Published in Math. Z. 213, 479-490 (1993).

Unipotent blocks of finite reductive groups of a given type.

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

Introduction

Let G be a connected reductive group defined over GF(q), let F be the corresponding Frobenius endomorphism of G. The center Z(G) is not necessarily connected. Following [L.2], one considers a regular imbedding i : G → G′ with (G′, F) a connected reductive group defined over GF(q), Z(G′) connected and i providing an isomorphism defined over GF(q) of G with a closed subgroup of G′ satisfying [i(G′), i(G′)] = [G, G]. Let

G ⟶^{i} G′
↓^{π}  ↓_{π′}
G_{a} ⟶^{∼} G′_{ad}

be the corresponding surjections onto adjoint groups. Then Lusztig has shown how the series E(G^{F} , s) for G, G′ and G_{a} correspond ([L.2] 5.1).

In the present paper, taking ℓ a prime not dividing q, we try to translate the above considerations to the level of the “unipotent ℓ-series”

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

(see [B–M]) and corresponding ℓ-blocks. This is our Theorem 12. It is stated in terms of an isomorphism between integral block algebras preserving defect.

Concerning groups G of classical type A, B, C, D, we then show how this allows us to reduce from G to G_{a} the problem of finding the partition of E_{ℓ}(G^{F} , 1) associated with ℓ-blocks (see 4.2). This leads to a generalization of Fong-Srinivasan results ([F–S.1], [F–S.2]) on G = GL(n, q), U(n, q^{2}), CSp(2n, q), CSO(2n, q), SO(2n + 1, q) to arbitrary groups of the same type (see §4.2).

Sections 1.2 and 1.3 are devoted to preparatory work on E_{ℓ}(G^{F} , 1) in connection with [L.2]. Section 2 gives the necessary background on block theory.

1. The groups and the characters.

1.1. The groups.

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

Let F be the algebraic closure of a finite field GF(q). 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 GF(q)-rational structure of G. Many facts involving unipotent characters of G^{F} depend only on the adjoint group G_{ad} of G and the associated Frobenius (see sections 1.2 and 1.3). To every orbit ω of F on the set of connected components of the Dynkin diagram Δ of G there corresponds a direct component G_{ω} of the finite group G_{ad}^{F}. This component G_{ω} is characterized by its simple type, twisted or not, X_{ω} ∈ {A_{n},^{2}A_{n}, B_{n}, C_{n}, (n ≥ 1), D_{n},^{2}D_{n}(n ≥ 2),^{3} D_{4}, E_{6},^{2} E_{6}, E_{7}, E_{8}, F_{4}, G_{2}} and an extension field GF(q^{a(ω)}) of GF(q), of degree the length a(ω) of the orbit ω. Call (X_{ω}, q^{a(ω)}) the “adjoint rational type” — or simply rational type — of G_{ω}, and (X_{ω}, q^{a(ω)})_{ω} the “adjoint rational type” of G ; for evident convenience the rational type of a unitary group of dimension (n+1) relative to the extension (q^{2}/q) will be denoted by (A_{n}, −q).

Of course, if G as above is given, there are several ways to obtain it as the group of fixed points of some Frobenius on some algebraic group. For convenience we frequently write G^{F} in place of (G, F) when, considering properties of the finite group G^{F}, we have to refer to its ambient definition.

1.2. Characters in good series and adjoint groups.

We use the standard properties of finite reductive groups : fundamental group, duality, Lang theorem, unipotent characters (one may take [Ca] as a reference).

It will also be convenient to refer to the following group theoretic lemma. Lemma 1. ([St] 4.5). Let H be a group, F an endomorphism of H, Z a central F -stable subgroup of H and i: H^{F} /Z^{F} → (H/Z)^{F} the natural injective morphism. The cokernel of i is isomorphic to (Z ∩ [H, F])/[Z, F].

Let G defined by (G, F) as in 1.1. We will assume given an isomorphism F^{×} → (Q/ZZ)_{p′} and an imbedding of (Q/ZZ)_{p′} in \overline{Q_{ℓ}}^{×}. We follow the notations and definitions of Deligne and Lusztig [D–L] : if (s) is a F -stable class of semi-simple elements of the dual G^{∗} of G, let E(G, s) be the (s)-series of irreducible characters of G. So E(G, 1) is the set of “unipotent characters” of G.

If s ∈ (G^{∗})_{ℓ′}^{F}, one considers

E_{ℓ}(G, s) := ∪_{(t)}E(G, t)

where (t) runs over F -stable conjugacy classes of semi-simple elements of G^{∗} such that t_{ℓ′} ∈ (s) (see [B–M]).

Notation. If (G, F) is a connected reductive algebraic group defined over some finite field, let Π(G, F) be the set of prime integers different from p and not dividing |Z(G_{sc})^{F} |, where G_{sc} is a simply connected covering of [G, G].

Note that for any endomorphism f of any finite commutative group A, one has |A| = |[A, f]|.|A^{f}|. We shall use frequently the following consequence : “if r ∈ Π(G, F), then for any section U of the fundamental group of G on which F acts trivially, the order of U is prime to r”.

Lemma 2. Let (G, F) be a connected reductive algebraic group defined over GF(q), θ: G_{sc} → [G, G] a simply connected covering, A any subgroup of G^{F} containing θ(G_{sc}^{F}), π: G → G_{ad} the natural epimorphism and π^{∗}: G_{ad}^{∗} → G^{∗} a morphism dual to π. Then the following hold :

(i) The groups Z(G)^{F} ∩[G, G], G_{ad}^{F} /π(G^{F}) and G^{F} /Z^{∘}(G)^{F} .A are commutative Π(G, F)′-groups. (ii) If χ ∈ E_{ℓ}(G^{F} , 1), then Z(G)_{ℓ′}^{F} ⊂ Ker(χ). If s is a Π(G, F)-element of (G^{∗})^{F} and χ ∈ E(G^{F} , s), there exists a unique z ∈ Z(G^{∗})_{Π(G,F)}^{F} corresponding by duality to λ ∈ Irr(G^{F} /θ(G_{sc}^{F})) such that Z(G)^{F} ⊂ Ker(λ^{−1}χ) and λ^{−1}χ ∈ E(G^{F} , z^{−1}s). (iii) For every semi-simple Π(G, F)-element s of (G^{∗})^{F} and χ ∈ E(G^{F} , s), the restriction of χ to A is an irreducible character of A ; if furthermore ℓ ∈ Π(G, F) and G/A is a ℓ′-group, then the restriction from G^{F} to A induces an injection from E_{ℓ}(G^{F} , 1) into Irr(A). (iv) π^{∗} induces a bijection from the set of conjugacy classes of Π(G, F)-elements of (G_{ad}^{∗})^{F} to the set of conjugacy classes of Π(G, F)-elements of (G^{∗})^{F} contained in the image of π^{∗}. Proof. Note that Π(G, F) = Π(G^{∗}, F). We write Π for Π(G, F).

(i) As [G, G] = θ(G_{sc}) and Z(G) ∩ [G, G] ⊂ θ(Z(G_{sc})), the group Z(G)^{F} ∩ [G, G] is a group of F -fixed points on a section of the fundamental group of G. By Lang’s theorem we have [G, F] = G, [Z^{∘}(G), F] = Z^{∘}(G) and, by Lemma 1 applied to G with Z = Z(G), G_{ad}^{F} /π(G^{F}) is isomorphic to the group Z(G)/[Z(G), F] which is a quotient of Z(G)/Z^{∘}(G) and so a section of the fundamental group of G on which the operation of F is trivial. As G = Z^{∘}(G)[G, G], one has a surjective morphism Z^{∘}(G) × G_{sc} → G, defined by inclusion and θ, the kernel of which is isomorphic to θ^{−1}(Z^{∘}(G) ∩ [G, G]). We conclude once more by Lemma 1 that G^{F} /Z^{∘}(G)^{F} θ(G_{sc}^{F}) is a commutative Π′-group. So is G^{F} /A.

(ii) Deligne-Lusztig classification in series is founded on duality between algebraic groups and their tori ; if T and T^{∗} are F -stable maximal tori of G and G^{∗} respectively, a duality between T and T^{∗} defines an isomorphism (s ↦ ŝ) from (T^{∗})^{F} onto Irr(T^{F}) such that for every z ∈ Z(G)^{F} and χ ∈ E(G^{F} , s), χ(z) = χ(1)ŝ(z) (see [F–S.1], Lemma (2H)). Clearly, if s is a ℓ-element, T_{ℓ′}^{F} ⊂ Ker(ŝ) so that Z(G)_{ℓ′}^{F} ⊂ Ker(χ).

Let Z (resp. Z′) be the Hall Π-subgroup (resp. Hall Π′-subgroup) of Z(G)^{F}. From the assertion (i) it follows that G^{F} /θ(G_{sc}^{F}) is commutative and that Z ∩ θ(G_{sc}^{F}) = 1 so that every irreducible character of Z(G)^{F} the kernel of which contains Z′ is restriction of some irreducible character of G^{F} /θ(G_{sc}^{F}). Duality between G and G^{∗} induces an isomorphism between Z(G^{∗})^{F} and Irr(G^{F} /θ(G_{sc}^{F})). If λ ∈ Irr(G^{F} /θ(G_{sc}^{F})) corresponds to z ∈ Z(G^{∗})^{F} and χ ∈ E(G^{F} , s), then λχ ∈ E(G^{F} , zs) (see [D–L] (5.10.1), [L.1] (7.4.2)). If s is a Π-element of (G^{∗})^{F} and χ ∈ E(G^{F} , s), then Z′ ⊂ Ker(χ). The existence of z and λ as in the second assertion of (ii) follows. By assertion (i) Z is isomorphic to the Hall Π-subgroup of G^{F} /θ(G_{sc}^{F}). Therefore λ and z are unique.

(iii) Let s ∈ (G^{∗})^{F} and z ∈ Z(G^{∗})^{F}. Assume that zs is conjugate to s in G^{∗}. Then z ∈ Z(G^{∗})^{F} ∩ [G^{∗}, G^{∗}]. This last group is a Π′-group by (i). If s is a Πelement, then zs and z are Π-elements, so that z = 1. This shows that the stabilizer of every χ ∈ E(G^{F} , s) in Irr(G^{F} /θ(G_{sc}^{F})) (acting by multiplication) is trivial. By Clifford theory, the restriction to θ(G_{sc}^{F}) of such a χ is irreducible. Therefore the restriction to A of χ is irreducible. Assume now that ℓ ∈ Π, G^{F} /A is a ℓ′-group and χ, χ′ ∈ E_{ℓ}(G^{F} , 1) have equal restrictions to A. There exists λ ∈ Irr(G^{F} /A) corresponding to z ∈ Z(G^{∗})^{F} such that χ′ = λχ. As χ and χ′ are in series defined by ℓ-elements, z is an ℓ-element. But z and λ have equal orders, so that z = 1, λ = 1_{G^{F} /A} and χ = χ′.

(iv) The algebraic group G_{ad}^{∗} is simply connected and the image of π^{∗} is the derived group of G^{∗}. Let x be a Π-element of [G^{∗}, G^{∗}]^{F}. Let Z be the kernel of π^{∗} and consider the morphism τ: (G_{ad}^{∗})^{F} → [G^{∗}, G^{∗}]^{F} induced by π^{∗}. The kernel of τ is Z^{F} and by Lemma 1 the cokernel of τ is isomorphic to Z/[Z, F]. These groups are of order prime to Π. Therefore, x is the image by τ of a unique Π-element x_{1} of (G_{ad}^{∗})^{F}. Denote C_{G_{ad}}^{∗}(x_{1}) by C(x_{1}) and C_{[G^{∗},G^{∗}]}(x) by C(x). We have C(x_{1})^{F} /Z^{F} = C(x) ∩ τ((G_{ad}^{∗})^{F}) and the conjugacy classes of x_{1} in (G_{ad}^{∗})^{F} and x in τ((G_{ad}^{∗})^{F}) are in bijection. Now apply Lemma 1 to the groups Z and C(x_{1}). By Steinberg theorem ([Sp–St] 3.9, [St] 9.1) C(x_{1}) is connected, so Z ⊂ [C(x_{1}), F]. We obtain the exact sequence 1 → C(x_{1})^{F} /Z^{F} → (C(x_{1})/Z)^{F} → Z/[Z, F] → 1. Lemma 1 applied to the groups Z and G_{ad}^{∗} with the inner automorphism defined by x_{1} gives an isomorphism between C(x)/(C(x_{1})/Z) and ([G_{ad}^{∗}, x_{1}]∩Z). This last group is a Π-group, subgroup of the fundamental group of G (in fact C(x_{1})/Z = C^{∘}(x), see [B–M] 2.1). Thus the hypothesis on the order of x implies that C(x)^{F} = (C(x_{1})/Z)^{F}. The last exact sequence becomes 1 → C(x)∩τ((G_{ad}^{∗})^{F}) → C(x)^{F} → Z/[Z, F] → 1. We compare it with 1 → τ((G_{ad}^{∗})^{F}) → [G^{∗}, G^{∗}]^{F} → Z/[Z, F] → 1 and conclude that C(x)^{F} τ((G_{ad}^{∗})^{F}) = [G^{∗}, G^{∗}]^{F}. This shows that the conjugacy class of x in τ((G_{ad}^{∗})^{F}) is exactly one conjugacy class in [G^{∗}, G^{∗}]^{F}.

1.3. Good series and regular imbeddings.

In this section we apply Lusztig’s results [L.2] on Irr(G^{F}) when Z(G) is disconnected.

Let us recall the existence of regular imbeddings i: G → G′ in the sense of ([L.2], 7). Here (G, F) and (G′, F) are connected reductive groups over GF(q), the center of G′ is connected, i is an isomorphism of G with a closed subgroup of G′, and i(G) and G′ have the same derived subgroup. So G and G′ have isomorphic adjoint quotient groups and we assume G_{ad} = G′_{ad} in the following proposition. Proposition 3. Let i: G → G′ be a regular imbedding and π: G → G_{ad}, π′: G′ → G′_{ad} = G_{ad} the canonical epimorphisms. Dually, let i^{∗}: (G′)^{∗} → G^{∗}, π^{∗}: G_{ad}^{∗} → G^{∗}, (π′)^{∗}: G_{ad}^{∗} → (G′)^{∗} be dual morphisms such that π′ ∘i = π and π^{∗} = i^{∗} ∘(π′)^{∗}.

Let (s) be a F -stable conjugacy class of semi-simple Π(G, F)-elements of G^{∗}, with s ∈ (G^{∗})^{F}.

If s_{1} ∈ (G_{ad}^{∗})^{F} and s′ ∈ (G^{′∗})^{F} are such that s = i^{∗}(s′) and s′ = (π′)^{∗}(s_{1}), then the commutative diagram of canonical maps

G^{F}  ⟶^{i} G′^{F}
↓^{π}  ↓_{π′}
π(G^{F}) ⟶^{j} G_{ad}^{F}

(where j is associated with the inclusion π(G^{F}) ⊂ π(G)^{F}) gives rise by the associated restrictions to the following commutative diagram of bijections

E(G^{F} , s)  ⟵^{r(i)} E(G′^{F} , s′)
↑^{r(π)}  ↑^{r(π′)}
E(π(G^{F}), π(s)) ⟵^{r(j)} E(G_{ad}^{F} , s_{1})

where E(π(G^{F}), π(s)) := r(j)(E(G_{ad}^{F} , s_{1})) is a set of irreducible characters of π(G^{F}).

Proof. We first use the equality s = i^{∗}(s′). The group G′^{F} operates on the representations of its normal subgroup G^{F}, leaves stable the subset E(G^{F} , s) and this operation induces an operation of K := G_{ad}^{F} /π(G^{F}) on E(G^{F} , s). By [L.2], Proposition 5.1, the cardinality of an orbit of K on the series E(G^{F} , s) is the order of some subgroup of C_{G^{∗}}(s)^{F} /C_{G^{∗}}^{∘}(s)^{F}. But the exponent of C_{G^{∗}}(s)^{F} /C_{G^{∗}}^{∘}(s)^{F} divides the order of s (see [B–M], 2.1). So, by Lemma 2(i), in our hypothesis on s the action of G′^{F} on E(G^{F} , s) is trivial. On the other hand, by [L.2], Proposition 4, we know that the restrictions to G^{F} of elements of E(G′^{F} , s′) are multiplicity free. As i^{∗}(s′) = s, restrictions to G^{F} of elements of E(G′^{F} , s′) decompose in E(G^{F} , s). So by Clifford theory, r(i)(E(G′^{F} , s′)) ⊂ E(G^{F} , s). Moreover we know that duality defines an isomorphism from Ker(i^{∗})^{F} onto the group of representations of G′^{F} /i(G^{F}) in such a way that if μ ∈ Irr(G′^{F} /i(G^{F})) corresponds to z ∈ Ker(i^{∗})^{F}, tensor multiplication by μ induces a bijection from E(G′^{F} , s′) onto E(G′^{F} , zs′). If s′ and zs′ are conjugate in (G^{′∗})^{F}, z belongs to [G^{′∗}, G^{′∗}] ∩ Z(G^{′∗})^{F} the order of which is, by Lemma 2 (i) and hypothesis on s, prime to the order of s′ ; then z = 1. If μ ≠ 1, z ≠ 1 and the series E(G′^{F} , s′) and E(G′^{F} , zs′) are distinct. This shows that r(i) injects E(G′^{F} , s′) into E(G^{F} , s).

The elements of other series E(G′^{F} , t′) of irreducible characters of G′^{F} decompose by restriction to G^{F} on elements of E(G^{F} , s) if and only if i^{∗}((t′)) = (s) ; but if i^{∗}(t′) = i^{∗}(s′), with t′, s′ ∈ (G^{′∗})^{F}, then s′t^{′−1} ∈ Ker(i^{∗}) and s′t^{′−1} defines ζ ∈ Irr(G′^{F} /i(G^{F})). As tensor multiplication by ζ preserves restrictions to G^{F}, one has r(i)(E(G′^{F} , s′)) = r(i)(E(G′^{F} , t′)), hence r(i)(E(G′^{F} , s′)) = E(G^{F} , s). So r(i) is a bijection as claimed.

Suppose now s′ = (π′)^{∗}(s_{1}), so that s = π^{∗}(s_{1}). As Z(G′) is connected, (π′)^{∗} is an injection. Then Z(G′)^{F} is in the kernel of every element of E(G′^{F} , s′) so that restriction via (G′^{F} → G_{ad}^{F}) induces a bijection r(π′) from E(G_{ad}^{F} , s_{1}) onto E(G′^{F} , s′). The restriction via π′ ∘ i = j ∘ π sends an element of E(G_{ad}^{F} , s_{1}) on an irreducible character of G^{F}. As π(G^{F}) is a quotient group of G^{F} and a normal subgroup of G_{ad}^{F}, the restriction via j of an element of E(G_{ad}^{F} , s_{1}) is irreducible.

2. Blocks.

In this part, G is any finite group, ℓ is any prime number. One takes O a finite complete extension of ZZ_{ℓ} with a primitive |G|-th root of 1, k its residue field. All ℓ-blocks are considered as primitive idempotents of the center of OG. We denote by Irr(G) the set of irreducible characters of OG. Central functions on G have values in O and are extended by linearity to the whole OG.

If χ ∈ Irr(G), one denotes by b_{G}(χ) (or simply b(χ)) the ℓ-block of G such that χ(b(χ)) = χ(1). Similarly, if B is a sum of distinct ℓ-blocks of G (i.e. a central idempotent of OG), one denotes Irr(B) := {χ ∈ Irr(G); χ(B) = χ(1)}. One often says that χ is in B whenever χ ∈ Irr(B). The notation B_{0}(G) := b_{G}(1) designates the principal ℓ-block of G. We recall that ℓ-blocks are in OG_{ℓ′}.

2.1. Characters in blocks.

A first proposition gathers standard facts relating blocks of G and G/Z(G) (see [N–T] chapter 5,§8).

Proposition 4. (i) Let m : OG → OG/Z be the reduction map mod.Z, for Z an ℓ-subgroup of Z(G). If B is an ℓ-block of G, then m(B) is an ℓ-block of G/Z and Irr(m(B)) = {Res_{m}χ ; χ ∈ Irr(B), χ(Z) = χ(1)}. (ii) Let m′ : OG → OG/Z′ be the reduction map mod.Z′, for Z′ an ℓ′-subgroup of Z(G). If B is an ℓ-block of G, then m′(B) ≠ 0 if, and only if, for all χ ∈ Irr(B), χ(Z′) = χ(1). Moreover, if m′(B) ≠ 0, then m′ : OGB → (OG/Z′)m′(B) is an isomorphism of algebras. (iii) If Z′′ ⊂ Z(G) and χ, χ′ ∈ Irr(G) are such that Z′′ ⊂ Ker(χ) ∩ Ker(χ′), then χ and χ′ are in the same ℓ-block of G if, and only if, they are in the same ℓ-block as characters of G/Z′′.

The next proposition is about normal subgroups (see [N–T] chapter 5,§5).

Proposition 5. Let A ◁ G.

(i) Let b (resp. a) be a ℓ-block of G (resp. A) with χ_{0} ∈ Irr(b) such that Res_{A}^{G}χ_{0} ∈ Irr(a), then for all χ ∈ Irr(b), Res_{A}^{G}χ ∈ ZZ[Irr(a)] and each element of Irr(a) occurs in such a restriction. (ii) If G/A is an ℓ-group, B_{0}(A) = B_{0}(G) in Z(OG).

The following is about isomorphic blocks (see [B.2] Theorem 0.1 and proof).

Proposition 6. Let A be a subgroup of G and a (resp. b) a central idempotent of OA (resp. OG). Assume that Res_{A}^{G} induces a bijection between Irr(b) and Irr(a). Then, the multiplication by b induces an isomorphism of O-algebras

m_{b} : OAa → OGb.

Moreover, if χ ∈ Irr(b), then χ ∘ m_{b} = Res_{A}^{G}(χ) and, if D is a defect group of b_{G}(χ), then D ⊂ A and it is a defect group of m_{b}(b_{G}(χ)) = b_{A}(Res_{A}^{G}(χ)).

2.2. Decomposition maps and Brauer second main theorem.

If χ is a central function on G with values in O and if x ∈ G_{ℓ}, one denotes by d^{x,G}χ (or simply d^{x}χ) the central function on C_{G}(x) such that d^{x}χ(s) = χ(xs) when s ∈ C_{G}(x)_{ℓ′}, and d^{x}χ vanishes outside C_{G}(x)_{ℓ′}. We give an easy consequence of Brauer’s second and third ”Main Theorems” (see [N–T] 5.4.2).

Proposition 7. Let x ∈ G_{ℓ} and χ ∈ Irr(G). Assume that B_{0}(C_{G}(x)).d^{x}χ ≠ 0, then b(χ) = B_{0}(G).

2.3. The case of finite reductive groups.

Let (G, F) be a connected reductive algebraic group defined over GF(q). A key result is the compatibility of the decomposition map with the *R functor (see [F–S.1], [B.1] 4.3, [Di–M.1] 3.13 (iii)) :

Proposition 8. Let (G, F) be a connected reductive algebraic group defined over GF(q), L a F -stable Levi subgroup of G, ℓ a prime not dividing q and x ∈ L_{ℓ}^{F}. Then C_{L}^{∘}(x) is a Levi subgroup in C_{G}^{∘}(x) and

d^{x,L^{F}} ∘^{∗}R_{L^{F}}^{G^{F}} = ^{∗}R_{C_{L}^{∘}(x)^{F}}^{C_{G}^{∘}(x)^{F}} ∘ d^{x,G^{F}} .

(One recalls that C_{G}(x)/C_{G}^{∘}(x) is a ℓ-group ([B–M], 2.1), so that the map d^{x,G^{F}} takes its values in the space of central functions on C_{G}(x)^{F} vanishing outside C_{G}^{∘}(x)^{F})

With the notations of 1.2, one has :

Proposition 9. ([B–M], 2.2) If s ∈ (G^{∗})_{ℓ′}^{F}, then E_{ℓ}(G^{F} , s) is a union of ℓ-blocks, that is, E_{ℓ}(G^{F} , s) = Irr(e_{ℓ,s}^{G^{F}}) where e_{ℓ,s}^{G^{F}} ∈ Z(OG) is an idempotent.

Proposition 10. ([H] 4.2.4) With the notations of the above proposition, if b is a ℓ-block of G^{F} such that Irr(b) ⊂ E_{ℓ}(G^{F} , s), then Irr(b) ∩ E(G^{F} , s) ≠ ∅.

When s = 1, such blocks are often called unipotent blocks.

3. Unipotent blocks and regular imbeddings.

In this section, G = G^{F} is a finite reductive group over GF(q) as in 1.1, and ℓ is a prime not dividing q. We recall the commutative diagram of section 1.5, where i : G → G′ is a regular imbedding and G^{′F} = G′

G ⟶^{i} G′
↓^{π}  ↓_{π′}
π(G) ⟶^{j} G_{ad}^{F}

and the associated restriction maps r(i), r(π), r(j), r(π′) going the other way around on the corresponding sets of central functions.

A first result shows that this behaves well with respect to unipotent blocks as partitions of characters.

Proposition 11. Let G be a finite reductive group defined over GF(q) with associated Frobenius endomorphism F, let ℓ be a prime not dividing q. Then E_{ℓ}(G^{F} , 1) and E_{ℓ}(G_{ad}^{F} , 1) have the same number of blocks and the map from ZZ[E_{ℓ}(G_{ad}^{F} , 1)] to ZZ[E_{ℓ}(G^{F} , 1)] induced by the natural projection G → G_{ad} preserves the orthogonal decomposition into ℓ-blocks.

Proof. The map r(π′) sends E_{ℓ}(G^{′F} , 1) into E_{ℓ}(G_{ad}^{F} , 1). Then Proposition 4 (iii) tells us that it preserves ℓ-blocks since each ℓ-block of E_{ℓ}(G^{′F} , 1) meets E(G^{′F} , 1), hence has a character with Z(G′)^{F} in its kernel.

Let us now look at the restriction map r(i) = Res_{G^{F}}^{G^{′F}} : ZZE_{ℓ}(G^{′F} , 1) → ZZE_{ℓ}(G^{F} , 1) and show that it preserves the orthogonal decomposition into blocks. We have well r(i)(E_{ℓ}(G^{′F} , 1)) ⊂ ZZE_{ℓ}(G^{F} , 1) (see for instance [L2] 5.1). Each ℓblock of E_{ℓ}(G^{F} , 1) and E_{ℓ}(G^{′F} , 1) contains a unipotent character (Proposition 10) and one has r(i)(E(G^{′F} , 1)) = E(G^{F} , 1) ([L2] 5.1). This implies that all blocks b of G^{F} such that Irr(b) ⊂ E_{ℓ}(G^{F} , 1) are G^{′F} -fixed. It now remains to show that such a block b of G^{F} cannot split (as an idempotent) into a sum of several blocks of G^{′F}. Those blocks {b′} of G^{′F} would all satisfy Irr(b′) ∩ E(G^{′F} , 1) ≠ ∅. So, to prove our claim, it suffices to check that if χ, χ′ ∈ E(G^{′F} , 1) and b_{G^{F}} (r(i)(χ)) = b_{G^{F}} (r(i)(χ′)), then b_{G^{′F}} (χ) = b_{G^{′F}} (χ′). Since r(i)(χ′) ∈ Irr(b_{G^{F}} (r(i)(χ))), there exists χ′′ ∈ Irr(b_{G^{′F}} (χ)) such that (r(i)(χ′′), r(i)(χ))_{G^{F}} ≠ 0 (Proposition 5 (i)). But r(i)(χ) ∈ Irr(G^{F}) and G^{′F} /G^{F} is abelian, so Clifford theory implies that χ′′ = λχ where λ is a linear character of G^{′F} with λ(G^{F}) = 1. If we prove that the order of λ is a ℓ-power, then we have b_{G^{′F}} (χ) = b_{G^{′F}} (χ′′), hence b_{G^{′F}} (χ) = b_{G^{′F}} (χ′) as claimed.

Since χ is unipotent, there exists an F -stable maximal torus T′ ⊂ G′ such that (χ, R_{T^{′F}}^{G^{′F}} (1))_{G^{′F}} ≠ 0. Then (λχ, R_{T^{′F}}^{G^{′F}} (Res_{T^{′F}}^{G^{′F}} λ))_{G^{′F}} = (χ, λ^{−1}R_{T^{′F}}^{G^{′F}} (Res_{T^{′F}}^{G^{′F}} λ))_{G^{′F}} = (χ, R_{T^{′F}}^{G^{′F}} (1))_{G^{′F}} ≠ 0 (use [Di–M] 3.8). We get (χ′′, R_{T^{′F}}^{G^{′F}} (Res_{T^{′F}}^{G^{′F}} λ))_{G^{′F}} ≠ 0, but on the other hand χ′′ ∈ Irr(b(χ)) ⊂ E_{ℓ}(G^{′F} , 1) (Proposition 9), so the definition of Lusztig series implies that the order of Res_{T^{′F}}^{G^{′F}} λ is a ℓ-power. But we have T^{′F} G^{F} = G^{′F} (use Lemma 1 and the fact that [T′ ∩ G, F] = T′ ∩ G since this is connected), so the order of λ is a ℓ-power as claimed.

The theorem we prove next shows that we have much more when ℓ ∈ Π(G, F).

Theorem 12. Let G be a connected reductive group defined over GF(q) with associated Frobenius endomorphism F, let ℓ ∈ Π(G, F) (see 1.2) and O be a finite complete extension of ZZ_{ℓ} with a primitive |G^{F} |-th root of 1.

Then, there exists an isomorphism of algebras (notations of 2.3)

β : OG^{F} e_{ℓ,1}^{G^{F}} → OZ(G)_{ℓ}^{F} ⊗ OG_{ad}^{F} e_{ℓ,1}^{G_{ad}^{F}}

satisfying the following :

(i) if χ ∈ E_{ℓ}(G^{F} , 1), χ′ ∈ E_{ℓ}(G_{ad}^{F} , 1), μ ∈ Irr(Z(G)_{ℓ}^{F}) and χ = (μ ⊗ χ′) ∘ β, there exists a unique λ ∈ Irr(G^{F} /θ(G_{sc}^{F}))_{ℓ}, in duality with a unique z ∈ Z(G^{∗})_{ℓ}^{F}, such that Res_{Z(G)_{ℓ}^{F}} (χ/χ(1)) = Res_{Z(G)_{ℓ}^{F}} (λ) = μ and r(j)(χ′) = r(π)(λ^{−1}χ) ∈ Irr(π(G^{F})), (ii) if b is a ℓ-block of OG^{F} e_{ℓ,1}^{G^{F}} with defect group D ⊂ G^{F}, β(b) is a ℓ-block of O(G_{ad})^{F} e_{ℓ,1}^{G_{ad}^{F}} with defect group j(D/Z(G)_{ℓ}^{F}). Proof. Let Z := Z(G)_{ℓ}^{F}, Z′ := Z(G)_{ℓ′}^{F} and A := Z(G)^{F} [G, G]^{F}.

By Proposition 9, Lemma 2 (iii) and Proposition 5 (i), there exists a sum of ℓ-blocks of A, write it e^{A}, such that the restriction from G^{F} to A induces a bijection from E_{ℓ}(G^{F} , 1) onto Irr(e^{A}). By Proposition 6 one gets an isomorphism

OG^{F} e_{ℓ,1}^{G^{F}} ≅ OAe^{A}

which induces Res_{A}^{G^{F}} on characters and preserves ℓ-blocks and defect groups. Furthermore by Lemma 2 (ii) Z′ is in the kernel of every element of E_{ℓ}(G^{F} , 1) or Irr(e^{A}). By Proposition 4, π(e_{ℓ,1}^{G^{F}}) and π(e^{A}) are sums of ℓ-blocks of π(G) and π(A) respectively. One gets as above an isomorphism inducing restriction on characters

Oπ(G)π(e_{ℓ,1}^{G^{F}}) ≅ Oπ(A)π(e^{A}).

By Proposition 4 (ii), e^{A} has a natural image e^{A/Z′} in OA/Z′ such that OAe^{A} ≅ OA/Z′e^{A/Z′}. By Lemma 2 (i), Z ∩ [G, G] = 1. Thus one has an isomorphism A/Z′ ≅ Z × π(A) by a map which sends aZ′ to (z, π(a)) where z ∈ Z is such that az^{−1}Z′ ∩ [G, G] ≠ ∅. As π(A) ≅ (A/Z′)/(ZZ′/Z′), we get an isomorphism

π̃ : OAe^{A} → OZ ⊗ Oπ(A)π(e^{A}).

ℓ-blocks and defect groups clearly correspond. It is also clear that π̃ maps Res_{A}(χ) (χ ∈ E_{ℓ}(G^{F} , 1)) to Res_{Z}(χ/χ(1)) ⊗ Res_{π(A)}(λ^{−1}χ), where λ ∈ Irr(G^{F} /θ(G_{sc}^{F})) is defined by χ as in Lemma 2 (ii), that is Res_{Z}λ = Res_{Z}(χ/χ(1)).

Now, let us show that r(j) bijects E_{ℓ}(G_{ad}^{F} , 1) and Irr(π(e_{ℓ,1}^{G^{F}})). In view of Proposition 6, this would imply that

OG_{ad}^{F} e_{ℓ,1}^{G_{ad}^{F}} ≅ Oπ(G^{F})π(e_{ℓ,1}^{G^{F}})

by a map which induces r(j) on characters and preserves ℓ-blocks and defect groups. This would complete our proof.

By Lemma 2 (iv) and Proposition 3, if s′ ∈ (G^{′∗})_{ℓ}^{F}, the series E(G^{F} , i^{∗}(s′)) and E(G^{′F} , s′) are in bijection by r(i). Since the kernel of i^{∗} is connected, i^{∗} restricts to a surjection between the groups of rational points. The group Z(G)^{F} (resp. Z(G′)^{F}) is in the kernel of an element of E(G^{F} , t) (resp. E(G^{′F} , s′)) if, and only if, t ∈ π^{∗}(G_{ad}^{∗}) (resp. s′ ∈ π^{′∗}(G′_{ad}^{∗})). If it is the case for t := i^{∗}(s′) and s′, let s_{1} ∈ (G_{ad}^{∗})^{F} such that (π′)^{∗}(s_{1}) = s′ and π^{∗}(s_{1}) = i^{∗}(s′). We obtain that r(j) is a bijection between E(G_{ad}^{F} , s_{1}) and r(π)(E(G^{F} , i^{∗}(s′))). This holds for every conjugacy class (s_{1}) of ℓ-elements of (G_{ad}^{∗})^{F}, so we get our claim by Lemma 2 (iv).

4. Groups of classical types.

4.1. The remaining case for classical types. We assume Hypothesis (H) : G = G^{F} involves only classical types. Moreover ℓ = 2, or each component of the rational type of G is of the form (A_{n_{i}}, ε_{i}q_{i}) with ℓ dividing (q_{i}−ε_{i}).

The following is known in the cases when G is GL(n, q), U(n, q^{2}) ([F–S.1], [B.1]), CSp(2n, q), CSO(2n, q), SO(2n + 1, q) ([J]). Our proof is quite different.

Theorem 13. Under hypothesis (H), E_{ℓ}(G, 1) is the set of irreducible characters of the principal ℓ-block of G. Proof of the Theorem. The proof is by induction on |G|. In abelian group cases, E_{ℓ}(G, 1) = Irr(B_{0}(G)) is the set of linear characters of G of ℓ-power order.

Assume G ≠ Z(G). The Theorem is equivalent to the fact that E_{ℓ}(G^{F} , 1) is a single block, so Proposition 11 allows us to assume that G = G_{ad} ≠ 1. We use the following Lemma ; its proof is given after the proof of the theorem.

Lemma. Assume G = G_{ad} ≠ 1 and hypothesis (H) is satisfied. Let χ ∈ E(G^{F} , 1), then there exists an F -stable maximal torus T of G such that T_{ℓ}^{F} ≠ 1 and ∗R_{T^{F}}^{G^{F}} (χ) ≠ 0.

By Proposition 10, it suffices to check that E(G, 1) ⊂ Irr(B_{0}(G)). Let χ ∈ E(G, 1) and let T be as in the Lemma. Let x ∈ T_{ℓ}^{F}, then C_{G}^{∘}(x) is a F -stable reductive group which satisfies hypothesis (H) (in case ℓ ≠ 2, note that ℓ is good, so C_{G}^{∘}(x) is a Levi subgroup by [G–H] 2.1). One has (d^{x}χ, R_{T^{F}}^{C_{G}^{∘}(x)^{F}} 1_{T^{F}})_{C_{G}^{∘}(x)^{F}} = (d^{x}^{∗}R_{T^{F}}^{G^{F}} (χ), 1_{T^{F}})_{T^{F}} by Proposition 8. But ^{∗}R_{T^{F}}^{G^{F}} (χ) is a non zero combination of unipotent characters of T^{F}, that is, a multiple of 1_{T^{F}}, so (d^{x}χ, R_{T^{F}}^{C_{G}^{∘}(x)^{F}} 1_{T^{F}})_{C_{G}^{∘}(x)^{F}} = (^{∗}R_{T^{F}}^{G^{F}} (χ), 1_{T^{F}})_{T^{F}} (d^{x}1_{T^{F}} , 1_{T^{F}})_{T^{F}} ≠ 0. This

implies that B_{0}(C_{G}^{∘}(x)^{F}).d^{x}χ ≠ 0 since by the induction hypothesis R_{T^{F}}^{C_{G}^{∘}(x)^{F}} 1_{T^{F}} is in B_{0}(C_{G}^{∘}(x)^{F}). One may also write B_{0}(C_{G}(x)^{F}).d^{x}χ ≠ 0 by Proposition 5 (ii) and [B–M] 2.1. Then Proposition 7 says b(χ) = B_{0}(G^{F}). Proof of the Lemma. Since the assertion is about G_{ad}, one may assume G = G_{ad} is of simple rational type (X, ϵq).

One knows that G^{F} -conjugacy classes of maximal F -stable torus of G correspond to F -classes in the Weyl group of G.

In case the rational type is (A_{n}, εq), if T is a F -stable maximal torus of G, then there exists a partition {λ_{i}}_{i} of (n + 1) such that |T^{F} | = ∏_{i}((εq)^{λ_{i}} − 1)/(εq − 1). If |T^{F} | is prime to ℓ under hypothesis (H) then T is a Coxeter torus. A unipotent character χ of G^{F} corresponds to an irreducible character ζ of W (symmetric group of degree (n + 1)) so that the inner products (R_{T^{F}}^{G^{F}} (1_{T^{F}}), χ)_{G^{F}} are the values of ζ on conjugacy classes of W (see [L–S]). As n ≥ 1, there exist at least two classes on which ζ is not zero. So there exists at least one class of tori as claimed in the lemma.

If the type is from B, C, D, ℓ is 2, so q is odd, and one sees easily that for every maximal F -stable torus T of G, |T^{F} | is even. Now, by the definition of χ ∈ E(G^{F} , 1), there exists an F -stable maximal torus T ⊂ G such that (^{∗}R_{T^{F}}^{G^{F}} (χ), 1_{T^{F}})_{T^{F}} = (χ, R_{T^{F}}^{G^{F}} (1_{T^{F}}))_{G^{F}} ≠ 0, hence ^{∗}R_{T^{F}}^{G^{F}} (χ) ≠ 0.

4.2. Concluding remarks.

Let us now look at the following problem : for G any finite group of non exceptional Lie type and any prime ℓ different from the defining characteristic of G, give the distribution into ℓ-blocks of E_{ℓ}(G, 1) = Irr(e_{ℓ,1}^{G^{F}}).

Assume G is a connected reductive group defined over GF(q) with irreducible non exceptional type, F is the corresponding Frobenius endomorphism of G, and G = G^{F}. Let O be a complete finite extension of ZZ_{ℓ} containing a primitive |G|-th root of 1.

Let first set apart the case when ℓ = 2 or G is of type (A, εq) and ℓ divides q − ε. This was studied in 4.1 above : Theorem 13 tells us that e_{ℓ,1}^{G} is a single ℓ-block, the principal block of G.

Then, let us look at the complementary case. It is clear that ℓ ∈ Π(G, F) : if (G, F) is of type (A_{n}, εq), then the fundamental group of G is cyclic with F acting by εq, if G is of type B, C or D, the fundamental group is a 2-group while ℓ ≠ 2. Then Theorem 12 tells us that OG^{F} e_{ℓ,1}^{G^{F}} ≅ OZ(G)_{ℓ}^{F} ⊗ O(G_{ad})^{F} e_{ℓ,1}^{(G_{ad})^{F}} as O-algebras with an explicit correspondence for characters and defect groups.

Once there, one has to know the distribution into ℓ-blocks of E_{ℓ}(G^{F} , 1) for at least one group G_{0}^{F} of the same rational type as G^{F}, in order to obtain the distribution for G^{F}. One may take a group G_{0}^{F} for which previous results ([F–S.1], [F–S.2], [B.1]) are relevant. A short unified proof of the distribution into ℓ-blocks of E_{ℓ}(G_{0}^{F} , 1) for G_{0}^{F} = GL(n, q), U(n, q^{2}), Sp(2n, q), SO^{±}(n, q) is also given in [C–E] 4.

The method used in [C–E] applies in fact to all groups G_{0}^{F} of classical type with connected Z(G_{0}) and all ℓ-series E_{ℓ}(G_{0}^{F} , s), s ∈ (G_{0}^{∗})_{ℓ′}^{F}, ℓ ≠ 2. A sketch of proof is given in [C–E] 5.3. The main ingredient is Shoji’s theorem (see [F–S.2] Appendix) on compatibility between Jordan decomposition of characters E(C_{G^{∗}}(s)^{F} , 1) → E(G^{F} , s) and Lusztig functor, which holds even in the case when C_{G^{∗}}(s) is not a Levi subgroups of G^{∗} (“isolated series”). This provides a unified approach and generalization of the main results of [F–S.1], [F–S.2], [B.1].

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