if, the kernel of λ contains G′_{χ}^{F}, so that Irr(G′^{F} /G′_{χ}^{F}), considered as a subgroup of Irr(G^{′F} /G^{F}), is the stabilizer of χ′ [L.3, 9]. Hence

(1)  m_{G}^{s}(K_{G}(s)_{γ}^{F}) = Irr(G′^{F} /G′_{χ}^{F})

Proposition 1.3. Let G ↪ G′, T′, A ⊆ (T′)^{F} and s′ ↦ s be as above. Let χ_{A} ∈ E(G^{F} A, s′). Let G′_{χ_{A}}^{F} be the stabilizer of χ_{A} in G′^{F}. Then

(i) The group G′^{F} /G′_{χ_{A}}^{F} is isomorphic to a subgroup of C_{G^{∗}}(s)^{F} /C_{G^{∗}}^{∘}(s)^{F}.

(ii) If G^{F} A/G^{F} is an ℓ-group, then Res_{G^{F}}^{G^{F} A} induces a bijection E(G^{F} A, ℓ′) → E(G^{F} , ℓ′).

Proof of the Proposition. (i) One may choose χ ∈ E(G^{F} , s) such that ⟨χ, Res_{G^{F}}^{G^{F} A}χ_{A}⟩_{G^{F}} ≠ 0. Since Clifford theory between G^{′F} and G^{F} and also between the intermediate groups must not introduce multiplicities, the stabilizer of χ_{A} in G′^{F} is G′_{χ}^{F} .A. Hence one has

(2)  Res_{G^{F} A}^{G^{′F}}χ′ = ∑_{g∈(G^{′F} /G′_{χ}^{F} A)}^{g}χ_{A}

The abelian group G′^{F} /G′_{χ}^{F} A is isomorphic to its group of linear characters, which is a subgroup of Irr(G′^{F} /G′_{χ}^{F}). By (1) this yields (i).

(ii) By definition, Res_{G^{F}}^{G^{F} A} sends E(G^{F} A, ℓ′) into ℤE(G^{F} , ℓ′). Now, if χ ∈ E(G^{F} , s) and s is of order prime to ℓ, then the quotient G^{′F} /G′_{χ}^{F} is of order prime to ℓ

by (i). Hence G^{F} A ⊆ G′_{χ}^{F}. Applying this to any irreducible component of Res_{G^{F}}^{G^{F} A}χ_{A}

for χ_{A} ∈ E(G^{F} A, ℓ′), one gets that Res_{G^{F}}^{G^{F} A} sends E(G^{F} A, ℓ′) into E(G^{F} , ℓ′). The map is surjective by definition of the sets E(−, ℓ′). Now, if χ_{A} ∈ E(G^{F} A, ℓ′) has same restriction as χ′_{A} ∈ Irr(G^{F} A), then χ_{A} = λχ′_{A} where λ is a linear character of the ℓ-group G^{F} A/G^{F}. If λ ≠ 1, then χ_{A} ∈ E(G^{F} A, zs′) for z ∈ Z(G^{′∗})_{ℓ}^{F} and ẑ(G^{F} A) ≠ 1. By Proposition 1.2 (ii), χ′_{A} ∉ E(G^{F} A, ℓ′). So the restriction is one-to-one from E(G^{F} A, ℓ′) to E(G^{F} , ℓ′).

Proposition 1.4. Hypotheses of Proposition 1.3. Assume L^{∗} is an F -stable Levi subgroup of G^{∗} such that C_{G^{∗}}^{∘}(s) ⊆ L^{∗}. Then

(i) There is ζ_{A} ∈ E(L^{F} A, s′) such that ⟨χ_{A}, R_{LA⊆PA}^{GA}ζ_{A}⟩_{G^{F} A} ≠ 0. (ii) When ζ_{A} satisfies (i), one has

R_{LA⊆PA}^{GA}ζ_{A} = ε_{L}ε_{G} ∑_{g} ^{g}χ_{A}

where the sum is over a subgroup of G′^{F} /G′_{χ_{A}}^{F} isomorphic to some section of C_{G^{∗}}(s)^{F} /C_{L^{∗}}(s)^{F}. (iii) If ζ_{A}, ζ′_{A} ∈ E(L^{F} A, s′) and ζ′_{A} ≠ ζ_{A} then R_{LA⊆PA}^{GA}ζ_{A} and R_{LA⊆PA}^{GA}ζ′_{A} have no irreducible component in common.

Proof of the Proposition. We keep notations introduced in equality (1) before Proposition 1.3. Let a Levi L of G be in duality with L^{∗}, and L′ = Z(G′)L such that L′^{∗} maps onto L^{∗}. Let χ′ ∈ E(G′^{F} , s′) be such that ⟨Res_{G^{F} A}^{G′^{F}}χ′, χ_{A}⟩_{G^{F} A} ≠ 0. As C_{G′}^{∗}(s′) ⊆ L′^{∗}, ε_{L′}ε_{G′}R_{L′}^{G′} induces a bijection between E(L^{′F} , s′) and E(G^{′F} , s′) which is independent of the parabolic for L′ (see [DM.1, 13.25.(ii), 13.28]). Let ζ′ ∈ E(L^{′F} , s′) such that χ′ = ε_{L′}ε_{G′}R_{L′}^{G′}ζ′. By Jordan decomposition the same γ ∈ E(C_{G^{∗}}^{∘}(s)^{F} , 1) corresponds to χ′ and ζ′ (see [L.1, 7.9; L.2, 6.20]). Let ζ ∈ Irr(L^{F}) such that ⟨Res_{L^{F}}^{L^{′F}} ζ′, ζ⟩_{L^{F}} ≠ 0, let us form L′_{ζ}^{F} the stabilizer of ζ in L′^{F}, and K_{L}(s) = C_{L^{∗}}(s)/C_{L^{∗}}^{∘}(s). By (1) applied to L one has an injective map m_{L}^{s} and

(3)  m_{L}^{s}: K_{L}(s)^{F} → Irr(L′^{F} /L^{F}), m_{L}^{s}(K_{L}(s)_{γ}^{F}) = Irr(L′^{F} /L′_{ζ}^{F})

One has G′^{F} = G^{F} L′^{F}, hence Irr(G′^{F} /G^{F}) is isomorphic to Irr(L′^{F} /L^{F}) and the natural injection of K_{L}(s) in K_{G}(s) implies that G′_{χ}^{F} ⊆ L′_{ζ}^{F} .G^{F}.

Then ⟨χ_{A}, R_{LA⊆PA}^{GA}(Res_{L^{F} A}^{L^{′F}}ζ′)⟩_{G^{F} A} = ⟨χ_{A}, Res_{G^{F} A}^{G′^{F}}χ′⟩_{G^{F} A} ≠ 0, while for each irreducible component ζ_{A} of Res_{L^{F} A}^{L^{′F}}ζ′, one has ζ_{A} ∈ E(L^{F} A, s′). So there is a ζ_{A} such that ⟨χ_{A}, R_{LA⊆PA}^{GA}ζ_{A}⟩_{G^{F} A} ≠ 0. This is (i).

Conversely, when ζ_{A} ∈ E(L^{F} A, s′) is such that ⟨χ_{A}, R_{LA⊆PA}^{GA}ζ_{A}⟩_{G^{F} A} ≠ 0, let ζ′′ ∈ E(L^{′F} , s′) such that ζ_{A} is an irreducible component of Res_{L^{F} A}^{L^{′F}}ζ′′, and let χ′′ = ε_{L′}ε_{G′}R_{L′}^{G′}ζ′′ ∈ E(G′^{F} , s′). Then Res_{G^{F} A}^{G^{′F}}χ′′ = ∑_{i} ε_{L′}ε_{G′}R_{LA}^{GA}ζ_{i} where the ζ_{i}’s are irreducible components of Res_{L^{F} A}^{L^{′F}}ζ′′ and ζ_{A} is among them. But, for all such ζ_{i}, ε_{L}ε_{G}R_{LA⊆PA}^{GA}ζ_{i} is an actual character (see [G, Proof of 1.5, (e)]). Hence χ_{A} appears in Res_{G^{F} A}^{G^{′F}}χ′′. So, to check the remainder of Proposition 1.4, one may assume that χ′′ = χ′ and ζ′′ = ζ′.

One may choose χ ∈ E(G^{F} , s) and ζ ∈ E(L^{F} , s) such that ⟨χ, Res_{G^{F}}^{G^{F} A}χ_{A}⟩_{G^{F}} ≠ 0 and ⟨ζ, Res_{L^{F}}^{L^{F} A}ζ_{A}⟩_{L^{F}} ≠ 0. The stabilizers of χ_{A} in G′^{F} and ζ_{A} in L′^{F} are respectively G′_{χ}^{F} .A and L′_{ζ}^{F} .A. One has (2) (Proof of Proposition 1.3) and

(4)  Res_{L^{F} A}^{L^{′F}}ζ′ = ∑_{h∈(L^{′F} /L′_{ζ}^{F} .A)}^{h}ζ_{A}

We compute C = ⟨Res_{G^{F} A}^{G^{′F}}χ′, Res_{G^{F} A}^{G^{′F}}χ′⟩_{G^{F} A} with respect to C′ = ⟨R_{LA}^{GA}ζ_{A}, R_{LA}^{GA}ζ_{A}⟩_{G^{F} A}. From (2) one has C = |G′^{F} /G′_{χ}^{F} .A|. Since G′^{F} = G^{F} L′^{F} and G′_{χ}^{F} ⊆ L′_{ζ}^{F} .G^{F}, by isomorphism theorem one has

C = |L^{′F} /L′_{ζ}^{F} .A|.|L′_{ζ}^{F} .G^{F} A/G′_{χ}^{F} .A|.

Recall that Res_{G^{F} A}^{G^{′F}}χ′ = R_{LA}^{GA}Res_{L^{F} A}^{L′^{F}}ζ′. From (4) one has

C = |L^{′F} /L′_{ζ}^{F} A|C′ + Σ_{h≠h′}⟨R_{LA}^{GA}(^{h}ζ_{A}), R_{LA}^{GA}(^{h′}ζ_{A})⟩_{G^{F} A}.

All scalar products in the last sum are non negative hence C′ ≤ |L′_{ζ}^{F} .G^{F} A/G′_{χ}^{F} .A|. Now, up to the multiplicative sign ε_{L}ε_{G}, R_{LA}^{GA}ζ_{A} is a sum of conjugate of χ_{A}. As ζ_{A} is fixed by L′_{ζ}^{F} .A, R_{LA}^{GA}ζ_{A} is fixed by L′_{ζ}^{F} .G^{F} A. Hence one has C′ ≥ |L′_{ζ}^{F} .G^{F} A/G′_{χ}^{F} .A|. We obtain C = |L^{′F} /L′_{ζ}^{F} .A|C′. This implies that ε_{L}ε_{G}R_{LA}^{GA}ζ_{A} is a sum of irreducible characters in an orbit under the regular action of L′_{ζ}^{F} .G^{F} A/G′_{χ}^{F} .A. Moreover in the above expansion of C one has ⟨R_{LA}^{GA}(^{h}ζ_{A}), R_{LA}^{GA}(^{h′}ζ_{A})⟩_{G^{F} A} = 0 whenever h, h′ ∈ G′^{F} and hL′_{ζ}^{F} .G^{F} A ≠ h′L′_{ζ}^{F} .G^{F} A, that is (iii). This gives the equality in (ii) with g running over L′_{ζ}^{F} .G^{F} A/G′_{χ}^{F} .A, a section of G′^{F} /G^{F} A. The group L′_{ζ}^{F} .G^{F} A/G′_{χ}^{F} .A is also an image of L′_{ζ}^{F} .G^{F} /G′_{χ}^{F}. By (1) and (2) L′_{ζ}^{F} .G^{F} /G′_{χ}^{F} is isomorphic to K_{G}(s)_{γ}^{F} /K_{L}(s)_{γ}^{F}. Clearly one has C_{G^{∗}}(s)^{F} /C_{L^{∗}}(s)^{F} ≅ K_{G}(s)^{F} /K_{L}(s)^{F}.

Proposition 1.5. We keep the notations and hypotheses of Proposition 1.4, in particular C_{G^{∗}}^{∘}(s) ⊆ L^{∗}.

(i) If ζ ∈ E(L^{F} , s), χ ∈ Irr(G^{F}), and ⟨χ, R_{L}^{G}ζ⟩_{G^{F}} ≠ 0, then there exists a divisor c of |C_{G^{∗}}(s)^{F} : C_{L^{∗}}(s)^{F} | such that c.χ(1) = |G^{F} : L^{F} |_{p′}ζ(1), where p denotes the prime dividing q. (ii) If C_{G^{∗}}^{∘}(s)C_{G^{∗}}^{F} (s) ⊆ L^{∗}, then ε_{L}ε_{G}R_{L}^{G} induces a bijection from E(L^{F} , s) to E(G^{F} , s). (iii) Let t = s_{ℓ}. Assume C_{G^{∗}}^{∘}(t) ⊆ L^{∗}. If |G^{′F} /G^{F} A| is of order prime to ℓ, then ε_{L}ε_{G}R_{LA}^{GA} induces a bijection from E(L^{F} A, s′) to E(G^{F} A, s′).

Proof of the Proposition. (i) follows from Proposition 1.4.(iii) and the degree formula for Lusztig’s twisted induction [DM.1, 12.17]. Point (ii) is [DM.2, 13.25.(ii)] (and a consequence of the formula of Proposition 1.4.(ii) in a special case, see below).

(iii) One has C_{G^{∗}}^{∘}(s) ⊆ C_{G^{∗}}^{∘}(t) ⊆ L^{∗}. Let ζ′ ∈ E(L′, s′), ζ_{A} ∈ E(L^{F} A, s′), χ′ ∈ E(G^{′F} , s′), χ_{A} ∈ E(G^{F} A, s′) be as in Proposition 1.4 and its proof. We have showed that Res_{G^{F} A}^{G^{′F}}χ′ is a disjoint sum of the ε_{L′}ε_{G′}R_{LA}^{GA}ζ_{A} when ζ_{A} runs over the set of irreducible components of Res_{L^{F} A}^{L′^{F}}ζ′. When χ′ runs over E(G′^{F} , s′), ζ′ runs over E(L′^{F} , s′) so that χ_{A} runs over E(G^{F} A, s′) and ζ_{A} over E(L^{F} A, s′). By (i) of Proposition 1.3 and (iii) of Proposition 1.4, the map χ_{A} ↦ ζ_{A} (with ⟨χ_{A}, R_{LA⊆PA}^{GA}ζ_{A}⟩_{G^{F} A} ≠ 0) is well defined and surjective. By (ii) of Proposition 1.4 and hypothesis, it is bijective when moreover C_{G^{∗}}(s)/C_{L^{∗}}(s) is an ℓ-group. This follows from the next Lemma.

Lemma. Let s be a semi-simple element in (G^{∗})^{F}, let L^{∗} be a Levi subgroup of G^{∗} such that C_{G^{∗}}^{∘}(s_{ℓ}) ⊆ L^{∗}. Then there is a natural injection C_{L^{∗}}(s)/C_{L^{∗}}^{∘}(s) → C_{G^{∗}}(s)/C_{G^{∗}}^{∘}(s) with quotient an ℓ-group. Proof. Let t = s_{ℓ}. One has C_{G^{∗}}(s)/C_{L^{∗}}(s) = C_{C_{G^{∗}} (t)}(s)/C_{C_{L^{∗}} (t)}(s) and C_{G^{∗}}^{∘}(s) = C_{L^{∗}}^{∘}(s). From hypothesis C_{G^{∗}}(s)/C_{L^{∗}}(s) is a quotient of C_{G^{∗}}(t)/C_{G^{∗}}^{∘}(t), and this last one is an ℓ-group [DM.1, 13.15.(i)]. The claim follows by isomorphism theorem.

Proposition 1.6. Assume that ℓ is good for G. When f is a central function on G^{F} A, denote by d^{1}f its restriction to ℓ′-elements. If G^{′F} /G^{F} A is of order prime to ℓ, then {d^{1}χ | χ ∈ E(G^{F} A, ℓ′)} form a generating set of Brauer characters of G^{F} A.

Proof of the Proposition. Consider χ ∈ Irr(G^{F} A). It suffices to check that d^{1}χ is an integral linear combination of the d^{1}θ for θ ∈ E(G^{F} A, ℓ′). We show this by induction on dim G. When G is a torus, it is clearly true.

Let us write χ ∈ E(G^{F} A, s′t′) with t′ the ℓ-part of s′t′ ∈ (G′^{∗})^{F}. Then t′ ∈ C_{G^{′∗}}^{∘}(s′)_{ℓ}^{F} and (iii) of Proposition 1.5 applies with L′^{∗} = C_{G′^{∗}}^{∘}(t′), a Levi subgroup of G^{∗} because ℓ is good [CE.1, 2.1.(i)]. Then χ = ε_{L}ε_{G}R_{LA}^{GA}ζ_{A} with ζ_{A} ∈ E(L^{F} A, s′t′). Now by [G, 1.2.(iii)] (see also Proposition 2.3 below), one has d^{1}χ = ε_{L}ε_{G}R_{LA}^{GA}(d^{1}ζ_{A}). If t′ is not central in G′, then the induction hypothesis applies to L. So d^{1}ζ_{A} may be replaced by a linear combination of d^{1}ζ’s with ζ ∈ E(L^{F} A, ℓ′). Then Proposition 1.1.(iv) and the commutation between d^{1} and R_{LA}^{GA} gives our claim.

If t′ is central, then E(G^{′F} , s′t′) = λ′E(G^{′F} , s′) and therefore E(G^{F} A, s′t′) = λ_{A}.E(G^{F} A, s′) for linear ℓ-characters λ′, λ_{A}. Then d^{1}χ = d^{1}(λ_{A}^{−1}χ) with λ_{A}^{−1}χ ∈ E(G^{F} A, ℓ′).

Theorem 1.7. Assume ℓ is good for G. Then the restrictions to (G^{F})_{ℓ′} of the elements in E(G^{F} , ℓ′) are distinct and linearly independent.

Proof of the Theorem. Denote B(G^{F} A) = (d^{1}χ)_{χ∈E(G^{F} A,ℓ′)}. Then B(G^{′F}) is a basic set of Brauer characters of G^{′F} by [G, Theorem A]. Let T′ be any F -stable maximal torus of G′. Let A = (T′)_{ℓ}^{F}, then G′^{F} /G^{F} A is of order prime to ℓ. Proposition 1.6 above allows us to apply [G, 4.2], so B(G^{F} A) is a basic set of Brauer characters of G^{F} A. On the other hand G^{F} A/G^{F} is an ℓ-group, so Proposition 1.3.(ii) implies B(G^{F}) = (Res_{G^{F}}^{G^{F} A}β)_{β∈B(G^{F} A)}. But we have (G^{F})_{ℓ′} = (G^{F} A)_{ℓ′}, so restriction preserves linear independence.

Remark.ÊThe \overline{ℚ_{ℓ}}-space of central functions on G^{F} generated by the d^{1}χ’s for χ ∈ E(G^{F} , ℓ′) is the set of functions on (G^{F})_{ℓ′} that are constant on H-orbits where H is the group G^{F} A of the above proof, that is G^{F} ⊆ H ⊆ G^{′F} and H/G^{F} is the Sylow ℓ-subgroup of G^{′F} /G^{F}.

1.2. Rational series and rational actions.

The following Proposition is useful to study certain group actions on series.

Notation 1.8. If H is a closed subgroup of G containing a maximal torus T of G, let W (G|H, T) denote the subgroup of W (G, T) generated by involutions associated with the roots of G relative to T that are orthogonal to all the roots of H.

Proposition 1.9. Let T ⊆ L ⊆ G and T^{∗} ⊆ L^{∗} ⊆ G^{∗} be in duality with L a Levi subgroup and T an F -stable maximal torus. The transitivity of L on its maximal tori induces an isomorphism from N_{G^{F}} (L)/L^{F} onto (N_{W (G,T)}(W (L, T))/W (L, T))^{F} defined by xL^{F} ↦ n(x)W (L, T) for x ∈ N_{G^{F}} (L), ^{x}T = ^{l}T with l ∈ L and n(x) = l^{−1}x.

Let s ∈ T^{∗} such that its L^{∗}-conjugacy class is F -stable. Denote by Ē(L^{F} , s) the associated geometric series of characters of L^{F} (see [DM.1,§ 13]). One has :

(i) ^{x}Ē(L^{F} , s) = Ē(L^{F} ,^{n(x)} s) for any x ∈ N_{G^{F}} (L), the action of N_{G}(T) on T^{∗} being given by the identification W (G, T) = W (G^{∗}, T^{∗}). (ii) The map xL^{F} ↦ n(x)W (L, T) induces an isomorphim from N_{G^{F}} (L, Ē(L^{F} , s))/L^{F} onto a subgroup of (N_{W (C_{G^{∗}} (s),T^{∗})}(W (C_{L^{∗}}^{∘}(s), T^{∗}))/W (C_{L^{∗}}(s), T^{∗}))^{F}. (iii) Let s ∈ (T^{∗})^{F} and x ∈ N_{G^{F}} (L) ∩ p_{T}^{−1}(W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗}). If x and s have coprime orders, then x fixes every element of E(L^{F} , s).

Proof of the Proposition. The definition of n(x) clearly yields an isomorphism between N_{G}(L)/L and N_{W (G,T)}(W (L, T))/W (L, T) which commutes with the action of F. This yields the claimed isomorphism between F -fixed points thanks to Lang’s theorem.

(i) Now the series Ē(L^{F} , s) contains the set of irreducible components of the Deligne-Lusztig character R_{T}^{L}θ where θ is considered as a character of Y (T) with (F −1)Y (T) in its kernel and corresponds with s by the duality isomorphism X(T)⊗ (ℚ/ℤ)_{p′} ≅ Y (T^{∗})⊗(ℚ/ℤ)_{p′} (see [DM.1, 13.16]). One has^{x}R_{T}^{L}θ = R_{^{x}T}^{L}^{x}θ and (^{x}T,^{x}θ) is in the geometric conjugacy class in L of (^{l^{−1}x}T,^{l^{−1}x}θ) (where θ is considered as an element of X(T)). But ^{n(x)}θ corresponds with ^{n(x)}s when θ corresponds with s.

(ii) When s, s′ ∈ T^{∗} have F -stable L^{∗}–conjugacy class, one has Ē(L^{F} , s) = Ē(L^{F} , s′) if and only if s and s′ are W (L^{∗}, T^{∗})-conjugate. So (i) and the identification W (G, T) = W (G^{∗}, T^{∗}) imply that N_{G^{F}} (L, Ē(L^{F} , s))/L^{F} injects into (N_{W (C_{G^{∗}}^{∘} (s),T^{∗})}(W (L^{∗}, T^{∗}))/W (C_{L^{∗}}(s), T^{∗}))^{F}. This implies the claimed injection.

(iii) Let ζ ∈ E(L^{F} , s) and let us show that ^{x}ζ = ζ.

Let us consider a regular embedding G ↪ G′ = Z^{∘}(G′)G as in § 1.1. Let T′ = Z^{∘}(G′)T = C_{G′}^{∘}(T) and L′ = Z^{∘}(G′)L = C_{G′}^{∘}(Z^{∘}(L)). Let s′ ∈ G′^{∗}^{F} be semi-simple, of image s in G^{∗}. By (4) (Proof of Proposition 1.4), one has ∑_{t∈L′^{F} /L′_{ζ}^{F}} ^{t}ζ = Res_{L^{F}}^{L′^{F}} ζ′ for some ζ′ ∈ E(L′^{F} , s′). The series E(L′^{F} , s′) is a series in a group with connected center, so by Lusztig’s theorems (see [L.3, 6.1]) the Jordan decomposition is a bijection λ ↦ χ_{λ} from E(C_{L′^{∗}}^{∘}(s′)^{F} , 1) = E(C_{L^{∗}}^{∘}(s)^{F} , 1) to E(L′, s′) satisfying a compatibility relation with Deligne-Lusztig characters. The action of x on χ_{λ} is given by the action of y ∈ N_{W (C_{G^{∗}}^{∘} (s),T^{∗})}(W (C_{L^{∗}}^{∘}(s), T^{∗}))^{F} on λ, where y has the same image as x in N_{W (G′,T′)}(W (L′, T′))/W (L′, T′) identified to N_{W (G′^{∗},T′^{∗})}(W (L′^{∗}, T′^{∗}))/W (L′^{∗}, T′^{∗}) (see [CE.2, 2.3.(d)]). Arguing as in the proof of [CE.2, Lemma 4.2], one sees that if moreover xT^{∗} ∈ W (C_{G^{∗}}^{∘}(s), T^{∗}) fixes all the roots of C_{L^{∗}}^{∘}(s), then λ is fixed by y. This is the case when xT^{∗} ∈ W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗}). Then x fixes ζ′ and therefore its restriction to L^{F}. But now x and L′^{F} /L′_{ζ}^{F} commute as outer automorphisms of L^{F} since [x, L′^{F}] ⊆ L′^{F} ∩[G′, G′] ⊆ L^{F}. By Proposition 1.3.(i), the group L′^{F} /L′_{ζ}^{F} is isomorphic with a subgroup of C_{L^{∗}}(s)^{F} /C_{L^{∗}}^{∘}(s)^{F} and it is therefore of order prime to the one of x. Then x must fix all the summands in Res_{L^{F}}^{L′^{F}} ζ′ = ∑_{t∈(L′^{F} /L′_{ζ}^{F})} ^{t}ζ, so it fixes ζ.

1.3. e-cuspidality and Jordan decomposition of characters.

We use the notion of polynomial order P_{(G,F)} of a connected reductive group (G, F) (see [BMM; CE.1, 1.2; CE.2, 1.2]). Let e ≥ 1 be an integer and ϕ_{e} the corresponding cyclotomic polynomial. A ϕ_{e}-subgroup of G is an F -stable torus whose polynomial order is a power of ϕ_{e}. Their centralizers are called the e-split Levi subgroups of G. For any F -stable torus T let T_{ϕ_{e}} be the maximal subtorus of T with polynomial order a power of ϕ_{e}.

Let now χ ∈ Irr(G^{F}). We define several conditions on χ with regard to e. Recall that χ is said to be e-cuspidal if and only if ^{∗}R_{L⊆P}^{G}χ = 0 for every proper e-split Levi subgroup L ≠ G and every parabolic P of G admitting L as Levi subgroup.

Let (U) denote the following condition (uniform criterion, see [BMM, 2.1]) : (U) for every F -stable maximal torus T ⊆ G such that T_{ϕ_{e}} ⊈ Z(G), one has ^{∗}R_{T}^{G}χ = 0.

Let (J) be the conjunction of the following conditions (Jordan criterion). Let χ ∈ E(G^{F} , s). (J_{1}) Z^{∘}(C_{G^{∗}}^{∘}(s))_{ϕ_{e}} = Z^{∘}(G^{∗})_{ϕ_{e}} and (J_{2}) χ corresponds to a C_{G^{∗}}(s)^{F} –orbit of e-cuspidal irreducible unipotent characters of C_{G^{∗}}^{∘}(s)^{F} by Lusztig’s parametrization [L.3, 5.1].

The following gives relations between those conditions.

Proposition 1.10. Let us fix χ ∈ Irr(G^{F}) and e ≥ 1.

(i) (U) and (J) are equivalent. (ii) If χ is e-cuspidal, then (U) is satisfied. (iii) If χ = λρ, with λ ∈ Irr(G^{F} /[G, G]^{F}) and ρ ∈ E(G^{F} , 1), then (U) and ecuspidality for χ or ρ are equivalent.

Proof of the Proposition. (ii) If ^{∗}R_{T}^{G}χ ≠ 0 for some F -stable maximal torus T ⊆ G such that T_{ϕ_{e}} ⊈ Z(G), then L = C_{G}(T_{ϕ_{e}}) is a proper e-split Levi subgroup of G. Now, take P any parabolic subgroup of G with Levi subgroup L. One may write ^{∗}R_{T}^{G}χ = ^{∗}R_{T}^{L}^{∗}R_{L⊆P}^{G}χ by transitivity of twisted induction (see [DM.1, 11.5]) and since the R_{T}^{G} map does not depend on the Borel subgroup chosen to define it. Then ∗R_{L⊆P}^{G}χ ≠ 0. Hence e-cuspidality of χ implies (U).

(iii) The equivalence between (U) and e-cuspidality is known for unipotent characters (see [BMM, 3.13] where condition (U) is taken as the definition of e-cuspidality). When χ = λρ, one has ^{∗}R_{L}^{G}χ = (Res_{L^{F}}^{G^{F}} λ)^{∗}R_{L}^{G}ρ by [DM.1, 12.6.(ii)]. Hence χ is ecuspidal if, and only if, ρ is e-cuspidal. Therefore (U) holds for χ if, and only if, it holds for ρ. So we get (iii).

(i) Let us consider now a regular embedding G ↪ G′ and a dual morphism G^{′∗} → G^{∗}. Assume χ ∈ E(G^{F} , s) for a semi-simple rational element s ∈ G^{∗}. One has Ind_{G^{F}}^{G^{′F}} χ = Σ_{μ∈E}μχ′, where E is a multiplicative group of linear characters of G^{′F} /G^{F} and χ′ ∈ E(G^{′F} , s′) for some rational semi-simple element s′ of G^{′∗} with image s in G^{∗}.

For brevity, we write (U), (J), (J_{1}), (J_{2}) the conditions for χ and (U′), (J′), (J′_{1}), (J′_{2}) the corresponding conditions for χ′. In what follows we show (i) by checking that (J), (J′), (U′) and finally (U) are equivalent.

Let T^{′∗} be any maximal F -stable torus of C_{G^{′∗}}^{∘}(s′), denote by T^{∗} its image in C_{G^{∗}}^{∘}(s), let T′ and T = T′ ∩ G be some dual tori in G′ and G.

(U′) implies (U). With standard notations one has R_{T^{∗}}^{G}s = Res_{G^{F}}^{G^{′F}} R_{T^{′∗}}^{G′}s′ so that ⟨R_{T^{∗}}^{G}s, χ⟩_{G^{F}} = Σ_{μ∈E}⟨R_{T^{′∗}}^{G′}s′, μχ′⟩_{G^{′F}}. As we have seen above, if (U’) holds for χ′, then it holds for any μχ′ (μ ∈ E). Furthermore T_{ϕ_{e}} ⊈ Z(G) is equivalent to T′_{ϕ_{e}} ⊈ Z(G′). Thus our claim.

(J) is equivalent to (J′). Let ρ ∈ E(C_{G^{∗}}^{∘}(s)^{F} , 1) be the unipotent character corresponding to χ′ by Jordan decomposition (we identify E(C_{G^{∗}}^{∘}(s)^{F} , 1) with E(C_{G^{′∗}}^{∘}(s′)^{F} , 1)). The fundamental property of Jordan decomposition of characters is that

(5)  ⟨R_{T^{∗}}^{G}s, χ⟩_{G^{F}} = Σ_{a}⟨R_{T^{∗}}^{C_{G^{∗}}^{∘} (s)}1, a.ρ⟩_{C_{G^{∗}}^{∘} (s)^{F}} , (a ∈ C_{G^{∗}}(s)^{F} /C_{G^{∗}}(s)_{ρ}^{F}).

Here a.ρ runs over the orbit of ρ under the natural action of C_{G^{∗}}(s)^{F} on E(C_{G^{∗}}^{∘}(s)^{F} , 1) [L.3, 5.1], precisely the orbit considered in (J_{2}) for χ. Since e-cuspidal characters in E(C_{G^{∗}}^{∘}(s)^{F} , 1) are preserved under the action of C_{G^{∗}}(s)^{F}, the conditions (J) for χ and χ′ are clearly equivalent.

(U) implies (J_{1}). Since the regular character is a uniform function (see [DM.1, 12.14]) and by the definition of rational series E(G^{F} , s), there is a maximal F -stable torus T^{∗} such that ⟨R_{T^{∗}}^{G}s, χ⟩_{G^{F}} ≠ 0. Now (U) implies T^{∗}_{ϕ_{e}} ⊆ Z(G^{∗}), while on the other hand Z(C_{G^{∗}}^{∘}(s)) ⊆ T^{∗}. Therefore Z(C_{G^{∗}}^{∘}(s))_{ϕ_{e}} ⊆ Z^{∘}(G^{∗}), that is (J_{1}).

(U′) is equivalent to (J′). The above implication may of course be applied to χ′, so we may assume (J′_{1}). Now, when (J′_{1}) is satisfied, T′_{ϕ_{e}} is central in G′ if, and only if, T_{ϕ_{e}}^{′∗} is central in C_{G^{′∗}}^{∘}(s′). Then, by (iii) and the equality ⟨R_{T^{′∗}}^{G′}s′, χ′⟩_{G^{′F}} =

⟨R_{T^{′∗}}^{C_{G^{′∗}}^{∘} (s′)}1, ρ⟩_{C_{G^{′∗}}^{∘} (s′)^{F}}, a special case of (5), (J′_{2}) is equivalent to (U′).

(U) implies (U′). Let C_{G^{∗}}^{∘}(s) = Π_{ω}C_{ω} be a decomposition in rationally irreducible components. For θ ∈ E(C_{G^{∗}}^{∘}(s)^{F} , 1) let θ = Π_{ω}θ_{ω} be the corresponding decomposition of θ, so that ⟨R_{T^{∗}}^{C_{G^{∗}}^{∘} (s)}1, θ⟩_{C_{G^{∗}}^{∘} (s)^{F}} = Π_{ω}⟨R_{T^{∗}∩C_{ω}}^{C_{ω}}1, θ_{ω}⟩_{C_{ω}^{F}}. We shall use this to find a maximal F -stable torus T^{∗} such that, when a runs over C_{G^{∗}}(s)^{F}, Σ_{a}⟨R_{T^{∗}}^{C_{G^{∗}}^{∘} (s)}1, a.ρ⟩_{C_{G^{∗}}^{∘} (s)^{F}} ≠ 0.

Let Ω be any orbit under the action of C_{G^{∗}}(s)^{F} on the set of components in Π_{ω}C_{ω}. The action of C_{G^{∗}}(s)^{F} induces isomorphisms defined on 𝔽_{q} between the components C_{ω} (ω ∈ Ω). Let C_{Ω} be one of the C_{ω} and consider ρ_{ω} as a character of C_{Ω}. It is known that automorphisms of a rationally irreducible finite reductive group fix unipotent characters, except possibly for the non twisted type D in even rank, where two irreducible unipotent characters τ and τ′ parametrized by a degenerate symbol, call them “twin characters”, are exchanged by an automorphism induced by the full orthogonal group (see [L.3, 5.1] and the remark following it). It follows that if a ∈ C_{G^{∗}}(s)^{F}, then the orbit Ω contributes to ⟨R_{T^{∗}}^{C_{G^{∗}}^{∘} (s)}1, a.ρ⟩_{C_{G^{∗}}^{∘} (s)^{F}} by a factor Π_{ω∈Ω}⟨R_{T_{ω}^{∗}}^{C_{Ω}}1, \overline{ρ}_{a.ω}⟩_{C_{Ω}^{F}} where T_{ω}^{∗} is the image of T^{∗} ∩C_{ω} under the isomorphism C_{ω} → C_{Ω} and \overline{ρ}_{a.ω} = ρ_{ω} or \overline{ρ}_{a.ω} is the twin of ρ_{ω}. We apply the following lemma to C_{Ω} and the ρ_{ω} ’s for ω ∈ Ω.

Lemma. Let (H, F) be a rationally irreducible connected reductive group defined over 𝔽_{q}. Let e ∈ ℕ. Let ρ_{1}, ρ_{2}, … ρ_{m} ∈ E(H^{F} , 1). Then there exist maximal F -stable tori T_{1}, T_{2}, … T_{m} of H such that

(a) π := Π_{j}⟨R_{T_{j}}^{H}1, ρ_{j}⟩_{H^{F}} is non-zero,

(b) if ρ_{1} is not e-cuspidal, then (T_{1})_{ϕ_{e}} is not central in H,

(c) if τ and τ′ are “twin characters” parametrized by a degenerate symbol, then ⟨R_{T_{j}}^{H}1, τ⟩_{H^{F}} = ⟨R_{T_{j}}^{H}1, τ′⟩_{H^{F}} for all j ∈ {1, …, m},

(d) for any permutation σ of {1, 2, …, m}, Π_{j}⟨R_{T_{j}}^{H}1, ρ_{σ(j)}⟩_{H^{F}} ∈ {0, π}.

Now we have a maximal F -stable torus S_{Ω} = Π_{ω∈Ω}S_{ω} of Π_{ω∈Ω}C_{ω} with properties (a), (b), (c) and (d) of the Lemma with respect to the characters ρ_{ω}. Doing this for each orbit Ω under the action of C_{G^{∗}}(s)^{F}, one gets an F -stable maximal torus S in C_{G^{∗}}^{∘}(s) such that ⟨R_{S}^{C_{G^{∗}}^{∘} (s)}1, ρ⟩_{C_{G^{∗}}^{∘} (s)^{F}} ≠ 0 by (a). By (c) and (d), one

has ⟨R_{S}^{C_{G^{∗}}^{∘} (s)}1, a.ρ⟩_{C_{G^{∗}}^{∘} (s)^{F}} ∈ {0, ⟨R_{S}^{C_{G^{∗}}^{∘} (s)}1, ρ⟩_{C_{G^{∗}}^{∘} (s)^{F}} } for any a ∈ C_{G^{∗}}(s)^{F}. Then

formula (5) gives ⟨R_{S}^{G}s, χ⟩_{G^{F}} = r⟨R_{S}^{C_{G^{∗}}^{∘} (s)}1, ρ⟩_{C_{G^{∗}}^{∘} (s)^{F}} ≠ 0 for some integer r ≥ 1. By (U), one may assume (J_{1}). Assume (U′) does not hold. Then there exists a maximal F -stable torus T^{∗} of C_{G^{∗}}^{∘}(s) such that T_{ϕ_{e}}^{∗} is not central in C_{G^{∗}}^{∘}(s) and

⟨R_{T^{∗}}^{C_{G^{∗}}^{∘} (s)}1, ρ⟩_{C_{G^{∗}}^{∘} (s)^{F}} ≠ 0. Then by (iii) ρ_{ω} is not e-cuspidal for at least one ω. By (b) we may assume that S_{ϕ_{e}} is not central in C_{G^{∗}}^{∘}(s). This contradicts (U).

This completes the proof of Proposition 1.10.

Proof of the Lemma. Since all the characters we consider are unipotent, we may assume H = H_{ad}.

The case m = 1. Our claim for m = 1 is that for a given ρ ∈ E(H^{F} , 1) one may find a maximal F -stable torus T of H such that (a) ^{∗}R_{T}^{H}ρ ≠ 0, (b) T_{ϕ_{e}} ≠ 1 if ρ is not e-cuspidal and (c) ^{∗}R_{T}^{H}τ = ^{∗}R_{T}^{H}τ′ if τ, τ′ are twin characters in type D_{2n}.

Using the fact that the regular character is a uniform function, one gets (a) for at least one T. When ρ is not e-cuspidal, (U) is not satisfied thanks to (iii). So there is a torus satisfying (a) and (b). We may now assume that H is of type D_{2n} with n ≥ 2. Then ρ is associated with a symbol Λ. In case e is odd (resp. even) the generalized Harish-Chandra theory [FS.1] associates with ρ an e-split Levi subgroup L_{c} = H_{c} × (Q_{1} × ... × Q_{c}) ⊆ H where c ≥ 0 is an integer, H_{c} is of type D_{2n−ce} and each Q_{i} is an F -stable torus with polynomial order (x^{e} −1) (resp. (x^{\frac{e}{2}} +1)) and cyclic group Q_{i}^{F} (see [FS.1 ; BMM, § 3]). Then ^{∗}R_{L_{c}}^{H}ρ is a non-zero multiple of ρ_{c} ⊗ 1, an exterior tensor product where ρ_{c} is a unipotent character of H_{c}^{F} associated with the e-core (resp. \frac{e}{2}-co-core) of Λ and 1 is the trivial character of Q. Take Q′_{c} a maximal torus of H_{c} such that ^{∗}R_{Q′_{c}}^{H_{c}}ρ_{c} ≠ 0 and let T := Q′_{c} × Q_{1} × … Q_{c}. Then one has (a) and (b). As for (c), the Asai-Fong-Srinivasan formulas [FS.1, (3.1), (3.2)] imply that ∗R_{L_{1}}^{H}τ = ^{∗}R_{L_{1}}^{H}τ′ for L_{1} = H_{1} × Q_{1} as above. If c ≥ 1, then L_{1} ⊇ T so the transitivity of twisted induction implies (c). If c = 0, this means that ρ is e-cuspidal and (b) is empty. But there is necessarily some e′ ≥ 1 such that ρ is not e′-cuspidal (for instance no unipotent character is at the same time 1- and 2-cuspidal, since no diagram of rank 2n ≠ 0 has neither any 1-hook nor any 1-co-hook), then take T associated with ρ and e′ by the above process.

Arbitrary m. Let T_{1} be associated with ρ_{1} as above. Define inductively a partition (M_{α})_{α≥1} of the set of indices M = {1, 2, …, m} and a finite sequence of maximal F -stable tori S_{α} in the following way. First S_{1} = T_{1} and M_{1} := {j ∈ M | ∗R_{S_{1}}^{H}ρ_{j} ≠ 0}. Then, if ∪_{1≤β<α}M_{β} ≠ M, pick any j_{α} ∈ (M \ ∪_{1≤β<α}M_{β}). Let S_{α} be associated with ρ_{j_{α}} as in the above case m = 1, that is such that ^{∗}R_{S_{α}}^{H}ρ_{j_{α}} ≠ 0 and (c) is satisfied. Let M_{α} = {j ∈ M | ^{∗}R_{S_{α}}^{H}ρ_{j} ≠ 0} \ ∪_{1≤β<α}M_{β}. When j ∈ M_{α}, put T_{j} = S_{α}. Then clearly ^{∗}R_{T_{k}}^{H}ρ_{j} = 0 whenever k ∈ M_{β}, j ∈ M_{α} and β < α. Then Π_{j}⟨R_{T_{j}}^{H}1, ρ_{σ(j)}⟩_{H^{F}} ≠ 0 only if the permutation σ fixes the partition {M_{α}}_{α}. In that case Π_{j}⟨R_{T_{j}}^{H}1, ρ_{σ(j)}⟩_{H^{F}} = Π_{j}⟨R_{T_{σ^{−1}(j)}}^{H}1, ρ_{j}⟩_{H^{F}} = Π_{α}Π_{j∈M_{α}}⟨R_{S_{α}}^{H}1, ρ_{j}⟩_{H^{F}} =

Π_{j}⟨R_{T_{j}}^{H}1, ρ_{j}⟩_{H^{F}} , thus (d).

In the remainder of the paper we use the following notation.

Notation 1.11. Suppose L, M are F -stable Levi subgroups of G and ζ ∈ Irr(L^{F}), ξ ∈ Irr(M^{F}). Write (M, ξ) ≤ (L, ζ) if and only if M ⊆ L and there is a parabolic subgroup P of L admitting M as Levi subgroup such that ⟨ζ, R_{M⊆P}^{L}ξ⟩_{L^{F}} ≠ 0. One calls ≤_{e} the same relation restricted to pairs L, M of e–split Levi subgroups of G. Let us define <<_{e} as the transitive closure of ≤_{e}.

Note that χ ∈ Irr(G^{F}) is e-cuspidal if and only if (G, χ) is <<_{e}-minimal. A reasonable conjecture is that <<_{e} coincides with ≤_{e}, that e-cuspidality is equivalent with (U) and (J), and that the <<_{e}-minimal pairs below a given pair (G, χ) are all G^{F} -conjugate. Part of this will follow from our study of blocks (Theorem 4.2, Proposition 5.2).

2. “Connected” subpairs and twisted block induction.

Let G be a finite group, let ℓ be a prime number and O a local complete finite extension of ℤ_{ℓ}. Assume that its fraction field K contains |G|-th roots of unity. Let k = O/J(O). We consider ℓ-blocks as primitive idempotents in the center Z(k[G]) of the group algebra k[G]. An important fact is that all ℓ-blocks are in the subspace of k[G] generated by G_{ℓ′} [NT, 3.6.22].

The decomposition of the unit 1 ∈ Z(k[G]) as the sum of the blocks, along with the lifting of idempotents between Z(k[G]) and Z(O[G]), induce an orthogonal decomposition of the space of central functions CF(G, K) with respect to the scalar product ⟨ , ⟩_{G}. For b a block of G and f ∈ CF(G, K), one denotes by b.f the corresponding projection of f, and writes b = b_{G}(f) when b.f = f. This defines a partition Irr(G) = ∪_{b}Irr(b), where Irr(b) = {χ ∈ Irr(G) | b.χ = χ}.

We also recall the notion of generalized decomposition map. If x is an ℓ-element of G, one defines d^{x,G}: CF(G, K) → CF(C_{G}(x), K) by d^{x,G}f(y) = f(xy) if y ∈ C_{G}(x)_{ℓ′} and d^{x,G}f(y) = 0 otherwise.

When U is an ℓ-subgroup of G, one has the Brauer map Br_{U} : (k[G])^{U} → k[C_{G}(U)] which sends ∑_{g∈G} λ_{g}g to ∑_{g∈C_{G}(U)} λ_{g}g (see [AB, 2.4]).

The definitions and main properties of subpairs (Q, b), inclusion of subpairs, selfcentralizing subpairs, defect groups, covering blocks are taken from [AB], [B], [Br.1], [NT].

2.1. Subpairs in finite reductive groups.

In the following, we define a slight generalization of the subpairs in the case where G = G^{F} for G and F as in the preceding section. For simplicity, we just consider (abelian) ℓ-subgroups of groups T^{F} where T is an F -stable torus of G.

Definition-Proposition 2.1. Let (G, F) be a connected reductive group defined over 𝔽_{q}. Let ℓ be a prime not dividing q. Let V ⊆ U be abelian ℓ-subgroups of G^{F} such that U ⊆ C_{G}^{∘}(U). Let b_{U} be an ℓ-block of C_{G}^{∘}(U)^{F}. Then there is a unique ℓ-block b_{V} of C_{G}^{∘}(V)^{F} satisfying the two equivalent conditions

(a) Br_{U}(b_{V}).b_{U} ≠ 0, and

(b) Br_{U}(b_{V}).b_{U} = b_{U}.

Then one writes (V, b_{V})^{∘} ◁ (U, b_{U})^{∘}.

Proof of the Proposition. The group C_{G}^{∘}(U) is F -stable and reductive, so the hypothesis U ⊆ C_{G}^{∘}(U) is equivalent to the fact that there is an F -stable torus T such that U ⊆ T^{F}. Then the same is satisfied by V and U ⊆ C_{G}^{∘}(V). Let b_{V} be some ℓ–block of C_{G}^{∘}(V)^{F}. Now b_{V} is fixed by U, so Br_{U}(b_{V}) makes sense and is in k[C_{G^{F}} (U)]. Moreover, b_{V} has non zero coefficients only on ℓ′-elements, so does Br_{U}(b_{V}) and therefore Br_{U}(b_{V}) ∈ k[C_{G}^{∘}(U)^{F}] since C_{G}^{∘}(U)^{F} ◁ C_{G^{F}} (U) with index a power of ℓ (see for instance [CE.1, 2.1.(i)] or Proposition 3.1 below). In fact, Br_{U}(b_{V}) ∈ Z(k[C_{G}^{∘}(U)^{F}]) since b_{V} is C_{G}^{∘}(V)^{F} -fixed and C_{G}^{∘}(U)^{F} ⊆ C_{G}^{∘}(V)^{F}.

Now Br_{U}(b_{V}) (resp. b_{V}) is an idempotent (resp. a primitive idempotent) in Z(k[C_{G}^{∘}(U)^{F}]) and the equivalence between (a) and (b) is clear. The existence of b_{V} satisfying (a) follows from Br_{U}(1).b_{U} = b_{U} and the decomposition of 1 in sum of blocks. The uniqueness of b_{V} is due to the fact that if b′_{V} is a block of C_{G}^{∘}(V)^{F} distinct from b_{V}, then b_{V} and b′_{V}, and therefore Br_{U}(b_{V}) and Br_{U}(b′_{V}), are orthogonal.

In the following proposition, we give some basic properties of the “connected” subpairs introduced in Proposition 2.1, along with the relation with ordinary subpair inclusion.

Proposition 2.2. Let (G, F), V ⊆ U, b_{U} be as in the above proposition. Let b_{V} be an ℓ-block b_{V} of C_{G}^{∘}(V)^{F}. One has the following properties.

(i) (V, b_{V})^{∘} ◁ (U, b_{U})^{∘} in G if and only if ({1}, b_{V})^{∘} ◁ (U, b_{U})^{∘} in C_{G}^{∘}(V). (ii) The relation ◁ is transitive. (iii) There is a unique ℓ-block b̂_{V} of C_{G^{F}} (V) that covers b_{V}. (iv) If S is an ℓ-subgroup of G^{F} normalizing (V, b_{V})^{∘} and such that Br_{S}(b_{V}) ≠ 0, then S normalizes b̂_{V} and Br_{S}(b̂_{V}) ≠ 0.

(v) One has (V, b̂_{V}) ⊆ (U, b̂_{U}) (inclusion of ordinary subpairs) if and only if there is c ∈ C_{G^{F}} (V) such that (V,^{c}b_{V})^{∘} ◁ (U, b_{U})^{∘}.

Proof of the Proposition. (i) and (ii) are clear from the definition.

(iii) As said in the proof of Proposition 2.1, C_{G}^{∘}(U)^{F} ◁ C_{G^{F}} (U) with index a power of ℓ, so b̂_{U} = ∑_{t} ^{t}b_{U}, an orthogonal sum of idempotents where t ranges over a representative system of C_{G^{F}} (U) mod. the inertial group of b_{U} (see [NT, 5.5.6]).

(iv) It is clear that S normalizes C_{G}^{∘}(V)^{F}, C_{G^{F}} (V) and therefore b̂_{V}. Now, if one defines b′_{V} = b̂_{V} −b_{V}, it is an idempotent of kC_{G}^{∘}(V)^{F}, it is fixed by S and orthogonal to b_{V}, so Br_{S}(b̂_{V}) = Br_{S}(b_{V}) + Br_{S}(b′_{V}), a sum of orthogonal idempotents which is non-zero if the first is non-zero.

(v) The above with S = U and b_{V} replaced with^{c}b_{V} tells us that (V, b̂_{V}) ⊆ (U, b̂_{U}) when (V,^{c}b_{V})^{∘} ◁ (U, b_{U})^{∘}. Conversely, let us write b̂_{V} = ∑_{s} ^{s}b_{V}, an orthogonal sum of idempotents where s ranges over a subset of C_{G^{F}} (V). Writing now the ordinary inclusion (V, b̂_{V}) ⊆ (U, b̂_{U}) as Br_{U}(∑_{s} ^{s}b_{V}). ∑_{t} ^{t}b_{U} ≠ 0 implies at once that there are s, t such that Br_{U}(^{s}b_{V}).^{t}b_{U} ≠ 0, that is (V,^{t^{−1}s}b_{V})^{∘} ◁ (U, b_{U})^{∘}.

Remark. The theory of subpairs remains almost the same when one replaces the centralizer functor Q ↦ C_{G}(Q) with other functors Q ↦ F(Q) satisfying certain properties (see [Le] where F(Q) is the smallest normal subgroup of C_{G}(Q) with index a power of ℓ). Here F(Q) = C_{G}^{∘}(Q)^{F}.

Proposition 2.3. ÊLet G ↪ G′ = Z(G′)G be a regular embedding, let F be a Frobenius endomorphism of G′ defining it over 𝔽_{q}, let L′ be an F -stable Levi subgroup associated with a parabolic subgroup P′ of G′. Let A be a subgroup of L′ ∩ G^{F} included in a maximal torus of L′. Denote GA = GA, LA = L′ ∩ GA, PA = P′ ∩ GA, L = L′ ∩ G, P = P′ ∩ G. Let ℓ be a prime not dividing q. If x ∈ A_{ℓ}, then

d^{x,L^{F} A} ^{∗}R_{LA⊆PA}^{GA} = ^{∗}R_{C_{L}^{∘}(x)A⊆(P∩C_{G}^{∘}(x))A}^{C_{G}^{∘}(x)A}d^{x,G^{F} A} .

Proof of the Proposition. The proof of [Br.1, 4.3] applies using the character formula for the ^{∗}R_{LA⊆PA}^{GA} map [DM.2, 2.6.(ii)].

2.2. A twisted induction for blocks.

As an application of Theorem 1.7 and the notion introduced above, we show that for a certain class of F -stable Levi subgroups L ⊆ G and for good ℓ, the R_{L}^{G} map restricted to E(L^{F} , ℓ′) induces a map from ℓ-blocks of L^{F} to ℓ-blocks of G^{F}.

Let E be a set of integers ≥ 1. A ϕ_{E}-subgroup is an F -stable torus of G whose polynomial order is a product of cyclotomic polynomials in {ϕ_{e} ; e ∈ E}. An E-split Levi subgroup is the centralizer C_{G}(T) of a ϕ_{E}-subgroup T.

Proposition 2.4. Let E_{q,ℓ} = {d; ℓ|ϕ_{d}(q)}.Assume ℓ is good for G. A Levi subgroup of G is E_{q,ℓ}-split if and only if its image in G/Z(G) = G_{ad} is so. If an F -stable Levi subgroup L of G satisfies L = C_{G}^{∘}(Z(L)_{ℓ}^{F}), then it is E_{q,ℓ}-split. If moreover Z(G) is connected, then the converse is true.

Proof of the Proposition. The first statement is checked for any set E in the same way as for a singleton (see [CE.1, § 1.2]). For the second and third statements, the proof of [CE.1, 2.2.(ii)] applies.

Theorem 2.5. ÊAssume ℓ is good for G. Let L be an E_{q,ℓ}-split Levi subgroup of G and let b_{L} be an ℓ-block of L^{F}. Then there is an ℓ-block b_{G} of G^{F} such that for any ζ ∈ Irr(b_{L}) ∩ E(L^{F} , ℓ′) and any parabolic subgroup P of G containing L as a Levi subgroup, one has R_{L⊆P}^{G}ζ ∈ ℤIrr(b_{G}). If moreover L = C_{G}^{∘}(Z(L)_{ℓ}^{F}), then ({1}, b_{G})^{∘} ◁ (Z(L)_{ℓ}^{F} , b_{L})^{∘} in the sense of Proposition 2.1.

Proof of the Theorem. The proof is by induction on dim G - dim L.

If L = G, everything is clear. Assume L ≠ G.

Let G ⊆ G′ be a regular embedding. Let L′ = C_{G′}^{∘}(Z^{∘}(L)) = Z(G′)L be the Levi subgroup of G′ containing L. Then L′ is also E_{q,ℓ}-split by Proposition 2.4. The center of G′ being connected, one has L′ = C_{G′}^{∘}(Z(L′)_{ℓ}^{F}) by Proposition 2.4. So Z(L′)_{ℓ}^{F} ⊈ Z(G′). Let then x ∈ Z(L′)_{ℓ}^{F} \ Z(G) with x ∈ Z(L)_{ℓ}^{F} if moreover L = C_{G}^{∘}(Z(L)_{ℓ}^{F}). Let C′ = C_{G′}^{∘}(x). Then x ∈ C′ and C′ is an E_{q,ℓ}-split Levi subgroup by Proposition 2.4. So is C := C′ ∩ G by the same proposition and one has L ⊆ C ⊆ G with C ≠ G. By induction hypothesis, there is an ℓ-block b_{C} of C^{F} such that for any ζ ∈ Irr(b_{L}) ∩ E(L^{F} , ℓ′) and any parabolic subgroup Q ⊆ C having L as Levi subgroup, all the irreducible components of R_{L⊆Q}^{C}ζ are in b_{C} and moreover ({1}, b_{C})^{∘} ◁ (Z(L)_{ℓ}^{F} , b_{L})^{∘} in C if L = C_{C}^{∘}(Z(L)_{ℓ}^{F}) (which is the case when L = C_{G}^{∘}(Z(L)_{ℓ}^{F}) ).

Let A = Z(L′)_{ℓ}^{F}. Let us consider the group algebra k[G^{F} A] in characteristic ℓ and every ℓ-block as an idempotent in that group algebra. Note first that G^{F} A/G^{F} is of order a power of ℓ, so Proposition 1.3.(ii) implies that restriction induces a bijection E(G^{F} A, ℓ′) → E(G^{F} , ℓ′). When χ ∈ E(G^{F} , ℓ′) let us denote by χ_{A} ∈ E(G^{F} A, ℓ′) its extension to G^{F} A. Then b_{G^{F}} (χ) is fixed by G^{F} A and since G^{F} A/G^{F} is an ℓgroup, b_{G^{F}} (χ) is an ℓ-block of k[G^{F} A] equal to b_{G^{F} A}(χ_{A}) [NT, 5.5.6]. Samely, each ζ ∈ E(L^{F} , ℓ′) extends as ζ_{A} ∈ E(L^{F} A, ℓ′), therefore b_{L} is fixed by L^{F} A and is an ℓ-block of L^{F} A.

Let us consider the Brauer map Br_{<x>} in the group algebra k[G^{F} A]. Let χ be an irreducible component of R_{L⊆P}^{G}ζ for some ζ ∈ Irr(b_{L})∩E(L^{F} , ℓ′) and P a parabolic subgroup with Levi subgroup L. Let us show

(B)  Br_{<x>}(b_{G^{F}} (χ)).b_{C} = b_{C}.

This may also be written Br_{<x>}(b_{G^{F} A}(χ_{A})).b_{C} = b_{C}. One has C_{G^{F} A}(x) = C_{G^{F}} (x)A and C^{F} = C_{G}^{∘}(x)^{F} is a normal subgroup in it with index a power of ℓ since C_{G}(x)/C_{G}^{∘}(x) and G^{F} A/G^{F} are ℓ-groups. Then the ℓ-block of C_{G^{F} A}(x) covering b_{C} is b′ = ∑_{g} gb_{C}g^{−1}, where g ranges over C_{G^{F} A}(x) mod. the stabilizer of b_{C}. The summands are orthogonal idempotents and one has b′b_{C} = b_{C}b′ = b_{C}. So to check (B) it suffices to check Br_{<x>}(b_{G^{F} A}(χ_{A})).b′ = b′ or equivalently the ordinary subpair inclusion ({1}, b_{G^{F} A}(χ_{A})) ⊆ (<x>, b′) in G^{F} A. By Brauer second main theorem [NT, 5.4.2], it suffices to check that b′.d^{x}(χ_{A}) ≠ 0. This in turn is implied by

(D)  b_{C}.d^{x}(χ_{A}) ≠ 0,

which makes sense since C_{G^{F} A}(x)_{ℓ′} = C_{ℓ′}^{F}. Let us check (D).

Let f = b_{L}.d^{1}^{∗}R_{L⊆P}^{G}χ be the projection on b_{L} of the restriction of ^{∗}R_{L⊆P}^{G}χ to ℓ′- elements of L^{F}. By the hypothesis on χ, one may write ^{∗}R_{L⊆P}^{G}χ = m_{1}ζ_{1} + … + m_{ν}ζ_{ν} with distinct ζ_{i} ∈ E(L^{F} , ℓ′) (see Proposition 1.2.(iv)), the m_{i}’s being non zero integers and ζ = ζ_{1}. Now b_{L^{F}} (ζ_{i}).d^{1}(ζ_{i}) = d^{1}(ζ_{i}) thanks to Brauer second main theorem, so

f = b_{L}.d^{1}(^{∗}R_{L⊆P}^{G}χ) = m_{1}d^{1}(ζ_{1}) + ∑_{i≥2 ; b_{L^{F}} (ζ_{i})=b_{L}} m_{i}d^{1}(ζ_{i}).

By Theorem 1.7, those d^{1}(ζ_{i})’s for 1 ≤ i ≤ ν are linearly independent, so b_{L}d^{1}(^{∗}R_{L⊆P}^{G}χ) = f ≠ 0. One sees clearly that f(x^{−1}) is the complex conjugate of f(x) for all x ∈ L^{F}, so ⟨f, f⟩_{L^{F}} ≠ 0. Then there is some ζ_{i} in the expression of f above, let us call it ζ′, such that ζ′ ∈ E(L^{F} , ℓ′) ∩ Irr(b_{L}) and ⟨f, d^{1}ζ′⟩_{L^{F}} ≠ 0.

Let us show that ⟨d^{x}χ_{A}, R_{L⊆P∩C}^{C}ζ′⟩_{C^{F}} = ⟨f, d^{1}ζ′⟩_{L^{F}}. First C_{GA}(x)_{ℓ′}^{F} = C_{ℓ′}^{F} since C_{G}(x)/C_{G}^{∘}(x) and G^{F} A/G^{F} are ℓ-groups. Then d^{x}χ_{A} is a central function on C^{F} and ⟨d^{x}χ_{A}, R_{L⊆P∩C}^{C}ζ′⟩_{C^{F}} makes sense. One has R_{L⊆P∩C}^{C}ζ′ = Res_{C^{F}}^{C^{F} A}R_{LA⊆(PA∩C)}^{CA}ζ′_{A} where ζ′_{A} extends ζ′ and ζ′_{A} ∈ E(L^{F} A, ℓ′). Therefore

⟨d^{x}χ_{A}, R_{L⊆P∩C}^{C}ζ′⟩_{C^{F}} = |A : A ∩ C| ⟨d^{x}χ_{A}, R_{LA⊆(PA∩C)}^{CA}ζ′_{A}⟩_{C^{F} A}.

Now, ⟨d^{x}χ_{A}, R_{LA⊆(PA∩C)}^{CA}ζ′_{A}⟩_{C^{F} A} = ⟨d^{x}^{∗}R_{LA⊆PA}^{GA}χ_{A}, ζ′_{A}⟩_{L^{F} A} by Proposition 2.3. Moreover, as x ∈ A ⊆ Z(L^{F} A), d^{x}^{∗}R_{LA⊆PA}^{GA}χ_{A} = d^{1}^{∗}R_{LA⊆PA}^{GA}χ_{A} by Proposition 1.2.(v). So

⟨d^{x}χ_{A}, R_{LA⊆(PA∩C)}^{CA}ζ′_{A}⟩_{C^{F} A} = ⟨d^{1}^{∗}R_{LA⊆PA}^{GA}χ_{A}, ζ′_{A}⟩_{L^{F} A} = ⟨χ_{A}, d^{1}R_{LA⊆PA}^{GA}ζ′_{A}⟩_{G^{F} A}.

Since A is an ℓ-group, one may again restrict the central functions of the last scalar product to G^{F}, thus giving

⟨χ_{A}, d^{1}R_{LA⊆PA}^{GA}ζ′_{A}⟩_{G^{F} A} = |A : A ∩ G|^{−1}⟨χ, d^{1}R_{L⊆P}^{G}ζ′⟩_{G^{F}} .

By Proposition 2.3 one has ⟨χ, d^{1}R_{L⊆P}^{G}ζ′⟩_{G^{F}} = ⟨d^{1}^{∗}R_{L⊆P}^{G}χ, d^{1}ζ′⟩_{L^{F}} = ⟨f, d^{1}ζ′⟩_{L^{F}}. On the other hand A ∩ G = A ∩ C = A ∩ L = Z(L)_{ℓ}^{F}, which yields our claim.

We have thus obtained that ⟨d^{x}χ_{A}, R_{L⊆P∩C}^{C}ζ′⟩_{C^{F}} ≠ 0. This implies that b_{C}.d^{x}χ_{A} ≠ 0 since all the irreducible components of R_{L⊆P∩C}^{C}ζ′ are in b_{C}. This is (D).

We have now checked (B) as soon as χ is an irreducible component of some R_{L⊆P}^{G}ζ for ζ ∈ Irr(b_{L}) ∩ E(L^{F} , ℓ′) and P a parabolic for L in G. If χ′ satisfies the same hypothesis as χ for other ζ and P, then Br_{<x>}(b_{G^{F}} (χ′)).b_{C} = b_{C} so b_{G^{F}} (χ′) = b_{G^{F}} (χ) since otherwise b_{G^{F}} (χ′)b_{G^{F}} (χ) = 0 and therefore 0 = Br_{<x>}(b_{G^{F}} (χ′)b_{G^{F}} (χ)).b_{C} = Br_{<x>}(b_{G^{F}} (χ′))Br_{<x>}(b_{G^{F}} (χ)).b_{C} = Br_{<x>}(b_{G^{F}} (χ′)).b_{C} = b_{C}, a contradiction. This yields the first statement of the Theorem.

If L = C_{G}^{∘}(Z(L)_{ℓ}^{F}, then x has been chosen in Z(L)_{ℓ}^{F}, so (B) implies ({1}, b_{G})^{∘} ◁ (<x>, b_{C})^{∘} in the sense of Proposition 2.1. But ({1}, b_{C})^{∘} ◁ (Z(L)_{ℓ}^{F} , b_{L})^{∘} in C by induction hypothesis. So ({1}, b_{G})^{∘} ◁ (Z(L)_{ℓ}^{F} , b_{L})^{∘} by Proposition 2.2.(ii)

This completes the proof of the Theorem.

Notation 2.6. Under the hypothesis of the above theorem on (G, F), ℓ, L and if b is an ℓ-block of L^{F}, one may define R_{L}^{G}b as the ℓ-block of G^{F} containing all the irreducible components of the R_{L⊆P}^{G}ζ’s for ζ ∈ Irr(b) ∩ E(L^{F} , ℓ′) and P a parabolic subgroup of G containing L as a Levi subgroup.

Remark 2.7. (a) As said in Theorem 2.5, R_{L}^{G}b does not depend on the choice of the parabolic subgroup having L as Levi subgroup. The transitivity R_{L}^{G}b = R_{M}^{G}(R_{L}^{M}b) holds as soon as L ⊆ M ⊆ G is an inclusion of E_{q,ℓ}-split Levi subgroups. As an easy consequence of this and Theorem 2.5, denoting K = C_{G}^{∘}(Z(L)_{ℓ}^{F}) ⊇ L, one has ({1}, R_{L}^{G}b)^{∘} ◁ (Z(L)_{ℓ}^{F} , R_{L}^{K}b)^{∘} for any E_{q,ℓ}-split Levi subgroup L and any ℓ-block b of L^{F}. Another easy consequence of Theorem 2.5 (by iteration) is that if e is the order of q modulo ℓ, χ ∈ E(G^{F} , ℓ′) and (L, ζ) <<_{e} (G, χ) as defined in Notation 1.11, then b_{G^{F}} (χ) = R_{L}^{G}(b_{L^{F}} (ζ)).

(b) Let G ↪ G′ be a regular embedding, let L′ be an E_{q,ℓ}-split Levi subgroup of G′, let b′ be an ℓ-block of L′^{F} and b an ℓ-block of L^{F} covered by b′. Then R_{L′}^{G′}b′ covers R_{L}^{G}b.

To check this, note first that Irr(b′) ∩ E(L′^{F} , ℓ′) is non-empty (a theorem of Hiss, which is also a consequence of Proposition 1.6). Let ζ′ ∈ Irr(b′) ∩ E(L′^{F} , ℓ′). Then Res_{L^{F}}^{L′^{F}} ζ′ has an irreducible component ζ ∈ Irr(b) (see [NT, 5.5.7.(i)]). Moreover Res_{G^{F}}^{G′^{F}} R_{L′}^{G′}ζ′ = R_{L}^{G}Res_{L^{F}}^{L′^{F}} ζ′ is non-zero and is a sum of G′^{F} -conjugate of R_{L}^{G}ζ, so it has a non-zero projection on a G′^{F} -conjugate of R_{L}^{G}b, hence on R_{L}^{G}b itself. But R_{L′}^{G′}ζ′ has all its irreducible components in R_{L′}^{G′}b′ by Theorem 2.5, so R_{L′}^{G′}b′ covers R_{L}^{G}b by [NT, 5.5.7.(ii)].

The following theorem shows that if b is an ℓ-block of a finite reductive group G^{F} and Irr(b) ∩ E(G^{F} , ℓ′) is known, the knowledge of Lusztig’s twisted induction is enough to determine the whole set Irr(b).

Theorem 2.8. Let (G, F) be a connected reductive group defined over 𝔽_{q}. Let ℓ be a prime good for G. Let χ ∈ E(G^{F} , st), with st ∈ G^{∗}^{F} and t = (st)_{ℓ}. Let G(t) ⊆ G be a Levi subgroup in duality with C_{G^{∗}}^{∘}(t), let P be a parabolic subgroup of G having G(t) as Levi subgroup. Then s ∈ C_{G^{∗}}^{∘}(t)^{F} and there is χ_{t} ∈ E(G(t)^{F} , s) such that ⟨χ, R_{G(t)⊆P}^{G}(t̂χ_{t})⟩_{G^{F}} ≠ 0, where t̂is the linear character of G(t)^{F} associated with t by duality. For any such (G(t), P, χ_{t}) associated with χ, all the irreducible components of R_{G(t)⊆P}^{G}χ_{t} are in Irr(b_{G^{F}} (χ)) ∩ E(G^{F} , ℓ′).

Proof. Since ℓ is good, C_{G^{∗}}^{∘}(t) is a Levi subgroup of G^{∗}. Applying Proposition 1.4.(i) with A = 1 to the series associated with st and L^{∗} = C_{G^{∗}}^{∘}(t), one gets the existence of some ξ ∈ E(G(t)^{F} , st) such that ⟨R_{G(t)⊆P}^{G}ξ, χ⟩_{G^{F}} ≠ 0. On the other hand, E(G(t)^{F} , st) = t̂E(G(t)^{F} , s), since t is central in C_{G^{∗}}^{∘}(t) (see for instance [DM.1, 13.30.(ii)] and proof), thus allowing to write ξ = t̂χ_{t} with χ_{t} ∈ E(G(t)^{F} , s).

Let b = b_{G^{F}} (χ). Assume now that χ ∈ E(G^{F} , st) is an irreducible component of R_{G(t)⊆P}^{G}(t̂χ_{t}) for some χ_{t} ∈ E(G(t), s). One must check that R_{G(t)⊆P}^{G}(χ_{t}) ∈ ℤIrr(b). First C_{G^{∗}}^{∘}(t) is E_{q,ℓ}-split by its definition and Proposition 2.4. Then G(t) is also E_{q,ℓ}-split by [CE.2, 1.3.(i)]. So Theorem 2.5 implies the existence of the block R_{G(t)}^{G}b_{G(t)^{F}} (χ_{t}) of G^{F} and it suffices to check that it equals b. Now the proof goes like the one of [CE.2, Lemma 4.5], replacing reference to [G, Theorem A] by the above Theorem 1.7.

In the following (tiny) case where twisted induction induces a perfect isometry on an ℓ-block, one may show that it does not depend on the parabolic subgroup used to define it (see also [DM.1, 13.28]).

Proposition 2.9. Assume ℓ is good for G. Let L be an E_{q,ℓ}-split Levi subgroup of G, let P, P′ be two parabolic subgroups of G having L as Levi subgroup. Let s ∈ (G^{∗})_{ℓ′}^{F} be a semi-simple element such that C_{G^{∗}}^{∘}(s)C_{G^{∗}}(s)^{F} ⊆ L^{∗}, let b_{L} be an ℓ-block of L^{F} such that Irr(b_{L}) ∩ E(L^{F} , s) has a single element ρ. Then R_{L⊆P}^{G}(ρ) = R_{L⊆P′}^{G}(ρ) and it is ε_{G}ε_{L} times the only element in Irr(R_{L}^{G}b_{L}) ∩ E(G^{F} , s).

Proof of the Proposition. Let R ∈ {ε_{G}ε_{L}R_{L⊆P}^{G}, ε_{G}ε_{L}R_{L⊆P′}^{G}}. The characters of the virtual bi-modules which define twisted inductions R_{L}^{G} are perfect by [Br.2, 2.1] (the proof there does not use the assumption that Z(G) is connected).

As in [BMi.1], let us define E_{ℓ}(G^{F} , s) as the union of rational series E(G^{F} , st) such that (st)_{ℓ} = t. Arguing as in [Br.2, 2.3] and using Proposition 1.5.(ii), it is easy to prove that R is an isometry from E_{ℓ}(L^{F} , s) to E_{ℓ}(G^{F} , s).

By [BMi.1, 2.2], E_{ℓ}(L^{F} , s) and E_{ℓ}(G^{F} , s) are unions of ℓ-blocks. Since R induces a perfect isometry between them, the partitions of ℓ-blocks are preserved [Br.2, 1.5]. By Theorem 2.5, R sends Irr(b_{L}) to Irr(R_{L}^{G}b_{L}). Since R(E(L^{F} , s)) = E(G^{F} , s), this implies that E(G^{F} , s) ∩ Irr(R_{L}^{G}b_{L}) has a single element equal to R(ρ). It is then clear that since R_{L}^{G}b_{L} does not depend on the parabolic chosen for L, R(ρ) is the same for R = ε_{G}ε_{L}R_{L⊆P}^{G} or ε_{G}ε_{L}R_{L⊆P′}^{G}.

Remark 2.10. The above Proposition applies in particular when L = C_{G}^{∘}(Z(L)_{ℓ}^{F}) and b_{L} is an ℓ-block of L^{F} of defect Z(L)_{ℓ}^{F}. In that case, E(L^{F} , ℓ′) ∩ Irr(b_{L}) contains only the canonical character of b_{L} thanks to Proposition 1.2.(v) and the fact that the canonical character is the only element of Irr(b_{L}) with Z(L)_{ℓ}^{F} in its kernel. In this special case of a block b_{L} with central defect, one may check easily that R_{L⊆P}^{G} and R_{L⊆P′}^{G} coincide on the whole Irr(b_{L}).

3. Centralizers of abelian ℓ-subgroups.

Let us recall some properties of centralizers of ℓ-subgroups from [CE.1] valid essentially when ℓ is good and does not divide the number of connected components of Z(G). Then we give some adapted statements in the other cases where ℓ is good.

Proposition 3.1. [CE.1, 2.1] Let Y be an abelian ℓ-subgroup of G^{F}.

(i) C_{G}(Y)/C_{G}^{∘}(Y) is an ℓ-group. (ii) If ℓ is good for G, then C_{G}^{∘}(Y) is a Levi subgroup of G. (iii) If ℓ is good for G and does not divide |(Z(G)/Z^{∘}(G))^{F} |, then C_{G}(Y)^{F} = C_{G}^{∘}(Y)^{F}.

Proposition 3.2. [CE.1, 2.1.(v), 2.2 and proofs] Assume ℓ is good for G and does not divide |(Z(G)/Z^{∘}(G))^{F} |. Let S be a ϕ_{E_{q,ℓ}}-subgroup of G. Let L = C_{G}(S) be the associated E_{q,ℓ}-split Levi subgroup of G. Then

(i) S_{ℓ}^{F} ⊆ Z(G) if, and only if, S ⊆ Z(G). (ii) Z^{∘}(L)_{ℓ}^{F} = Z(L)_{ℓ}^{F} (iii) L = C_{G}^{∘}(S_{ℓ}^{F}) = C_{G}^{∘}(Z(L)_{ℓ}^{F}) and L^{F} = C_{G}^{∘}(Z(L)_{ℓ}^{F})^{F} = C_{G^{F}} (Z(L)_{ℓ}^{F}).

Proposition 3.3. [CE.1, 1.2] Let H be an F -stable reductive subgroup of G of maximal rank. If ℓ is good for G, then ℓ is good for H. If |(Z(G)/Z^{∘}(G))^{F} | is prime to ℓ, then so is |(Z(H)/Z^{∘}(H))^{F} |.

By Proposition 3.2 the equality L = C_{G}^{∘}(Z(L)_{ℓ}^{F}) may fail only in rational components of G of type (A_{n}, εq^{d}) where ε = ±1 and q^{d} = ε modulo ℓ. The central product of the rational components of [G, G] of type as above, along with Z^{∘}(G), is denoted by G_{a}. The remaining components of [G, G] generate the subgroup we denote by G_{b}. Then, for any F -stable reductive subgroup J ⊆ G of maximal rank, one has J = (J ∩ G_{a})(J ∩ G_{b}) (central product) and J ∩ G_{a} ⊆ J_{a} [CE.2, Notation 1.5.(i)]. In this situation one has, by Lang’s theorem, Z(J)_{ℓ}^{F} = Z(J ∩ G_{a})_{ℓ}^{F} Z(J ∩ G_{b})_{ℓ}^{F}.

Assuming that L is a Levi subgroup, on dual side one has G^{∗} = (G^{∗})_{a}(G^{∗})_{b} and L^{∗} = (L^{∗} ∩ (G^{∗})_{a})(L^{∗} ∩ (G^{∗})_{b}). Note that LG_{a} is in duality with L^{∗}(G^{∗})_{a}. But L^{∗} ∩(G^{∗})_{a} (resp. L^{∗} ∩(G^{∗})_{b}) is not necessarily in duality with L∩G_{a} (resp. L∩G_{b}). As (G_{a} ∩ G_{b})_{ℓ}^{F} = {1}, L^{∗} ∩ (G^{∗})_{a} differs from (L ∩ G_{a})^{∗} by a finite isogeny without any rational element of order ℓ in its kernel. Such an isogeny leaves polynomial orders of connected subgroups unchanged, and ℓ-orders of groups of rational points are also unchanged.

More conveniently we write G^{∗}_{a} instead of (G^{∗})_{a}.

Many difficulties arise from the case G = G_{a}. In that case we call diagonal Levi subgroups of G_{a} the F -stable Levi subgroups of G_{a} that contain a diagonal maximal torus (see [CE.1, 2.3; DM.1, § 15.4]). By [CE.1, 2.4], the e-split Levi subgroups of G_{a} are diagonal (we leave the converse as an exercise, see also the description given in [BMM, p. 46]). We also use diagonal tori to describe the types of maximal tori and Levi subgroups. So, as in [CE.2, 1.4], we call Coxeter torus a maximal F -stable torus whose distance to a diagonal torus T is given by a Coxeter element of W (G_{a}, T)^{F}. Coxeter tori are G_{a}^{F} -conjugate. The following proposition gives information on cases where statement (iii) of Proposition 3.2 fails.

Proposition 3.4. Assume ℓ is good for G, different from 3 if G^{F} has components of type ^{3}D_{4}. Let L be an e-split Levi subgroup of G and let K = C_{G}^{∘}(Z(L)_{ℓ}^{F}). Then (i) N_{K}(L) = L. (ii) Assume G = G_{a}, let T be a Coxeter torus of L, then {g ∈ G^{F} ; T^{g} ⊆ K} = N_{G^{F}} (T).N_{G^{F}} (K). (iii) Let I be an F –stable closed reductive subgroup of maximal rank of G, assume I ∩ G_{a} is a torus and L = C_{G}(Z^{∘}(I)_{ϕ_{e}}). Let M = C_{G}^{∘}(Z(I)_{ℓ}^{F}). Let T be any F -stable maximal torus of I. One has M ⊆ K and |W (M, T)^{F} |_{ℓ′} = |W (K, T)^{F} |_{ℓ′}.

Proof of the Proposition. By the properties of the decomposition G = G_{a}.G_{b} recalled before, and the equality G^{F} = (G_{a}^{F} .G_{b}^{F}).T^{F}, it is clear that it suffices to prove the proposition for G_{a} and G_{b}.

Assume G = G_{b}. Then (ii) is empty and L = K by Proposition 3.2. Then (i) follows at once. Moreover Z(I)_{ℓ}^{F} = Z^{∘}(I)_{ℓ}^{F} by Propositions 3.3 and 3.2.(1). By Proposition 3.3 Z^{∘}(I)_{ϕ_{e}} is central in C_{G}^{∘}(Z^{∘}(I)_{ℓ}^{F}). Then by [CE.1, 1.6], Z^{∘}(I)_{ϕ_{d}} is central in C_{G}^{∘}(Z^{∘}(I)_{ϕ_{e}}) for any d ∈ E_{q,ℓ} . As Z^{∘}(I)_{ℓ}^{F} ⊆ Π_{d∈E_{q,ℓ}}Z^{∘}(I)_{ϕ_{d}}, C_{G}^{∘}(Z^{∘}(I)_{ℓ}^{F}) = C_{G}^{∘}(Z^{∘}(I)_{ϕ_{e}}). So L = M = K and everything in (iii) is clear.

We now assume G = G_{a}. Then in (iii) I = T. The following describes the inclusion L ⊆ K, the proof reduces easily to a computation in general linear groups.

Lemma 3.5. Notations and hypothesis of Proposition 3.4. Assume in addition G = G_{a}. Let [G, G] = <G_{ω}>_{ω} be the decomposition into rationally irreducible components and L_{ω} = L ∩ G_{ω} , K_{ω} = K ∩ G_{ω}. For each ω, if G_{ω} is of type (A_{n−1}, r) where ±r is a power of q, then either L_{ω} = K_{ω} or L_{ω} belongs to the conjugacy class of diagonal Levi subgroups defined by a partition n = ∑_{i} n_{i}, where n_{1} is prime to ℓ, n_{1} + n_{2}, n_{3}, …, n_{k} are divisible by |r − 1|_{ℓ} and K_{ω} belongs to the conjugacy class of diagonal Levi subgroups of G_{ω} defined by (n_{1} + n_{2}, n_{3}, …, n_{k}).

Let S be a diagonal maximal torus of L.

To get (i) of Proposition 3.4, one notes that, in the above lemma, n_{1} ≠ n_{2} since ℓ is odd and divides n_{1} + n_{2} but neither n_{1} nor n_{2}. This implies N_{W (K,S)}(W (L, S)) = W (L, S) by the corresponding property in symmetric groups, hence (i).

Let us check (ii). Let g ∈ G^{F} such that ^{g}T ⊆ K. With notations of Lemma 3.5, set [G, G] = <G_{ω}>_{ω}. One has G^{F} = T^{F} Π_{ω}G_{ω}^{F} and this allows us to assume G = G_{a} = G_{ω}. Lemma 3.5 gives parametrizations of K as diagonal Levi subgroup and of T as maximal F -stable torus with respect to S. As g ∈ G^{F}, T and ^{g}T have equal polynomial orders, hence equal parametrizations in G (this is true in type A since any two elements of a symmetric group are conjugate if and only if they have the same characteristic polynomial in the standard representation). If L = K, then ^{g}T is a Coxeter torus of K and Coxeter tori of K are all K^{F} -conjugate, hence g ∈ N_{G^{F}} (T)K^{F}.

Assume L ≠ K, so we are in the situation described in Lemma 3.5. Let K = Π_{2≤i≤k}K_{i} be the decomposition of K into rationally irreducible components where K_{i}^{F} is of type A_{m_{i}−1} with m_{2} = n_{1} +n_{2}, m_{i} = n_{i} if i ≥ 3. The maximal tori of K are of K^{F} -type some subpartition of (n_{1} +n_{2}, n_{3}, …, n_{k}). The maximal torus ^{g}T has the same G^{F} -type (n_{1}, n_{2}, …, n_{k}) as T. So we must look at how those can be permuted to give a subpartition of (n_{1} + n_{2}, n_{3}, …, n_{k}). As (n_{1} + n_{2}), n_{3}, …, n_{k} are multiples of |r − 1|_{ℓ}, while n_{1} and n_{2} are not, the permutation on (n_{1}, n_{2}, …, n_{k}) preserves the partition (n_{1} + n_{2}, n_{3}, …, n_{k}). The corresponding permutation of components K_{j} is clearly induced by N_{G^{F}} (K). Now if ^{g}(T ∩ K_{i}) ⊆ K_{i} for all i, then T ∩ K_{i} and gT ∩ K_{i} = ^{g}(T ∩ K_{i}) are conjugate under K_{i}^{F}. This proves (ii).

Concerning (iii), note first that Z(L) ⊆ Z(I) since L ⊇ I. Therefore M ⊆ K and everything in (iii) takes place inside K. So we may replace (iii) by the same statement with the additional hypothesis that G = K. Recall that I is a maximal torus, so I = T. Let π: G → G/Z(G) = G_{ad} be the canonical surjection. As π(T_{ℓ}^{F}) ⊆ π(T)_{ℓ}^{F}, C_{G_{ad}}^{∘}(π(T)_{ℓ}^{F}) ⊆ π(M), whereas W (G, T) = W (G_{ad}, π(T)) and W (M, T) = W (π(M), π(T)). Hence it suffices to check (iii) when G is adjoint. If the morphism π above had connected kernel then the inclusion π(T_{ℓ}^{F}) ⊆ π(T)_{ℓ}^{F} would be an equality, and therefore (iii) for G would be equivalent to (iii) for G_{ad}. We may therefore assume that G is a direct product of general linear groups and that (G, F) is of irreducible rational type, thus G^{F} ≅ GL(n, r) (with ±r a power of q, and the convention that GL(n, −q) = U(n, q^{2})). The condition T_{ϕ_{e}} = Z^{∘}(L)_{ϕ_{e}} implies that T is a Coxeter torus in L (see [CE.2, 1.5.(ii)] and proof). If L = G, then T^{F} ≅ GL(1, r^{n}) and M^{F} ≅ GL(n_{ℓ′}, r^{n_{ℓ}}). Hence one has W (G, T)^{F} ≅ ℤ/nℤ and W (M, T)^{F} ≅ ℤ/n_{ℓ′}ℤ. If L ≠ G, then n = n_{1} + n_{2}, n_{1} and n_{2} are prime to ℓ and ℓ divides n (Lemma 3.5). In that case L = M and W (L, T)^{F} = W (G, T)^{F}, because n_{1} ≠ n_{2}.

Remark. Proposition 3.2 and the proof of (iii) show that one has K ∩ G_{b} ⊆ L and, if K_{0} is a rationally irreducible component of K, then K_{0} ⊆ L or K_{0} ⊆ M.

4. The ℓ-blocks.

Here is our main theorem on ℓ-blocks of finite groups G^{F} in connection with twisted induction. Our study of ℓ-blocks also allows to complete the description of e-cuspidal characters started in Proposition 1.10.

Theorem 4.1. Assume ℓ ≥ 7 if some component of G is of type E_{8} , ℓ ≥ 5 otherwise.

(a) The map (L, ζ) ↦ R_{L}^{G}(b_{L^{F}} (ζ)) defined by Theorem 2.5, Notation 2.6, yields a bijection between the ℓ-blocks of G^{F} and the G^{F} -conjugacy classes of pairs (L, ζ) where L is an e-split Levi subgroup of G and ζ ∈ E(L^{F} , ℓ′) is e-cuspidal.

(b) If b is an ℓ-block of G^{F} associated with a pair (L, ζ) as above, one has Irr(b) ∩ E(G^{F} , ℓ′) = {χ ∈ Irr(G^{F}) ; (L, ζ) <<_{e} (G, χ)} (see Notation 1.11).

Theorem 4.2. Assume ℓ ≥ 7 if some component of G is of type E_{8} , ℓ ≥ 5 otherwise. Then an element of E(G^{F} , ℓ′) is e-cuspidal if and only if it satisfies condition (J) of § 1.3.

The remainder of the section is devoted to a common proof of Theorem 4.1 and Theorem 4.2. The proof starts with a pair (L, ζ) where ζ satisfies condition (J) of § 1.3, and it is only near the end of the proof that we check that ζ is indeed e-cuspidal.

Let us fix the following notations. Definition-Notation 4.3. As in [CE.1, 1.1] write ℓ ∈ Γ(G, F) if, with hypotheses of Theorem 4.1, |(Z(G)/Z^{∘}(G))^{F} |.|(Z(G^{∗})/Z^{∘}(G^{∗}))^{F} | is of order prime to ℓ.

Let L be an e–split Levi subgroup of G, let ζ ∈ E(L^{F} , ℓ′) satisfying condition (J) of § 1.3 in L for the integer e. Denote s ∈ (L^{∗}_{ss}^{F})_{ℓ′} such that ζ ∈ E(L^{F} , s).

We abbreviate C_{G^{∗}}^{∘}(s) = G^{∗}(s), C_{L^{∗}}^{∘}(s) = L^{∗}(s), … and so on.

By condition (J_{1}) and the fact that L^{∗} is e-split one has L^{∗} = C_{G^{∗}}^{∘}(Z^{∘}(L^{∗}(s))_{ϕ_{e}}) [CE.1, 1.4].

Let T^{∗} be a maximally split torus in L^{∗}(s) and let T ⊆ L be in duality with T^{∗}. We abbreviate W (G, T) = W (G), W (G^{∗}, T^{∗}) = W (G^{∗}), W (G | H, T) = W (G | H) in Notation 1.8, and so on for all Weyl groups with respect to T or T^{∗}. Let K = C_{G}^{∘}(Z(L)_{ℓ}^{F}), T^{∗} ⊆ K^{∗} ⊆ G^{∗} in duality with T ⊆ K ⊆ G. Let M be the Levi subgroup of G containing T and defined by M ∩ G_{a} = C_{G_{a}}^{∘}((T ∩ G_{a})_{ℓ}^{F}) and M^{∗} ∩ G^{∗}_{b} = C_{G^{∗}_{b}}^{∘}(Z(L^{∗}(s) ∩ G^{∗}_{b})_{ℓ}^{F}) (duality with respect to T, T^{∗}).

Lemma 4.4. One has

(i) L^{∗}(s) ∩ G^{∗}_{a} = T^{∗} ∩ G^{∗}_{a}, a diagonal torus of G^{∗}(s) ∩ G^{∗}_{a}. (ii) L ∩ G_{a} = C_{G_{a}}(T_{ϕ_{e}}) and T ∩ G_{a} is a Coxeter torus of L ∩ G_{a}, (iii) M ∩ G_{b} ⊆ L ∩ G_{b} = K ∩ G_{b} and L ∩ G_{b} is the smallest e–split Levi subgroup of G_{b} containing M ∩ G_{b}. (iv) M ⊆ K and N_{G^{F}} (M) ⊆ N_{G^{F}} (K).

(v) K^{∗}(s) = M^{∗}(s) = L^{∗}(s).

Proof. By the hypothesis L^{∗}(s) has an e-cuspidal unipotent character. On the other hand L^{∗}(s) ∩ G^{∗}_{a} is a Levi subgroup of G^{∗}(s) ∩ G^{∗}_{a} = (G^{∗}(s) ∩ G^{∗}_{a})_{a}. By [CE.1, 3.3.(i)] this implies that L^{∗}(s) ∩ G^{∗}_{a} is a diagonal torus of G^{∗}(s) ∩ G^{∗}_{a}. It is therefore also equal to T^{∗} ∩ G^{∗}_{a}, whence (i). By condition (J_{1}), (T^{∗} ∩ G^{∗}_{a})_{ϕ_{e}} = Z^{∘}(L^{∗} ∩ G^{∗}_{a})_{ϕ_{e}}, so T^{∗} ∩ G^{∗}_{a} is a Coxeter torus of L^{∗} ∩ G^{∗}_{a}. Dually and by equality of polynomial orders, T ∩ G_{a} is a Coxeter torus of L ∩ G_{a} and, since L is e–split, one gets (ii).

By properties of the decomposition G_{a}G_{b}, one has L = (L ∩ G_{a})(L ∩ G_{b}) and Z(L) = Z(L ∩ G_{a})Z(L ∩ G_{b}). The quotient Z(L)^{F} /Z(L ∩ G_{a})^{F} Z(L ∩ G_{b})^{F} is an F -fixed section of Z(G_{b}), so it is ℓ′ [CE.1, 2.3]. Therefore Z(L)_{ℓ}^{F} = Z(L ∩ G_{a})_{ℓ}^{F} Z(L ∩ G_{b})_{ℓ}^{F} and K = C_{G_{a}}^{∘}(Z(L ∩ G_{a})_{ℓ}^{F})C_{G_{b}}^{∘}(Z(L ∩ G_{b})_{ℓ}^{F}). One has ℓ ∈ Γ(G_{b}, F) hence ℓ ∈ Γ(G^{∗}_{b}, F) (see [CE.1, Table 1]). By Proposition 3.2, L ∩ G_{b} = C_{G_{b}}^{∘}(Z(L ∩ G_{b})_{ℓ}^{F}) = K ∩ G_{b}, thus one has (iii) on the relationship between L and K, hence the dual equality L^{∗} ∩ G^{∗}_{b} = K^{∗} ∩ G^{∗}_{b}. One has L^{∗}(s) = (T^{∗} ∩ G^{∗}_{a})(L^{∗}(s) ∩ G^{∗}_{b}). The equality L^{∗} = C_{G^{∗}}^{∘}(Z^{∘}(L^{∗}(s))_{ϕ_{e}}) implies N_{G^{∗}}(L^{∗}(s))^{F} ⊆ N_{G^{∗}}(L^{∗}). So Proposition 3.4 (i) implies that N_{K^{∗}∩G^{∗}(s)}(L^{∗}(s))^{F} = L^{∗}(s)^{F}. In other words the external Weyl group N_{K^{∗}(s)}(L^{∗}(s))/L^{∗}(s) has no other rational point than 1. This implies the same property on the G^{∗}_{a}-side : the Weyl group of the torus T^{∗} ∩ G^{∗}_{a}, a maximal one in K^{∗}(s) ∩ G^{∗}_{a}, has no rational point other than 1. Hence T^{∗} ∩ G^{∗}_{a} = K^{∗}(s) ∩ G^{∗}_{a}. Finally L^{∗}(s) = K^{∗}(s), that is (v) on K and L.

By definition of M, one has M ∩ G_{a} ⊆ K ∩ G_{a}. As ℓ ∈ Γ(G^{∗}_{b}, F), ℓ ∈ Γ(L^{∗} ∩ G^{∗}_{b}, F) (Notation 4.3), L^{∗} ∩ G^{∗}_{b} = C_{G^{∗}_{b}}^{∘}(Z(L^{∗} ∩ G^{∗}_{b})_{ℓ}^{F}) by Proposiition 3.2, so that M^{∗} ∩ G^{∗}_{b} ⊆ L^{∗} ∩ G^{∗}_{b} = K^{∗} ∩ G^{∗}_{b}. Hence M ⊆ K and L^{∗}(s) ⊆ M^{∗} ⊆ K^{∗}. As L^{∗}(s) = K^{∗}(s), L^{∗}(s) = M^{∗}(s). So we have (v).

One has Z^{∘}(L^{∗} ∩ M^{∗})_{ϕ_{e}} ⊆ Z^{∘}((L^{∗} ∩ M^{∗})(s))_{ϕ_{e}} = Z^{∘}(L^{∗}(s))_{ϕ_{e}}. But the latter is Z^{∘}(L^{∗})_{ϕ_{e}} by 4.3. Therefore Z^{∘}(L^{∗} ∩ M^{∗})_{ϕ_{e}} = Z^{∘}(L^{∗})_{ϕ_{e}} and dually Z^{∘}(L ∩ M)_{ϕ_{e}} = Z^{∘}(L)_{ϕ_{e}}, see [Ca, 4.4]. The intersections with G_{b} yield Z^{∘}(M∩G_{b})_{ϕ_{e}} = Z^{∘}(L∩G_{b})_{ϕ_{e}}, so L ∩ G_{b} = C_{G_{b}}(Z^{∘}(L ∩ G_{b})_{ϕ_{e}}) = C_{G_{b}}(Z^{∘}(M ∩ G_{b})_{ϕ_{e}}) is the smallest e–split Levi subgroup of G_{b} containing M ∩ G_{b}. So we have (iii).

Concerning (iv), since G^{F} = G_{a}^{F} G_{b}^{F} T^{F}, it suffices to check the inclusions on the G_{a} and G_{b} sides. On the latter, (iii) gives the result. On the side G_{a}, one has N_{G_{a}}^{F} (M) ⊆ {g ∈ G_{a}^{F} ; ^{g}(T ∩ G_{a}) ⊆ K ∩ G_{a}}. It suffices to check that the latter is in N_{G_{a}}^{F} (K ∩ G_{a}). One has L ∩ G_{a} = C_{G_{a}}((T ∩ G_{a})_{ϕ_{e}}) by (ii) and K ∩ G_{a} = C_{G_{a}}^{∘}(Z(L ∩ G_{a})_{ℓ}^{F}). So Proposition 3.4.(ii) gives {g ∈ G_{a}^{F} ; ^{g}(T ∩ G_{a}) ⊆ K ∩ G_{a}} ⊆ N_{G_{a}}^{F} (T∩G_{a})N_{G_{a}}^{F} (K∩G_{a}). But N_{G_{a}}^{F} (T∩G_{a}) ⊆ N_{G_{a}}^{F} (L∩G_{a}) ⊆ N_{G_{a}}^{F} (K∩G_{a}) by the above definitions of L ∩ G_{a} and K ∩ G_{a} from T ∩ G_{a}. This yields (iv).

Lemma 4.5. C_{K^{∗}}^{∘}(s)C_{K^{∗}}(s)^{F} ⊆ L^{∗} ∩ M^{∗}.

Proof. Let us check first C_{K^{∗}}^{∘}(s)C_{K^{∗}}(s)^{F} ⊆ L^{∗}. By Lemma 4.4.(ii), one has N_{K^{F}} (T ∩ G_{a}) ⊆ N_{K^{F}} (L). Then Proposition 3.4.(i) implies N_{K^{F}} (T ∩ G_{a}) = N_{L^{F}} (T ∩ G_{a}), or in terms of Weyl groups W (T.(K ∩ G_{a}))^{F} = W (T.(L ∩ G_{a}))^{F}. The same holds for duals, so N_{L^{∗}}(T^{∗} ∩ G^{∗}_{a})^{F} = N_{K^{∗}}(T^{∗} ∩ G^{∗}_{a})^{F}. By the properties of K^{∗}(s) given in Lemma 4.4 (i), (v), where (T^{∗} ∩ G^{∗}_{a}) = C_{G^{∗}_{a}}^{∘}(s), and Lemma 4.4 (iii) then C_{W (K^{∗})}(s)^{F} = C_{W (L^{∗})}(s)^{F}. This implies C_{K^{∗}}^{∘}(s)C_{K^{∗}}(s)^{F} ⊆ L^{∗}.

Let us turn to the inclusion C_{K^{∗}}^{∘}(s)C_{K^{∗}}(s)^{F} ⊆ M^{∗}. We have T ⊆ M ⊆ K by Lemma 4.4.(iv). As T^{∗} is maximally split in C_{K^{∗}}^{∘}(s) = K^{∗}(s) = L^{∗}(s) the Frattini argument shows that C_{K^{∗}}(s)^{F} = K^{∗}(s)^{F} (N_{G^{∗}}(T^{∗}) ∩ C_{K^{∗}}(s)^{F}). So the sought inclusion is equivalent to an inclusion in a Weyl group over T^{∗}, W (K^{∗}(s))C_{W (K^{∗})}(s)^{F} ⊆ W (M^{∗}). First K^{∗}(s) = M^{∗}(s) (Lemma 4.4.(v)) implies W (K^{∗}(s)) ⊆ W (M^{∗}). The inclusion of C_{W (K^{∗})}(s)^{F} in W (M^{∗}) may be verified independently in G^{∗}_{a} and G^{∗}_{b}. As s is of order prime to ℓ, the quotient K^{∗}(s)C_{K^{∗}}(s)^{F} /K^{∗}(s) is of order prime to ℓ by Proposition 3.1.(i). Now Proposition 3.4 (iii) yields the claimed inclusion. On the G^{∗}_{a}-side, apply it to G_{a} with I = T ∩ G_{a}, knowing that L∩G_{a} = C_{G_{a}}^{∘}(Z^{∘}(L∩G_{a})_{ϕ_{e}}). On the G^{∗}_{b}-side apply it in G^{∗}_{b}, with I = L^{∗}(s)∩G^{∗}_{b}, knowing that M^{∗} ∩ G^{∗}_{b} = C_{G^{∗}_{b}}^{∘}(Z(L^{∗}(s) ∩ G^{∗}_{b})_{ℓ}^{F}) and L^{∗} ∩ G^{∗}_{b} = K^{∗} ∩ G^{∗}_{b} = C_{G^{∗}}_{b}(Z^{∘}(L^{∗}(s) ∩ G^{∗}_{b})_{ϕ_{e}}) = C_{G^{∗}}_{b}(Z(L^{∗} ∩ G^{∗}_{b})_{ℓ}^{F}).

Definition 4.6. Choose a parabolic subgroup P (resp. Q) of K containing L (resp. M) as a Levi subgroup (see Lemma 4.4.(iv)). Denote by R_{M}^{K} and R_{L}^{K} the associated twisted induction maps.

Let ζ_{K} = ε_{L}ε_{K}R_{L}^{K}ζ and ζ_{M} be the orthogonal projection of ε_{M}ε_{K}^{∗}R_{M}^{K}ζ_{K} on ℤE(M^{F} , s) ⊆ ℤIrr(M^{F}).

Let Z = Z(M)_{ℓ}^{F}.

Lemma 4.7. ζ_{K} ∈ E(K^{F} , s), ζ_{M} ∈ E(M^{F} , s), and ζ_{K} = ε_{M}ε_{K}R_{M}^{K}ζ_{M}.

Proof. By Proposition 1.5.(ii) and Lemma 4.5, ε_{L}ε_{K}R_{L}^{K}ζ ∈ E(K^{F} , s). For the same reasons between M and K, ε_{M}ε_{K}R_{M}^{K} induces a bijection between E(M^{F} , s) and E(K^{F} , s). Its inverse is therefore ε_{M}ε_{K}P_{s}^{∗}R_{M}^{K} where P_{s} is the orthogonal projection on ℤE(M^{F} , s) in ℤIrr(M^{F}). Then ζ_{M}, as the image of ζ_{K}, is in E(M^{F} , s) and ζ_{K} = ε_{M}ε_{K}R_{M}^{K}ζ_{M}.

Lemma 4.8. Z = (Z ∩ G_{a})(Z ∩ G_{b}) (direct product), Z ∩ G_{a} = (T ∩ G_{a})_{ℓ}^{F} and M = C_{G}^{∘}(Z).

Proof. By properties of the G_{a}G_{b} decomposition, one has Z = (Z(M)∩G_{a})_{ℓ}^{F} (Z(M)∩ G_{b})_{ℓ}^{F} and Z = (Z ∩ G_{a})(Z ∩ G_{b}), a direct product.

The inclusion T ∩ G_{a} ⊆ M ∩ G_{a} of Levi subgroups of G_{a} implies the reverse inclusion of centers, whence Z ∩ G_{a} ⊆ (T ∩ G_{a})_{ℓ}^{F}. The equality follows by the definition of M ∩ G_{a}.

To check that M = C_{G}^{∘}(Z), by definition of M it suffices to show that M ∩ G_{b} = C_{G_{b}}^{∘}(Z∩G_{b}). As ℓ ∈ Γ((G_{b})_{ad}, F) (see [CE.1, Table 1]) we may apply Proposition 2.4 and Proposition 3.2, that is M ∩ G_{b} = C_{G_{b}}^{∘}(Z(M ∩ G_{b})_{ℓ}^{F}) if, and only if, M ∩ G_{b} is E_{q,ℓ}-split. By definition of M^{∗} ∩ G^{∗}_{b}, this property is satisfied by (M ∩ G_{b})^{∗} in (G_{b})^{∗}, hence it is satisfied by M ∩ G_{b} [CE.2, 1.3.(i)].

Definition 4.9. Denote b_{G} = R_{L}^{G}(b_{L^{F}} (ζ)). Denote b_{Z} = b_{M^{F}} (ζ_{M}), let (Z, b̂_{Z}) be the subpair of G^{F} associated to b_{Z} by Proposition 2.2.(iii).

Lemma 4.10. One has ({1}, b_{G})^{∘} ◁ (Z, b_{Z})^{∘} in G^{F}, in the sense of Definition 2.1.

Proof. We have b_{K^{F}} (ζ_{K}) = R_{L}^{K}(b_{L^{F}} (ζ)) by definition of ζ_{K}, Theorem 2.5 and Notation 2.6. Now Remark 2.7 yields ({1}, b_{G})^{∘} ◁ (Z(L)_{ℓ}^{F} , b_{K^{F}} (ζ_{K}))^{∘}. Samely, in K^{F}, b_{K^{F}} (ζ_{K}) = R_{M}^{K}b_{Z} by the definition of ζ_{M}, so ({1}, b_{K^{F}} (ζ_{K}))^{∘} ◁ (Z, b_{Z})^{∘} in K^{F} by Theorem 2.5. By Proposition 2.2.(i), the last relation implies (Z(L)_{ℓ}^{F} , b_{K^{F}} (ζ_{K}))^{∘} ◁ (Z, b_{Z})^{∘} in G^{F}. Thus our claim by combining the two relations we have in G^{F} (Proposition 2.2.(ii)).

Lemma 4.11. ζ_{M}(1)_{ℓ} = |M^{F} : Z|_{ℓ}.

Proof. By the definition of ζ_{K} from ζ and the equality K^{∗}(s) = L^{∗}(s) of Lemma 4.4.(v), the degree of ζ_{K} is given by Proposition 1.5.(i), therefore ζ_{K}(1)_{ℓ} = |K^{F} : L^{F} |_{ℓ}ζ(1)_{ℓ}. Samely the equality ζ_{K} = ε_{M}ε_{K}R_{M}^{K}ζ_{M} (Lemma 4.7) yields ζ_{M}(1)_{ℓ} = |M^{F} |_{ℓ}|L^{F} |_{ℓ}^{−1}ζ(1)_{ℓ}. Since ζ satisfies (J_{2}), ζ is associated with an ecuspidal λ ∈ E(L^{∗}(s)^{F} , 1) by Lusztig’s parametrization. Now, by [L.3, 5.1], ζ(1)_{ℓ} differs from |L^{F} |_{ℓ}|L^{∗}(s)|_{ℓ}^{−1}λ(1)_{ℓ} by a divisor of |C_{L^{∗}}(s)^{F} : L^{∗}(s)^{F} |. This last integer is prime to ℓ, while λ(1)_{ℓ} = |L^{∗}(s)^{F} : Z(L^{∗}(s))^{F} |_{ℓ} by [CE.1, 4.3]. Thus ζ_{M}(1)_{ℓ} = |M^{F} |_{ℓ}|Z(L^{∗}(s))^{F} |_{ℓ}^{−1}. To complete our proof, it therefore suffices to check that |Z| = |Z(L^{∗}(s))^{F} |_{ℓ}.

One has |Z| = |(T ∩ G_{a})_{ℓ}^{F} | |Z ∩ G_{b}| by Lemma 4.8. Now, |Z ∩ G_{b}| = |Z(M ∩ G_{b})_{ℓ}^{F} | = |Z^{∘}(M ∩ G_{b})_{ℓ}^{F} | because ℓ ∈ Γ(G_{b}, F) and so ℓ ∈ Γ(M ∩ G_{b}, F) (Notation 4.3). Furthermore |Z^{∘}(M∩G_{b})_{ℓ}^{F} | = |Z^{∘}(M^{∗} ∩G^{∗}_{b})_{ℓ}^{F} | = |Z(L^{∗}(s)∩G^{∗}_{b})_{ℓ}^{F} | because the tori Z^{∘}(M∩G_{b}) and Z^{∘}(M^{∗} ∩G^{∗}_{b}) have equal polynomial orders and by definition of M^{∗}∩G^{∗}_{b}. On the other hand, Z(L^{∗}(s))_{ℓ}^{F} = (T^{∗}∩G^{∗}_{a})_{ℓ}^{F} Z(L^{∗}(s)∩G^{∗}_{b})_{ℓ}^{F} by Lemma 4.4.(i) and |(T^{∗} ∩ G^{∗}_{a})^{F} | = |(T ∩ G_{a})^{F} | by equality of polynomial orders, hence our claim.

Lemma 4.12. C_{G^{F}} (Z, ζ_{M}) = M^{F}.

Proof. As Z = (Z ∩ G_{a})(Z ∩ G_{b}), C_{G}(Z) = C_{G_{a}}(Z ∩ G_{a})C_{G_{b}}(Z ∩ G_{b}) and C_{G^{F}} (Z) = M^{F} C_{G_{a}^{F}} (Z ∩ G_{a})C_{G_{b}^{F}} (Z ∩ G_{b}). By Proposition 3.2 C_{G_{b}^{F}} (Z ∩ G_{b}) = M^{F} ∩ G_{b} and G^{∗}(s) ∩ G^{∗}_{a} is a Levi subgroup of G^{∗}_{a}. If x ∈ C_{G^{F}} (Z) and x fixes ζ_{M}, then x fixes the geometric series Ē(M^{F} , s). By Proposition 1.9.(ii), C_{G^{F}} (Z, ζ_{M})/M^{F} has therefore an isomorphic image, say E, in the subgroup of N_{W (G)}(W (M))/W (M,) = N_{W (G^{∗})}(W (M^{∗}))/W (M^{∗}) corresponding to C_{W (G^{∗})}(s)W (M^{∗}). But C_{W (G)}(s)/W (T^{∗}C_{G^{∗}_{a}}^{∘}(s)) ≅ C_{G^{∗}}_{a}(s)/C_{G^{∗}_{a}}^{∘}(s) is an ℓ′-group while C_{G^{F}} (Z, ζ_{M})/M^{F} is an ℓ-group (Proposition 3.1.(i)), hence E is isomorphic with its projection in W (T^{∗}C_{G^{∗}_{a}}^{∘}(s)). Let G(s) denote the Levi subgroup of G containing T and in duality with C_{T^{∗}G^{∗}_{a}}^{∘}(s) ⊇ T^{∗}. Then W (T^{∗}C_{G^{∗}_{a}}^{∘}(s)) = W (G(s)). On the other hand, L^{∗}(s) ∩ G^{∗}_{a} is a diagonal torus of G^{∗}(s) ∩ G^{∗}_{a} (Lemma 4.4.(i)), so T is a diagonal torus of G(s) and C_{G^{F}} (Z) ∩ G(s) = T^{F} [CE.1, 2.4]. Since E must act trivially on Z, one gets E = {1} and C_{G^{F}} (Z, ζ_{M}) = M^{F}.

Lemma 4.13. ({1}, b_{G}) ⊆ (Z, b̂_{Z}) and the latter is a self-centralizing subpair in G^{F}.

Proof. The ordinary subpair inclusion ({1}, b_{G}) ⊆ (Z, b̂_{Z}) follows by Lemma 4.10 and Proposition 2.2.(v).

One has ζ_{M}(1)_{ℓ} = |M^{F} : Z|_{ℓ} by Lemma 4.11 and Z is central in M^{F} (Lemma 4.8), moreover Z is in the kernel of ζ_{M} (Proposition 1.2.(v)), so b_{Z} is an ℓ-block of central defect group of M^{F} (see [NT, 3.6.29]), with ζ_{M} as canonical character. Now C_{G^{F}} (Z)/M^{F} is an ℓ-group (Proposition 3.1.(i) again) and therefore b̂_{Z} is of defect group Z in C_{G^{F}} (Z) if, and only if, M^{F} is the inertial group of ζ_{M} in C_{G^{F}} (Z) [NT, 5.5.16]. This is Lemma 4.12.

Lemma 4.14. The ℓ-elements of p_{T}^{−1}(W (G^{∗}(s)|L^{∗}(s)))^{F} normalize (Z, b_{Z}) and (Z, b̂_{Z}). One has |N_{G^{F}} (Z, b̂_{Z})/C_{G^{F}} (Z)|_{ℓ} ≤ |(N_{W (G^{∗}(s))}(W (L^{∗}(s)))/W (L^{∗}(s)))^{F} |_{ℓ}.

Proof. By definition in Notation 1.8, the elements of p_{T}^{−1}(W (G^{∗}(s)|L^{∗}(s)))^{F} fix the roots of L^{∗}(s) and therefore, by the definition of M, they normalize M and Z. One has M^{∗}(s) = L^{∗}(s), so Proposition 1.9.(iii) implies that the ℓ-elements of p_{T}^{−1}(W (G^{∗}(s)|L^{∗}(s)))^{F} normalize (Z, ζ_{M}). They also normalize C_{G^{F}} (Z), so they normalize (Z, b̂_{Z}).

Concerning the second statement, one has N_{G^{F}} (Z, b̂_{Z})/C_{G^{F}} (Z) = N_{G^{F}} (Z, b_{Z})/M^{F} by Proposition 2.2.(iii) and Lemma 4.12. Now, ζ_{M} being the canonical character of b_{Z}, N_{G^{F}} (Z, b_{Z}) = N_{G^{F}} (M, ζ_{M}). The latter injects into (N_{W (C_{G^{∗}} (s))}(W (L^{∗}(s)))/W (C_{M^{∗}}(s)))^{F} by Proposition 1.9.(ii). The indexes |C_{G^{∗}}(s)/G^{∗}(s)| and |C_{M^{∗}}(s)/M^{∗}(s)| are prime to ℓ, so one gets the sought inequality.

To simplify notations let us denote in the following Lemma Φ = ϕ_{E_{q,ℓ}}, then the notation S_{Φ} applies to any torus S defined on 𝔽_{q}. One has S_{ℓ}^{F} ⊆ S_{Φ}.

Lemma 4.15. Assume (Z, b̂_{Z}) ⊆ (E, b_{E}) is a subpair inclusion in G^{F}.

(a) If Z(E) ⊆ G_{a}, then E ⊆ G_{a}, Z = (T ∩ G_{a})_{ℓ}^{F}, and G_{b} ⊆ M ∩ L. If moreover the center of G is connected, one has TG_{a} ∩ M = C_{TG_{a}}^{∘}(T_{Φ}).

(b) If 1 ≠ z ∈ Z(E)G_{a} ∩ G_{b}, then C_{G}^{∘}(z) contains E, M, C_{G^{F}} (Z) and G_{a}. There is at least one proper e–split Levi subgroup of G containing C_{G}^{∘}(z), and any such Levi subgroup also contains L.

Proof. The subpair (Z, b̂_{Z}) being self-centralizing (Lemma 4.13), one has Z(E) ⊆ Z(C_{G^{F}} (E))_{ℓ}^{F} ⊆ Z (see [B, 6]).

(a) Assume first that Z(E) ⊆ G_{a}. Then by [CE.1, 2.3 (ii)], Z ⊆ E ⊆ G_{a} and therefore M ⊇ G_{b}. Then M^{∗} ⊇ G^{∗}_{b} and therefore Z^{∘}(L^{∗}(s) ∩ G^{∗}_{b})_{ℓ}^{F} = 1 by the definition of M^{∗} ∩ G^{∗}_{b}. Then Z^{∘}(L^{∗}(s) ∩ G^{∗}_{b})_{ϕ_{e}} = 1 and L^{∗} ⊇ G^{∗}_{b}, L ⊇ G_{b}. The equality Z = (T ∩ G_{a})_{ℓ}^{F} comes from Lemma 4.8 and the inclusion Z ⊆ G_{a}. Then M ⊇ G_{b} since M = C_{G}^{∘}(Z) (Lemma 4.8).

If the center of G is connected, all its Levi subgroups have the same property, so Z = Z^{∘}(M)_{ℓ}^{F} = Z^{∘}(M ∩ G_{a})_{ℓ}^{F} = (Z^{∘}(M ∩ G_{a})_{Φ})_{ℓ}^{F}. Let S = (T ∩ G_{a})_{Φ}, one has Z = (T ∩ G_{a})_{ℓ}^{F} = S_{ℓ}^{F}. But Z^{∘}(M ∩ G_{a})_{Φ} ⊆ S with same rational ℓ-elements, so by the definition of E_{q,ℓ}, Z^{∘}(M ∩ G_{a})_{Φ} and S have equal polynomial orders. Therefore Z^{∘}(M ∩ G_{a})_{Φ} = S and M ∩ G_{a} ⊇ C_{G}^{∘}(S).

(b) Assume now that Z(E) ⊈ G_{a}. The intersection (G_{a}∩G_{b})^{F} being an ℓ′-group, there exist non-trivial elements z in the projection of Z(E) in G_{b}. Note that for any z ∈ (G_{b})_{ℓ}^{F}, C_{G}(z)^{F} = C_{G}^{∘}(z)^{F}. This is because C_{G}(z)^{F} = (G_{a}C_{G_{b}}(z))^{F} contains G_{a}^{F} C_{G_{b}}(z)^{F} as a normal subgroup with index prime to ℓ, but C_{G_{b}}(z)^{F} = C_{G_{b}}^{∘}(z)^{F} since ℓ ∈ Γ(G_{b}, F) (apply Proposition 3.1.(iii)) so C_{G}^{∘}(z)^{F} ⊇ G_{a}^{F} C_{G_{b}}(z)^{F} and therefore C_{G}(z)^{F} = C_{G}^{∘}(z)^{F} since C_{G}(z)^{F} /C_{G}^{∘}(z)^{F} is an ℓ-group. Then for any 1 ≠ z ∈ Z(E)G_{a} ∩ G_{b}, C_{G}^{∘}(z) contains G_{a}, M, E and C_{G^{F}} (Z).

By [CE.1, 2.5], there is a proper e-split Levi subgroup H of G containing C_{G}^{∘}(z). Then H ⊇ L since H ⊇ G_{a} and L ∩ G_{b} is the smallest e–split Levi subgroup of G_{b} containing M ∩ G_{b} (Lemma 4.4.(iii)).

Lemma 4.16. There exists a maximal b_{G}-subpair (D, b_{D}) such that

• D ⊆ p_{T}^{−1}(W (G^{∗}(s)|L^{∗}(s)))^{F} with D ∩ T = Z and p_{T}(D) is a Sylow ℓ-subgroup of W (G^{∗}(s)|L^{∗}(s))^{F},

• D normalizes (Z, b̂_{Z}), and (Z, b̂_{Z}) ⊆ (D, b_{D}),

• the extension 1 → Z → D → D/Z → 1 is split, and

• Z is the unique maximal abelian normal subgroup of D.

Proof. The statement of the Lemma is clearly associated with the datum (G, F, ℓ, L, ζ), which in turn gives rise to b_{G} = R_{L}^{G}(b_{L^{F}} (ζ)) and M. So we may use induction on the dimension of G_{ad}.

Let (D, b_{D}) be a maximal subpair of N_{G^{F}} (Z, b̂_{Z}) containing (Z, b̂_{Z}), let (E, b_{E}) be a subpair of G^{F} containing and normalizing (D, b_{D}). It clearly suffices to show that D = E to obtain that D is a defect group of b_{G}. We apply the dychotomy of Lemma 4.15 to E.

Assume case (a) of that Lemma.

Assume moreover that Z(G) is connected. One may now argue as in [CE.2]. All the statements (i) to (iii) of [CE.2, Lemma 4.3] are satisfied, with same notations, thanks to Lemma 4.15 for statement (i), Lemma 4.4.(i) for statement (ii), Lemma 4.15 and [CE.2, 1.5.(ii)] for statement (iii). By [C, 4.2] there is an ℓ-subgroup S ⊆ p_{T}^{−1}(W (T^{∗}G^{∗}_{a}(s))) ∩ G_{a}^{F} such that S ∩ T = 1, |S| = |W (T^{∗}G^{∗}_{a}(s))^{F} |_{ℓ}. Applying the same proof as in [CE.2, 4.4] implies that Br_{S}(b_{Z}) ≠ 0. By Proposition 2.2.(iv), this implies that there is a subpair inclusion (Z, b̂_{Z}) ⊆ (ZS, b_{ZS}) in G^{F}. This group ZS has order |Z| |W (T^{∗}G^{∗}_{a}(s))^{F} |_{ℓ} = |Z| |(N_{W (G^{∗}(s))}(W (L^{∗}(s)))/W (L^{∗}(s)))^{F} |_{ℓ} in that case since L^{∗} ⊇ G^{∗}_{b} and L^{∗}(s) ∩ G^{∗}_{a} = T^{∗} ∩ G^{∗}_{a} (Lemma 4.4.(i)). But [CE.2, 3.1] implies that |D/Z| = |N_{G^{F}} (Z, b̂_{Z})/C_{G^{F}} (Z)|_{ℓ} while |N_{G^{F}} (Z, b̂_{Z})/C_{G^{F}} (Z)|_{ℓ} ≤ |N_{W (G^{∗}(s))^{F}} (W (L^{∗}(s)))/W (L^{∗}(s))^{F} |_{ℓ} by Lemma 4.14. This implies that one may take D = ZS.

Let us check that Z is the unique maximal abelian normal subgroup of D, or better that S has no quadratic element on Z = (T ∩ G_{a})_{ℓ}^{F} [C, 2.3]. To see that, it suffices to check that T ∩ G_{a} is a diagonal torus in a connected reductive group of types A containing S viewed as subgroup of the associated Weyl group (apply [CE.2, 1.9]). By Lemma 4.4.(i), it suffices to take a group dual to G^{∗}(s) ∩ G^{∗}_{a}.

Now E ⊆ N_{G^{F}} (Z, b̂_{Z}), so E = D by the definition of D. Note that S ⊆ [G, G] since S is taken in any F -stable complement of the extension 1 → T/T_{2} → N_{G}(T)/T_{2} → W (G) → 1 and such a complement exists in [G, G].

If Z(G) is not connected, let us consider a regular embedding G ↪ G′ = T′G where T ⊆ T′, T′ a maximal torus defined on 𝔽_{q} in G′. Let L′ = T′L and ζ′ an irreducible character of (L′)^{F} with ζ in its restriction to L^{F}. Then ζ′ satisfies condition (J) of § 1.3 in L′ since ζ satisfies the same in L and this condition is clearly unaffected by regular embedding. Then R_{L′}^{G′}(b_{L′}^{F} (ζ′)) covers R_{L}^{G}(b_{L^{F}} (ζ)) by Remark 2.7.(b). By what we have just checked, R_{L′}^{G′}(b_{L′}^{F} (ζ′)) satisfies the Lemma. So it has a defect group D′ which is a split extension of (T′)_{ℓ}^{F} by S ⊆ N_{G^{F}} (T′) such that S ∩ T′ = 1 and S is isomorphic with a Sylow ℓ-subgroup of W (G^{∗}(s)|L^{∗}(s))^{F} (this group does not change when going from G to G′ since in the surjection G′^{∗} → G^{∗} the connected centralizers correspond). By [NT, 5.5.16.(ii)], D′ ∩G^{F} is a defect group of b_{G}. But this is T_{ℓ}^{F} S since S ⊆ [G′, G′] ⊆ G. Moreover S normalizes (Z, b̂_{Z}) by Proposition 1.9.(iii), so D′ ∩G^{F} normalizes (Z, b̂_{Z}). On the other hand, we must have |D| = |N_{G^{F}} (Z, b̂_{Z})/C_{G^{F}} (Z)|_{ℓ} by [CE.2, 3.1]. This forces |E| ≤ |D′ ∩ G^{F} | = |D|, so both are equal.

The action of S is without quadratic elements on T_{ℓ}^{F} by the same argument as in the connected center case since [CE.2, 1.9] applies to derived groups when ℓ > 3.

Assume now case (b) of Lemma 4.15. Let z be a non-trivial element in the projection of Z(E) in G_{b} and H a proper e-split Levi subgroup of G containing C_{G}^{∘}(z). All we have proved in G about (L, ζ) is also true in H. The block R_{L}^{H}(b_{L^{F}} (ζ)) is defined, the group M is the same in H and G, the subpair (Z, b̂_{Z}) is a subpair of H^{F}, and ({1}, R_{L}^{H}b_{L^{F}} (ζ)) ⊆ (Z, b̂_{Z}) ⊆ (D, b_{D}) ⊆ (E, b_{E}) in H^{F} by Lemma 4.13 for the first inclusion. But the defect group of R_{L}^{H}(b_{L^{F}} (ζ)) is of order |Z| |W (H^{∗}(s)|L^{∗}(s))^{F} |_{ℓ}, so |D| ≤ |Z| |W (G^{∗}(s)|L^{∗}(s))^{F} |_{ℓ}. So by Lemma 4.14, D = E and is a defect group of R_{L}^{H}(b_{L^{F}} (ζ)) in H^{F}. The induction hypothesis then gives all our claims.

Let us now prove the injectivity of the map (L, ζ) ↦ R_{L}^{G}(b_{L^{F}} (ζ)).

Lemma 4.17. Let L′ be another e-split Levi subgroup of G and ζ′ ∈ E(L′^{F} , ℓ′) satisfying (J) of § 1.3 in L′. Then R_{L}^{G}(b_{L^{F}} (ζ)) = R_{L′}^{G}(b_{L′}^{F} (ζ′)) if and only if (L, ζ) and (L′, ζ′) are G^{F} -conjugate.

Proof. The “if” part is clear. We now check the converse by induction on the dimension of G.

With (L′, ζ′) are associated as in the preceding lemmas and definitions T′, K′, M′, ζ′_{K′}, ζ′_{M′}, Z′ = Z(M′)_{ℓ}^{F}, b̂′_{Z′}, giving rise to a maximal subpair (D′, b′_{D′}). Since R_{L}^{G}(b_{L^{F}} (ζ)) = R_{L′}^{G}(b_{L′}^{F} (ζ′)), the conjugacy of maximal subpairs allows to assume, up to changing (L′, ζ′) into a G^{F} -conjugate, that the above choices lead to (D, b_{D}) = (D′, b_{D′}).

We now apply the dychotomy of Lemma 4.15 to E = D.

Assume case (b). Let 1 ≠ z ∈ Z(D)G_{a} ∩ G_{b}. Let H be a proper e–split Levi subgroup of G containing C_{G}^{∘}(z). Then H contains L, L′, M, M′, Z = Z(M)_{ℓ}^{F}, C_{G^{F}} (Z), Z′ = Z(M′)_{ℓ}^{F}, C_{G^{F}} (Z′) by Lemma 4.15.(b) and therefore, the inclusions (Z, b̂_{Z}) ⊆ (D, b_{D}) and (Z′, b̂_{Z′}) ⊆ (D, b_{D}) take place in H^{F}. One has also ({1}, R_{L}^{H}(b_{L^{F}} (ζ))) ⊆ (Z, b̂_{Z}) and ({1}, R_{L′}^{G}(b_{L′}^{F} (ζ′))) ⊆ (Z′, b̂_{Z′}) since Lemma 4.13 applies with the same definition of M in H and G. Then R_{L}^{H}(b_{L^{F}} (ζ)) = R_{L′}^{H}(b_{L′}^{F} (ζ′)) and the induction hypothesis implies that (L, ζ) and (L′, ζ′) are H^{F} -conjugate.

Assume now case (a) of Lemma 4.15. Then D ⊆ G_{a} and L, L′, K, K′, M, M′ all contain G_{b}. Therefore L = C_{G}^{∘}((T ∩ G_{a})_{ϕ_{e}}), L′ = C_{G}^{∘}((T′ ∩ G_{a})_{ϕ_{e}}), M = C_{G}^{∘}((T ∩ G_{a})_{ℓ}^{F}) and M′ = C_{G}^{∘}((T′ ∩ G_{a})_{ℓ}^{F}) (Lemma 4.4.(ii) and Lemma 4.15.(a)).

Denote s, s′ such that ζ ∈ E(L^{F} , s) and ζ′ ∈ E(L′^{F} , s′). Then Irr(R_{L}^{G}(b_{L^{F}} (ζ))) ∩ E(G^{F} , s) ≠ ∅ since this set contains by definition the irreducible components of R_{L}^{G}ζ and the latter is in ℤE(G^{F} , s) by Proposition 1.2.(i). Samely Irr(R_{L′}^{G}(b_{L′}^{F} (ζ′))) ∩ E(L′^{F} , s′) ≠ ∅. Both blocks being equal, [BMi.1, 2.2] implies that s and s′ are G^{∗}^{F} - conjugate. Then G^{∗}(s)∩G^{∗}_{a} and G^{∗}(s′)∩G^{∗}_{a} are G^{∗}^{F} -conjugate, and consequently by Lemma 4.4.(i), so are T^{∗} ∩G^{∗}_{a} and T′^{∗} ∩G^{∗}_{a}. In dual groups, T∩G_{a} and T′ ∩G_{a} are G^{F} -conjugate by parametrization of maximal tori. So, up to G^{F} -conjugacy, we assume that T∩G_{a} = T′ ∩G_{a}. We may not have any longer (D, b_{D}) = (D′, b_{D′}), but we are still in case (a) of Lemma 4.15 and the above equalities concerning L, L′, M and M′ still hold. Then T∩G_{a} = T′ ∩G_{a} implies M = M′, L = L′ and consequently K = K′ by its definition. We must show that ζ and ζ′ are N_{G^{F}} (L)-conjugate.

Denote by ζ′_{K}, ζ′_{M}, (Z, b̂′_{Z}) ⊆ (D′, b′_{D′}) the outcome of the constructions of the preceding lemmas for (L, ζ′). The conjugacy of maximal subpairs [AB, 3.10] implies that (D, b_{D}) and (D′, b′_{D′}) are G^{F} -conjugate. By the uniqueness of Z in D as maximal normal abelian subgroup (Lemma 4.16), (Z, b̂_{Z}) and (Z, b̂′_{Z}) are G^{F} -conjugate. By the definition of b̂_{Z}, this implies (Z, b_{M^{F}} (ζ_{M})) = ^{g}(Z, b_{M^{F}} (ζ′_{M})) for some g ∈ G^{F}. Then (M, ζ_{M}) = ^{g}(M, ζ′_{M}) by M = C_{G}^{∘}(Z) (Lemma 4.8) and uniqueness of canonical characters in blocks with central defect (Lemma 4.11). Then ^{g}K = K by Lemma 4.4.(iv).

We have ζ_{K} = ε_{M}ε_{K}R_{M⊆Q}^{K}ζ_{M} and ζ′_{K} = ε_{M}ε_{K}R_{M⊆Q′}^{K}ζ′_{M} by Lemma 4.7, so (ζ_{K})^{g} = R_{M⊆Q^{g}}^{K}ζ′_{M}. By Lemma 4.11, ζ′_{M} defines a block of M^{F} with central defect, so it is the only character in E(M^{F} , s′)∩Irr(b_{M^{F}} (ζ′_{M})) (see Remark 2.10). Then Proposition 2.9 applies (see also Remark 2.10) and therefore R_{M⊆Q^{g}}^{K}(ζ′_{M}) = R_{M⊆Q′}^{K}(ζ′_{M}), that is ζ_{K} = ^{g}ζ′_{K}.

Now the pairs (L, ζ) and ^{g}(L, ζ′) both satisfy condition (J) (or (T) by Proposition 1.10.(i)) and share the same associated pair (K, ζ_{K}) = ^{g}(K, ζ_{K}). It now suffices to show that (L, ζ) and ^{g}(L, ζ′) are K^{F} -conjugate. Note that this amounts to Lemma 4.17 in the case when K = K′ = G. So let us denote by (L, ζ) and (L′, ζ′) two e–split Levi subgroups of G with ζ ∈ E(L^{F} , ℓ′) and ζ′ ∈ E(L′^{F} , ℓ′) satisfying condition (J). Assume Z(L)_{ℓ}^{F}, Z(L′)_{ℓ}^{F} are both in Z(G)^{F} and ζ_{G} = ε_{L}ε_{G}R_{L⊆P}^{G}ζ = ε_{L}ε_{G}R_{L⊆P′}^{G}ζ′. We must show that (L, ζ) and (L′, ζ′) are G^{F} -conjugate.

As we have seen before, L and L′ must be G^{F} -conjugate, so we may indeed assume L = L′. Let s′ be a semi-simple rational element of L^{∗} such that ζ′ ∈ E(L^{F} , s′). As ζ_{G} is an irreducible component in both R_{L}^{G}ζ and R_{L′}^{G}^{g}ζ′, then ζ_{G} ∈ E(G^{F} , s)∩E(G^{F} , s′), so s and s′ are (G^{∗})^{F} -conjugate. Let x ∈ G^{∗}^{F} such that s′ = s^{x}. Then by the same arguments as above, (G^{∗}(s) ∩ G^{∗}_{a})^{x} = G^{∗}(s′) ∩ G^{∗}_{a} and therefore x ∈ N_{G^{∗}}(L^{∗})^{F} (recall that K = G, so that L ⊇ G_{b} thanks to Lemma 4.4.(iii)). By Proposition 3.4.(i), N_{W (G)}(W (L)) = W (L) hence N_{G^{∗}}(L^{∗}) = L^{∗}. Then x ∈ (L^{∗})^{F}, which means that we may assume s = s′.

Assume that we know moreover that ζ is the only element of E(L^{F} , s) ∩ Irr(b_{L^{F}} (ζ)). Then Lemma 4.5 and Proposition 2.9 tell us that R_{L⊆P}^{G}ζ = R_{L⊆P′}^{G}ζ. But the first is ε_{G}ε_{L}ζ_{G} by definition. So we get R_{L⊆P′}^{G}ζ = R_{L⊆P′}^{G}ζ′ = ε_{G}ε_{L}ζ_{G}. But ζ, ζ′ ∈ E(L^{F} , s) implies ζ = ζ′ by Proposition 1.5.(ii), which completes the proof of the lemma.

To see that ζ is the only element of E(L^{F} , s)∩Irr(b_{L^{F}} (ζ)), one may clearly assume L = G. Then M ⊆ L and ζ = ε_{M}ε_{L}R_{M}^{L}ζ_{M}. By Lemma 4.11, ζ is the only element in E(M^{F} , s) ∩ Irr(b_{Z}). Now Lemma 4.5 allows to apply Proposition 2.9 between M and L, so we get that ζ is the only element of E(L^{F} , s) ∩ Irr(b_{L^{F}} (ζ)).

Let us check Theorem 4.2. One may assume G = L. By Proposition 1.10.(i) and (ii), it suffices to check that if χ ∈ E(G^{F} , ℓ′) satisfies (J), then it is e-cuspidal. One may apply the above lemmas to the block b_{G^{F}} (χ) with (L, ζ) = (G, χ). There is clearly (L′, ζ′) which is e-cuspidal and such that (L′, ζ′) <<_{e} (G, χ), as defined in Notation 1.11. But then b_{G^{F}} (χ) = R_{L′}^{G}(b_{L′}^{F} (ζ′)) by Remark 2.7.(a). By Proposition 1.10.(i) and (ii), the pair (L′, ζ′) satisfies condition (J). Then Lemma 4.17 applies, so (G, χ) and (L′, ζ′) are G^{F} -conjugate hence equal. So χ = ζ′ is e-cuspidal, which completes the proof of Theorem 4.2.

Now, Lemma 4.17 tells us that the map (L, ζ) ↦ R_{L}^{G}(b_{G^{F}} (ζ)) is injective from the set of G^{F} -conjugacy classes of pairs (L, ζ) with e-split L and e-cuspidal ζ ∈ E(L^{F} , ℓ′), to the set of ℓ-blocks of G^{F}. Let us check that it is also surjective.

Let χ ∈ E(G^{F} , ℓ′). Arguing as above, there is a pair (L, ζ) such that (L, ζ) <<_{e} (G, χ) and ζ is e-cuspidal. Therefore χ ∈ Irr(R_{L}^{G}(b_{L^{F}} (ζ))) by Remark 2.7.(a). By a theorem of Hiss or as a consequence of Theorem 2.8, any ℓ-block of G^{F} has at least one character in E(G^{F} , ℓ′), so we get the surjectivity of the map (L, ζ) ↦ R_{L}^{G}(b_{G^{F}} (ζ)), whence (a) of Theorem 4.1. This also yields (b) since one has the inclusion Irr(R_{L}^{G}(b_{G^{F}} (ζ))) ∩ E(G^{F} , ℓ′) ⊇ {χ ∈ Irr(G^{F}) ; (L, ζ) <<_{e} (G, χ)} and, for varying (L, ζ), this clearly exhausts E(G^{F} , ℓ′).

5. Concluding remarks.

Here are several remarks and by-products of the proof of Theorem 4.1. Let (G, F), q, ℓ, (L, ζ), G^{∗}, s ∈ L^{∗}^{F} , T, G^{∗}(s), L^{∗}(s) be as in the section above, Definition 4.3. 5.1. Defect groups.

Lemma 4.16 gives a description of the defect groups of the ℓ-block R_{L}^{G}(b_{L^{F}} (ζ)). One may compare the outcome in the general case with the one when s = 1. Let (G_{s}, F) be in duality with (G^{∗}(s), F), let L_{s} be in duality with L^{∗}(s). Let λ ∈ E(L^{∗}(s)^{F} , 1) be any unipotent e-cuspidal character associated with ζ by Lusztig’s parametrization as in (J_{2}). Applying Lemma 4.16 to b_{G^{F}} (L, ζ) and b_{(G_{s})^{F}} (L_{s}, λ) (see also [CE.2, 3.6]), one gets

Proposition 5.1. The defect groups of b_{G^{F}} (L, ζ) in G^{F} are isomorphic with the ones of b_{(G_{s})^{F}} (L_{s}, λ) in (G_{s})^{F}.

In the unipotent case, that is s = 1, the structure of defect groups given by Lemma 4.16 shows that [CE.1, 4.4.(ii); CE.2, 3.5] hold in that case.

5.2. Other primes. The primes that are bad for G, along with ℓ = 3 for ^{3}D_{4} and ℓ = 2 for ^{2}A_{n}, provide counterexamples to Theorem 4.1 with s = 1.

When ℓ = 3, the proof used in section 4 shows that the map (L, ζ) ↦ R_{L}^{G}(b_{L^{F}} (ζ)) is surjective with defect groups as described above. Injectivity may be proved when moreover 3 ∈ Γ(G, F), since then Z is the unique maximal abelian normal subgroup of D (see [CE.2, 1.9]) and Lemma 4.16 therefore holds. This shows that Theorem 4.1 and Theorem 4.2 are true for types B, C, D as soon as ℓ is odd.

5.3. Isomorphic blocks. Let (G, F), q, ℓ, satisfy the hypotheses of Theorem 4.1. Let G ↪ G′ be a regular embedding. Let L be an e-split Levi subgroup of G, let s be a semi-simple ℓ′-element of L^{∗}^{F} such that Z(L^{∗})_{ϕ_{e}} = Z(L^{∗}(s))_{ϕ_{e}}. Let λ ∈ E(L^{∗}(s)^{F} , 1) be an e-cuspidal unipotent character. We are interested in the blocks R_{L}^{G}(b_{L^{F}} (ζ)) where ζ is associated with (s, λ) by Lusztig parametrization (the ζ’s are e-cuspidal by Theorem 4.2. Let s′ ∈ (L^{′∗})^{F} be a rational semi-simple ℓ′-element of image s in L^{∗} and let ζ′_{0} ∈ E(L^{′F} , s′) corresponding with λ by Jordan decomposition. By Lusztig’s theorem [L.3, 5.1], the characters ζ ∈ E(L^{F} , s) corresponding with λ are the irreducible components of Res_{L^{F}}^{L^{′F}} ζ′_{0} and are L^{′F} -conjugate. So the corresponding blocks b_{L^{F}} (ζ) of L^{F}, resp. R_{L}^{G}(b_{L^{F}} (ζ)) of G^{F}, are G^{′F} -conjugate, hence isomorphic. By Remark 2.7.(b), the latters are all covered by the block b′ = R_{L′}^{G′}(b_{L^{′F}} (ζ′_{0})) of G^{′F}. Let b be one of those R_{L}^{G}(b_{L^{F}} (ζ)). Then E(G^{F} , ℓ′) ∩ Irr(b) is included in the set of irreducible components of the restrictions to G^{F} of the elements in E(G^{′F} , ℓ′)∩Irr(b′): Indeed b′ covers b, so, if χ ∈ E(G^{F} , ℓ′) ∩ Irr(b), there is some χ′′ ∈ Irr(b′) such that ⟨χ, Res_{G^{F}}^{G^{′F}} χ′′⟩_{G^{F}} ≠ 0. If s′′ ∈ (G^{′∗})^{F} is such that χ′′ ∈ E(G^{′F} , s′′), then χ ∈ E(G^{F} , s) where s is the image of s′′ by the surjection G^{′∗} → G^{∗} dual of the regular embedding. Since χ ∈ E(G^{F} , ℓ′), this implies that s′′_{ℓ} is central and therefore χ′′ = μχ′ for a linear ℓ-character μ ∈ Irr(G^{′F} /G^{F}) and χ′ ∈ E(G^{′F} , ℓ′). Then clearly χ′ ∈ Irr(b′) and χ is among the components of Res_{G^{F}}^{G^{′F}} χ′ = Res_{G^{F}}^{G^{′F}} χ′′.

5.4. <<_{e} and ≤_{e}. Many groups G′ with connected center satisfy the condition (2.1.R) of [CE.2, 2.1], that is a reasonable compatibility between Jordan decomposition and twisted induction of characters (theorems due to Asai, Fong, Kawanaka, Shoji, Srinivasan, see the Appendix of [FS.2] and its references). If G is regularly embedded in such a G′ and ζ ∈ E(L^{F} , s) ⊆ E(L^{F} , ℓ′) is an e-cuspidal character of an e-split Levi subgroup L of G, then we show below that Irr(R_{L}^{G}b_{L^{F}} (ζ)) ∩ E(G^{F} , s) coincides with the set of irreducible components of R_{L}^{G}ζ. For instance, this applies to blocks of finite special linear or unitary groups (taking G′ a general linear group), or the groups G^{F} such that G is regularly embedded in the conformal groups associated with symplectic or orthogonal forms. The cases of Sp_{2n}(q) and SO_{2n}^{±}(q) are not new (see [S]).

Proposition 5.2. Let ℓ satisfy the hypotheses of Theorem 4.1 with respect to G and q. Let G ↪ G′ be a regular embedding. Let L be an e-split Levi subgroup with an e-cuspidal ζ in E(L^{F} , ℓ′). Then Irr(R_{L}^{G}b_{L^{F}} (ζ)) ∩ E(G^{F} , ℓ′) = {χ ∈ Irr(G^{F}) ; (L, ζ) ≤ (G, χ)} in the following cases

• e = 1.

• G = G′ and the condition (2.1.R) of [CE.2, 2.1] is satisfied by the rational series of G^{′F} containing the irreducible components of R_{L′}^{G′}ζ.

• The relations <<_{e} and ≤_{e} coincide in G′, and the R_{L′}^{G′} map (where L′ = Z(G′)L) does not depend on the parabolic subgroups used to define it.

Proof of the Proposition. The first point stems from Theorem 4.1.(b) and the fact that ≤_{1} coincides with <<_{1} by classical Harish-Chandra theory (see [DM.1, 6.4]). Samely, ≤_{e} and <<_{e} coincide on the rational series satisfying the condition (2.1.R) of [CE.2, 2.1] by [CE.2, 2.7].

We now verify the last point. Let χ ∈ Irr(R_{L}^{G}b_{L^{F}} (ζ)) ∩ E(G^{F} , ℓ′), or equivalently by Theorem 4.1.(b) (L, ζ) <<_{e} (G, χ). Let us show (L, ζ) ≤ (G, χ).

Let ζ′ ∈ E(L^{′F} , ℓ′) be associated with ζ as in the preceding section. Then by the property of covering described above, R_{L′}^{G′}(b_{L^{′F}} (ζ′)) covers R_{L}^{G}(b_{L^{F}} (ζ)) and ⟨χ, Res_{G^{F}}^{G^{′F}} χ′⟩_{G^{F}} ≠ 0 for a χ′ ∈ Irr(R_{L′}^{G′}(b_{L^{′F}} (ζ′))) ∩ E(G^{′F} , ℓ′). Theorem 4.1.(b) then says that (L′, ζ′) <<_{e} (G′, χ′). Since <<_{e} coincides with ≤_{e} in G′, one has (L′, ζ′) ≤ (G′, χ′).

Let us check ⟨Ind_{G^{F}}^{G′^{F}} χ, R_{L′}^{G′}ζ′⟩_{G^{′F}} ≠ 0. One has Ind_{G^{F}}^{G′^{F}} χ = ∑_{λ∈J} λχ′ where J ⊆

Irr(G′^{F} /G^{F}). Then ⟨Ind_{G^{F}}^{G′^{F}} χ, R_{L′}^{G′}ζ′⟩_{G^{′F}} = ∑_{λ∈J}⟨χ′, R_{L′}^{G′}λ^{−1}ζ′⟩_{G^{′F}}. In this sum, the non-zero terms are for λ such that λ^{−1}ζ′ ∈ E(L^{′F} , ℓ′) (use Proposition 1.2.(iv)) and b_{G′}^{F} (χ′) = R_{L′}^{G′}(b_{G′}^{F} (λ^{−1}ζ′)). For those λ, (L′, λ^{−1}ζ′) and (L′, ζ′) are G′^{F} -conjugate by Theorem 4.1.(a). But then R_{L′}^{G′}λ^{−1}ζ′ = R_{L′}^{G′}ζ′ since the twisted induction only depends on the Levi subgroup. So ⟨Ind_{G^{F}}^{G′^{F}} χ, R_{L′}^{G′}ζ′⟩_{G^{′F}} is a non-zero multiple of ⟨χ′, R_{L′}^{G′}ζ′⟩_{G^{′F}} ≠ 0.

On the other hand, ⟨Ind_{G^{F}}^{G′^{F}} χ, R_{L′}^{G′}ζ′⟩_{G^{′F}} = ⟨χ, Res_{G^{F}}^{G′^{F}} R_{L′}^{G′}ζ′⟩_{G^{F}} = ⟨χ, R_{L}^{G}Res_{L^{F}}^{L^{′F}} ζ′⟩_{G^{F}} = ⟨χ, R_{L}^{G}(∑_{h∈L^{′F} /L_{ζ}^{′F})} ζ^{h})⟩_{G^{F}}. So there is h such that (L, ζ^{h}) ≤

(G, χ). But then R_{L}^{G}(b_{L^{F}} (ζ)) = R_{L}^{G}(b_{L^{F}} (ζ^{h})) = b_{G^{F}} (χ), while ζ^{h} is e-cuspidal. So (L, ζ) and (L, ζ^{h}) are G^{F} -conjugate and therefore (L, ζ) ≤ (G, χ).

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