Published in Progress in Math., 141 (1996), p. 141-164.

LOCAL METHODS FOR BLOCKS OF REDUCTIVE GROUPS OVER A FINITE FIELD.

Marc Cabanes, Michel Enguehard

Introduction The aim of this paper is to show for general blocks of reductive groups over a finite

field some analogues of the results in [CE.1] about unipotent blocks. This includes the distribution of ordinary characters into blocks (Theorem 3.3) and the structure of defect groups (Theorem 3.5), thus yielding the main results of [FS.1; FS.2; Br.1; CE.1]. It should be noted that, by a theorem of Broué [Br.2, 2.3], most blocks are in some sense “equivalent” to a unipotent block. This theorem provides an important idea one must keep in mind, but, in addition to the restrictions on the blocks concerned, Broué’s equivalence (a perfect isometry) does not imply the isomorphism of defect groups (our Remark 3.6) but just the equality of their orders.

On the other hand, our statements do include a restriction on the “rational series” we consider, and an assumption on the “Jordan decomposition” of ordinary characters of reductive groups over a finite field (2.1.R), part of which is yet conjectural in the present state of the theory (see Definition 2.1).

Let G be a connected reductive algebraic group endowed with a Frobenius endomorphism F defining a rational structure over the finite field IF_{q}. Let (G^{∗}, F) be a dual group. We study the ℓ-blocks of G^{F} for ℓ a prime not dividing q. In order to avoid technicalities in this introduction, assume Z(G) is connected and ℓ > 5 does not divide the order of (Z(G^{∗})/Z^{∘}(G^{∗}))^{F}. (For more details, see the introduction to section 3.)

Denote by e the order of q (mod ℓ). Theorem 3.3 shows that the ℓ-blocks are parametrized by the G^{F} -conjugacy classes of certain e-cuspidal pairs (L, ζ). Those pairs generalize the unipotent case described by [BMM]. We show that the defect groups write as a semi-direct product D = Z.S where Z is the unique maximal abelian normal subgroup of D, Z = Z(C_{G}^{∘}(Z))_{ℓ}^{F}, and S is a Sylow ℓ-subgroup of a finite reflection group acting on Z. The “local” method to determine the blocks and their defect groups consists essentially in including a given ℓ-block of G^{F} in a “maximal subpair” (see [AB]). Thus we combine (and partly develop) some techniques belonging to block theory (see Proposition 3.2) and to the theory of reductive groups over a finite field (sections 1 and 2). We also draw heavily on the preparatory material of [CE.1], e.g. the decomposition G = G_{a}.G_{b}. Nevertheless our proof applies to reductive groups of any type.

Notations. When H is an algebraic group, H^{∘} denotes its identity component. We use the notations C_{H}^{∘}(X) and Z^{∘}(H) for the identity components of the centralizer of X in H and of the center of H. If H is reductive and connected, then H_{ad} = H/Z(H) is its adjoint group.

If K is an algebraically closed field of characteristic zero, and G is a finite group, then Irr(G) is the set of irreducible characters of the group algebra K[G].

1. Reductive groups over a finite field.

1.1. The groups.

The groups G^{F} we consider in this paper are defined as follows.

Let IF be the algebraic closure of a finite field IF_{q}. Let G be a connected reductive affine algebraic group over IF and let F: G → G be an endomorphism which is a Frobenius morphism for an IF_{q}-rational structure of G. We abbreviate this by saying that (G, F) is “a connected reductive group defined over IF_{q}”. The group of points of G on IF_{q} is denoted by G^{F}.

Such a group (G, F) may be defined by a root datum with Frobenius endomorphism F, say ((X, R, Y, R^{∨}), F). Here X = X(T) and Y = Y (T) are free ZZ-modules in duality over ZZ, and are identified respectively with the group of characters and the group of one-parameter subgroups of some maximal F -stable torus T of G. Then R ⊆ X (resp. R^{∨} ⊆ Y) is the set of roots (resp. coroots) of G relative to T. The sets R and R^{∨} are in bijection by (r ↦ r^{∨}). Two root data with Frobenius endomorphism ((X, R, Y, R^{∨}), F) and ((X^{∗}, R^{∗}, Y^{∗}, (R^{∗})^{∨}), F^{∗}) are said to be in duality if ((X^{∗}, R^{∗}, Y^{∗}, (R^{∗})^{∨}), F^{∗}) is isomorphic to ((Y, R^{∨}, X, R), F). Then the corresponding reductive groups (G, F) and (G^{∗}, F^{∗}) are said to be in duality and the set of G^{F} -conjugacy classes of maximal tori of (G, F) is in bijection with the analogous set in (G^{∗}, F^{∗}) in such a way that tori T and T^{∗} in corresponding classes may be used to define the given duality between (G, F) and (G^{∗}, F^{∗}) (see [Ca, Chapter 4] for more details and properties of reductive groups in duality). In this situation we usually identify ((X(T^{∗}), R^{∗}, Y (T^{∗}), (R^{∗})^{∨}), F^{∗}) with ((Y (T), R^{∨}, X(T), R), F). Furthermore an F -stable Levi subgroup L of G that contains T defines a Levi subgroup L^{∗} of G^{∗}, such that (L, F) and (L^{∗}, F) are in duality : T^{∗} is a maximal torus of L^{∗} and the sets of roots of L relative to T and of L^{∗} relative to T^{∗} are in bijection by r ↦ r^{∨}. One obtains finally a bijection between the set of G^{F} -conjugacy classes of F -stable Levi subgroups of G and the set similarly defined in G^{∗}, see [DM.2, p.113].

Let us recall the following notations [CE.1, 1.1].

Notation 1.1. If (G, F) is a connected reductive group defined over IF_{q}, then let γ(G, F) be the set of prime integers ℓ satisfying all of the following conditions : ℓ does not divide q, ℓ is good for G, ℓ ≠ 2, ℓ does not divide |(Z(G)/Z^{∘}(G))^{F} |.

Let Γ(G, F) = (γ(G, F) ∩ γ(G^{∗}, F)) \ {3} when G_{ad}^{F} has a component isomorphic to some ^{3}D_{4}(q^{m}) ; let Γ(G, F) = γ(G, F) ∩ γ(G^{∗}, F) otherwise.

Concerning the condition that ℓ does not divide |(Z(G)/Z^{∘}(G))^{F} |, we mention the following. Let A be a finite abelian group and F an endomorphism of A, then ℓ does not divide |A^{F} | if and only if, for every section S = B/B′ with F -stable subgroups B and B′, ℓ does not divide |S^{F} | (assume S = S^{F} and use |B| = |B^{F} |.|[B, F]|).

Lemma 1.2. Assume G is a central product of closed connected F -stable subgroups H and J. If Z^{∘}(G) ⊆ H, then Γ(G, F) ⊆ Γ(H, F).

Proof. A prime is good for a central product if and only if it is good for every component.

First, Z(H)/Z^{∘}(H) is a section of Z(G)/Z^{∘}(G) since Z(H) ⊆ Z(G) and Z^{∘}(H) ⊇ Z^{∘}(G). Concerning dual groups, the quotient Z(G^{∗})/Z^{∘}(G^{∗}) is isomorphic, with F -action, to the kernel of the simply connected covering (G_{sc} → [G, G]) [Ca, 4.5.8]. One has [G, G] = [H, H].[J, J] and G_{sc} = H_{sc} × J_{sc}, hence Z(G^{∗})/Z^{∘}(G^{∗}) ⊇ Z(H^{∗})/Z^{∘}(H^{∗}) × Z(J^{∗})/Z^{∘}(J^{∗}) and therefore Z(H^{∗})/Z^{∘}(H^{∗}) is an F -stable section of Z(G^{∗})/Z^{∘}(G^{∗}). The Lemma follows.

1.2. Polynomial orders.

We use the notion of polynomial order of a connected reductive group (G, F) defined over IF_{q} [BM], we denote it by P_{(G,F)}.

Let now E be a non-empty set of positive integers. We call ϕ_{E}-subgroups of G the F -stable tori S such that P_{(S,F)} is a product of powers of the cyclotomic polynomials ϕ_{m} for m ∈ E. Their centralizers in G are called E-split Levi subgroups of G. Note that, if S is an F -stable torus of G, then S contains a unique maximal ϕ_{E}-subgroup, we denote it by S_{ϕ_{E}}, see [BM, 3.1.(2)]. The case when E is a singleton has remarkable properties [BM, 3.4].

Proposition 1.3. Let (G, F) be a connected reductive group defined over IF_{q}, let (G^{∗}, F) be dual to (G, F).

(i) The bijection between the sets of G^{F} -conjugacy classes of F -stable Levi subgroups of G and of (G^{∗})^{F} -conjugacy classes of F -stable dual Levi subgroups of G^{∗} induces a bijection between the sets of conjugacy classes of E-split Levi subgroups of G and G^{∗} for each non-empty set E of positive integers. (ii) Let ℓ ∈ Γ(G, F) and denote E_{q,ℓ} = {d ; ℓ divides ϕ_{d}(q)}. If S is a ϕ_{E_{q,ℓ}}-subgroup of G, then C_{G}(S) = C_{G}^{∘}(S_{ℓ}^{F}) and C_{G}(S)^{F} = C_{G}(S_{ℓ}^{F})^{F}. Conversely, if L is an F -stable Levi subgroup of G such that L = C_{G}^{∘}(Z(L)_{ℓ}^{F}), then L is E_{q,ℓ}-split.

Proof. (i) Just replace e by E in the proof of [CE.1, 1.4]. (ii) The proof of [CE.1, 2.2.(ii)] applies.

1.3. On type A.

The induction method used in section 4 essentially reduces the study of blocks to the case of type A. We need some ad hoc results in this situation.

We freely use the notions of “rationally irreducible components”, corresponding to the orbits of F on the irreducible components of a reductive group G with Frobenius endomorphism F, and associated “rational type”, see [CE.1, § 1.1].

Definition 1.4. If (G, F) is of rational type ×_{ω}(A_{n_{ω}}, ε_{ω}q^{m_{ω}}), then a maximal F -stable torus T of G of polynomial order P_{(Z^{∘}(G),F)}Π_{ω}(x^{m_{ω}} −ε_{ω})^{n_{ω}} will be called a diagonal torus of G. All diagonal tori are G^{F} -conjugate. If a Levi subgroup L contains a diagonal torus T of G, then T is a diagonal torus of L. An F -stable maximal torus of polynomial order P_{(Z^{∘}(G),F)}Π_{ω}((x^{m_{ω}(n_{ω}+1)} − ε_{ω}^{(n_{ω}+1)})/(x^{m_{ω}} − ε_{ω})) will be called a Coxeter torus of G. All Coxeter tori of G are G^{F} -conjugate.

Notation 1.5. Let (G, F) be a connected reductive group defined over IF_{q}. Let ℓ be a prime not dividing q and good for G. As in [CE.1, 2.3] define G_{a} as the central product of Z^{∘}(G) and of the rationally irreducible components of [G, G] whose rational type is of the form (A_{n}, εq^{m}) with ℓ dividing q^{m} − ε. Let G_{b} be the central product of the components of [G, G] not included in G_{a}. Then G = G_{a}.G_{b} (central product). Moreover

(i) If H is an F -stable Levi subgroup of G_{a}, then H = H_{a}.

(ii) If T is a maximal F -stable torus of G such that T_{ℓ}^{F} ⊆ Z^{∘}(G), then T is a Coxeter torus of TG_{a}.

Proofs. (i) This is clear from the definition of G_{a}.

(ii) One may clearly assume G = G_{a} is of irreducible rational type (A_{n}, εq^{m}) with εq^{m} ≡ 1 (mod ℓ). The polynomial order of a maximal torus is of the form P_{(T,F)} = P_{(Z^{∘}(G),F)}(x^{m} − ε)^{−1}Π_{i}(x^{n_{i}m} − ε^{n_{i}}). If T_{ℓ}^{F} ⊆ Z^{∘}(G), then (T/Z^{∘}(G))^{F} is ℓ′. This readily implies that in P_{(T/Z^{∘}(G),F)} = (P_{(Z^{∘}(G),F)})^{−1}P_{(T,F)} = (x^{m} − ε)^{−1}Π_{i}(x^{n_{i}m} − ε^{n_{i}}) there is just one n_{i}.

We need an elementary property of pairs of Levi subgroups of groups of type A sharing a maximal torus which is diagonal in one and Coxeter in the other.

Proposition 1.6. Let (G, F) be a connected reductive group defined over IF_{q} of irreducible rational type (A_{n−1}, r) where r = ϵq^{b}, ϵ ∈ {−1, 1}. Let E be a non-empty set of positive integers. Denote ϕ_{E}(x) = Π_{m∈E}ϕ_{m}(x). Let C be an F -stable Levi subgroup of G of rational type Π_{k}(A_{n_{k}−1}, r^{a_{k}}) (n_{k} ≥ 1, Σ_{k}n_{k}a_{k} = n) and let T be a diagonal torus of C. Let K = C_{G}(T_{ϕ_{E}}) and assume that T is a Coxeter torus of K. For any k, let b_{k} be the smallest of the divisors α of a_{k} such that g.c.d.((ϵx^{b})^{α} − 1, ϕ_{E}(x)) = g.c.d.((ϵx^{b})^{a_{k}} − 1, ϕ_{E}(x)) and set a_{k} = b_{k}m_{k}. Then K is of rational type Π_{k}(A_{m_{k}−1}, r^{b_{k}})^{n_{k}}. If w ∈ (N_{C}(T)/T)^{F} stabilizes a rationally irreducible component of K, then w fixes any root relative to T of this component.

Proof. Both hypotheses and conclusions are independent of isogenies and of the center of G. So one may assume that G is a general linear group GL(n)^{b} acting on its standard space. The proof reduces to an elementary verification.

Corollary 1.7. In hypothesis and notations of Proposition 1.6, if E = E_{q,ℓ} (see Proposition 1.3.(ii)), ℓ ∈ Γ(G, F) and ℓ divides (r − 1), then b_{k} is the ℓ-part of a_{k} and the center of any semi-simple group isogeneous to [K, K] is of order prime to ℓ.

Proof. Denote by e the order of q mod ℓ. Then E_{q,ℓ} is the set of all eℓ^{s} (s ≥ 0). As ℓ divides (εq^{b} −1), ϕ_{eℓ^{s}} divides (ϵx^{b} −1) for some s and this implies that ϕ_{e} divides (ϵx^{b} −1) (recall that ℓ ≠ 2). Then, for s ≤ t, ϕ_{eℓ^{t}} divides ((ϵx^{b})^{α} − 1) if, and only if, ϕ_{eℓ^{t}} divides ((ϵx^{b})^{α_{ℓ}} −1). This in turn occurs if, and only if, ϕ_{eℓ^{t}/α_{ℓ}} divides (ϵx^{b} −1). Hence b_{k} = (a_{k})_{ℓ} (the ℓ-part of a_{k}), and m_{k} is prime to ℓ. The order of the center of a simply-connected covering of [K, K] is Π_{k}m_{k}.

We recall a useful property of defect groups in reductive groups over a finite field.

Notation 1.8. [C, 2.2] If A is an abelian normal subgroup of a group G, then one writes A ˙◁ G if and only if, for any subgroup H ⊆ G such that A ⊆ H, A is the unique maximal abelian normal subgroup of H.

Proposition 1.9. Let (G, F) be a connected reductive group defined over IF_{q}. Let ℓ be a prime ≥ 5 and not dividing q or ℓ = 3 ∈ Γ(G, F). Assume G = G_{a} and let T be a diagonal torus of G. Let Q be a Sylow ℓ-subgroup of N_{G^{F}} (T), then T_{ℓ}^{F} ˙◁ Q.

Proof. Note that if G = G_{a}, the diagonal torus T satisfies T = C_{G}(T_{ϕ_{e}}) by [CE.1, 2.4] and T_{ϕ_{e}} is thus a maximal ϕ_{e}-subgroup of G. For ℓ ≥ 5, Proposition 1.9 is therefore a consequence of [C, 4.4]. Moreover the relation T_{ℓ}^{F} ˙◁ Q fails only when some non trivial ℓ-element v of N_{G}(T)^{F} satisfies [v, [v, T_{ℓ}^{F}]] = {1} [C, 2.3]. By [C, 4.5], this has to happen in a rationally irreducible component H of G of type (A_{2}, εq^{d}) for some ε = ±1, with w := vT of order ℓ = 3 in a symmetric group of degree 3 acting on S = T ∩ H [C, 4.1]. We may assume Z^{∘}(G) ⊆ H. Then 3 ∈ Γ(H, F) (Lemma 1.2). Denote by π: H → H_{ad} the reduction map mod. Z(H). By a direct computation in H_{ad} = PGL(n)^{d}, and since π(C_{H}^{∘}(h)) = C_{H_{ad}}^{∘}(π(h)) for any semi-simple element h ∈ H [Ca, 3.5.3], one sees that C_{H}^{∘}(v) ∩ S ⊆ Z(H). We have also C_{H}^{∘}(v)^{F} = C_{H}(v)^{F} by [CE.1, 2.1.(iii)], so the relation [v, [v, S_{3}^{F}]] = {1} implies [v, S_{3}^{F}] ⊆ Z(H)^{F}. Then w, a cycle of order 3, acts trivially on π(S_{3}^{F}). But π(S_{3}^{F}) = π(S)_{3}^{F} [CE.1, 2.1.(ii)], so w fixes all the rational 3-elements of the diagonal torus of H_{ad}. This is easily seen to be impossible in H_{ad}^{F} = PGL(3, εq^{d}) (recall that 3 divides εq^{d} − 1 since G = G_{a} relatively to ℓ = 3).

2. The irreducible characters

2.1. Lusztig series and Jordan decomposition of characters.

When (G, F) is a connected reductive group defined over IF_{q}, we consider the irreducible representations, characters and central functions of G^{F} with values in an algebraically closed field K of characteristic zero.

When L ⊆ G is an F -stable Levi subgroup of G, let us recall the existence of Lusztig map R_{L}^{G} : CF(L^{F} , K) → CF(G^{F} , K) and its adjoint (for the usual inner product) ^{∗}R_{L}^{G} : CF(G^{F} , K) → CF(L^{F} , K), between the spaces of central functions on G^{F} and L^{F} (see [DM.2, §§11–12] for a systematic account). Let us recall also the partition of the set of irreducible characters of G^{F} into Lusztig’s rational series Irr(G^{F}) = ∪ E(G^{F} , s) where s ranges over the semi-simple elements of (G^{∗})^{F} modulo (G^{∗})^{F} -conjugacy, see [L.2,§2; DM.2, 14.41]. When s = 1, the elements of E(G^{F} , 1) are called unipotent characters.

One has the so-called “Jordan decomposition” of characters. For any semi-simple element s of G^{∗}^{F}, Lusztig has shown the existence of a bijection γ ↦ χ_{s,γ}^{G} from E(C_{G^{∗}}(s)^{F} , 1) to E(G^{F} , s), [L.2, 5.1]. In what follows, we shall impose C_{G^{∗}}^{∘}(s)^{F} = C_{G^{∗}}(s)^{F} (otherwise the bijection holds with an appropriate definition of E(C_{G^{∗}}(s)^{F} , 1)). Certain natural conditions force the uniqueness of the bijection, see [DM.1, 7.1].

We define a property of compatibility between such a “Jordan decomposition” and Lusztig’s functors.

Definition 2.1. Let (G, F) be a connected reductive group defined over IF_{q}. Let s be a semi-simple element of (G^{∗})^{F} such that C_{G^{∗}}^{∘}(s)^{F} = C_{G^{∗}}(s)^{F}. For any pair (L, L^{∗}) of dual Levi subgroups by duality between maximal tori T and T^{∗} of G and G^{∗} and such that s ∈ T^{∗}, there exists a bijection λ ↦ χ_{s,λ}^{L} from E(C_{L^{∗}}^{∘}(s)^{F} , 1) to E(L^{F} , s).

When T ⊆ M ⊆ L and T^{∗} ⊆ M^{∗} ⊆ L^{∗} are in duality and λ ∈ E(C_{L^{∗}}^{∘}(s)^{F} , 1), μ ∈ E(C_{M^{∗}}^{∘}(s)^{F} , 1), we will denote by (2.1.R) the following equality

(2.1.R)  ε_{L}ε_{M}⟨R_{M}^{L}χ_{s,μ}^{M}, χ_{s,λ}^{L}⟩_{L^{F}} = ε_{C_{L^{∗}}^{∘} (s)}ε_{C_{M^{∗}}^{∘} (s)}⟨R_{C_{M^{∗}}^{∘} (s)}^{C_{L^{∗}}^{∘} (s)}μ, λ⟩_{C_{L^{∗}}^{∘} (s)^{F}} .

(We put as usual ε_{G} = (−1)^{IF_{q}−rank(G)}, see [DM.2, 8.3].)

Remark 2.2. If L^{∗} is a Levi subgroup of G^{∗} and s is semi-simple in L^{∗}, then C_{G^{∗}}^{∘}(s)∩L^{∗} = C_{L^{∗}}^{∘}(s) (write L^{∗} = C_{G^{∗}}(S^{∗}) = C_{G^{∗}}^{∘}(S^{∗}) for a torus S^{∗} and note that S^{∗} ⊆ C_{G^{∗}}^{∘}(s)). A consequence is that, if s is a rational semi-simple element of G^{∗} which satisfies the condition C_{G^{∗}}^{∘}(s)^{F} = C_{G^{∗}}(s)^{F}, then it satisfies it in any F -stable Levi subgroup containing s. By [CE.1, 2.1.(ii)], this applies to C_{G^{∗}}^{∘}(t) when t ∈ C_{G^{∗}}(s)_{Γ(G,F)}^{F} : one gets C_{G^{∗}}^{∘}(<s, t>)^{F} = C_{G^{∗}}(<s, t>)^{F}.

It is shown in [FS.2, Appendix] that for some classical groups G, the equality (2.1.R) holds for all M, L, μ and λ. The bijections λ ↦ χ_{s,λ}^{L} defined by Lusztig satisfy the projection of (2.1.R) on the spaces of uniform functions, that is (2.1.R) for M = T a torus, μ = 1 and any λ [L.2, 5.1]. In the next proposition, we draw several consequences.

Proposition 2.3. Let (G, F) and s ∈ (G^{∗})_{ss}^{F} such that C_{G^{∗}}^{∘}(s)^{F} = C_{G^{∗}}(s)^{F}. Let L be some F -stable Levi subgroup of G, of dual L^{∗} with s ∈ L^{∗}. Let λ ↦ χ_{s,λ}^{L} be a bijection from E(C_{L^{∗}}^{∘}(s)^{F} , 1) to E(L^{F} , s) such that the equalities (2.1.R) hold whenever M is a torus. Let λ ∈ E(C_{L^{∗}}^{∘}(s)^{F} , 1).

(a) Let p be the characteristic of IF_{q} and let L = G. Then χ_{s,λ}^{G}(1) = λ(1).|(G^{∗})^{F} : C_{G^{∗}}(s)^{F} |_{p′}.

(b) Let L = G. If t ∈ Z(G^{∗})^{F} and t̂ ∈ Irr(G^{F}) is defined by t and the duality, then χ_{st,λ}^{G} = t̂χ_{s,λ}^{G}.

(c) If C_{G^{∗}}^{∘}(s) ⊆ L^{∗}, then χ_{s,λ}^{G} = ε_{G}ε_{L}R_{L}^{G}χ_{s,λ}^{L}.

(d) Let x ∈ N_{G}(L)^{F} and y ∈ N_{C_{G^{∗}}^{∘} (s)}(L^{∗})^{F} having the same image in N_{W (G)}(W (L))/W (L). Then ^{x}χ_{s,λ}^{L} = χ_{s,^{y}λ}^{L}.

Proof. The equality of (a) stems from the expression of the regular representation in terms of Deligne-Lusztig characters, see for instance [DM.2, 13.24]. As for (b), it is clear that the condition (2.1.R) for L = G and M a torus implies that χ_{st,λ}^{G} and t̂χ_{s,λ}^{G} have the same inner product with all Deligne-Lusztig characters R_{T}^{G}θ. But this implies that they are equal since, by [L.2, 5.1], [L.1, 4.23] applies to any series E(G^{F} , s) such that C_{G^{∗}}^{∘}(s)^{F} = C_{G^{∗}}(s)^{F}. The assertion (c) is proved by a similar argument.

In (d), assuming T ⊆ L ⊆ G and T^{∗} ⊆ L^{∗} ⊆ G^{∗} in duality, one identifies W (G) := W (G, T) with W (G^{∗}, T^{∗}) and W (L) := W (L, T) with W (L^{∗}, T^{∗}). Note that, by conjugacy of maximal tori in a reductive group, N_{G}(L)/L is isomorphic in a natural way with N_{W (G)}(W (L))/W (L). This gives sense to the claimed equality. The characters xχ_{s,λ}^{L} and χ_{s,^{y}λ}^{L} are identified by their scalar products with Deligne-Lusztig characters. Then with usual notations (see for instance [DM.2, 13.13]) one has ^{x}R_{S}^{L}(s) = R_{^{y}S}^{L}(s) for any maximal F -stable torus S of C_{L^{∗}}^{∘}(s). Now ⟨^{x}χ_{s,λ}^{L}, R_{S}^{L}(s)⟩_{L^{F}} = ⟨χ_{s,λ}^{L}, R_{^{y^{−1}} S}^{L}(s)⟩_{L^{F}} =

⟨λ, R_{^{y^{−1}} S}^{C_{L^{∗}}^{∘} (s)}1⟩_{C_{L^{∗}}^{∘} (s)^{F}} = ⟨^{y}λ, R_{S}^{C_{L^{∗}}^{∘} (s)}1⟩_{C_{L^{∗}}^{∘} (s)^{F}} = ⟨χ_{s,^{y}λ}^{L}, R_{S}^{L}(s)⟩_{L^{F}}.

2.2. Generalized Harish-Chandra theory.

Definition 2.4. Fix d ≥ 1 an integer. Let (G, F) be a connected reductive group defined over a finite field. An irreducible character χ of G^{F} is said to be d-cuspidal if and only if for every proper d-split Levi subgroup L of G, one has ^{∗}R_{L}^{G}ζ = 0. If L is a d-split Levi subgroup of G and ζ ∈ Irr(L^{F}) is d-cuspidal, then (L, ζ) is called a d-cuspidal pair of (G, F).

When d = 1, the above is just the usual concept of cuspidality in “Harish-Chandra theory” for reductive groups over a finite field, see for instance [DM.2, chapter 6].

Notation 2.5. Let (G, F) be a connected reductive group defined over IF_{q}. When L_{1}, L_{2} are F -stable Levi subgroups of G and χ_{i} ∈ Irr(L_{i}^{F}), one writes (L_{1}, χ_{1}) ≤ (L_{2}, χ_{2}) if and only if L_{1} ⊆ L_{2} and ⟨χ_{2}, R_{L_{1}}^{L_{2}}χ_{1}⟩_{L_{2}}^{F} ≠ 0.

The equality (2.1.R) on (L, M, λ, γ) implies that (M, χ_{s,μ}^{M}) ≤ (L, χ_{s,λ}^{L}) is equivalent to (C_{M^{∗}}^{∘}(s), μ) ≤ (C_{L^{∗}}^{∘}(s), λ).

The following relates d-cuspidality with Jordan decomposition.

Proposition 2.6. Let (G, F) a connected reductive group defined over a finite field and s ∈ (G^{∗})_{ss}^{F} such that C_{G^{∗}}^{∘}(s)^{F} = C_{G^{∗}}(s)^{F}. Let d ≥ 1. Assume there is a family of bijections λ → χ_{s,λ}^{L} (see Definition 2.1) satisfying (2.1.R) for any torus M. Let γ ∈ E(C_{G^{∗}}^{∘}(s)^{F} , 1) and χ = χ_{s,γ}^{G}. If χ is d-cuspidal, then

(i) γ is d-cuspidal and

(ii) Z^{∘}(C_{G^{∗}}^{∘}(s))_{ϕ_{d}} = Z^{∘}(G^{∗})_{ϕ_{d}}.

Assume now that (2.1.R) holds for any d-split Levi subgroup M, L = G and any irreducible characters λ, μ. Then (i) and (ii) imply that χ is d-cuspidal.

Proof. If Z^{∘}(C_{G^{∗}}^{∘}(s))_{ϕ_{d}} ≠ Z^{∘}(G^{∗})_{ϕ_{d}}, then C_{G^{∗}}(Z^{∘}(C_{G^{∗}}^{∘}(s))_{ϕ_{d}}) is a proper d-split Levi subgroup of G^{∗}. For a dual Levi subgroup L of G, one has^{∗}R_{L}^{G}χ ≠ 0 by Proposition 2.3.(c), hence χ is not d-cuspidal.

If χ is d-cuspidal, then one has ^{∗}R_{T}^{G}χ = 0 whenever T is an F -stable maximal torus of G such that T_{ϕ_{d}} ⊈ Z(G). By (2.1.R), this implies that ^{∗}R_{T^{∗}}^{C_{G^{∗}}^{∘} (s)}γ = 0 for all F -stable maximal torus of C_{G^{∗}}^{∘}(s) such that (T^{∗})_{ϕ_{d}} ⊈ Z(G^{∗}). Hence γ is d-cuspidal. Moreover there exists a torus T^{∗} such that ^{∗}R_{T^{∗}}^{C_{G^{∗}}^{∘} (s)}γ ≠ 0, so that (T^{∗})_{ϕ_{d}} = Z(G^{∗}) and that implies Z^{∘}(C_{G^{∗}}^{∘}(s))_{ϕ_{d}} = Z^{∘}(G^{∗})_{ϕ_{d}}.

If χ is not d-cuspidal, then there exists a proper d-split Levi subgroup M of G, in duality with M^{∗} containing s, and some μ ∈ E(C_{M^{∗}}^{∘}(s)^{F} , 1), such that (M, χ_{s,μ}^{M}) ≤ (G, χ). By (2.1.R) we get that ^{∗}R_{C_{M^{∗}}^{∘} (s)}^{C_{G^{∗}}^{∘} (s)}γ ≠ 0. On the other hand, the subgroups of the form C_{M^{∗}}^{∘}(s) = M^{∗}∩C_{G^{∗}}^{∘}(s) for d-split M^{∗} with s ∈ M^{∗} are precisely the d-split Levi subgroups of (C_{G^{∗}}^{∘}(s), F). So either γ is not d-cuspidal, or there is a proper d-split M^{∗} in G^{∗} such that C_{M^{∗}}^{∘}(s) = C_{G^{∗}}^{∘}(s). But the latter may happen only if Z^{∘}(C_{G^{∗}}^{∘}(s)) contains a ϕ_{d}-subgroup which is non central in G^{∗}.

The following is standard in the case d = 1.

Proposition 2.7. Let (G, F) be a connected reductive group defined over IF_{q}. Let d ≥ 1. Let s ∈ (G^{∗})^{F} semi-simple and such that C_{G^{∗}}^{∘}(s)^{F} = C_{G^{∗}}(s)^{F}. Assume that (2.1.R) holds for any pair (L, M) of d-split Levi subgroups of G and any (λ, μ). We then have the following :

(i) If χ ∈ E(G^{F} , s), then there exists a d-cuspidal pair (L, ζ) of (G, F) such that (L, ζ) ≤ (G, χ). Moreover (L, ζ) is unique up to G^{F} -conjugacy.

(ii) If (G, χ), (L, ζ) are as in (i) above, then

∗R_{L}^{G}χ = ⟨^{∗}R_{L}^{G}χ, ζ⟩_{L^{F}} ∑_{g∈N_{G^{F}} (L)/N_{G^{F}} (L,ζ)}^{g}ζ.

Proof. Let γ ∈ E(C_{G^{∗}}^{∘}(s)^{F} , 1) such that χ = χ_{s,γ}^{G} and let (L^{∗}_{s}, λ) be a d-cuspidal pair of (C_{G^{∗}}^{∘}(s), F) such that (L^{∗}_{s}, λ) ≤ (C_{G^{∗}}^{∘}(s), γ). One has L^{∗}_{s} = C_{C_{G^{∗}}^{∘} (s)}(Z^{∘}(L^{∗}_{s})_{ϕ_{d}}) and L_{s}^{∗} is uniquely defined modulo C_{G^{∗}}^{∘}(s)^{F} -conjugacy. Let T^{∗} be a maximal F -stable torus in L_{s}^{∗}, T a dual torus in G and L a d-split Levi subgroup of G in duality around (T, T^{∗}) with C_{G^{∗}}(Z^{∘}(L_{s}^{∗})_{ϕ_{d}}). Then ζ := χ_{s,λ}^{L} is defined and, by Proposition 2.6 and (2.1.R), (L, ζ) is a d-cuspidal pair such that (L, ζ) ≤ (G, χ). By our hypothesis and [BMM, 3.A], (L′, ζ′) ≤ (G, χ) is equivalent to the existence of a dual F -stable Levi subgroup L′^{∗} in G^{∗} such that s ∈ L′^{∗}, ζ′ = χ_{s,λ′}^{L′} (λ′ ∈ E(C_{L′^{∗}}^{∘}(s)^{F} , 1)) and, up to rational conjugacy, (L_{s}^{∗}, λ) ≤ (C_{L′^{∗}}^{∘}(s), λ′). This implies that Z^{∘}(C_{L′^{∗}}^{∘}(s))_{ϕ_{d}} ⊆ Z^{∘}(L_{s}^{∗})_{ϕ_{d}} up to G^{∗}^{F} -conjugacy. By Proposition 2.6, Z^{∘}(L_{s}^{∗})_{ϕ_{d}} = Z^{∘}(L^{∗})_{ϕ_{d}}. If (L′, ζ′) is also a d-cuspidal pair of (G, F), then Z^{∘}(L′^{∗})_{ϕ_{d}} ⊆ Z^{∘}(L^{∗})_{ϕ_{d}} up to G^{∗}^{F} -conjugacy so that L ⊆ L′ up to G^{F} -conjugacy.

If (L, ζ_{i}) are d-cuspidal pairs such that ζ_{i} = χ_{s,λ_{i}}^{L} and (L, ζ_{i}) ≤ (G, χ) (i = 1, 2), then (L_{s}^{∗}, λ_{i}) are d-cuspidal pairs such that (L_{s}^{∗}, λ_{i}) ≤ (C_{G^{∗}}^{∘}(s), γ). Therefore λ_{1} and λ_{2} are conjugate under N_{C_{G^{∗}}^{∘} (s)^{F}} (L_{s}^{∗}). By conjugacy of maximal tori in L^{∗}, λ_{1} and λ_{2} are conjugate under some y ∈ (N_{C_{G^{∗}}^{∘} (s)^{F}} (T^{∗}) ∩ N(L^{∗})) and p_{T^{∗}}(y) = w ∈ W (C_{G^{∗}}^{∘}(s), T^{∗})^{F}. Since L_{s}^{∗} is d-split in C_{G^{∗}}^{∘}(s), one has N_{C_{G^{∗}}^{∘} (s)^{F}} (L_{s}^{∗}) = N_{C_{G^{∗}}^{∘} (s)^{F}} (Z^{∘}(L_{s}^{∗})_{ϕ_{d}}) ⊆ N_{(G^{∗})^{F}} (Z^{∘}(L_{s}^{∗})_{ϕ_{d}}) = N_{(G^{∗})^{F}} (L^{∗}). With notations of Proposition 2.3.(d), w ∈ N_{W (G)}(W (L))/W (L). By the equality in Proposition 2.3.(d), ζ_{1} and ζ_{2} are conjugate under N_{G^{F}} (L). Thus one has ⟨R_{L}^{G}ζ_{1}, χ⟩_{G^{F}} = ⟨R_{L}^{G}ζ_{2}, χ⟩_{G^{F}}.

3. Blocks of reductive groups over a finite field We state below our main theorems on blocks of reductive groups over a finite field.

The hypothesis of Theorem 3.3 and Theorem 3.5 bring two main restrictions.

The first restriction is on the series E(G^{F} , s) for which our theorems are stated. This stems from the hypothesis (2.1.R) in which we assume commutation between Lusztig’s map R_{L}^{G} and a “Jordan decomposition” of E(G^{F} , s), this only when C_{G^{∗}}^{∘}(s)^{F} = C_{G^{∗}}(s)^{F}. When C_{G^{∗}}(s) is not connected, one may expect that some analogue of Jordan decomposition satisfies similar properties (see [L.2; DM.1]), and that the method described below can be adapted.

Also, our results are about ℓ-blocks of G^{F} for ℓ ∈ Γ(G, F) (see Notation 1.1). Bad primes actually provide exceptions, even for unipotent blocks : one has numerous examples of cuspidal unipotent characters of G_{ad}^{F} that are not of ℓ-defect zero for a bad prime ℓ dividing q − 1 (see the tables of degrees of unipotent characters in [Ca; L.1]).

The case treated in [CE.2] is one where the series does not necessarily satisfy C_{G^{∗}}^{∘}(s)^{F} = C_{G^{∗}}(s)^{F} and ℓ is good but not in Γ(G, F). 3.1. Some general results on blocks.

Let G be a finite group, let ℓ be a prime number and O a local complete finite extension of ZZ_{ℓ}. Let K denote its fraction field. Assume that K is algebraically closed. We consider ℓ-blocks as primitive idempotents in the center Z(O[G]) of the group algebra O[G]. Let k = O/J(O) and a ↦ \overline{a} the reduction map modulo J(O) from O[G] to k[G]. We recall the Brauer morphism associated with an ℓ-subgroup Q of G (see [AB, 2.4]) :

Br_{Q} : (O[G])^{Q} → k[C_{G}(Q)],

∑_{g∈G} λ_{g}g ↦ \overline{∑_{g∈C_{G}(Q)} λ_{g}g}.

The definitions and main properties of subpairs (Q, b), inclusion of subpairs, defect groups, self-centralizing subpairs and their canonical character are taken from [AB], [B], [Br.1].

The decomposition of the unit 1 ∈ Z(O[G]) as the sum of the blocks induces an orthogonal decomposition of the space of central functions CF(G, K) with respect to the usual inner product ⟨ , ⟩_{G}. For b a block of G and f ∈ CF(G, K), one denotes by b.f the corresponding projection of f. One also has the partition Irr(G) = ∪_{b}Irr(b). If χ ∈ Irr(G), then one denotes by b_{G}(χ) the ℓ-block of G such that χ ∈ Irr(b_{G}(χ)).

The following statement is well-known, see [B, 6], [Br.1, 1.C].

Proposition 3.1. If (P, b_{P}) is a subpair in G, then b_{P} is a block of N_{G}(P, b_{P}). Let (Q, b_{Q}) be a maximal b_{P} -subpair in N_{G}(P, b_{P}). Then (P, b_{P}) ◁ (Q, b_{Q}) in G and |Q : C_{Q}(P)| = |N_{G}(P, b_{P}) : C_{G}(P)|_{ℓ}. If (P, b_{P}) is self-centralizing, then C_{Q}(P) = Z(P).

The following describes some subpair inclusions in a situation reminiscent of wreath products.

Proposition 3.2. Let S, H, H̃, H_{1}, …, H_{m} be subgroups of a finite group. Assume the following :

• H ◁ H̃ and the quotient H̃/H is of order prime to ℓ,

• H = Z(H).H_{1} … H_{m} is a central product,

• S is an ℓ-group that normalizes H, H̃ and permutes the set {H_{1}, …, H_{m}},

• N_{S}(H_{i}) = C_{S}(H_{i}) for all 1 ≤ i ≤ m.

Then :

(i) If b is an S-fixed ℓ-block of H, then Br_{S}(b) ≠ 0. (ii) If b̃ is an S-fixed ℓ-block of H̃ with central defect groups (in H̃), then Br_{S}(b̃) ≠ 0.

Proof . (i) One may assume that the H_{i}’s are distinct. Define the action of S on {1, …, m} by H_{σ.i} = ^{σ}H_{i} for σ ∈ S, 1 ≤ i ≤ m. Denote S_{i} = {σ ∈ S ; σ.i = i} = N_{S}(H_{i}) = C_{S}(H_{i}) by the hypotheses. If 1 ≤ i ≤ m and h ∈ H_{i}, then Π_{σ∈S/S_{i}}^{σ}h ∈ H^{S}.

Since H is a central quotient of Z(H) × H_{1} × … × H_{m} and by the well-known results on blocks of direct products and central quotients [NT, § 5.8], one has b = e_{λ}b_{1} … b_{m} where λ ∈ Irr(Z(H)_{ℓ′}), e_{λ} = |Z(H)_{ℓ′}|^{−1} ∑_{z∈Z(H)_{ℓ′}} λ(z^{−1})z and each b_{i} is an ℓ-block of H_{i} which satisfies zb_{i} = λ(z)b_{i} for all z ∈ Z(H)_{ℓ′} ∩ H_{i} = Z(H_{i})_{ℓ′} (compute zb). If σ ∈ S then σb_{i} = b_{σ.i} : otherwise ^{σ}b_{i}.b_{σ.i} = 0 and ^{σ}b.b = 0.

Let us write b = ∑_{h∈H_{ℓ′}} β(h)h, b_{i} = ∑_{h∈(H_{i})_{ℓ′}} β_{i}(h)h in O[H] [NT, 3.6.22.(ii)]. For each i, choose h_{i} ∈ (H_{i})_{ℓ′} such that β_{i}(h_{i}) ∈ O \ J(O). This is possible since b_{i} is an idempotent. One may assume that h_{σ.i} = ^{σ}h_{i} for all σ,i since S_{i} fixes h_{i} and ^{σ}b_{i} = b_{σ.i}. Define h = h_{1} … h_{m}. Then h ∈ H^{S}, so it suffices to check that β(h) ∈ O \ J(O). As b = e_{λ}b_{1} … b_{m}, β(h) is the sum of all the products λ(z^{−1})β_{1}(x_{1}) … β_{m}(x_{m}) for z ∈ Z(H)_{ℓ′} and x_{i} ∈ (H_{i})_{ℓ′} such that zx_{1} … x_{m} = h_{1} … h_{m}. This equality implies x_{i} = h_{i}z_{i} with z_{i} ∈ Z(H_{i})_{ℓ′} such that zz_{1} … z_{m} = 1. But then λ(z^{−1})β_{1}(x_{1}) … β_{m}(x_{m}) = β_{1}(h_{1}) … β_{m}(h_{m}) since β_{i}(h_{i}z_{i}) = β_{i}(h_{i})λ(z_{i}^{−1}) for all i. Thus β(h) = β_{1}(h_{1}) … β_{m}(h_{m})Π_{i=1}^{m}|Z(H_{i})_{ℓ′}| ∈ O \ J(O) as claimed.

(ii) Let θ̃ ∈ Irr(b̃) denote the canonical character of b̃. By Clifford theory, ResH̃ ; H θ̃ = u ∑_{θ∈(H̃/H).θ_{0}} θ where u divides |H̃ : H| [NT, 3.5.12], θ_{0} ∈ Irr(H), and H̃/H acts on Irr(H) by conjugacy. Since S permutes the θ’s and H̃/H is of order prime to ℓ, one may assume that θ_{0} is S-fixed. Then Br_{S}(b_{H}(θ_{0})) ≠ 0 by (i).

The fact that H̃_{ℓ} ⊆ H also implies Z(H̃)_{ℓ} ⊆ H, so Z(H̃)_{ℓ} ⊆ Z(H). Moreover Z(H̃)_{ℓ} acts trivially on the representation space of θ̃, so each of the θ’s also has Z(H̃)_{ℓ} in its kernel. The fact that u is prime to ℓ gives θ(1)_{ℓ} = θ̃(1)_{ℓ} = |H : Z(H̃)|_{ℓ}. Therefore θ is the canonical character in a block of defect Z(H̃)_{ℓ} in H, and Z(H̃)_{ℓ} = Z(H)_{ℓ}.

Blocks with central defect groups are easy to express from one of their irreducible characters [NT, 3.6.22, 5.8.14]. One has b̃ = \frac{θ̃(1)}{|H̃:Z(H̃)_{ℓ}|} ∑_{h∈H̃_{ℓ′}} θ̃(h^{−1})h and b_{H}(θ) =

\frac{θ(1)}{|H:Z(H)_{ℓ}|} ∑_{h∈H_{ℓ′}} θ(h^{−1})h for all θ. Now, our claim is equivalent to θ̃(H̃^{S}) ⊈ J(O). It

suffices to check that (ResH̃ ; H θ̃)(H^{S}) ⊈ J(O), or equivalently Br_{S}(∑_{θ∈(H̃/H).θ_{0}} b_{H}(θ)) ≠ 0. Assume Br_{S}(∑_{θ∈(H̃/H).θ_{0}} b_{H}(θ)) = 0 and multiply this by Br_{S}(b_{H}(θ_{0})). One obtains Br_{S}(∑_{θ∈(H̃/H).θ_{0}} b_{H}(θ)b_{H}(θ_{0})) = 0 since Br_{S} is an algebra morphism on (O[H])^{S}. But the b_{H}(θ)’s are distinct since they are defined by distinct canonical characters. We get Br_{S}(b_{H}(θ_{0})) = 0, a contradiction.

3.2. Blocks of reductive groups over a finite field and their ordinary characters.

In Theorem 3.3, we describe the ℓ-blocks of G^{F} by means of e-cuspidal pairs of (G, F). Theorem 3.5 gives the structure of defect groups. The proofs are given in section 4.

Theorem 3.3. Let (G, F) be a connected reductive group defined over IF_{q}. Let ℓ ∈ Γ(G, F) (see Notation 1.1) and e be the order of q mod ℓ. Let s ∈ (G^{∗})_{ℓ′}^{F} be semi-simple and such that C_{G^{∗}}(s)^{F} = C_{G^{∗}}^{∘}(s)^{F}. Assume that (2.1.R) holds for any pair (L, M) of e-split Levi subgroup of G and any (λ, μ). Let (L, ζ) be an e-cuspidal pair such that L is in duality with a Levi subgroup L^{∗} of G^{∗}, s ∈ L^{∗} and ζ ∈ E(L^{F} , s). Then

(1) There is an ℓ-block of G^{F}, denoted by b_{G^{F}} (L, ζ), such that Irr(b_{G^{F}} (L, ζ))∩E(G^{F} , s) = {χ; ⟨R_{L}^{G}ζ, χ⟩_{G^{F}} ≠ 0}.

(2) If (L′, ζ′) is another e-cuspidal pair such that b_{G^{F}} (L′, ζ′) is defined, then b_{G^{F}} (L′, ζ′) = b_{G^{F}} (L, ζ) if and only if (L′, ζ′) and (L, ζ) are G^{F} -conjugate.

(3) Let t ∈ C_{G^{∗}}(s)_{ℓ}^{F}, let γ ∈ E(C_{G^{∗}}^{∘}(st)^{F} , 1). Let G(t) be a Levi subgroup of G in duality with C_{G^{∗}}^{∘}(t). Then χ_{st,γ}^{G} ∈ Irr(b_{G^{F}} (L, ζ)) if, and only if, for some χ ∈ Irr(G^{F}), one has ⟨R_{L}^{G}ζ, χ⟩_{G^{F}} ≠ 0 and ⟨R_{G(t)}^{G}χ_{s,γ}^{G(t)}, χ⟩_{G^{F}} ≠ 0.

Concerning maximal tori and Weyl groups, we use the following notations. Notation 3.4. When T ⊆ G is a maximal torus in a connected reductive group, one denotes by p_{T}: N_{G}(T) → W (G, T) the reduction map mod T. If H is a connected subgroup of G normalized by T, then we denote by W (G|H, T) the subgroup of W (G, T) generated by reflections associated with the roots of G relative to T that are orthogonal to all the roots of H relative to T.

Theorem 3.5. We keep the hypotheses and notations of Theorem 3.3. In addition, let M^{∗} = C_{G^{∗}}^{∘}(Z^{∘}(C_{L^{∗}}^{∘}(s))_{ℓ}^{F}). Let T^{∗} be a maximal F -stable torus of C_{L^{∗}}^{∘}(s). Then T^{∗} ⊆ M^{∗} ⊆ L^{∗} are all Levi subgroups. Let T ⊆ M ⊆ L be in duality with them. Then there is a subgroup N such that

(⋆) N ⊆ N_{G}(T), p_{T}(N) = W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗}) (see Notation 3.4), F (N) = N, and N ∩ T = T_{2}.

Take any N satisfying (⋆) and any Sylow ℓ-subgroup S in N^{F}. Denote Z = Z^{∘}(M)_{ℓ}^{F}. Then

(1) S ≅ p_{T}(S) and the latter is a Sylow ℓ-subgroup of W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F},

(2) Z.S is a semi-direct product with Z ˙◁ Z.S (see Notation 1.8), (3) ZS is a defect group of b_{G^{F}} (L, ζ).

Remark 3.6. We keep the same notations and hypotheses as above. The defect groups of b_{G^{F}} (L, ζ) are isomorphic to the semi-direct product of Z = Z^{∘}(M)_{ℓ}^{F} by a Sylow ℓ-subgroup of W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F}, the action being defined by restriction of the natural one of W (G^{∗}, T^{∗}) ≅ W (G, T) on T.

Let (G(s), F) be a connected reductive group defined over IF_{q} and in duality with C_{G^{∗}}^{∘}(s) around dual tori T and T^{∗}. One may assume that G(s) is defined by a root datum ((X(T), R_{s}, Y (T), R(C_{G^{∗}}^{∘}(s), T^{∗})), F), but G(s) is not necessarily a subgroup of G. Let L(s) be the e-split Levi subgroup of G(s) containing T and in duality with C_{L^{∗}}^{∘}(s). Then λ ∈ E(C_{L^{∗}}^{∘}(s)^{F} , 1) defines an element of E(L(s)^{F} , 1) that we still call λ and which is e-cuspidal. Theorems 3.3 and 3.5 apply to (L(s), λ) in G(s) (one has ℓ ∈ Γ(G(s), F) by [CE.1, 1.2]). We claim that the defect group of the unipotent block b_{G(s)^{F}} (L(s), λ) is isomorphic to the defect group of b_{G^{F}} (L, ζ).

By Proposition 1.3, C_{L^{∗}}^{∘}(s) = C_{C_{G^{∗}}^{∘} (s)}^{∘}(Z^{∘}(C_{L^{∗}}^{∘}(s))_{ℓ}^{F}). Hence the defect group of b_{G(s)^{F}} (L(s), λ) is the semi-direct product of Z^{∘}(L(s))_{ℓ}^{F} by a Sylow ℓ-subgroup of W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F} (Theorem 3.5). In those descriptions of the defect groups of b_{G^{F}} (L, ζ) and b_{G(s)^{F}} (L(s), λ), the Weyl group acting is the same with same action on T^{F}. By [Ca, 4.4.5] one has |Z^{∘}(M)_{ℓ}^{F} | = |Z^{∘}(M^{∗})_{ℓ}^{F} |, |Z^{∘}(C_{L^{∗}}^{∘}(s))_{ℓ}^{F} | = |Z^{∘}(L(s))_{ℓ}^{F} |. We show in Lemma 4.1 that C_{M^{∗}}^{∘}(s) = C_{L^{∗}}^{∘}(s) and Z^{∘}(M^{∗})_{ℓ}^{F} = Z^{∘}(C_{L^{∗}}^{∘}(s))_{ℓ}^{F}. Hence one has W (L(s), T) = W (C_{L^{∗}}^{∘}(s), T^{∗}) ⊆ W (M^{∗}, T^{∗}) = W (M, T), which implies Z(M) ⊆ Z(L(s)). Therefore Z^{∘}(L(s))_{ℓ}^{F} = Z^{∘}(M)_{ℓ}^{F}.

This isomorphism of defect groups gives further evidence to the conjecture made by several authors that the Jordan decomposition of characters induces a Morita equivalence between block algebras (see [Br.2, 2.3]), since, by a recent theorem of Puig proved under certain additional hypotheses, any two Morita equivalent block algebras must have isomorphic defect groups.

Concerning perfect isometries (see [Br.2, § 1]), our theorems easily imply the following. Suppose T^{∗} ⊆ C_{G^{∗}}^{∘}(s) ⊆ K^{∗} where K^{∗} is an F -stable Levi subgroup of G^{∗}. Let T ⊆ K where the latter is an F -stable Levi subgroup of G in duality with K^{∗}. Then a defect group of b_{G^{F}} (L, χ_{s,λ}^{L}) may be taken in K^{F}, it is a defect group of b_{K^{F}} (L ∩ K, χ_{s,λ}^{(L∩K)}), and the functor R_{K}^{G} defines a perfect isometry between those blocks (use [Br.2, 2.1]).

Remark 3.7. The statement (3) of Theorem 3.3 might be stated in terms of e-cuspidal pairs of G(t), as in [CE.1, 4.4.(iii)].

4. Proof of the Theorems 4.1. Three main steps.

We fix s ∈ G^{∗} and (L, ζ) as in Theorem 3.3.

The first inclusion step already allows to denote b_{G^{F}} (χ) = b_{G^{F}} (L, ζ) for any χ occuring in R_{L}^{G}ζ, see [AB, 3.4]. Step 1. If χ ∈ Irr(G^{F}) and (L, ζ) ≤ (G, χ) (see Notation 2.5), then ({1}, b_{G^{F}} (χ)) ⊆ (Z^{∘}(L)_{ℓ}^{F} , b_{L^{F}} (ζ)). Proof of Step 1. One has L = C_{G}^{∘}(Z^{∘}(L)_{ℓ}^{F}) and L^{F} = C_{G^{F}} (Z^{∘}(L)_{ℓ}^{F}) by Proposition 1.3.(ii), so the subpairs above make sense. The inclusion follows from [CE.1, 4.2] and Proposition 2.7 (ii).

Let now L^{∗} be an F -stable Levi subgroup of G^{∗}, containing s, in the dual class of the G^{F} -conjugacy class of L. Let T^{∗} ⊆ C_{L^{∗}}^{∘}(s) be a maximal F -stable torus, then s ∈ T^{∗} and T^{∗} is a maximal torus of G^{∗}. There is a dual torus T in L and a duality between L and L^{∗}, defined by root data relative to T, T^{∗}. Then, by Proposition 2.6, ζ = χ_{s,λ}^{L}, where λ is e-cuspidal unipotent in E(C_{L^{∗}}^{∘}(s)^{F} , 1), and Z^{∘}(L^{∗})_{ϕ_{e}} = Z^{∘}(C_{L^{∗}}^{∘}(s))_{ϕ_{e}}. Defining M^{∗} := C_{G^{∗}}^{∘}(Z^{∘}(C_{L^{∗}}^{∘}(s))_{ℓ}^{F}), this is a Levi subgroup of G by [CE.1, 2.1.(ii)].

Since T^{∗} ⊆ M^{∗} ⊆ L^{∗}, we can define an F -stable Levi subgroup M of G, dual to M^{∗} and such that T ⊆ M ⊆ L. Let Z = Z^{∘}(M)_{ℓ}^{F}. With these definitions one has

Lemma 4.1. (i) M and M^{∗} are E_{q,ℓ}-split and M = C_{G}^{∘}(Z). (ii) C_{M^{∗}}^{∘}(s) = C_{L^{∗}}^{∘}(s). (iii) Z^{∘}(M^{∗})_{ℓ}^{F} = Z^{∘}(C_{L^{∗}}^{∘}(s))_{ℓ}^{F} and it is of order |Z|. (iv) Z^{∘}(M)_{ϕ_{e}} = Z^{∘}(L)_{ϕ_{e}}.

(v) Denote ξ = χ_{s,λ}^{M} ∈ E(M^{F} , s). One has ζ = ε_{L}ε_{M}R_{M}^{L}ξ.

Proof. (i) follows from Proposition 1.3.(i)–(ii).

(ii) By (i) and (ii) of Proposition 1.3, L^{∗} is e-split and L^{∗} = C_{G^{∗}}^{∘}((Z^{∘}(L^{∗})_{ϕ_{e}})_{ℓ}^{F}) = C_{G^{∗}}^{∘}((Z^{∘}(C_{L^{∗}}^{∘}(s))_{ϕ_{e}})_{ℓ}^{F}) ⊇ M^{∗} ⊇ T^{∗}. We also have C_{M^{∗}}^{∘}(s) = C_{L^{∗}}^{∘}(s) since both are the connected centralizers in C_{G^{∗}}^{∘}(s) of Z^{∘}(C_{L^{∗}}^{∘}(s))_{ℓ}^{F}.

(iii) We have ℓ ∈ Γ(M^{∗}, F) ([CE.1, 1.2]) , so Z^{∘}(C_{L^{∗}}^{∘}(s))_{ℓ}^{F} ⊆ Z^{∘}(M^{∗}) ⊆ Z^{∘}(C_{M^{∗}}^{∘}(s)), and therefore by (ii) Z^{∘}(C_{L^{∗}}^{∘}(s))_{ℓ}^{F} = Z^{∘}(M^{∗})_{ℓ}^{F}.

(iv) We also get Z^{∘}(M^{∗})_{ϕ_{e}} ⊆ Z^{∘}(C_{L^{∗}}^{∘}(s))_{ϕ_{e}} = Z^{∘}(L^{∗})_{ϕ_{e}} while Z^{∘}(L^{∗}) ⊆ Z^{∘}(M^{∗}), so Z^{∘}(M^{∗})_{ϕ_{e}} = Z^{∘}(L^{∗})_{ϕ_{e}}. By [Ca, 4.4.5] and the definition of polynomial order in [CE.1, § 1.2], this equality implies that Z^{∘}(M)_{ϕ_{e}} and Z^{∘}(L)_{ϕ_{e}} have same polynomial order. They are therefore equal since one has an inclusion.

(v) Since ζ = χ_{s,λ}^{L} and C_{L^{∗}}^{∘}(s) ⊆ M^{∗}, one has ζ = ε_{L}ε_{M}R_{M}^{L}ξ by Proposition 2.3.(c).

Here is the second step on subpair inclusions : Step 2. (Z^{∘}(L)_{ℓ}^{F} , b_{L^{F}} (ζ)) ⊆ (Z, b_{M^{F}} (ξ)) and the latter is self-centralizing with canonical character ξ.

Proof of Step 2. By Lemma 4.1.(i) and Proposition 1.3, M^{F} = C_{G^{F}} (Z), so the above subpairs make sense.

The claimed inclusion is clearly equivalent to the inclusion ({1}, b_{L^{F}} (ζ)) ⊆ (Z, b_{M^{F}} (ξ)) in L^{F}. To prove it, we use a slight generalization of [CE.1, 4.2]. As in [BMi, 2.3], denote by P_{s} (resp. P_{s,ℓ}) the orthogonal projection associated with the subset E(M^{F} , s) ⊆ Irr(M^{F}) (resp. ∪_{t}E(M^{F} , t) ⊆ Irr(M^{F}) where t runs over the t ∈ (M^{∗})^{F} such that t_{ℓ′} = s). Let A be a subgroup of Z, H = C_{L}^{∘}(A), and χ_{H} ∈ Irr(H^{F}) be such that (M, ξ) ≤ (H, χ_{H}). Then [CE.1, 4.2] assumes the equality ^{∗}R_{M}^{H}χ_{H} = ⟨ξ,^{∗}R_{M}^{H}χ_{H}⟩_{M^{F}} _{∑g∈N_{H}(M)_{F} /N_{H}(M,ξ)_{F}} ^{g}ξ. We do not have this equality in general, but it is easily seen that the proof of [CE.1, 4.2] applies as soon as the image by P_{s,ℓ} of this equality holds (this is due to the fact that P_{s,ℓ} is the projection on a union of blocks by [BMi, 2.2], and therefore commutes with the decomposition map). Now the group H is a Levi subgroup [CE.1, 2.1.(ii)] and χ_{H} = χ_{s,λ}^{H} by Proposition 2.3.(c). In fact, Proposition 2.3.(c) implies that R_{M}^{H} is an isometry from ZZE(M^{F} , s) onto ZZE(H^{F} , s) and its inverse is given by P_{s} ∘^{∗}R_{M}^{H} on ZZE(H^{F} , s) (see also [G, 1.5]). Now, P_{s,ℓ}^{∗}R_{M}^{H}χ_{s,λ}^{H} = P_{s}^{∗}R_{M}^{H}χ_{s,λ}^{H} = ε_{M}ε_{H}ξ, whence our claim.

There remains to check that (Z, b_{M^{F}} (ξ)) is self-centralizing. One may apply [CE.1, 4.3] to λ since ℓ ∈ Γ(C_{L^{∗}}^{∘}(s), F) by [CE.1, 1.2]. So λ is in a block of defect group {1} of the quotient C_{L^{∗}}^{∘}(s)^{F} /Z(C_{L^{∗}}^{∘}(s))^{F}, hence λ(1)_{ℓ} = |C_{L^{∗}}^{∘}(s)^{F} /Z(C_{L^{∗}}^{∘}(s))^{F} |_{ℓ} = |C_{L^{∗}}^{∘}(s)^{F} /Z^{∘}(C_{L^{∗}}^{∘}(s))^{F} |_{ℓ}. By Proposition 2.3.(a), one has ξ(1)_{ℓ} = λ(1)_{ℓ}.|(M^{∗})^{F} : C_{M^{∗}}^{∘}(s)^{F} |_{ℓ}. Using (ii) and (iii) of Lemma 4.1, one gets ξ(1)_{ℓ} = |M^{F} /Z|_{ℓ}. As s is of order prime to ℓ, Z is in the kernel of ξ, so the above equality shows that ξ is of central defect group Z. Therefore Z = Z(M^{F})_{ℓ} and the pair (Z, b_{M^{F}} (ξ)) is self-centralizing with canonical character ξ as claimed.

The third step allows to reach a maximal subpair. As a first information on the defect groups, we have the following.

Lemma 4.2.  Let M, Z = Z^{∘}(M)_{ℓ}^{F}, ξ = χ_{s,λ}^{M} be as before.  Then

p_{T}^{−1}(W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗}))^{F} normalizes (Z, b_{M^{F}} (ξ)). So the defect groups of b_{G^{F}} (L, ζ) have order at least |Z|.|W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F} |_{ℓ}.

Proof of Lemma 4.2. Recall the identification W (G, T) = W (G^{∗}, T^{∗}). Take v ∈ N_{G^{F}} (T) and v^{∗} ∈ (N_{C_{G^{∗}}^{∘} (s)}(T^{∗}))^{F} such that p_{T^{∗}}(v^{∗}) = p_{T}(v) ∈ W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F}. Then p_{T}(v) fixes each root of C_{L^{∗}}^{∘}(s) relative to T^{∗}, so v normalizes Z^{∘}(C_{L^{∗}}^{∘}(s))_{ℓ}^{F} and M^{∗}. Therefore v permutes the roots of M, and it normalizes M and Z. We have ^{v}χ_{s,λ}^{M} = χ_{s,^{v^{∗}} λ}^{M} by Proposition 2.3.(d). Since v^{∗} fixes the roots of C_{M^{∗}}^{∘}(s), it acts on [C_{M^{∗}}^{∘}(s), C_{M^{∗}}^{∘}(s)] by an interior automorphism. Hence v^{∗} ∈ (C_{G^{∗}}([C_{M^{∗}}^{∘}(s), C_{M^{∗}}^{∘}(s)])[C_{M^{∗}}^{∘}(s), C_{M^{∗}}^{∘}(s)])^{F}. The natural F -isomorphism sends v^{∗}C_{G^{∗}}([C_{M^{∗}}^{∘}(s), C_{M^{∗}}^{∘}(s)]) into ([C_{M^{∗}}^{∘}(s), C_{M^{∗}}^{∘}(s)]_{ad})^{F}. Then v^{∗} fixes λ since restriction induces a bijection between the sets of irreducible unipotent characters of C_{M^{∗}}^{∘}(s)^{F} and of ([C_{M^{∗}}^{∘}(s), C_{M^{∗}}^{∘}(s)]_{ad})^{F} [CE.1, 3.1]. This proves the first assertion. On the other hand, we have C_{G^{∗}}^{∘}(s) ∩ M^{∗} = C_{M^{∗}}^{∘}(s) = C_{L^{∗}}^{∘}(s) by Remark 2.2 and Lemma 4.1.(ii), so W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗}) ∩ W (M^{∗}, T^{∗}) = 1 and p_{T}^{−1}(W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})) ∩ M = T. Now, applying Proposition 3.1 to the selfcentralizing subpair (Z, b_{M^{F}} (ξ)), we get that the defect groups of b_{G^{F}} (L, ζ) have order at least |Z|.|p_{T}^{−1}(W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗}))^{F} : T^{F} |_{ℓ}. By Lang theorem, this gives the second statement of the Lemma.

By [C, 4.2], there is a subgroup N′ ⊆ N_{G}(T) such that N_{G}(T) = T.N′, F (N′) = N′ and N′ ∩ T^{F} = T_{2}^{F}. Now, N′ ∩ p_{T}^{−1}(W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})) clearly satisfies condition (⋆) of Theorem 3.5. Step 3. Take any N satisfying condition (⋆) of Theorem 3.5 and any Sylow ℓ-subgroup S of N^{F}. Then S normalizes Z and S ≅ p_{T}(S), a Sylow ℓ-subgroup of W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F}. Moreover, Z ˙◁ ZS (Notation 1.8) and there is a maximal subpair in G^{F} of the form (ZS, b_{ZS}) such that (Z, b_{M^{F}} (ξ)) ◁ (ZS, b_{ZS}) in G^{F}.

Proof of Step 3. S normalizes (Z, b_{M^{F}} (ξ)) thanks to Lemma 4.2. We have S ≅ p_{T}(S) since T ∩ N^{F} is a finite 2-group and ℓ ≠ 2. To prove that p_{T}(S) is a Sylow ℓ-subgroup of W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F} = p_{T}(N)^{F}, it suffices to check that p_{T}(N)^{F} = p_{T}(N^{F}) or equivalently (1 − F)(T ∩ N) = T ∩ N. One has T ∩ N = T_{2} and by Lang theorem T_{2} = (1 − F)(T)_{2}. But the latter is (1 − F)(T_{2}) since the elements of finite order in T correspond by (1 − F) whose kernel is finite.

We prove the remaining part of Step 3 by induction on dim G. Having fixed N and S, take (D, b_{D}) a maximal subpair of G^{F} containing (Z, b_{M^{F}} (ξ)). One has Z(D) ⊆ Z by a standard property of self-centralizing subpairs.

Assume that Z(D) ⊈ G_{a} (see Notation 1.5).

Then there exists a proper e-split Levi subgroup H of G such that C_{G}^{∘}(Z(D)) ⊆ H [CE.1, 2.5]. One has M ⊆ H since M = C_{G}^{∘}(Z) by Lemma 4.1.(i). Then L ⊆ H since H is e-split and Z^{∘}(H)_{ϕ_{e}} ⊆ Z^{∘}(M)_{ϕ_{e}} = Z^{∘}(L)_{ϕ_{e}} by Lemma 4.1.(iv). There is a Levi subgroup H^{∗} of G^{∗}, dual to H, such that L^{∗} ⊆ H^{∗}, and C_{H^{∗}}^{∘}(s)^{F} = C_{H^{∗}}^{F} (s) (see Remark 2.2). Now (L, ζ) is an e-cuspidal pair of (H, F), M is defined from L in H just like in G, b_{H^{F}} (L, ζ) is well defined, and the inclusions ({1}, b_{H^{F}} (L, ζ)) ⊆ (Z^{∘}(L)_{ℓ}^{F} , b_{L^{F}} (ζ)) ⊆ (Z, b_{M^{F}} (ξ)) (Steps 1 and 2) and (Z, b_{M^{F}} (ξ)) ⊆ (D, b_{D}) hold in H^{F}. The group N ∩H satisfies the condition (⋆) of Theorem 3.5 in H. Note that the hypotheses of Definition 2.1 transfer to any e-split Levi subgroup of G (Remark 2.2). By induction, Step 3 is true in H^{F}. A first consequence is |D| = |Z|.|W (C_{H^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F} |_{ℓ}. Along with Lemma 4.2, this implies that the index of W (C_{H^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F} in W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F} is prime to ℓ and |D| = |Z|.|S|. Then |N^{F} : N^{F} ∩ H| is prime to ℓ, so there exists v ∈ N^{F} such that ^{v}S ⊆ N^{F} ∩ H. Step 3 in H also tells us that we have a subpair inclusion (Z, b_{M^{F}} (ξ)) ◁ (Z.^{v}S, b) in H^{F} with Z ˙◁ Z.^{v}S. Using Lemma 4.2, this implies (Z, b_{M^{F}} (ξ)) ◁ (ZS,^{v^{−1}}b) in G^{F} and the latter is maximal since |D| = |Z|.|S|. Moreover Z ˙◁ ZS, whence Step 3 for G. 4.2. The “minimal” case.

Assume now that Z(D) ⊆ G_{a}. We prove Step 3 in that case. The condition Z(D) ⊆ G_{a} imposes strong constraints to the situation.

Lemma 4.3. Assume Z(D) ⊆ G_{a}. Then

(i) D ⊆ G_{a}, L ⊇ M ⊇ G_{b}, (ii) C_{L^{∗}}^{∘}(s) ∩ T^{∗}(G^{∗})_{a} = T^{∗} and it is a diagonal torus of C_{T^{∗}(G^{∗})_{a}}^{∘}(s), (iii) T^{∗}(G^{∗})_{a} ∩ M^{∗} = C_{T^{∗}}_{(G^{∗})_{a}}(T^{∗}_{ϕ_{E_{q,ℓ}}}) and T^{∗} is a Coxeter torus of T^{∗}(G^{∗})_{a} ∩ M^{∗}, (iv) Z = (T ∩ G_{a})_{ℓ}^{F}, S ⊆ G_{a}^{F} and Z ˙◁ ZS (see Notation 1.8).

Proof. By [CE.1, 2.3 (ii)], D ⊆ G_{a}, so one obtains Z ⊆ G_{a}, M = C_{G}^{∘}(Z) ⊇ G_{b}, and G^{∗}_{b} ⊆ M^{∗} ⊆ L^{∗}. This yields (i). The pair (C_{L^{∗}}^{∘}(s), λ) is unipotent e-cuspidal in C_{G^{∗}}^{∘}(s) (Proposition 2.6). This takes place in the central product C_{(G^{∗})_{a}}^{∘}(s)C_{(G^{∗})_{b}}^{∘}(s), so one has a unipotent e-cuspidal pair ((G^{∗})_{a}∩C_{L^{∗}}^{∘}(s), λ_{a}) in C_{(G^{∗})_{a}}^{∘}(s) = C_{(G^{∗})_{a}}^{∘}(s)_{a} (Notation 1.5.(i)) and therefore (G^{∗})_{a} ∩ C_{L^{∗}}^{∘}(s) is a diagonal torus in C_{(G^{∗})_{a}}^{∘}(s) [CE.1, 3.3.(i)]. It contains hence equals T^{∗} ∩ (G^{∗})_{a}, whence also (ii). This implies Z(M^{∗} ∩ T^{∗}(G^{∗})_{a})_{ℓ}^{F} = (T^{∗})_{ℓ}^{F} by definition of M^{∗}, hence M^{∗} ∩ T^{∗}(G^{∗})_{a} = C_{T^{∗}}_{(G^{∗})_{a}}((T^{∗})_{ϕ_{E_{q,ℓ}}}) by Proposition 1.3 (ii). By Lemma 4.1.(iii), one has (T^{∗} ∩ (G^{∗})_{a})_{ℓ}^{F} ⊆ Z^{∘}(M^{∗}). Thanks to Notation 1.5.(ii), this implies that T^{∗} ∩ (G^{∗})_{a} is a Coxeter torus of M^{∗} ∩ (G^{∗})_{a}, whence (iii).

Using the inclusion (T^{∗} ∩ (G^{∗})_{a})_{ℓ}^{F} ⊆ Z^{∘}(M^{∗}), the fact that T^{∗} ∩ (G^{∗})_{b} is connected and centralizes M^{∗} ∩ T^{∗}(G^{∗})_{a}, along with the definition of (G^{∗})_{a} and (G^{∗})_{b}, one gets (T^{∗})_{ℓ}^{F} = (T^{∗} ∩ (G^{∗})_{a})_{ℓ}^{F} (T^{∗} ∩ (G^{∗})_{b})_{ℓ}^{F} ⊆ Z^{∘}(M^{∗} ∩ T^{∗}(G^{∗})_{a}) and |(T^{∗})_{ℓ}^{F} | ≤ |Z^{∘}(M^{∗} ∩ T^{∗}(G^{∗})_{a})_{ℓ}^{F} |. On the other hand, M^{∗} ∩T^{∗}(G^{∗})_{a} and M∩TG_{a} are in duality, so [Ca, 4.4.5] |T_{ℓ}^{F} | ≤ |Z^{∘}(M ∩ TG_{a})_{ℓ}^{F} |. Also T ⊇ Z^{∘}(M ∩ TG_{a}) and therefore T_{ℓ}^{F} = Z^{∘}(M ∩ TG_{a})_{ℓ}^{F}. Then (T ∩ G_{a})_{ℓ}^{F} centralizes M, so Z = (T ∩ G_{a})_{ℓ}^{F}.

As L^{∗} ⊇ (G^{∗})_{b}, then W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗}) ⊆ W (T^{∗}(G^{∗})_{a}, T^{∗}) and N ⊆ TG_{a}. Thus N∩G_{a}◁N with abelian quotient. On the other hand, N/[N, N] is generated by its elements whose orders are finite powers of 2 since N/T_{2} = N/T∩N ≅ W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗}) is generated by involutions. So the finite subgroups of N/N ∩ G_{a} are of order prime to ℓ and therefore S ⊆ N ∩ G_{a}.

Let us check Z ˙◁ ZS. For groups of type A every prime number is good, so C_{T^{∗}(G^{∗})_{a}}^{∘}(s) is a Levi subgroup of T^{∗}(G^{∗})_{a} containing T^{∗} [CE.1, 2.1.(ii)]. So, one may define G(s), the F -stable Levi subgroup of TG_{a} containing T and in duality with C_{T^{∗}(G^{∗})_{a}}^{∘}(s). Then N ⊆ G(s). Moreover, defining G_{a}(s) := G(s) ∩ G_{a}, this is a Levi subgroup of G_{a} and T.G_{a}(s) = G(s). Then T (resp. T ∩ G_{a}) is diagonal in TG_{a}(s) (resp. G_{a}(s)) by (ii) and ZS ⊆ G_{a}(s) as we have just established. Then (T ∩ G_{a})_{ℓ}^{F} = Z ⊆ ZS in N_{G_{a}(s)^{F}} (T). Now Proposition 1.9 gives Z ˙◁ ZS as claimed, since ℓ ∈ Γ(G_{a}, F) ⊆ Γ(G_{a}(s), F) (Lemma 1.2 and [CE.1, 1.2]).

Lemma 4.4. Assume Z(D) ⊆ G_{a}, then there is a subpair (ZS, b_{ZS}) such that (Z, b_{M^{F}} (ξ))◁ (ZS, b_{ZS}).

Proof. Thanks to Lemma 4.3.(ii), (iii), we may apply Proposition 1.6 and Corollary 1.7 in T^{∗}(G^{∗})_{a} with C = C_{T^{∗}(G^{∗})_{a}}^{∘}(s) and K = M^{∗} ∩ T^{∗}(G^{∗})_{a} both containing T^{∗}. This describes the type of M^{∗} ∩ T^{∗}(G^{∗})_{a} and the action of W (T^{∗}C_{(G^{∗})_{a}}^{∘}(s), T^{∗})^{F} on the roots of M^{∗} ∩ T^{∗}(G^{∗})_{a}. Let Π_{ω}M_{ω} be the decomposition of [M, M] ∩ G_{a} into its rationally irreducible components, as in Proposition 1.6. We apply Proposition 3.2 with H̃ = M^{F}, b̃ = b_{M^{F}} (ξ), H = Z^{∘}(G_{a})^{F} .Π_{ω}M_{ω}^{F} .G_{b}^{F}, the H_{i}’s being the subgroups M_{ω}^{F} and G_{b}^{F}. By Corollary 1.7, Z(M_{ω}) is an ℓ′-group for any component M_{ω}, and so is Z(G_{b})^{F}. Now it follows from Lang’s theorem and [St, 4.5] that H̃/H is an ℓ′-group. As S ⊆ G_{a} by Lemma 4.3.(iv), S centralizes G_{b}. By Lemma 4.2, S normalizes M and b_{M^{F}} (ξ), so S permutes the M_{ω}’s and we may conclude once we have checked N_{S}(M_{ω}) ⊆ C_{S}(M_{ω}) for all ω. By Proposition 1.6, if x ∈ N_{S}(M_{ω}) then p_{T}(x) fixes all the coroots of M_{ω}, so it fixes all the roots of M_{ω}. Since M_{ω} is semi-simple, this implies that each element of N_{S}(M_{ω}) acts on M_{ω} in the same way as an element of T∩M_{ω}. So N_{S}(M_{ω}) ⊆ ((T∩M_{ω})C_{G}(M_{ω}))_{ℓ}^{F}. By [St, 4.5] again, ((T ∩ M_{ω})C_{G}(M_{ω}))^{F} /(T ∩ M_{ω})^{F} C_{G}(M_{ω})^{F} is a quotient of Z(M_{ω}), so it is of order prime to ℓ and therefore N_{S}(M_{ω}) ⊆ (T∩M_{ω})_{ℓ}^{F} C_{G}(M_{ω})_{ℓ}^{F}. But (T∩M_{ω})_{ℓ}^{F} ⊆ Z(M) (Lemma 4.3.(iv)), so N_{S}(M_{ω}) ⊆ C_{S}(M_{ω}). Proposition 3.2 applies and we get our claim by the definition of normal inclusion.

Step 3 now amounts to prove that (ZS, b_{ZS}) is maximal. Since Z ˙◁ ZS (Lemma 4.3.(iv)), we may apply [C, 3.1.(1)], so it suffices to prove that if (E, b_{E}) satisfies (Z, b_{M^{F}} (ξ)) ◁ (E, b_{E}), then |E| ≤ |ZS|. As (Z, b_{M^{F}} (ξ)) is self-centralizing with canonical character ξ (Step 2) and E ⊆ G_{a} (Lemma 4.3.(i)), then E ∩ M^{F} = Z and E/Z injects in N_{(TG_{a})^{F}} (M, ξ)/M^{F}. One has N_{G}(M)/M ≅ N_{G}(M) ∩ N_{G}(T)/N_{M}(T) by conjugacy of maximal tori. Let w ∈ W (G, T) be the image modulo T of some ℓ-element of N_{(TG_{a})^{F}} (M, ξ) ∩ N_{G}(T). Then w fixes ξ = χ_{s,λ}^{M}, so it fixes the Lusztig series defined by s. This implies that w stabilizes the intersection with T^{∗} of the M^{∗}-conjugacy class of s. By [DM.2, 0.12.(iv)], w ∈ W (T^{∗}(G^{∗})_{a}, T^{∗}) ∩ (W (M^{∗}, T^{∗})C_{W (G^{∗},T^{∗})}(s))^{F}. As s is of order prime to ℓ, so is C_{G^{∗}}(s)/C_{G^{∗}}^{∘}(s) ≅ C_{W (G^{∗},T)}(s)/W (C_{G^{∗}}^{∘}(s), T^{∗}) [CE.1, 2.1.(i)]. Hence w ∈ (W (M^{∗}, T^{∗})W (C_{G^{∗}}^{∘}(s), T^{∗}))^{F}. By Remark 2.2, W (C_{G^{∗}}^{∘}(s), T^{∗}) ∩ W (M^{∗}, T^{∗}) = W (C_{M^{∗}}^{∘}(s), T^{∗}). Using projection on W (T^{∗}(G^{∗})_{a}, T^{∗}), one sees that E/Z injects in the group of F -fixed points of N_{W (T^{∗}C_{(G^{∗})a}^{∘} (s),T^{∗})}(W (T^{∗}C_{M^{∗}∩(G^{∗})_{a}}^{∘}(s), T^{∗}))/W (T^{∗}C_{M^{∗}∩(G^{∗})_{a}}^{∘}(s), T^{∗}). But W (T^{∗}C_{M^{∗}∩(G^{∗})_{a}}^{∘}(s), T^{∗}) = {1} by Lemma 4.1.(ii) and Lemma 4.3.(ii). Then E/Z injects in W (T^{∗}C_{(G^{∗})_{a}}^{∘}(s), T^{∗})^{F}, a subgroup of W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F} (Lemma 4.3.(ii) again). On the other hand, |S| is the order of a Sylow ℓ-subgroup of W (C_{G^{∗}}^{∘}(s)|C_{L^{∗}}^{∘}(s), T^{∗})^{F} by what we have already proved of Step 3. So |E| ≤ |ZS| and the proof of Step 3 is now complete.

4.3. End of the proof.

Steps 1, 2, 3 have allowed us to define blocks b_{G^{F}} (L, ζ) such that (L, ζ) ≤ (G, χ) implies χ ∈ Irr(b_{G^{F}} (L, ζ)). With this definition of b_{G^{F}} (L, ζ), Theorem 3.5 follows by Step 3.

Let us now prove the injectivity of (L, ζ) ↦ b_{G^{F}} (L, ζ). If (L, ζ) and (L′, ζ′) are two e-cuspidal pairs defining the blocks b_{G^{F}} (L, ζ) and b_{G^{F}} (L′, ζ′) which satisfy Steps 1, 2, 3, then we have ({1}, b_{G^{F}} (L, ζ)) ⊆ (Z, b_{M^{F}} (ξ)) ◁ (ZS, b_{ZS}) and ({1}, b_{G^{F}} (L′, ζ′)) ⊆ (Z′, b_{(M′)^{F}} (ξ′)) ◁ (Z′S′, b_{Z′S′}) with evident notations. If b_{G^{F}} (L, ζ) = b_{G^{F}} (L′, ζ′), then, by conjugacy of maximal subpairs, there is x ∈ G^{F} such that ^{x}(Z′S′, b_{Z′S′}) = (ZS, b_{ZS}). But then ^{x}(Z′, b_{(M′)^{F}} (ξ′)) ◁ (ZS, b_{ZS}), while Z ˙◁ ZS. So ^{x}Z′ ⊆ Z. Whence by symmetry xZ′ = Z and therefore ^{x}(Z′, b_{(M′)^{F}} (ξ′)) = (Z, b_{M^{F}} (ξ)) (see [AB, 3.4]). By Step 2, this implies ^{x}(M′, ξ′) = (M, ξ). Then ^{x}L′ = L by Lemma 4.1.(iv) and ^{x}ζ′ = ζ by (v) of the same Lemma, whence Theorem 3.3.(2).

The equality in Theorem 3.3.(1) also follows since we already had an inclusion.

There now remains to check that Irr(b_{G^{F}} (L, ζ)) is as stated in Theorem 3.3.(3). By [BMi, 2.1], Irr(b_{G^{F}} (L, ζ)) ⊆ ∪_{t}E(G^{F} , st), the union being on the ℓ-elements t ∈ C_{G^{∗}}^{∘}(s)^{F}.

Note that, for such a t, C_{G^{∗}}^{∘}(st) = C_{C_{G^{∗}}^{∘} (s)}^{∘}(t) is a Levi subgroup of C_{G^{∗}}^{∘}(s) (Remark 2.2 and [CE.1, 2.1 (ii)]). Assume t ∈ T^{∗} and consider χ′ = χ_{st,γ}^{G} ∈ E(G^{F} , st) with γ ∈ E(C_{G^{∗}}^{∘}(st)^{F} , 1). Then χ_{s,γ}^{G(t)} ∈ E(G(t)^{F} , s) and R_{G(t)}^{G}χ_{s,γ}^{G(t)} is a non-zero element of ZZE(G^{F} , s). So, the following Lemma completes the proof of Theorem 3.3.(3).

Lemma 4.5. Keep hypotheses and notations as above. Then for any χ ∈ E(G^{F} , s), such that ⟨R_{G(t)}^{G}χ_{s,γ}^{G(t)}, χ⟩_{G^{F}} ≠ 0, one has b_{G^{F}} (χ_{st,γ}^{G}) = b_{G^{F}} (χ) Proof . The proof is very similar to the one in [CE.1] about unipotent characters. Since C_{G^{∗}}^{∘}(t) is a Levi subgroup of G^{∗} [CE.1, 2.1 (ii)], let G(t) ⊇ T be an F -stable Levi subgroup of G in duality with C_{G^{∗}}^{∘}(t). By Proposition 2.3.(b) and (c), one has χ_{st,γ}^{G} = ε_{G}ε_{G(t)}R_{G(t)}^{G}χ_{st,γ}^{G(t)} and χ_{st,γ}^{G(t)} = t̂χ_{s,γ}^{G(t)}. Then d^{1}χ_{st,γ}^{G(t)} = d^{1}χ_{s,γ}^{G(t)} where d^{1} is the “decomposition map” which multiplies the central functions with the characteristic function of the elements of order prime to ℓ. Using the compatibility between d^{1} and the Lusztig map [BMi, 3.5], one obtains d^{1}χ_{st,γ}^{G} = d^{1}(R_{G(t)}^{G}χ_{s,γ}^{G(t)}). By [G, Theorem A] the d^{1}χ for χ ∈ E(G^{F} , s) are linearly independent. Applying Brauer’s second Main Theorem [NT, 5.4.2], this yields b_{G^{F}} (χ_{s,γ}^{G}) = b_{G^{F}} (χ) whenever ⟨R_{G(t)}^{G}χ_{s,γ}^{G(t)}, χ⟩_{G^{F}} ≠ 0.

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