Published in Invent. math. 117, 149-164 (1994).

On unipotent blocks and their ordinary characters.

Marc Cabanes, Michel Enguehard DMI-ENS, 45 rue d’Ulm, F-75005 PARIS

Introduction

There is a great deal of evidence suggesting that block theory for finite reductive groups should lead to an enrichment of the Harish-Chandra theory for those groups (see [S] 5.1, [FS.2], [P.2] 3.4, [H] 6.1.10).

An e-generalized Harish-Chandra theory (for e a positive integer) is introduced in [BMM] 1.A. The context is as follows. Let G be a connected reductive group defined over IF_{q} with associated Frobenius endomorphism F, let e ≥ 1 an integer. When L is an F -stable Levi subgroup of G, one denotes by R_{L}^{G} and ^{∗}R_{L}^{G} the Lusztig functor and its adjoint. A unipotent e-pair (L, χ) is defined by an e-split Levi subgroup L of G and a unipotent character χ ∈ E(L^{F} , 1). One writes (L_{1}, χ_{1}) ≤ (L_{2}, χ_{2}) if and only if L_{1} ⊂ L_{2} and ⟨χ_{1},^{∗}R_{L_{1}}^{L_{2}}χ_{2}⟩_{L_{1}^{F}} ≠ 0 (see 3.1). The notion of unipotent e-cuspidal pairs and the resulting partition of E(G^{F} , 1) are defined as in the classical case (corresponding to e = 1).

If ℓ is a prime not dividing q and e is the order of q mod. ℓ, it seems natural in connection with blocks, to ask if the partition of the unipotent characters E(G^{F} , 1) resulting from “e-theory” coincides with ℓ-blocks. The blocks of G^{F} which are defined by unipotent characters are called unipotent ℓ-blocks. We prove

Theorem. Let (G, F) be a connected reductive group defined over IF_{q} with a dual group (G^{∗}, F). Let ℓ be a prime not dividing q. Assume ℓ is odd, good for G, and ℓ ≠ 3 if ^{3}D_{4} is involved in (G, F). Let e be the order of q mod. ℓ. There is a bijection (L, λ) ↦ b_{G^{F}} (L, λ) between the G^{F} -conjugacy classes of unipotent e-cuspidal pairs of (G, F) and the set of unipotent ℓ-blocks of G^{F}, which satisfies the following :

(i) Irr(b_{G^{F}} (L, λ)) ∩ E(G^{F} , 1) = {χ ; (L, λ) ≤ (G, χ)}. (ii) Any Sylow ℓ-subgroup of C_{G}^{∘}([L, L])^{F} is a defect group of b_{G^{F}} (L, λ).

(iii) If t ∈ (G^{∗})_{ℓ}^{F}, then denote by G(t) ⊂ G an F -stable Levi subgroup in duality with C_{G^{∗}}^{∘}(t) and by t̂the associated linear character of G(t)^{F}. With those notations, the elements of Irr(b_{G^{F}} (L, λ)) are the characters occuring in some R_{G(t)}^{G}(t̂χ_{t}), where t ∈ (G^{∗})_{ℓ}^{F}, χ_{t} ∈ E(G(t)^{F} , 1), and (L_{t}, λ_{t}) ≤ (G(t), χ_{t}) for an e-cuspidal pair (L_{t}, λ_{t}) of G(t) such that [L, L] = [L_{t}, L_{t}] and Res_{[L,L]^{F}}^{L^{F}} λ = Res_{[L_{t},L_{t}]^{F}}^{L_{t}^{F}} λ_{t}.

The restriction on ℓ actually corresponds to counter-examples : bad primes will be treated in a forthcoming publication (see also [CE] 4.1). One can also see that, except for a few cases, assertions (i) and (ii) of the Theorem could be derived from existing results on particular groups ([FS.1], [FS.3] for classical groups, [S], [H] for exceptional groups) suitably combined with [CE] Proposition 11. But the present proof is more intrinsic.

We use extensively local block theory as introduced in [AB] and used in [P.1],

[Br] (see also the introduction of [C]). This method leads us to pay attention to several unnoticed properties which provide essential ingredients. The central product decomposition G = G_{a}.G_{b} (Notation 2.3), defined with respect to a good prime ℓ not dividing q, separates two distinct phenomena. In (G_{a}, F) there is only one e-cuspidal pair and all unipotent characters are in the principal ℓ-block. In (G_{b}, F) no rational ℓ-element is central and the centralizers of rational ℓ-elements embed in proper e-split Levi subgroups (Proposition 2.5). This last fact is in turn related to a property of centers of subgroups of maximal rank of G (Proposition 1.6). The decomposition G = G_{a}.G_{b} plays a major rôle in many inductive proofs, the case G = G_{a} appearing as “minimal”. The derived subgroups of Levi subgroups and their centralizers also have remarkable properties (Proposition 1.7, Proposition 3.5).

Our work is almost self-contained concerning block structure and uses no combinatorics.

Notations. If σ is an endomorphism of a group G, then G^{σ} := {g ∈ G ; σ(g) = g} denotes the set of fixed points of σ in G, and [G, σ] the subgroup generated by the set of g^{−1}σ(g) (g ∈ G). We use as well the notation ^{g}a to denote the action of g ∈ G on a ∈ A when G acts (on the left) on A, especially when the action is derived from the action of G on itself by conjugation.

If n, m ≥ 1 are integers, one denotes by (n, m) their g.c.d. When π is a set of prime numbers (or a prime number), π′ is the complement in the set of all prime numbers. Then the π-contribution of a natural integer n is denoted by n_{π}. A πgroup is any finite group such that its order equals its π-contribution, a π-element is any element which generates a π-group and Y_{π} is the set of π-elements of Y.

When G is a finite group and K is an algebraically closed field of characteristic zero, one denotes Irr(G) ⊂ CF(G, K) the sets of irreducible characters of the group algebra KG and class functions endowed with the usual inner product ⟨ , ⟩_{G} (see §4 for notations relative to blocks).

Algebraic groups are usually denoted by boldface letters. To each connected reductive affine algebraic group G is associated a dual group G^{∗}, a simply connected group G_{sc} with an isogeny G_{sc} → [G, G] and an adjoint group G_{ad}, isomorphic to G/Z(G). The identity component (i.e. the connected component containing the identity) of a non-connected algebraic group H is denoted by H^{∘}.

1. The groups.

1.1. Reductive groups and good primes.

In the following we fix some vocabulary. Let F be the algebraic closure of a finite field IF_{q} of q elements and characteristic p. Let G be a connected reductive affine algebraic group over F and let F: G → G be an endomorphism which is a Frobenius morphism for a IF_{q}-rational structure of G. We abbreviate this by saying that (G, F) is a connected reductive group defined over IF_{q}.

To every orbit ω of F on the set of connected components of the Dynkin diagram of G there corresponds a well defined F -stable subgroup G′_{ω} of [G, G] and a component G_{ω} = Z^{∘}(G)G′_{ω} of G. The finite group (G_{ω}/Z(G_{ω}))^{F} is characterized by its simple type X_{ω} ∈ {A_{n},^{2}A_{n}, B_{n}, C_{n}, (n ≥ 1), D_{n},^{2}D_{n}(n ≥ 2),^{3}D_{4}, G_{2}, F_{4}, E_{6},^{2}E_{6}, E_{7}, E_{8}} and an extension field IF_{q^{m(ω)}} of IF_{q} of degree m(ω) equal to the length of the orbit ω. Call (X_{ω}, q^{m(ω)}) the irreducible rational type of G_{ω}, and ×_{ω}(X_{ω}, q^{m(ω)}) the rational type of G. As usual (A_{n}, −q^{m}) denotes (^{2}A_{n}, q^{m}).

In the following table the group is of irreducible rational type (X, εq^{m}) with ε = ±1. We recall the list of bad primes ([SSt] II.4) ; |Z(G_{sc})^{F} |_{g} denotes the contribution of the good primes for G to the number of rational points in the fundamental group. We also recall the restriction on ℓ of our main theorem.

Table I

type  (A_{n}, εq^{m})  B, C, D,^{2}D ^{3}D_{4} E_{6,7},^{2}E_{6}, F_{4}, G_{2}  E_{8}
bad ℓ’s  ∅  {2}  {2}  {2, 3}  {2, 3, 5}
|Z(G_{sc})^{F} |_{g} (n + 1, q^{m} − ε)  1  1  1  1
Th. 4.4  ℓ ≥ 3  ℓ ≥ 3  ℓ ≥ 5  ℓ ≥ 5  ℓ ≥ 7

When working with centralizers, one may find it convenient to leave out elements whose orders involve bad primes, but also certain divisors of |Z(G_{sc})^{F} |.

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 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 of type (^{3}D_{4}, q^{m}) ; let Γ(G, F) := γ(G, F) ∩ γ(G^{∗}, F) otherwise.

For any endomorphism f of any finite commutative group A, one has |A| = |[A, f]|.|A^{f}|. This implies the following when ℓ ∈ γ(G, F) : For any section U/V of Z(G)/Z^{∘}(G) on which F acts trivially, |U/V | is prime to ℓ.

The set γ(G, F) behaves well with respect to certain subgroups (see also [BMM] 1).

Proposition 1.2. Let (G, F) be a connected reductive group defined over IF_{q}. If H is an F -stable reductive subgroup of G which contains an F -stable maximal torus of G, then the following holds : γ(G, F) ⊂ γ(H, F) and γ(G^{∗}, F) ⊂ γ(H^{∗}, F). Proof. Let T be a maximal F -stable torus of G contained in H. Let R ⊂ X(T) (resp. R^{∨} ⊂ Y (T)) be the set of roots (resp. coroots) of G relatively to T. We may assume that (H^{∗}, F) is in duality with (H, F) and is defined by the root datum (Y (T), X(T), (R_{H})^{∨}, R_{H}), where R_{H} is a subsystem of R and (R_{H})^{∨} is the set of roots of H^{∗} relatively to T^{∗}.

A prime ℓ is good for G if and only if (ZZR/ZZA)_{ℓ} = {1} for every subset A ⊂ R ([SSt] II.4). Thus, if ℓ is good for R, then it is good for any subsystem. So, if it is good for G (resp G^{∗}), it is good for H (resp H^{∗}).

The groups Z(G)/Z^{∘}(G), Z(H)/Z^{∘}(H), Z(G^{∗})/Z^{∘}(G^{∗}) and Z(H^{∗})/Z^{∘}(H^{∗}) are isomorphic, with F -action, to the p′-torsion-group of X(T)/ZZR, X(T)/ZZR_{H}, Y (T)/ZZR^{∨} and Y (T)/ZZ(R_{H})^{∨} respectively ([Ca.2] 4.5.8). Let ℓ ∈ γ(G, F). Then (X(T)/ZZR)_{ℓ}^{F} = {1}, so (X(T)/ZZR_{H})_{ℓ}^{F} = (ZZR/ZZR_{H})_{ℓ}^{F}. But the latter is {1} since ℓ is good for G. This implies ℓ ∈ γ(H, F).

If ℓ ∈ γ(G^{∗}, F), then (Y (T)/ZZR^{∨})_{ℓ}^{F} = {1}. As ℓ is good for G^{∗} ([SSt] II.4), one has (Y (T)/ZZ(R_{H})^{∨})_{ℓ}^{F} = (ZZR^{∨}/ZZ(R_{H})^{∨})_{ℓ}^{F} = {1}. This proves that γ(G^{∗}, F) ⊂ γ(H^{∗}, F).

1.2. Generic orders.

For each connected reductive group (G, F) defined over IF_{q}, there exists a “generic order” P_{(G,F)}(x) ∈ ZZ[x] such that |G^{F} | = P_{(G,F)}(q) (see [Ca.2] 2.9 or [BM] 1.12). A defining property is as follows. There is a ≥ 1 such that |G^{F^{m}}| = P_{(G,F)}(q^{m}) for all m ≥ 1 such that m ≡ 1 (mod. a) (in the notations of [BM], take a the order of ϕ^{∗}). The polynomial P_{(G,F)} has the form x^{N} ∏_{d≥1} ϕ_{d}^{v_{d}} for v_{d}, N ∈ IN and ϕ_{d} the d-th cyclotomic polynomial. By a standard application of Lang’s theorem ([DM.2] 3.13), one has P_{(G,F)} = P_{([G,G],F)}P_{(Z^{∘}(G),F)}. Generic orders are unchanged by isogenies. Using the above definition of generic order, one sees easily that, if H is a connected F -stable reductive subgroup of G, then P_{(H,F)} divides P_{(G,F)}, with equality if and only if H = G.

In the following proposition, we show that an F -orbit of length m on the set of irreducible components of G induces the substitution x ↦ x^{m} in generic orders.

Proposition 1.3. Let (G, F) be a connected reductive group defined over IF_{q}. Assume G = G_{1}.G_{2} … G_{m} is a central product of connected reductive subgroups G_{i} such that F (G_{i}) = G_{i+1} (i is taken mod. m). If the product is direct or if G is semi-simple, then P_{(G,F)}(x) = P_{(G_{1},F^{m})}(x^{m}). The F -stable Levi subgroups of (G, F) are the L_{1}.F (L_{1}) … F^{m−1}(L_{1})’s, where L_{1} is any F^{m}-stable Levi subgroup of (G_{1}, F^{m}). Proof. By the properties of generic orders recalled above, it is enough to prove this for direct products. It is clear that the projection π_{1}: G → G_{1} bijects G^{F^{k}} and G_{1}^{F^{mk}} for any k ≥ 1, so P_{(G,F)}(q^{k}) = P_{(G_{1},F^{m})}(q^{mk}) for infinitely many k’s, whence the claimed equality.

Consider L = C_{G}(S) where S is a torus of G. One has L = C_{G_{1}}(π_{1}(S)) .F (C_{G_{1}}(π_{1}(S)) … F^{m−1}(C_{G_{1}}(π_{1}(S)), this gives the second statement with L_{1} = C_{G_{1}}(π_{1}(S)).

The notions of ϕ_{d}-subgroup and of d-split Levi subgroup of G were introduced in [BM]. A ϕ_{d}-subgroup of G is an F -stable torus S such that P_{(S,F)} is a power of ϕ_{d}. The maximal ϕ_{d}-subgroups under inclusion for a given d ≥ 1 are G^{F} -conjugate ([BM] 3.4(3)). When G is a torus, there is a unique maximal ϕ_{d}-subgroup, we denote it by G_{ϕ_{d}}. A d-split Levi subgroup of G is the centralizer in G of some ϕ_{d}- subgroup of G. If π: G → G_{ad} denotes the reduction map mod. Z(G), it induces a bijection between the d-split Levi subgroups of G and of G_{ad} (see [BMM] 1).

If (G, F) and (G^{∗}, F) are in duality, then P_{(G,F)} = P_{(G^{∗},F)} and P_{(Z^{∘}(G),F)} = P_{(Z^{∘}(G^{∗}),F)} ([Ca.2] §4.4). We show that G and G^{∗} also have the same d-split Levi subgroups.

Proposition 1.4. Let (G, F) be a connected reductive group defined over IF_{q}, let (G^{∗}, F) be in duality with (G, F). Then the corresponding bijection L ↦ L^{∗} between G^{F} -conjugacy classes of F -stable Levi subgroups of G and (G^{∗})^{F} - conjugacy classes of F -stable Levi subgroups of G^{∗} (see [DM.2] p 113) induces a bijection between the classes of d-split Levi subgroups of G for each integer d ≥ 1. Proof. By symmetry, it suffices to check that if L is d-split, then L^{∗} is d-split. Let S^{∗} := Z^{∘}(L^{∗})_{ϕ_{d}}, then M^{∗} := C_{G^{∗}}(S^{∗}) ⊃ L^{∗} is in duality with M such that G ⊃ M ⊃ L. One has Z^{∘}(M^{∗})_{ϕ_{d}} = Z^{∘}(L^{∗})_{ϕ_{d}}, while P_{Z^{∘}(L^{∗}),F} = P_{Z^{∘}(L),F} and P_{Z^{∘}(M^{∗}),F} = P_{Z^{∘}(M),F}, so Z^{∘}(M)_{ϕ_{d}} = Z^{∘}(L)_{ϕ_{d}} and M ⊂ C_{G}(Z^{∘}(L)_{ϕ_{d}}) = L since L is d-split. So L = M and L^{∗} = M^{∗}, which is d-split.

1.3. Further results on generic orders.

Concerning the values of cyclotomic polynomials on integers, recall that if ℓ is a prime and q, d ≥ 1 are integers then ℓ divides ϕ_{d}(q) if and only if q_{ℓ} = 1 and d_{ℓ′} is the order of q mod. ℓ.

Here are two useful properties of generic orders of maximal tori.

Lemma 1.5. Let (G, F) be a connected reductive group defined over IF_{q}.

(i) The g.c.d. of the generic orders of the F -stable maximal tori of G is the generic order of Z^{∘}(G). (ii) Let d ≥ 1 be an integer and ℓ ∈ Γ(G_{ad}, F) (see Notation 1.1) be a prime divisor of ϕ_{d}(q). If T is an F -stable maximal torus of G such that T_{ϕ_{dℓ}} ⊄ Z(G), then T_{ϕ_{d}} ⊄ Z(G). Proof. The various G^{F} -conjugacy classes of F -stable maximal tori in (G, F) are parametrized by F -conjugacy classes in the Weyl group W := W (G, T_{0}) where T_{0} is some reference torus. If T is obtained from T_{0} by twisting with w ∈ W, and if F_{0} is defined by F = qF_{0} where F and F_{0} act on Y (T_{0}) ⊗ IR, then P_{(T,F)} is the characteristic polynomial of F_{0}^{−1}w on Y (T_{0}) ⊗ IR ([Ca.2], 3.3.5).

For every F -stable torus S of G which contains Z(G), one has P_{(S/Z(G),F)} = P_{(S,F)}/P_{(Z^{∘}(G),F)}. So, in view of the statements to prove, we may assume that G = G_{ad}. Then it is a direct product of its components, so we may also assume that G is rationally irreducible.

(i) The substitution (x ↦ x^{m}) in a set of coprime polynomials cannot produce common divisors. So, thanks to Proposition 1.3, we assume G is irreducible. If G is split, the generic orders of a split maximal torus T_{0} and of a torus obtained from T_{0} by twisting by a Coxeter element of W (G, T_{0}) are coprime (Coxeter elements have no fixed points in the standard representation). If (G, F) is of type (^{2}A_{n}, q) = (A_{n}, −q), (^{2}D_{2k+1}, q) = (D_{2k+1}, −q) or (^{2}E_{6}, q) = (E_{6}, −q) the property is deduced from the split case by the substitution (x ↦ −x). If (G, F) is of type (^{2}D_{2k}, q) the Coxeter tori have generic order (x^{2k} + 1), prime to the generic order of the maximal split tori. If (G, F) is of type (^{3}D_{4}, q), the maximally split tori have generic order ϕ_{1}^{2}ϕ_{3}, while the maximal ϕ_{12}-subgroup, being of rank 4, is a maximal torus.

(ii) Using [Ca.1], one sees by an easy inspection that a triple (T, d, ℓ) such that T_{ϕ_{dℓ}} ≠ {1} and T_{ϕ_{d}} = {1} may occur only when

• ℓ = 2 or

• ℓ is bad for G or

• ℓ = 3 and (G, F) is of type ^{3}D_{4} or

• (G, F) is of rational type (A_{n}, q^{m}), where d divides m, ℓ divides (n + 1) and is prime to m/d, (so, if ℓ divides ϕ_{d}(q), then ℓ divides q^{m} − 1), or

• (G, F) is of rational type (A_{n}, −q^{m}), where 2m/d is an odd integer prime to ℓ (so, if ℓ divides ϕ_{d}(q), then ℓ divides q^{m} + 1) and ℓ divides n + 1.

Under our assumptions, ℓ is good and |Z(G_{sc})^{F} |_{ℓ} = 1, so all the above cases are impossible (see Table I).

Proposition 1.6. Let (G, F) be a connected reductive group defined over IF_{q}. Let d ≥ 1 an integer and ℓ ∈ Γ(G_{ad}, F) (see Notation 1.1) a prime divisor of ϕ_{d}(q). Let H be an F -stable connected reductive subgroup containing a maximal torus of G and such that Z^{∘}(H)_{ϕ_{dℓ}} ⊄ Z(G). Then Z^{∘}(H)_{ϕ_{d}} ⊄ Z(G). Proof. Suppose Z^{∘}(H)_{ϕ_{dℓ}} ⊄ Z(G) and Z^{∘}(H)_{ϕ_{d}} ⊂ Z(G). Applying Lemma 1.5 (i) to (H, F), one finds an F -stable maximal torus T ⊂ H such that T_{ϕ_{d}} ⊂ Z^{∘}(H). Since T is an F -stable maximal torus of G, Lemma 1.5 (ii) yields T_{ϕ_{dℓ}} ⊂ Z(G). But Z^{∘}(H) ⊂ T and this contradicts Z^{∘}(H)_{ϕ_{dℓ}} ⊄ Z(G).

Centralizers of derived groups of Levi subgroups play an important rôle in block theory for finite reductive groups.

Proposition 1.7. Let (G, F) be a connected reductive group defined over IF_{q}. Let L be a Levi subgroup of G. Let K be a closed subgroup of G such that [L, L] ⊂ K ⊂ L.

(i) C_{G}^{∘}(K) is a reductive subgroup of G, Z^{∘}(L) is a maximal torus in it. (ii) If L is d-split and K is F -stable, then Z^{∘}(L)_{ϕ_{d}} is a maximal ϕ_{d}-subgroup of C_{G}^{∘}(K). (iii) If L and M are d-split Levi subgroups of G and [L, L] ⊂ M, then there exists c ∈ C_{G}^{∘}(L ∩ M)^{F} such that ^{c}L ⊂ M, with equality ^{c}L = M when [L, L] = [M, M]. Proof. Let J = C_{G}^{∘}(K) and let T be a maximal torus in L.

(i) Clearly K = [L, L].(K ∩ T), so T normalizes K and J. For J, the corresponding roots clearly form a symmetric set. So ([BT] 3.13 (b)) J is a reductive subgroup of G. One has L = C_{G}(Z^{∘}(L)), so C_{J}^{∘}(Z^{∘}(L)) ⊂ C_{L}^{∘}([L, L]) = Z^{∘}(L), whence the maximality of Z^{∘}(L).

(ii) One has L = C_{G}(Z^{∘}(L)_{ϕ_{d}}), so C_{J}(Z^{∘}(L)_{ϕ_{d}}) ⊂ Z^{∘}(L), whence the maximality of Z^{∘}(L)_{ϕ_{d}} as ϕ_{d}-subgroup of J.

(iii) Take K := L ∩ M, then Z^{∘}(M) ⊂ J. By (ii) and conjugacy of maximal ϕ_{d}-subgroups, there is c ∈ J^{F} such that ^{c}Z^{∘}(L)_{ϕ_{d}} ⊃ Z^{∘}(M)_{ϕ_{d}}. Taking centralizers in G yields ^{c}L ⊂ M since L and M are d-split. When [L, L] = [M, M], ^{c}L = M by symmetry.

2. Centralizers of ℓ-subgroups and e-split Levi subgroups.

Proposition 2.1. Let (G, F) be a connected reductive group defined over IF_{q}. Let π: G → G_{ad} be the reduction map mod. Z(G). Let ℓ be a prime not dividing q. 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 ∈ γ(G^{∗}, F) (see Notation 1.1), then C_{G}^{∘}(Y)^{F} = C_{G}(Y)^{F}.

Let S be an F -stable torus of G. Assume ℓ does not divide |(Z(G)/Z^{∘}(G))^{F} |. (iv) π(S_{ℓ}^{F}) = π(S)_{ℓ}^{F}.

(v) Let d ≥ 1 an integer. Assume in addition that ℓ divides ϕ_{d}(q) and S is a ϕ_{d}-subgroup. Then S_{ℓ}^{F} ⊂ Z(G) if and only if S ⊂ Z(G). Proof. (i) and (ii). The case of a cyclic Y is [BMi] 2.1, [GH] 2.1, respectively. The general case is easily deduced by induction.

(iii) Let us prove the statement by induction on |Y | and dim G. For every y ∈ G_{ℓ}^{F}, the quotient group C_{G}(y)/C_{G}^{∘}(y) is isomorphic, with F -action, to some section of the kernel of the simply connected covering G_{sc} → [G, G] ([St] 9.1 (a)). This kernel is in duality with and abstractly isomorphic to Z(G^{∗})/Z^{∘}(G^{∗}). So C_{G}(y)^{F} /C_{G}^{∘}(y)^{F} = (C_{G}(y)/C_{G}^{∘}(y))^{F} is a γ(G^{∗}, F)′-group. On the other hand, it is an ℓ-group by (i), so it is {1}.

Suppose now that Y = ⟨A, y⟩ is abelian and induction hypothesis holds on A : then C_{G}^{∘}(A)^{F} = C_{G}(A)^{F}. By Proposition 1.2, ℓ ∈ γ(C_{G}^{∘}(A)^{∗}, F). The cyclic case implies that C_{C_{G}^{∘}(A)}^{∘}(y)^{F} = C_{C_{G}^{∘}(A)}(y)^{F}. But it is clear that C_{G}^{∘}(Y) = C_{C_{G}^{∘}(A)}^{∘}(y) and finally C_{G}^{∘}(Y)^{F} = C_{G}(Y)^{F}.

(iv) By Lemma 1 in [CE], π(S)^{F} = (SZ(G)/Z(G))^{F} is an extension of ([SZ(G), F] ∩ Z(G))/[Z(G), F] by (SZ(G))^{F} /Z(G)^{F}. By Lang’s theorem, Z^{∘}(G) ⊂ [Z(G), F]. Therefore the hypothesis on ℓ implies that ([SZ(G), F] ∩ Z(G))/[Z(G), F] is ℓ′ ; thus π(S)_{ℓ}^{F} ⊂ π(S^{F}). Moreover, if s is of finite order, then π(s_{ℓ}) = π(s)_{ℓ}. This implies π(S)_{ℓ}^{F} = π(S_{ℓ}^{F}).

(v) The “if” part is clear, so assume π(S) ≠ {1}. In G_{ad}, π(S) is a non trivial ϕ_{d}-subgroup, so |π(S)^{F} | = ϕ_{d}(q)^{r} for some r > 0 and, using (iv), π(S)_{ℓ}^{F} = π(S_{ℓ}^{F}) ≠ {1}. Then S_{ℓ}^{F} ⊄ Z(G).

Proposition 2.2. Let (G, F) be a connected reductive group over IF_{q}, let d ≥ 1, let ℓ be a prime divisor of ϕ_{d}(q). Assume ℓ ∈ γ(G, F) (see Notation 1.1). Let S be a ϕ_{d}-subgroup of G, and L := C_{G}(S) be the associated d-split Levi subgroup. Then

(i) Z^{∘}(L)_{ℓ}^{F} = Z(L)_{ℓ}^{F} (ii) 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}). Proof. (i) By Proposition 1.2, ℓ ∈ γ(L, F), so Z(L)^{F} /Z^{∘}(L)^{F} = (Z(L)/Z^{∘}(L))^{F} is an ℓ′-group.

(ii) The second equality follows from the first by application of Proposition 2.1 (iii). Now, it clearly suffices to check C_{G}(S) = C_{G}^{∘}(S_{ℓ}^{F}). We prove this by induction on dim G. If S is central in G, then everything is clear. Otherwise, Proposition 2.1 (v) implies S_{ℓ}^{F} ⊄ Z(G). Then one may apply the induction hypothesis to (C_{G}^{∘}(S_{ℓ}^{F}), F) since ℓ ∈ γ(C_{G}^{∘}(S_{ℓ}^{F}), F) by Proposition 1.2. It provides the equality sought.

In several inductive proofs, we have to isolate those rational components G′_{ω} of [G, G] (notations of 1.1) whose type ensures that Z(G′_{ω})^{F} is an ℓ′-group (see Table I).

Notation 2.3. Let (G, F) be a connected reductive group defined over IF_{q}. Let ℓ be a prime not dividing q and good for G. Define G_{a} as the central product of the subgroups G_{ω}’s (see 1.1) 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 G′_{ω} (see 1.1) of [G, G] not included in G_{a}. Then G = G_{a}.G_{b} (central product).

(i) Z(G_{b})^{F} and G^{F} /G_{a}^{F} .G_{b}^{F} are abelian ℓ′-groups. (ii) If Y is an ℓ-subgroup of G^{F} such that Z(C_{G^{F}} (Y))_{ℓ} ⊂ Z(G)G_{a}, then Y ⊂ G_{a}.

If G = G_{a} is of rational type ×_{ω}(A_{n_{ω}}, ε_{ω}q^{m_{ω}}), a maximal F -stable torus T of G of generic 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. Note also that any F -stable Levi subgroup L of G_{a} satisfies L = L_{a}, and if L contains a diagonal torus T of G_{a}, then T is a diagonal torus of L. Proof. (i) First Z(G_{b})_{ℓ}^{F} = {1} by the definition of G_{b} (see Table I). By Lemma 1 in [CE], G^{F} /G_{a}^{F} .G_{b}^{F} is isomorphic to a section of Z(G_{b}) on which F acts trivially, so it is an ℓ′-group since Z(G_{b})_{ℓ}^{F} = {1}.

(ii) By (i) above, Y ⊂ G_{a}^{F} .G_{b}^{F}. Let Y_{a} = {y ∈ (G_{a})_{ℓ}^{F} | yG_{b}^{F} ∩ Y ≠ ∅} and Y_{b} be similarly defined. One has C_{G^{F}} (Y) = C_{G_{a}}(Y_{a})^{F} .C_{G_{b}}(Y_{b})^{F} and Z(Y_{b}) is clearly central in it. The hypothesis then implies that Z(Y_{b}) ⊂ Z(G_{b})^{F}, but the latter is an ℓ′-group by (i). So Z(Y_{b}) = {1}, hence Y_{b} = {1} and Y ⊂ Y_{a} ⊂ G_{a}.

In case ℓ ∉ γ(G, F) but G = G_{a}, one replaces Proposition 2.2 above with the following alternative property : Lemma 2.4. Let (G, F) be a connected reductive group over IF_{q}, let ℓ be an odd prime not dividing q, let e be the order of q mod. ℓ. Assume G = G_{a} and let T be a diagonal torus of G. Then one has T = C_{G}(T_{ϕ_{e}}) = C_{G}^{∘}(T_{ℓ}^{F}) and T^{F} = C_{G^{F}} (T_{ℓ}^{F}). Proof. Let ×_{ω}(A_{n_{ω}}, ε_{ω}q^{m_{ω}}) be the rational type of G, one has P_{(T,F)} = P_{(Z^{∘}(G),F)}Π_{ω}(x^{m_{ω}} − ε_{ω})^{n_{ω}}. For each ω, ℓ divides q^{m_{ω}} − ε_{ω}, so ϕ_{e} divides x^{m_{ω}} − ε_{ω}.

One sees first by induction on dim G/T that T = C_{G}^{∘}(T_{ϕ_{e}}) = C_{G}^{∘}(T_{ℓ}^{F}). If G is a torus, there is nothing to prove. On the other hand, when q^{m} − ε ≡ 0 (mod ℓ) and n ≥ 1, then ℓ divides (q^{m} − ε)^{n}/(q^{m} − ε, n + 1) (here we use the fact that ℓ ≠ 2 : consider SL(2, εq), with εq ≡ 3 (mod 4)). This implies |T_{ℓ}^{F} | > |Z(G)_{ℓ}^{F} | provided G is not a torus, so T_{ℓ}^{F} is not central in G. Then C_{G}^{∘}(T_{ℓ}^{F}) is a proper Levi subgroup (Proposition 2.1 (ii)) with diagonal torus T and one obtains T = C_{G}^{∘}(T_{ℓ}^{F}) by induction. Similarly, T_{ϕ_{e}} ⊄ Z^{∘}(G), and one gets T = C_{G}(T_{ϕ_{e}}) by considering C_{G}(T_{ϕ_{e}}) and applying induction.

Now we prove that C_{G^{F}} (T_{ℓ}^{F}) ⊂ T, again by induction on dim G/T. By a theorem of Steinberg ([St] §9), if [G, G] is simply connected then C_{G}(T_{ℓ}^{F}) is connected. Let G′ := Z^{∘}(G) × G_{sc}, let π′: G′ → G be the natural epimorphism and let K be the kernel of π′. The isomorphism Z(G) ≅ Z^{∘}(G) × Z(G_{sc})/K gives Z(G)/Z^{∘}(G) ≅ Z(G_{sc})/K. So one has |K^{F} |(Z(G)^{F} : Z^{∘}(G)^{F}) = |Z(G_{sc})^{F} | = Π_{ω}(q^{m_{ω}} − ε_{ω}, n_{ω} + 1) (see Table I).

Let g ∈ C_{G^{F}} (T_{ℓ}^{F}), one must check g ∈ T. Let S = π′^{−1}(T_{ℓ}^{F}) and g′ ∈ S such that g = π′(g′). If s ∈ S, then g^{′−1}F (g′) ∈ K, s^{−1}F (s) ∈ K and [g′, s] ∈ K, so that [g, s] ∈ K^{F}. The map (s ↦ [g′, s]) thus defines a morphism from S to K_{ℓ}^{F}. The kernel Z of this morphism contains K and is of order ≥ |S|/|K^{F} |_{ℓ}. One obtains (Z : K) ≥ |T^{F} |_{ℓ}/|K^{F} |_{ℓ} = (|Z(G)^{F} | Π_{ω}(q^{m_{ω}} − ε_{ω})^{n_{ω}}/(q^{m_{ω}} − ε_{ω}, n_{ω} + 1))_{ℓ} > |Z(G^{F})|_{ℓ} if G is not a torus. Then π′(Z) is not central. On the other hand, by definition of Z, g′ ∈ C_{G′}(Z) = C_{G′}^{∘}(Z), so g ∈ π′(C_{G′}^{∘}(Z)) ⊂ C_{G}^{∘}(π′(Z)). But the group H := C_{G}^{∘}(π′(Z)) is a proper Levi subgroup of G. In this group, T is a diagonal torus and, by induction hypothesis, T^{F} = C_{H^{F}} (T_{ℓ}^{F}). Then g ∈ T as claimed.

Proposition 2.5. Let (G, F) be a connected reductive group defined over IF_{q}. Let ℓ be an odd prime, good for G and ℓ ≠ 3 if ^{3}D_{4} is involved in (G, F). Let e ≥ 1 be the order of q mod. ℓ. Let h ∈ G_{ℓ}^{F} \ (G_{a})_{ℓ}^{F} (Notation 2.3), then there exists a proper e-split Levi subgroup L ⊂ G, L ≠ G such that C_{G}^{∘}(h) ⊂ L. Proof. In view of Notation 2.3 (i), one has h = h_{a}h_{b} with h_{a} ∈ (G_{a})_{ℓ}^{F}, h_{b} ∈ (G_{b})_{ℓ}^{F} \ {1}. Then C_{G}^{∘}(h) = C_{G_{a}}^{∘}(h_{a}).C_{G_{b}}^{∘}(h_{b}). So we may assume that G = G_{b}. In that case ℓ ∈ Γ(G, F) = Γ(G_{ad}, F) (see Table I), so Proposition 1.6 applies to H = C_{G}^{∘}(h). One has h ∈ H^{F} by Proposition 2.1 (iii), so h ∈ Z(H)^{F}. On the other hand, (Z(H)/Z^{∘}(H))^{F} is an ℓ′-group by Proposition 1.2. Then h ∈ Z^{∘}(H)^{F} and Z^{∘}(H)_{ℓ}^{F} ⊄ Z(G). So there exists some a ≥ 0 such that Z^{∘}(H)_{ϕ_{eℓ^{a}}} ⊄ Z(G), and therefore Z^{∘}(H)_{ϕ_{e}} ⊄ Z(G) thanks to Proposition 1.6. Now take L := C_{G}(Z^{∘}(H)_{ϕ_{e}}).

3. Unipotent characters. One recalls the partition Irr(G^{F}) = ∪_{(s)}E(G^{F} , s) in Lusztig series ((s) ranges

over conjugacy classes of semi-simple elements of (G^{∗})^{F}). One calls unipotent characters the elements of E(G^{F} , 1). Proposition 3.1. Let (G, F) be a connected reductive group defined over a finite field. Let π: G → G_{ad} be the reduction map mod. Z(G). Then the restriction maps associated with the inclusions π(G^{F}) ⊂ G_{ad}^{F} and [G, G]^{F} ⊂ G^{F} induce bijections E(G_{ad}^{F} , 1) → E(G^{F} , 1) → E([G, G]^{F} , 1). They commute with the Lusztig functor and its adjoint. Proof. The first statement is well-known (apply [DM.2] 13.20). For the commutation with R and ^{∗}R, apply [DM.2] 13.22 to the inclusion [G, G] → G, to a regular imbedding G → G′ (see [L]) and to G′ → → G′_{ad} = G_{ad}. 3.1. Generalized “Harish-Chandra theory”.

If L_{1}, L_{2} are F -stable Levi subgroups of G and χ_{i} ∈ E(L_{i}^{F} , 1), one writes (L_{1}, χ_{1}) ≤ (L_{2}, χ_{2}) if, and only if, L_{1} ⊂ L_{2} and ⟨χ_{1},^{∗}R_{L_{1}}^{L_{2}}χ_{2}⟩_{L_{1}^{F}} ≠ 0.

Let d ≥ 1 an integer. One considers the unipotent d-pairs (L, χ) where L is a d-split Levi subgroup of G and χ ∈ E(L^{F} , 1). A d-pair (L, λ) is said to be d-cuspidal if and only if for every proper d-split Levi subgroup M of L, one has ^{∗}R_{M}^{L}λ = 0 (see [BMM]). A unipotent character χ ∈ E(G^{F} , 1) is said to be d-cuspidal if and only if (G, χ) is a d-cuspidal pair. By Proposition 3.1, it is clear that the unipotent d-cuspidal pairs are the same in G, G_{ad} and [G, G]. If (L, λ) is d-cuspidal in G, then (L, λ) is d-cuspidal in any F -stable closed connected reductive subgroup of G containing L. Here is the fundamental result of the “d-theory”.

Proposition 3.2 ([BMM] 3.A). Let (G, F) be a connected reductive group defined over IF_{q} and d ≥ 1, then ≤ on the set of d-pairs induces an ordering. Given χ ∈ E(G^{F} , 1), there exists a d-cuspidal pair (L, λ) such that (L, λ) ≤ (G, χ). Moreover ^{∗}R_{L}^{G}χ = ⟨λ,^{∗}R_{L}^{G}χ⟩_{L^{F}} ∑_{g∈N_{G}(L)^{F} /N_{G}(L,λ)^{F}} ^{g}λ.

3.2. e-cuspidal pairs and centralizers of ℓ-elements. Proposition 3.3. Let (G, F) be a connected reductive group defined over IF_{q}. Let ℓ be an odd prime not dividing q and good for G. Let G = G_{a}.G_{b} be the decomposition of Notation 2.3 relative to ℓ. Let e be the order of q mod ℓ. Let (L, λ) be an e-cuspidal unipotent pair of G. Then

(i) L ∩ G_{a} is a diagonal torus of G_{a} (see Notation 2.3), (ii) Z(L)_{ℓ}^{F} = Z^{∘}(L)_{ℓ}^{F} , L = C_{G}^{∘}(Z(L)_{ℓ}^{F}) and L^{F} = C_{G^{F}} (Z(L)_{ℓ}^{F}). Proof. First L = (L ∩ G_{a}).(L ∩ G_{b}) and L ∩ G_{a}, L ∩ G_{b} are e-split Levi of G_{a}, G_{b} with an e-cuspidal unipotent character, call λ_{a} the one of (L ∩ G_{a})^{F}.

(i) Let T be a diagonal torus of G_{a}. By Lemma 2.4 T = C_{G}^{∘}(T_{ϕ_{e}}), so T is esplit in G_{a} and T_{ϕ_{e}} is a maximal ϕ_{e}-subgroup of G_{a}. If one writes L∩G_{a} = C_{G_{a}}(S) for S a ϕ_{e}-subgroup of G_{a}, then S is in a G_{a}^{F} -conjugate of T_{ϕ_{e}}, so one may assume L ∩ G_{a} ⊃ T. But now [LS] ensures that ^{∗}R_{T}^{L∩G_{a}}λ_{a} ≠ 0. This implies T = L ∩ G_{a} since λ_{a} is e-cuspidal.

(ii) Let L = T.M, where T is a diagonal torus of G_{a} and M is an e-split Levi subgroup of G_{b}. Then Z(L) = T.Z(M) and Z^{∘}(L) = T.Z^{∘}(M). As G_{a}∩G_{b} is central and (G_{a} ∩ G_{b})^{F} is an ℓ′-group, Lemma 1 in [CE] implies Z(L)_{ℓ}^{F} = T_{ℓ}^{F} .Z(M)_{ℓ}^{F} and Z^{∘}(L)_{ℓ}^{F} = T_{ℓ}^{F} .Z^{∘}(M)_{ℓ}^{F}. By Proposition 2.2 (i), Z^{∘}(M)_{ℓ}^{F} = Z(M)_{ℓ}^{F}, so Z^{∘}(L)_{ℓ}^{F} = Z(L)_{ℓ}^{F}.

Now C_{G}(Z(L)_{ℓ}^{F}) = C_{G_{a}}(T_{ℓ}^{F}).C_{G_{b}}(Z(M)_{ℓ}^{F}), so C_{G}^{∘}(Z(L)_{ℓ}^{F}) = C_{G_{a}}^{∘}(T_{ℓ}^{F}).C_{G_{b}}^{∘}(Z(M)_{ℓ}^{F}). We have ℓ ∈ γ(G_{b}, F) since ℓ is good for G_{b} (see Table I). Then Proposition 2.2 (ii) in G_{b} and Lemma 2.4 in G_{a} yield C_{G}^{∘}(Z(L)_{ℓ}^{F}) = L. Furthermore, C_{G}(Z(L)_{ℓ}^{F})/C_{G}^{∘}(Z(L)_{ℓ}^{F}) is isomorphic to C_{G_{a}}(T_{ℓ}^{F})/T. By the last equality of Lemma 2.4, F fixes only 1 in this quotient, so that C_{G^{F}} (Z(L)_{ℓ}^{F}) = L^{F}.

Definition 3.4. If L_{1}, L_{2} are F -stable Levi subgroups of G and χ_{i} ∈ E(L_{i}^{F} , 1), one writes (L_{1}, χ_{1}) ∼ (L_{2}, χ_{2}) if, and only if, [L_{1}, L_{1}] = [L_{2}, L_{2}] and Res_{[L_{1},L_{1}]^{F}}^{L_{1}^{F}} χ_{1} =

Res_{[L_{2},L_{2}]^{F}}^{L_{2}^{F}} χ_{2}. Note that ∼ is an equivalence relation. If there is an integer d ≥ 1 such that (L_{1}, χ_{1}) and (L_{2}, χ_{2}) are both d-pairs, then they are ∼-equivalent if and only if they are C_{G}^{∘}(L_{1} ∩ L_{2})^{F} -conjugate.

The last statement above is proved as follows : if (L_{1}, χ_{1}) ∼ (L_{2}, χ_{2}) and the L_{i} are d-split, then, by Proposition 1.7 (iii), there exists c ∈ C_{G}^{∘}(L_{1} ∩ L_{2})^{F} such that L_{1} = ^{c}L_{2}. But then Res_{[L_{1},L_{1}]^{F}}^{L_{1}^{F}} λ_{1} = Res_{[L_{2},L_{2}]^{F}}^{L_{2}^{F}} λ_{2} = ^{c}(Res_{[L_{2},L_{2}]^{F}}^{L_{2}^{F}} λ_{2}) =

Res_{[L_{1},L_{1}]^{F}}^{L_{1}^{F}} (^{c}λ_{2}), which in turn implies λ_{1} = ^{c}λ_{2} by Proposition 3.1.

The following relates the ≤ orderings for e-pairs of G and for e-pairs of centralizers of ℓ-elements.

Proposition 3.5. Let (G, F) be a connected reductive group defined over IF_{q}. Let ℓ be an odd prime, good for G, ℓ ≠ 3 if ^{3}D_{4} is involved in (G, F). Let e be the order of q mod. ℓ. Let t ∈ (G^{∗})_{ℓ}^{F} and H be an F -stable Levi subgroup of G in duality with C_{G^{∗}}^{∘}(t).

(i) The condition (L, λ) ∼ (L_{H}, λ_{H}) defines a unique bijection between the ∼-classes of e-cuspidal pairs (L, λ) in G such that [L, L] ⊂ H, and the ∼-classes of e-cuspidal pairs (L_{H}, λ_{H}) of H. (ii) Let (L, λ) (resp. (L_{H}, λ_{H})) be an e-cuspidal unipotentÂ pair of G (resp. H) such that (L, λ) ∼ (L_{H}, λ_{H}). If χ ∈ E(G^{F} , 1), χ_{H} ∈ E(H^{F} , 1) satisfy (L_{H}, λ_{H}) ≤ (H, χ_{H}) and (H, χ_{H}) ≤ (G, χ), then (L, λ) ≤ (G, χ). Proof. First, C_{G^{∗}}^{∘}(t) is a Levi subgroup by Proposition 2.1 (iii).

The proof of (i)(ii) is by induction on dim G − dim H. If G = H, everything is clear. Another easy case is when G = G_{a} (Notation 2.3) and H is any F -stable Levi subgroup : then H = H_{a} and, by Proposition 3.3 (i), there is only one conjugacy class of e-cuspidal pairs in G and H, and it corresponds to diagonal tori and trivial character. Then (i) and (ii) are clear. From now on, we separate two cases. First case. t ∈ (G^{∗})_{a}^{F}. Then C_{G^{∗}}^{∘}(t) ⊃ (G^{∗})_{b} , so H ⊃ G_{b} and H = (H∩G_{a}).G_{b}. Now, by Proposition 3.1, one may consider everything in G_{ad} where the central products are direct products. But then (i) and (ii) are clear on each side since they reduce to the cases G = G_{a} and G = H. Second case. t ∉ (G^{∗})_{a}^{F}. Then, by Proposition 2.5, there is a proper e-split Levi subgroup M^{∗} ⊂ G^{∗} such that C_{G^{∗}}^{∘}(t) ⊂ M^{∗}. By Proposition 1.4 one has H ⊂ M with M a proper e-split Levi subgroup of G. By induction, (i) and (ii) are satisfied by H in M. Then, if (L_{H}, λ_{H}) is an e-cuspidal pair of H, there exists in M an e-cuspidal pair (L, λ) such that (L, λ) ∼ (L_{H}, λ_{H}). Conversely, let (L, λ) be ecuspidal in G with [L, L] ⊂ H. Then [L, L] ⊂ M, so, by Proposition 1.7 (iii), there is c ∈ C_{G}^{∘}([L, L])^{F} such that ^{c}L ⊂ M. By induction hypothesis, there exists an e-cuspidal pair (L_{H}, λ_{H}) in H such that (L_{H}, λ_{H}) ∼ ^{c}(L, λ). But ^{c}(L, λ) ∼ (L, λ) is clear, so we get (i).

Assume the hypotheses of (ii), take c ∈ C_{G}^{∘}([L, L])^{F} as above such that ^{c}L ⊂ M. Then ^{c}(L, λ) ∼ (L_{H}, λ_{H}). One has ⟨^{∗}R_{M}^{G}χ, R_{H}^{M}χ_{H}⟩_{M^{F}} = ⟨χ, R_{H}^{G}χ_{H}⟩_{G^{F}} ≠ 0, so there exists χ_{M} ∈ E(M^{F} , 1) such that (H, χ_{H}) ≤ (M, χ_{M}) and (M, χ_{M}) ≤ (G, χ). The induction hypothesis then gives ^{c}(L, λ) ≤ (M, χ_{M}). Now the transitivity of ≤ on e-pairs in G (Proposition 3.2) implies ^{c}(L, λ) ≤ (G, χ), hence (L, λ) ≤ (G, χ) as claimed.

4. Unipotent blocks.

4.1. Subpairs, Brauer second main theorem.

Let G be a finite group, let ℓ be a prime number. Let O be a complete local ring of characteristic zero and residual field k := O/J(O) of characteristic ℓ. We assume its field of fractions K is algebraically closed. All ℓ-blocks are considered as primitive idempotents in the center Z(OG) of the group algebra OG. Let a ↦ \overline{a} be the reduction map mod. J(O) from OG to kG. Then Z(kG) is the image of Z(OG) and primitive idempotents are in bijection. The definitions of subpairs (Q, b), Brauer morphism Br_{Q} , inclusion of subpairs, along with the relationship with defect groups are taken from [AB].

One denotes by Irr(G) the set of irreducible characters of KG. All central functions χ ∈ CF(G, K) are extended as linear forms on KG ; if b is an ℓ-block of G, one defines b.χ by b.χ(g) = χ(bg). If χ ∈ Irr(G) , one denotes by b_{G}(χ) the ℓ-block of G such that b_{G}(χ).χ = χ.

If χ ∈ CF(G, K) and if h ∈ G_{ℓ}, one denotes by d^{h,G}χ the central function on C_{G}(h) such that d^{h,G}χ(t) = χ(th) when t ∈ C_{G}(h)_{ℓ′}, and d^{h,G}χ vanishes outside C_{G}(h)_{ℓ′}. Here is a reformulation of R. Brauer’s second main theorem.

Proposition. ([NT] §5.4) If h ∈ G_{ℓ}, χ ∈ Irr(G) and b is an ℓ-block of C_{G}(h) such that b.d^{h,G}χ ≠ 0, then ({1}, b_{G}(χ)) ⊂ (< h >, b).

4.2. Some basic results in finite reductive groups.

A key-fact concerning finite reductive groups is the compatibility of the decomposition map with the *R functor (see [FS.1], [Br] 4.3, [DM.1] 3.13 (iii)) :

Proposition 4.1. Let (G, F) be a connected reductive algebraic group defined over IF_{q}, L an F -stable Levi subgroup of G, ℓ a prime not dividing q and h ∈ L_{ℓ}^{F}. Then C_{L}^{∘}(h) is a Levi subgroup in C_{G}^{∘}(h) and d^{h,L^{F}} ∘^{∗}R_{L}^{G} = ^{∗}R_{C_{L}^{∘}(h)}^{C_{G}^{∘}(h)} ∘ d^{h,G^{F}}.

(Recall that C_{G}(h)/C_{G}^{∘}(h) is an ℓ-group (Proposition 2.1 (i)), so d^{h,G^{F}} maps CF(G^{F} , K) to the space of central functions of C_{G}(h)^{F} vanishing outside C_{G}^{∘}(h)^{F})

If π is a set of primes, denote E_{π}(G^{F} , 1) := ∪_{(s)}E(G^{F} , s) where s ranges over the semi-simple π-elements of (G^{∗})^{F} (see [BMi]). Proposition 4.2. Let (G, F) be a connected reductive group defined over IF_{q}. Let ℓ be a prime not dividing q. Let L be an F -stable Levi subgroup of G such that L^{F} = C_{G^{F}} (Z(L)_{ℓ}^{F}). Let λ ∈ E_{ℓ′}(L^{F} , 1). Assume that ^{∗}R_{L}^{H}χ_{H} = ⟨λ,^{∗}R_{L}^{H}χ_{H}⟩_{L^{F}} ∑_{g∈N_{H}(L)^{F} /N_{H}(L,λ)^{F}} ^{g}λ for all subgroups H = C_{G}^{∘}(A) where A ⊂ Z(L)_{ℓ}^{F} and all χ_{H} ∈ Irr(H^{F}) such that (L, λ) ≤ (H, χ_{H}). If χ ∈ Irr(G^{F}) and (L, λ) ≤ (G, χ), then ({1}, b_{G^{F}} (χ)) ⊂ (Z(L)_{ℓ}^{F} , b_{L^{F}} (λ)). Proof. First note that the hypotheses satisfied by (L, λ) in G are in fact satisfied in any subgroup of the form H = C_{G}^{∘}(A) for A ⊂ Z(L)_{ℓ}^{F}.

The proof of the proposition is by induction on |G^{F} : L^{F} |. If L^{F} = G^{F}, then χ = λ and everything is clear. If L^{F} ≠ G^{F}, then Z(L)_{ℓ}^{F} ⊄ Z(G^{F}). Let z ∈ Z(L)_{ℓ}^{F} \ Z(G^{F}), let H := C_{G}^{∘}(z) ⊃ L. Then (H, F) is connected, reductive and defined over IF_{q}, moreover H^{F} ≠ G^{F}. One has H^{F} ◁ C_{G^{F}} (z) and the index is a power of ℓ (Proposition 2.1 (i)). Then ⟨d^{z,G^{F}} χ, R_{L}^{H}λ⟩_{H^{F}} = ⟨d^{z,L^{F}} (^{∗}R_{L}^{G}χ), λ⟩_{L^{F}} by Proposition 4.1. But λ ∈ E_{ℓ′}(L^{F} , 1), so χ ∈ E_{ℓ′}(G^{F} , 1) and ^{∗}R_{L}^{G}χ ∈ ZZE_{ℓ′}(L^{F} , 1). Then d^{z,L^{F}} ^{∗}R_{L}^{G}χ = d^{1,L^{F}} ^{∗}R_{L}^{G}χ and ⟨d^{z,G^{F}} χ, R_{L}^{H}λ⟩_{H^{F}} = ⟨ψ, λ⟩_{L^{F}} where ψ := d^{z,L^{F}} ^{∗}R_{L}^{G}χ = d^{1,L^{F}} ^{∗}R_{L}^{G}χ = ⟨λ,^{∗}R_{L}^{H}χ⟩_{L^{F}} ∑_{g∈N_{H}(L)^{F} /N_{H}(L,λ)^{F}} d^{1,L^{F}} (^{g}λ) ≠ 0 (value at 1). For all y ∈ L^{F}, ψ(y) is an algebraic number and ψ(y^{−1}) is the complex conjugate of ψ(y), therefore ⟨ψ, ψ⟩_{L^{F}} > 0 in IR. On the other hand, ⟨ψ, ψ⟩_{L^{F}} = ⟨λ,^{∗}R_{L}^{H}χ⟩_{L^{F}} ∑_{g∈N_{H}(L)^{F} /N_{H}(L,λ)^{F}} ⟨ψ,^{g}λ⟩_{L^{F}} = |N_{H}(L)^{F} : N_{H}(L, λ)^{F} |⟨λ,^{∗}R_{L}^{H}χ⟩_{L^{F}} ⟨ψ, λ⟩_{L^{F}}. Thus ⟨ψ, λ⟩_{L^{F}} ≠ 0 and ⟨d^{z,G^{F}} χ, R_{L}^{H}λ⟩_{H^{F}} ≠ 0. This implies that there is χ′ ∈ Irr(H^{F}) such that (L, λ) ≤ (H, χ′) and b_{H^{F}} (χ′).d^{z,G^{F}} χ ≠ 0. By induction, ({1}, b_{H^{F}} (χ′)) ⊂ (Z(L)_{ℓ}^{F} , b_{L^{F}} (λ)) in H^{F}, or equivalently Br_{Z(L)_{ℓ}^{F}} (\overline{b_{H^{F}} (χ′)})\overline{b_{L^{F}} (λ)} = \overline{b_{L^{F}} (λ)} in kG^{F}. Let E be a representative system of C_{G^{F}} (z) mod. the inertial group of b_{H^{F}} (χ′). Then b′ := ∑_{t∈E} tb_{H^{F}} (χ′)t^{−1} is a block of C_{G^{F}} (z) (see [NT] 5.5.11). One has b′.d^{z,G^{F}} (χ) = |E|b_{H^{F}} (χ′).d^{z,G^{F}} (χ) ≠ 0, so Brauer Second main Theorem implies ({1}, b_{G^{F}} (χ)) ⊂ (⟨z⟩, b′) in G^{F}. Let us also check (⟨z⟩, b′) ⊂ (Z(L)_{ℓ}^{F} , b_{L^{F}} (λ)) in G^{F}. Otherwise, one would get Br_{Z(L)_{ℓ}^{F}} (\overline{b′})\overline{b_{L^{F}} (λ)} = 0 in (kG^{F})^{Z(L)_{ℓ}^{F}}. Multiplying by Br_{Z(L)_{ℓ}^{F}} (\overline{b_{H^{F}} (χ′)}) yields Br_{Z(L)_{ℓ}^{F}} (\overline{b_{H^{F}} (χ′)})\overline{b_{L^{F}} (λ)} = 0, but this contradicts the inclusion ({1}, b_{H^{F}} (χ′)) ⊂ (Z(L)_{ℓ}^{F} , b_{L^{F}} (λ)) in H^{F}.

We get our claim by combining the two inclusions we have.

The following relates cuspidality and blocks with central defect group.

Proposition 4.3. Let (G, F) be a connected reductive group defined over IF_{q}. Let ℓ be an odd prime, good for G and ℓ ≠ 3 if ^{3}D_{4} is involved in (G, F). Let e be the order of q mod. ℓ. Assume that λ ∈ E(G^{F} , 1) is e-cuspidal. Then b_{G^{F}} (λ) has a central defect group and λ is canonical in its ℓ-block. Proof. Since Z(G^{F}) is in the kernel of λ, it suffices to check that if h ∈ G_{ℓ}^{F} \ Z(G^{F}) then d^{h,G^{F}} λ = 0 ([NT] 3.6.29). By Proposition 3.3 (i), G_{a} is a torus, so G_{a} = Z^{∘}(G) and one may apply Proposition 2.5 : there exists a proper e-split Levi subgroup L such that C_{G}^{∘}(h) ⊂ L. Then Proposition 4.1 implies d^{h,G^{F}} λ = d^{h,L^{F}} ^{∗}R_{L}^{G}λ = 0 since λ is e-cuspidal.

4.3. Unipotent blocks and generalized Harish-Chandra theory.

In the following theorem, unipotent ℓ-blocks of G^{F} are parametrized by ecuspidal pairs of (G, F). We describe defect groups and the characters of unipotent ℓ-blocks.

Theorem 4.4. Let (G, F) be a connected reductive group defined over IF_{q}. Let ℓ be a prime not dividing q. Assume ℓ is odd, good for G and ℓ ≠ 3 if ^{3}D_{4} is involved in (G, F). Let e be the order of q mod. ℓ. There is a bijection (L, λ) ↦ b_{G^{F}} (L, λ) between the G^{F} -conjugacy classes of unipotent e-cuspidal pairs of (G, F) and the set {b_{G^{F}} (χ) ; χ ∈ E(G^{F} , 1)} satisfying the following :

(i) Irr(b_{G^{F}} (L, λ)) ∩ E(G^{F} , 1) = {χ; (L, λ) ≤ (G, χ)}.

Moreover, (ii) Any Sylow ℓ-subgroup of C_{G}^{∘}([L, L])^{F} is a defect group of b_{G^{F}} (L, λ). (iii) If t ∈ (G^{∗})_{ℓ}^{F}, denote by G(t) ⊂ G an F -stable Levi subgroup in duality with C_{G^{∗}}^{∘}(t) and by t̂the associated linear character of G(t)^{F}. With those notations, the elements of Irr(b_{G^{F}} (L, λ)) are the characters occuring in some R_{G(t)}^{G}(t̂χ_{t}), where t ∈ (G^{∗})_{ℓ}^{F}, χ_{t} ∈ E(G(t)^{F} , 1), and (L_{t}, λ_{t}) ≤ (G(t), χ_{t}) for an e-cuspidal pair (L_{t}, λ_{t}) of G(t) such that (L, λ) ∼ (L_{t}, λ_{t}) (this forces [L, L] ⊂ G(t)).

Proof. First, let us fix an e-cuspidal pair (L, λ) of G. Denote Z := Z^{∘}(L)_{ℓ}^{F} = Z(L)_{ℓ}^{F}, then L = C_{G}^{∘}(Z) and L^{F} = C_{G^{F}} (Z) by Proposition 3.3 (ii). If χ ∈ E(G^{F} , 1) is such that (L, λ) ≤ (G, χ), then the inclusion ({1}, b_{G^{F}} (χ)) ⊂ (Z, b_{L^{F}} (λ)) holds by Proposition 4.2 and Proposition 3.2. By the standard properties of subpairs ([AB] 3.4), this implies that b_{G^{F}} (χ) = b_{G^{F}} (χ′) for all χ′ ∈ E(G^{F} , 1) such that (L, λ) ≤ (G, χ′). So one may define b_{G^{F}} (L, λ) := b_{G^{F}} (χ).

Our next task is to construct a maximal subpair. Denote J := C_{G}^{∘}([L, L]). By Proposition 1.7 (ii), Z^{∘}(L)_{ϕ_{e}} is a maximal ϕ_{e}-subgroup in (J, F), so |J^{F} : N_{J^{F}} (Z^{∘}(L)_{ϕ_{e}})| ≡ 1 mod. ϕ_{e}(q) ([BM] 3.4.(4)). But ℓ divides ϕ_{e}(q), so N_{J^{F}} (Z^{∘}(L)_{ϕ_{e}}) contains a Sylow ℓ-subgroup D of J^{F}. Since L = C_{G}(Z^{∘}(L)_{ϕ_{e}}) = C_{G}^{∘}(Z), one has N_{J^{F}} (Z^{∘}(L)_{ϕ_{e}}) = N_{J^{F}} (L) = N_{J^{F}} (Z), so Z ◁ D.

Lemma 4.5. (Z, b_{L^{F}} (λ)) is a self-centralizing subpair (in the sense of [B] 6,

[Br] 1.C) and (D, b_{C_{G^{F}} (D)}(Res_{C_{G^{F}} (D)}^{L^{F}}λ)) is a maximal subpair with canonical char-

acter Res_{C_{G^{F}} (D)}^{L^{F}}λ. Moreover

({1}, b_{G^{F}} (L, λ)) ◁ (Z, b_{L^{F}} (λ)) ◁ (D, b_{C_{G^{F}} (D)}(Res_{C_{G^{F}} (D)}^{L^{F}}λ)).

Proof of the lemma. By Proposition 4.3, b_{L^{F}} (λ) has central defect group Z(L^{F})_{ℓ} in L^{F}. By [Ca.2] 3.6.8, Z(L^{F})_{ℓ} = Z(L)_{ℓ}^{F}, so Z = Z(L^{F})_{ℓ} and (Z, b_{L^{F}} (λ)) is self-centralizing as claimed.

One has C_{G^{F}} (D) ⊂ C_{G^{F}} (Z) = L^{F}, so the restriction Res_{C_{G^{F}} (D)}^{L^{F}}λ makes sense.

On the other hand [L, L] ⊂ C_{G}(D) and Res_{[L,L]^{F}}^{L^{F}} λ is irreducible (Proposition 3.1),

so Res_{C_{G^{F}} (D)}^{L^{F}}λ ∈ Irr(C_{G^{F}} (D)). Denote λ_{D} := Res_{C_{G^{F}} (D)}^{L^{F}}λ.

We check the normal inclusion (Z, b_{L^{F}} (λ)) ◁ (D, b_{C_{G^{F}} (D)}(λ_{D})) by applying [B] 6D. First, D normalizes (Z, b_{L^{F}} (λ)). If g ∈ D, then λ, λ^{g} ∈ E(L^{F} , 1) and they have the same restriction to [L, L]^{F} since g centralizes [L, L]. By Proposition 3.1, this implies λ = λ^{g}. Let us show that (D, b_{C_{G^{F}} (D)}(λ_{D})) is selfcentralizing. Since Z(L)^{F} is in the kernel of λ, λ_{D} has Z ∩C_{G^{F}} (D) in its kernel and λ_{D}(1)_{ℓ} = λ(1)_{ℓ} = |L^{F} : Z|_{ℓ} ≥ |C_{G^{F}} (D) : Z ∩ C_{G^{F}} (D)|_{ℓ} since b_{L^{F}} (λ) is of defect Z. Then b_{C_{G^{F}} (D)}(λ_{D}) has central defect group Z∩C_{G^{F}} (D) = Z(C_{G^{F}} (D))_{ℓ} with canonical character λ_{D}. In addition, D ∩ C_{G^{F}} (Z) = D ∩ L^{F} ⊂ C_{G}([L, L]) ∩ L^{F} = Z(L)^{F}, whence D ∩ C_{G^{F}} (Z) ⊂ Z. So all the necessary conditions of [B] 6D hold and (Z, b_{L^{F}} (λ)) ◁ (D, b_{C_{G^{F}} (D)}(λ_{D})).

It remains to check that (D, b_{C_{G^{F}} (D)}(λ_{D})) is maximal. We prove this by induction on dim G.

Assume (D, b_{C_{G^{F}} (D)}(λ_{D})) ⊂ (E, b_{E}), we prove D = E. Then (E, b_{E}) contains (Z, b_{L^{F}} (λ)), so it is self-centralizing and Z(E) = Z(C_{G^{F}} (E))_{ℓ} = E ∩ L^{F} ⊂ Z ([B] 6A, 6B). Let H := C_{G}^{∘}(Z(E)). Then (H, F) clearly satisfies the hypotheses of the Theorem, (L, λ) is a unipotent e-cuspidal pair of (H, F), and one has (Z, b_{L^{F}} (λ)) ⊂ (D, b_{C_{G^{F}} (D)}(λ_{D})) ⊂ (E, b_{E}) in H^{F}.

If H ≠ G, the induction hypothesis implies D = E.

If H = G, then Z(E) = Z(C_{G^{F}} (E))_{ℓ} ⊂ Z(G), so Notation 2.3 (ii) implies E ⊂ G_{a}. Then Z ⊂ G_{a}, so L ⊃ G_{b}. But L ∩ G_{a} is a torus (Proposition 3.3 (i)), whence [L, L] = G_{b}. Then D is a Sylow ℓ-subgroup of C_{G}^{∘}([L, L])^{F} = G_{a}^{F}, therefore D = E.

We now have (ii). In order to get (i), it just remains to prove that if (L, λ) and (L′, λ′) are e-cuspidal and if b_{G^{F}} (L, λ) = b_{G^{F}} (L′, λ′), then (L, λ) and (L′, λ′) are G^{F} -conjugate. Suppose b_{G^{F}} (L, λ) = b_{G^{F}} (L′, λ′) and construct maximal subpairs as in Lemma 4.5 : ({1}, b_{G^{F}} (L, λ)) ◁ (Z, b_{L^{F}} (λ)) ◁ (D, b) := (D, b_{C_{G^{F}} (D)}(Res_{C_{G^{F}} (D)}^{L^{F}}λ))

and ({1}, b_{G^{F}} (L′, λ′))◁(Z′, b_{L′}^{F} (λ′))◁(D′, b′) := (D′, b_{C_{G^{F}} (D′)}(Res_{C_{G^{F}} (D′)}^{L′^{F}}λ′)). By conjugacy of maximal subpairs ([AB] 3.10), one has (D′, b′) = ^{g}(D, b) for some g ∈ G^{F}. One has [L, L] ⊂ C_{G}^{∘}(D) ⊂ L, so [L, L] = [C_{G}^{∘}(D), C_{G}^{∘}(D)] and [L′, L′] = g[L, L]. The equality b′ = ^{g}b implies Res_{C_{G^{F}} (D′)}^{L′^{F}}λ′ = ^{g}(Res_{C_{G^{F}} (D)}^{L^{F}}λ) (canonical

characters), whence Res_{[L′,L′]^{F}}^{L′^{F}} λ′ = ^{g}(Res_{[L,L]^{F}}^{L^{F}} λ). So we have ^{g}(L, λ) ∼ (L′, λ′). But both are e-pairs, so they are G^{F} -conjugate (see Definition 3.4). Then (L, λ) and (L′, λ′) are G^{F} -conjugate and we get our claim.

(iii) By [BMi] 2.2, every element of Irr(b_{G^{F}} (L, λ)) is in some E(G^{F} , t) with t ∈ (G^{∗})_{ℓ}^{F}. Consider χ ∈ E(G^{F} , t). First C_{G^{∗}}^{∘}(t) is a Levi subgroup by Proposition 2.1 (ii), so one may denote by G(t) an F -stable Levi subgroup of G in duality with it and by t̂ the associated linear character of G(t)^{F} (see [DM.2] 13.20). By the “Jordan-decomposition” of the elements in E(G^{F} , t) in the case of non connected center, there exists some χ_{t} ∈ E(G(t)^{F} , 1) such that ε_{G}ε_{G(t)}R_{G(t)}^{G}(t̂χ_{t}) = Σ_{χ′∈C.χ}χ′ where χ′ runs over the orbit of χ under the action of C := C_{G^{∗}}(t)^{F} /C_{G^{∗}}^{∘}(t)^{F} described in [L]. Let (L_{t}, λ_{t}) be an e-cuspidal pair of G(t) such that (L_{t}, λ_{t}) ≤ (G(t), χ_{t}). Now Proposition 3.5 (i) says that there exists a unipotent e-cuspidal pair (L′, λ′) of G such that (L_{t}, λ_{t}) ∼ (L′, λ′).

Assume χ ∈ Irr(b_{G^{F}} (L, λ)). It suffices to check that (L, λ) and (L′, λ′) are G^{F} -conjugate, or equivalently, by what we have already proved of the Theorem, that b_{G^{F}} (L, λ) = b_{G^{F}} (L′, λ′).

Let us prove b_{G^{F}} (L, λ).R_{G(t)}^{G}χ_{t} ≠ 0. By applying Brauer second main theorem and Proposition 4.1, one gets d^{1,G^{F}} (b_{G^{F}} (L, λ).R_{G(t)}^{G}χ_{t}) = b_{G^{F}} (L, λ).R_{G(t)}^{G}(d^{1,G(t)^{F}} χ_{t}) =

b_{G^{F}} (L, λ).R_{G(t)}^{G}(d^{1,G(t)^{F}} (t̂χ_{t})) = b_{G^{F}} (L, λ).d^{1,G^{F}} R_{G(t)}^{G}(t̂χ_{t}) =

ε_{G}ε_{G(t)}b_{G^{F}} (L, λ).d^{1,G^{F}} (Σ_{χ′∈C.χ}χ′) = ε_{G}ε_{G(t)}d^{1,G^{F}} (Σ_{χ′∈C.χ}χ′) ≠ 0, where the non-vanishing follows by evaluation at 1.

Now there exists χ_{1} ∈ Irr(b_{G^{F}} (L, λ)) such that (G(t), χ_{t}) ≤ (G, χ_{1}). But then Proposition 3.5 (ii) implies (L′, λ′) ≤ (G, χ_{1}), hence χ_{1} ∈ Irr(b_{G^{F}} (L′, λ′)) by (i).

Then b_{G^{F}} (L, λ) = b_{G^{F}} (L′, λ′) as claimed.

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