Representations of Hecke algebras and finite groups of Lie type

Richard Dipper^{1}, Meinolf Geck^{2}, Gerhard Hiss^{3}, and Gunter Malle^{4}

1 Mathematisches Institut B Universität Stuttgart Postfach 801140 70550 Stuttgart Germany 2 U.F.R. de Mathématiques de l’Université Paris 7 2 Place Jussieu 75251 Paris Cedex 05 France 3 Lehrstuhl D für Mathematik RWTH Aachen Templergraben 64 52062 Aachen Germany 4 IWR der Universität Heidelberg Im Neuenheimer Feld 368 69120 Heidelberg Germany

1 Introduction

The aim of this article is to lead the reader on a journey through the representation theory of finite groups of Lie type and Hecke algebras. We will present some basic results obtained in recent years, explain the ideas behind them, and give lots of examples; proofs are usually omitted but we provide explicit references to an extensive bibliography.

Let G be a finite group of Lie type. Examples are G = GL_{n}(q), where q is a prime power, or any finite classical group. Hecke algebras arise naturally as endomorphism algebras of certain induced modules for G. This shows their importance for studying representations of G. By a result of Iwahori [80], they can be obtained as deformations of group algebras of finite Coxeter groups. Here, the standard example is the symmetric group 𝔖_{n}.

The group G is defined over a finite field, whose characteristic we call the defining characteristic of G. The case of representations over an algebraically closed field of characteristic 0 is relatively well-understood. The irreducible representations are classified by Lusztig [93], and Lusztig’s subsequent work on character sheaves has provided the framework in which the problem of computing the corresponding character values can be attacked; see Shoji [115].

Therefore, the emphasis in this article will be on modular representations over a field of positive characteristic, which is different from the defining characteristic of G. The overall aim of the theory, as we present it here, is to find a classification of the simple modules and to determine properties of the simple modules such as their degrees, their Brauer characters, projective covers etc.

On our journey, we will see our group G comes along with a number of rich additional structures (like ’generic’ objects and ’Hom-functors’). It is the interplay between these structures which is responsible for most of the progress made in recent years. It should also be pointed out that explicit computations played an important role in ’guessing’ the correct form of a result or providing hints for further research.

Let us now describe in somewhat more detail the contents of this article. Chapter 2 introduces two basic principles for studying modular represen-

tations of G: Harish-Chandra philosophy and Brauer’s theory of blocks. The Harish-Chandra philosophy, i.e., the idea of inducing representations from subgroups and studying their endomorphism algebras, has been successfully applied to the classification problem. It may look surprising that, as far as the formulation of the statements is concerned, this theory bears many similarities with that in characteristic 0, but a number of new features do arise. The main results are stated in Corollary 2.4 (classification of irreducible representations of G by certain triples) and Theorem 2.9 (structure of endomorphism algebras). The unknown quantities in this set-up are the representations which are not covered by the induction process; these are the cuspidal representations. We do not yet have a satisfactory theory for them; however, explicit results have been obtained in various examples, see Section 4.7.

A basic part of Brauer’s theory is the idea of relating modular representations with those in characteristic 0 via the process of modular reduction. In this way it is possible to define a partition of the representations in characteristic 0 into blocks. In the case of groups of Lie type the right approach to this question seems to be via generalized Harish-Chandra theories of generic groups. We describe this set-up in Section 2.2 and then state the main results in Section 2.3.

Chapter 3 deals with Hecke algebras. Recall that these algebras arise as endomorphism algebras of induced cuspidal representations. Up to possible twists and extensions by automorphisms, they can also be described abstractly as deformations of group algebras of finite Weyl groups. The connections between the representations of these ‘Iwahori-Hecke algebras’ and those of G will be explained in Section 3.1 and, in a refined form, in Section 3.3. The remaining sections give a survey on the known results about Iwahori-Hecke algebras. Conjecture 3.12 formulates in precise terms a conjectural ’generic’ behavior of modular representations of Iwahori-Hecke algebras (originally due to James in type A). The most striking recent result is Ariki’s solution of the Lascoux-Leclerc-Thibon conjecture, which we explain in Section 3.6.

In Chapter 4, we come back to Brauer’s theory of modular representations. Note that knowing the characters of the modular representations of G is equivalent to knowing the characters in characteristic 0 plus the decomposition matrix determined by the process of modular reduction. Thus, the determination of decomposition matrices (or, at least, certain properties of them) is of central importance. All the known examples indicate that the decomposition matrices we are looking at (i.e., the non-defining characteristic case) have a lower uni-triangular shape. The results in this chapter can be seen to support this observation. We introduce the notion of basic sets in Section 4.1. In Section 4.2 we formulate some general conjectures, in particular on a Jordan decomposition of Brauer characters. The remaining sections give a survey on the present knowledge about ℓ-modular decomposition matrices of groups of Lie type, where ℓ is different from the defining characteristic. The general linear groups are considered in Section 4.3, without any restriction on ℓ. Section 4.4 treats the other classical groups under some restrictions on ℓ. We give a short survey on the exceptional groups in Section 4.6. Our final Section 4.7 presents the known results on the distribution of the irreducible Brauer characters into Harish-Chandra series.

2 Harish-Chandra philosophy

2.1 Finite groups with BN-pairs

Many important results of Harish-Chandra theory can be formulated and proved within the framework of groups with split BN-pairs. Thus let G be a finite group with a split BN-pair of characteristic p, satisfying the commutator relations (cf. [16, §§ 65,69]). Without further notice we shall use the letters B and N to denote the Borel subgroup and the monomial subgroup of G, respectively. We also fix an integral domain R and work in the category mod-RG of finitely generated right RG-modules.

Harish-Chandra induction and restriction. We begin by introducing the functors Harish-Chandra induction, and Harish-Chandra restriction. There is one pair of such functors for each Levi subgroup of G.

Let L be a Levi subgroup of G and P a parabolic subgroup with Levi complement L, i.e., P = UL, where U is the largest normal p-subgroup of P, and U ∩ L = {1}. Then the functor

R_{L,P}^{G} : mod-RL → mod-RG, M ↦ Ind_{P}^{G}(Infl_{L}^{P}(M)),

is called Harish-Chandra induction (from L to G), and the functor

∗R_{L,P}^{G} : mod-RG → mod-RL, M ↦ Inv_{U}(M),

is called Harish-Chandra restriction (from G to L). Here, Ind_{P}^{G} is the usual induction functor from P to G, and Infl_{L}^{P} denotes inflation from L to P via the epimorphism P → L. Furthermore, Inv_{U}(M) denotes the set of U-invariants of the RG-module M, considered as RL-module.

Harish-Chandra restriction from G to L is right adjoint, and, in case p is invertible in R, also left adjoint, to Harish-Chandra induction from L to G. The collection of functors R_{L}^{G} and their adjoints satisfy a transitivity property and a Mackey formula. For more details we refer to Dipper and Fleischmann [24], where these functors are investigated in a much more general context.

The following result is of fundamental importance in Harish-Chandra theory.

Theorem 2.1. (Deligne, Dipper/Du, Howlett/Lehrer) Suppose that p is invertible in R. Let P and Q be two parabolic subgroups of G such that L is a Levi complement in P and in Q. Then the two functors R_{L,P}^{G} and R_{L,Q}^{G} are naturally isomorphic. The same is then true for their adjoint functors ^{∗}R_{L,P}^{G} and ^{∗}R_{L,Q}^{G}.

A proof of this theorem in the special case that R is a field of characteristic 0, was given by Deligne (see Lusztig and Spaltenstein [95]). In the general case it was proved independently by Dipper and Du [22] and by Howlett and Lehrer [79].

In the applications we have in mind, the assumption of Theorem 2.1 is always satisfied. In these cases we may, and will, omit the subscript P from the notation for Harish-Chandra induction and restriction.

Let us consider a simple example which shows that the assumption on the characteristic of R cannot be omitted. Let G = SL_{2}(p), where p is odd. For B and N we take the upper triangular matrices and the monomial matrices in G, respectively. Let R = k denote the prime field of characteristic p. Then the group L of diagonal matrices in G is isomorphic to the multiplicative group of k. Let M denote the 1-dimensional kL-module affording this k-representation of L. Then if B′ denotes the subgroup of lower triangular matrices in G, the two kGmodules R_{L,B}^{G}(M) = Ind_{B}^{G}(Infl_{L}^{B}(M)) and R_{L,B′}^{G}(M) = Ind_{B′}^{G}(Infl_{L}^{B′}(M)) are not isomorphic. To see this, observe that B′ = s^{−1}Bs where s is an element in N \ L. Hence Ind_{B′}^{G}(Infl_{L}^{B′}(M)) ≅ Ind_{B}^{G}(Infl_{L}^{B}(M^{s})). But Infl_{L}^{B}(M^{s}) is not isomorphic to Infl_{L}^{B}(M) and so, by Green correspondence, the induced modules are not isomorphic either.

Definition 2.2. Suppose that p is invertible in R. An RG-module M is called cuspidal, if ^{∗}R_{L}^{G}(M) = {0} for all Levi subgroups L properly contained in G.

The notion of cuspidal modules was introduced by Harish-Chandra in [68, Section 3] in the case that R is a field of characteristic 0, and by Dipper and Fleischmann in general [24]. We remark that this notion only makes sense if p is invertible in R. For example, if R = k is a field of characteristic p, then the unipotent radical of every parabolic subgroup of G has non-trivial fixed points on every non-zero kG-module M, in other words ^{∗}R_{L,P}^{G} (M) ≠ {0} for all Levi subgroups L and all parabolic subgroups P with Levi complement L.

To illustrate the importance of the concept of cuspidal modules let us assume for the moment that R = k is a field of characteristic ℓ ≠ p. If M is a kGmodule, let P (M) denote its projective cover. Then a simple kG-module M is not cuspidal if and only if there is a proper Levi subgroup L of G and a projective kL-module Q such that P (M) is a direct summand of R_{L}^{G}(Q).

Harish-Chandra series. In this section we let k denote a field of characteristic distinct from p. The following theorem gives some structural information about modules which are Harish-Chandra induced from simple cuspidal modules.

Theorem 2.3. (Geck/Hiss/Malle, Geck/Hiss) Let L be a Levi subgroup of G. Suppose that X is a simple cuspidal kL-module, and that Y is an indecomposable direct summand of R_{L}^{G}(X). Then

(a) Y has a unique simple quotient and a unique simple submodule which are isomorphic to each other.

If L′ is another Levi subgroup, X′ a cuspidal simple kL′-module and Y′ an indecomposable direct summand of R_{L′}^{G}(X′), then

(b) Y is isomorphic to Y′ if and only if the simple quotient of Y is isomorphic to the simple quotient of Y′.

(c) If Y and Y′ are isomorphic, then the pairs (L, X) and (L′, X′) are conjugate in G.

Statement (c) follows from [72, Theorem 5.8]. Proofs of (a) (without the statement on the socles) and (b) are given in [54, Theorem 2.4], based on Dipper’s theory of Hom-functors [19]. The full version of (a) and an alternative proof of (b) is established in [51, Theorem 2.9], on the basis of the arguments of Sawada, Green and Cabanes, see [12]. Dipper has now shown that the above theorem holds in the general set-up of Hom-functors, see [20, Theorem 1.8, Lemma 2.2].

As a corollary we obtain a classification of the simple kG-modules.

Corollary 2.4. (Geck/Hiss/Malle [54, Theorem 2.4]) There is a bijection between the set of isomorphism classes of simple kG-modules and the set of equivalence classes of triples (L, X, ψ), where L is a Levi subgroup of G, X is a simple cuspidal kL-module and ψ a simple End_{kG}(R_{L}^{G}(X))-module (where the pair (L, X) has to be taken modulo conjugation in G and ψ modulo isomorphism).

This classification allows to partition the set of isomorphism classes of simple kG-modules.

Definition 2.5. (Dipper/Du [22], Hiss [72]) Let M be a simple kG-module labelled, via Corollary 2.4, by the triple (L, X, ψ). Then L and X are called semisimple vertex and semisimple source of M, respectively.

The Harish-Chandra series corresponding to L and X is the set of isomorphism classes of simple kG-modules labelled by the triples (L, X, ψ), where ψ runs through the simple End_{kG}(R_{L}^{G}(X))-modules. Thus two simple kG-modules are in the same Harish-Chandra series if and only if they have a common semisimple vertex and a common semisimple source.

In Section 4.7 we shall give examples in classical groups where the distribution of the simple kG-modules into Harish-Chandra series is completely known.

There are alternative characterizations of Harish-Chandra series.

Remark 2.6. (Dipper/Du [22], Hiss [72]) (a) Let K denote the set of pairs

{(L, X) | L Levi subgroup of G, X a simple kL-module (up to isomorphism)}.

Define a partial order on K by (L, X) ≤ (L′, X′) if and only if L ≤ L′ and X is isomorphic to a composition factor of ^{∗}R_{L}^{L′}(X′).

Then a pair (L, X) ∈ K is minimal with respect to this partial ordering, if and only if X is a cuspidal simple kL-module. In this case, the Harish-Chandra series corresponding to (L, X) is the set of simple kG-modules M such that (L, X) ≤ (G, M).

If M is a simple kG-module, there is a unique (up to conjugacy and isomorphism) minimal pair (L, X) with (L, X) ≤ (G, M).

(b) Let (L, X) be a minimal pair in K. Then the set of isomorphism classes of simple kG-modules in the head of R_{L}^{G}(X) equals the set of isomorphism classes of simple kG-modules in the socle of R_{L}^{G}(X). This set of simple kG-modules equals the Harish-Chandra series corresponding to (L, X).

(c) Let (L, X) be a minimal pair in K and let M be a simple kG-module not lying in the Harish-Chandra series corresponding to (L, X).

If M is a composition factor of R_{L}^{G}(X), the semisimple vertex of M′ strictly contains L. If P (M) is a direct summand of R_{L}^{G}(P (X)), then the semisimple vertex of M is strictly contained in L.

It may in particular happen that, in the notation of the above remark, some composition factor of R_{L}^{G}(X) is cuspidal, even is L is strictly contained in G. Simple kG-modules which do not arise in this way are called supercuspidal.

Definition 2.7. (Vigneras [117]) A simple kG-module is called supercuspidal if it is not contained as a composition factor in a kG-module which is Harish-Chandra induced from a proper Levi subgroup.

Remark 2.8. (a) Let M be a simple kG-module. Then M is supercuspidal if and only if P (M) is cuspidal. (This follows from the adjointness of Harish-Chandra induction and restriction, see [73].)

(b) We may also order the set K of Remark 2.6(a) as follows. Define (L, X) ≤′ (L, X′) if and only if L ≤ L′ and X′ is isomorphic to a composition factor of R_{L}^{L′}(X). Then a pair (L, X) ∈ K is minimal with respect to ≤′, if and only if X is a supercuspidal simple kL-module.

Endomorphism algebras. Again let R = k be a field of characteristic not dividing p. We next turn to the question of finding the structure of the endomorphism algebras End_{kG}(R_{L}^{G}(X)), where L is a Levi subgroup of G and X is a simple cuspidal kL-module. We put H(L, X) := End_{kG}(R_{L}^{G}(X)).

Let W (L) := (N_{G}(L) ∩ N)L/L denote the relative Weyl group of L and let W (L, X) be the stabilizer of X in W (L). Then, by a result of Dipper and Fleischmann [24], the dimension of H(L, X) equals the order of W (L, X).

Theorem 2.9. (Howlett/Lehrer, Geck/Hiss/Malle) The group W (L, X) has a semidirect product decomposition

W (L, X) = R(L, X)C(L, X)

with normal subgroup R(L, X) and complement C(L, X) such that R(L, X) is a Coxeter group and the following hold:

(a) H(L, X) is a C(L, X) graded k-algebra, i.e., there are subspaces H_{x} of H(L, X), x ∈ C(L, X), such that

H(L, X) = ⨁_{x∈C(L,X)} H_{x},

H_{1} is a subalgebra of H(L, X) and H_{x}H_{y} = H_{xy} for all x, y ∈ C(L, X).

(b) H_{1} is an Iwahori-Hecke algebra associated to the Coxeter group R(L, X), with multiplication possibly twisted by a 2-cocycle.

(c) H(L, X) is a symmetric k-algebra.

For the definition of an Iwahori-Hecke algebra associated to a Coxeter system, we refer the reader to Section 3.2.

Special cases of Theorem 2.9 for char(k) = 0 were proved by Iwahori, Harish-Chandra, Lusztig and Kilmoyer. The general case was settled by Howlett and Lehrer in characteristic 0 [78] and, building on their work, by Geck, Hiss and Malle [54, Section 3] in positive characteristic. The fact that H(L, X) is symmetric was proved in [51].

If R_{L}^{G}(X) has an indecomposable direct summand with multiplicity 1, or if X can be extended to its inertia subgroup in (N_{G}(L) ∩ N)L, then the 2-cocycle appearing in Theorem 2.9(c) is trivial (see [54, Corollary 3.13]). In this case we call H(L, X) an extended Iwahori-Hecke algebra associated to W (L, X).

In Section 4.7 we shall give some examples for the types and parameters of Iwahori-Hecke algebras occurring in groups of Lie type as endomorphism algebras of Harish-Chandra induced cuspidal modules.

2.2 Generic finite reductive groups

In the case of finite groups of Lie type the Harish-Chandra theory introduced in the previous section has a surprising generalization to a whole family of such theories, at least in the case where the ground field has characteristic 0. These provide a powerful tool to determine the distribution into blocks of the irreducible characters. The most convenient framework for explaining the theory and to prove results is that of generic groups. This notion captures much of the generic behavior of finite reductive groups in a series.

Generalized Harish-Chandra theories for groups of Lie type. Let G be a connected reductive algebraic group over the algebraic closure of a finite field of characteristic p. Assume that G is already defined over a finite field and let F : G → G be the corresponding Frobenius morphism. The finite group of Lie type G^{F} then has a split BN-pair of characteristic p satisfying the commutator relations. Thus the Harish-Chandra theory described in Section 2.1 applies to the ordinary and modular representations of G^{F}. In the particular case considered here the Harish-Chandra theory for ordinary characters can be regarded as a special case of a whole family of generalized Harish-Chandra theories, as will be described in this section.

Let T ≤ G be an F -stable maximal torus of G. Let X, Y be the character and the cocharacter group of T with respect to some Borel subgroup, R ⊂ X the set of roots and R^{∨} ⊂ Y the set of coroots. Then F acts as qϕ on Y_{ℝ} := Y ⊗_{ℤ} ℝ, where q is a power of p and ϕ is an automorphism of finite order. A different choice of Borel subgroup containing T changes ϕ by an element of the Weyl group W of G with respect to T. Hence ϕ is uniquely determined as automorphism of Y_{ℝ} up to an element of W. Thus to (G, T, F) is naturally associated the quintuple

(X, R, Y, R^{∨}, W ϕ) .

Here, X, Y are free ℤ-modules of equal finite rank, endowed with a duality X × Y → ℤ, (x, y) ↦ ⟨x, y⟩, R ⊂ X and R^{∨} ⊂ Y are root systems with a bijection R → R^{∨}, α ↦ α^{∨}, such that ⟨α, α^{∨}⟩ = 2. Moreover, W is the Weyl group of the root system R^{∨} in Y_{ℝ} and ϕ is an automorphism of Y of finite order stabilizing R^{∨}. A quintuple (X, R, Y, R^{∨}, W ϕ) satisfying these properties is called a generic finite reductive group, or generic group for short.

Let 𝔾 = (X, R, Y, R^{∨}, W ϕ) be a generic finite reductive group. Then for any choice of a prime number p, 𝔾 determines a pair (G, T) as above, up to inner automorphisms of G induced by T. Moreover, the additional choice of a power q of p determines a triple (G, T, F) as above. In this way 𝔾 gives rise to a whole series of groups 𝔾(q) := G^{F} of groups of Lie type.

A generic Levi subgroup of 𝔾 = (X, R, Y, R^{∨}, W_{𝔾}ϕ) is by definition a generic finite reductive group of the form 𝕃 = (X, R′, Y, R′^{∨}, W_{𝕃}wϕ), where R′^{∨} is a wϕ-stable parabolic subsystem of R^{∨} with Weyl group W_{𝕃}. A generic torus is a generic finite reductive group with R = R^{∨} = ∅. Thus a generic torus of 𝔾 has the form (X′, Y′, wϕ) where Y′ is a wϕ-stable direct summand of Y and X′ is its dual. For any choice of p and q (and hence of (G, T, F)) there exists a natural bijection between the W_{𝔾}-classes of generic Levi subgroups of 𝔾 and the G^{F} - classes of F -stable Levi subgroups of G, and similarly between the W_{𝔾}-classes of generic tori of 𝔾 and the G^{F} -classes of F -stable tori of G ([7, Theorem 2.1]). This result allows us to switch freely between the language of generic finite reductive groups and that of actual groups of Lie type.

For any generic finite reductive group 𝔾 there exists a polynomial |𝔾|(x) ∈ ℤ[x] such that for any choice of a prime power q the order of the corresponding finite group 𝔾(q) = G^{F} is given by |𝔾(q)| = |𝔾|(q). Moreover, |𝔾| has the form

|𝔾|(x) = x^{N} ∏_{d} Φ_{d}^{a_{d}}

for suitable integers a_{d} ≥ 0, where Φ_{d} denotes the dth cyclotomic polynomial over ℚ. For any d such that Φ_{d} divides |𝔾| we will define a d-Harish-Chandra theory as follows. An F -stable torus S ≤ G is called a Φ_{d}-torus if the order polynomial |𝕊|(x) is a power of Φ_{d}. Here 𝕊 denotes the generic finite reductive group corresponding to the F -stable torus S of G. The centralizers of Φ_{d}-tori of G are called d-split Levi subgroups (they are F -stable Levi subgroups of G not necessarily contained in an F -stable parabolic subgroup). These notions were introduced by Broué and Malle [7]. The set of F -fixed points of d-split Levi subgroups L of G will replace the set of standard Levi subgroups in the set-up of Section 2.1.

For any F -stable Levi subgroup L of G Deligne and Lusztig defined functors

R_{L}^{G} : ℤIrr(L^{F}) → ℤIrr(G^{F}), ^{∗}R_{L}^{G} : ℤIrr(G^{F}) → ℤIrr(L^{F}),

between the character groups of L^{F} and G^{F}, adjoined to each other with respect to the usual scalar product of characters. In the case that L is contained in an F - stable parabolic subgroup of G this is just ordinary Harish-Chandra induction and restriction. In the general case R_{L}^{G} is defined as an alternating trace on ℓ-adic cohomology groups of a certain algebraic variety attached to L and G (see for example [15]). (In fact, the definition also depends on the choice of a parabolic subgroup containing L, but in the situation considered below, both functors are independent of this choice.) We note that R_{L}^{G} takes characters to generalized characters. This Lusztig induction will replace ordinary Harish-Chandra induction in the set-up of the previous section.

The following generalization of Definition 2.5 first appeared in [9]:

Definition 2.10. An irreducible character γ ∈ Irr(G^{F}) is called d-cuspidal if ∗R_{L}^{G}(γ) = 0 for all d-split Levi subgroups L properly contained in G.

An irreducible character γ ∈ Irr(G^{F}) is called unipotent if there exists an F - stable maximal torus S ≤ G such that γ is a constituent of R_{S}^{G}(1). Let Uch(G^{F}) denote the set of unipotent characters of G^{F}. It was shown by Lusztig that there exists a set Uch(𝔾) and a map

Deg : Uch(𝔾) → ℚ[x], γ ↦ Deg(γ),

such that for any choice of p and q (and hence of G and F) there is a bijection ψ_{q}^{𝔾} : Uch(𝔾) → Uch(G^{F}) such that ψ_{q}^{𝔾}(γ) has degree ψ_{q}^{𝔾}(γ)(1) = Deg(γ)(q). Furthermore, it follows from results of Shoji that the maps R_{L}^{G} and ^{∗}R_{L}^{G} are generic, i.e., there exist maps

R_{𝕃}^{𝔾} : ℤUch(𝕃) → ℤUch(𝔾), ^{∗}R_{𝕃}^{𝔾} : ℤUch(𝔾) → ℤUch(𝕃),

satisfying ψ_{q}^{𝔾} ∘ R_{𝕃}^{𝔾} = R_{L}^{G} ∘ ψ_{q}^{𝕃} for all q (see [9, Ths. 1.26 and 1.33]). In this sense, the set of unipotent characters and the functors R_{L}^{G} and ^{∗}R_{L}^{G} are generic for a series of groups of Lie type.

Therefore, and because the unipotent characters play a prominent role in Lusztig’s classification of the irreducible characters of G^{F} (see Section 2.3), we will now restrict our attention to the subset Uch(G^{F}) ⊂ Irr(G^{F}), respectively to its generic version Uch(𝔾). In this case we have the following characterization of d-cuspidality (see [9, Prop. 2.9]):

Proposition 2.11. A unipotent character γ ∈ Uch(𝔾) is d-cuspidal if and only if Deg(γ)_{d} = |𝔾_{ss}|_{d}.

Here, we write f_{d} for the Φ_{d}-part of f ∈ ℚ[x]. Also, 𝔾_{ss} denotes the semisimple quotient of 𝔾, i.e., the generic finite reductive group attached to G/Rad(G).

A pair (𝕄, μ) consisting of a d-split Levi subgroup 𝕄 of 𝔾 and a unipotent character μ ∈ Uch(𝕄) is called d-split. It is called d-cuspidal if moreover μ is d-cuspidal. We introduce the following relation on the set of d-split pairs:

Definition 2.12. Let (𝕄_{1}, μ_{1}), (𝕄_{2}, μ_{2}) be d-split. We say that (𝕄_{1}, μ_{1}) ≤_{d} (𝕄_{2}, μ_{2}) if 𝕄_{1} is a d-split Levi subgroup of 𝕄_{2} and μ_{2} occurs in R_{𝕄_{1}}^{𝕄_{2}}(μ_{1}).

For a d-cuspidal pair (𝕃, λ) of 𝔾 we write Uch(𝔾, (𝕃, λ)) for the set of γ ∈ Uch(𝔾) such that (𝕃, λ) ≤_{d} (𝔾, γ). We call Uch(𝔾, (𝕃, λ)) the d-Harish Chandra series above (𝕃, λ) because of the following fundamental result, which shows that for any d we obtain a generalized Harish-Chandra theory:

Theorem 2.13. (Broué/Malle/Michel) Let 𝔾 be a generic finite reductive group and d ≥ 1 such that Φ_{d} divides |𝔾|.

(a) (Disjointness) The sets Uch(𝔾, (𝕃, λ)) (where (𝕃, λ) runs over a system of representatives of the W_{𝔾}-conjugacy classes of d-cuspidal pairs) form a partition of Uch(𝔾).

(b) (Transitivity) Let (𝕃, λ) be d-cuspidal and (𝕄, μ) be d-split such that (𝕃, λ) ≤_{d} (𝕄, μ) and (𝕄, μ) ≤_{d} (𝔾, γ). Then (𝕃, λ) ≤_{d} (𝔾, γ).

This was shown in special cases by Schewe [113] and Fong/Srinivasan [38]. The general statement is given in [9, Ths. 3.2 and 3.11]. Note that part (b) is far from obvious since R_{L}^{G} does not necessarily take characters to actual characters. For d = 1 we recover the ordinary Harish-Chandra theory for characters.

The statement of Theorem 2.13 has been checked case by case. In the ordinary case d = 1, Harish-Chandra series may be indexed by irreducible modules of endomorphism algebras of induced cuspidal modules. A similar statement is conjectured to hold in the d-case [8]. This should also provide a more conceptual proof of the preceding theorem. At present, only some consequences of this deep conjecture have been verified. To formulate these results we need to introduce one further invariant. Let (L, λ) be d-cuspidal. Then W_{𝔾}(𝕃, λ) := N_{G}(L, λ)/L is called the relative Weyl group of (L, λ) in G.

Theorem 2.14. (Broué/Malle/Michel) For each d there exists a collection of isometries

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

such that for all 𝕄 and all (𝕃, λ) we have

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

Here 𝕄 runs over the d-split Levi subgroups of 𝔾 and (𝕃, λ) over the set of d-cuspidal pairs of 𝕄.

This result was proved in [9, Theorem 3.2]. It seems to indicate that the constituents of R_{𝕃}^{𝔾}(λ) can be parameterized by the irreducible characters of an algebra which is a deformation of the group algebra of the relative Weyl group W_{𝔾}(𝕃, λ). It can be checked that all relative Weyl groups occurring in groups of Lie type are complex reflection groups. Suitable deformations of group algebras of complex reflection groups, called cyclotomic Hecke algebras, have been introduced by Broué and Malle [8]. We cannot present the details here.

Generic blocks. The importance of d-Harish-Chandra series lies in their connection with ℓ-blocks of finite groups of Lie type for primes ℓ not dividing q.

We fix 𝔾 and a choice of a prime power q, hence a pair (G, F). A prime ℓ not dividing q is called large for 𝔾 if ℓ does not divide the order of W ⟨ϕ⟩. If ℓ is large there exists a unique d such that ℓ|Φ_{d}(q) and Φ_{d}||𝔾|.

Theorem 2.15. (Broué/Malle/Michel) Let ℓ be large for 𝔾, and assume that ℓ|Φ_{d}(q). Then the partition of Uch(G^{F}) into ℓ-blocks coincides with the image under ψ_{q}^{𝔾} of the partition of Uch(𝔾) into d-Harish-Chandra series. In particular, the distribution of unipotent characters of G^{F} into ℓ-blocks is generic.

This is proved in [9, Theorem 5.24]. We write b_{ℓ}(L, λ) for the unipotent ℓblock of G^{F} indexed by (𝕃, λ). The unipotent blocks and their defect groups can be described more precisely. For this we need:

Definition 2.16. For γ ∈ Uch(𝔾) we denote by S_{d}(γ) the set of Φ_{d}-tori of 𝔾 contained in a maximal torus 𝕊 of 𝔾 such that ^{∗}R_{𝕊}^{𝔾}(γ) ≠ 0. The maximal elements of S_{d}(γ) are called the Φ_{d}-defect tori of γ.

The Φ_{d}-defect tori of a unipotent character can be characterized as follows (see [9, Theorem 4.8]):

Proposition 2.17. Let (𝕃, λ) be a d-cuspidal pair and γ ∈ Uch(𝔾, (𝕃, λ)). Then the Φ_{d}-defect tori of γ are conjugate to Rad(𝕃)_{d}.

Here Rad(𝕃) is the generic finite reductive group associated to the radical of L (this is in fact a torus), and Rad(𝕃)_{d} denotes its maximal Φ_{d}-subtorus.

For an F -stable Levi subgroup L of G denote by Ab_{ℓ}Irr(L^{F}) the group of characters (over a splitting field of characteristic 0) of ℓ-power order of the abelian group L^{F} /[L^{F} , L^{F}]. Then [9, Theorem 5.24] gives the following sharpening of Theorem 2.15:

Theorem 2.18. (Broué/Malle/Michel) Let ℓ be large for 𝔾, and assume that ℓ|Φ_{d}(q). Let (L, λ) be a d-cuspidal pair of G.

(a) The ℓ-block b_{ℓ}(L, λ) of G^{F} consists of the irreducible constituents of the virtual characters R_{L}^{G}(θλ), where θ ∈ Ab_{ℓ}Irr(L^{F}).

(b) The defect groups of b_{ℓ}(L, λ) are the Sylow ℓ-subgroups of the groups of F -fixed points of the Φ_{d}-defect tori of unipotent characters in b_{ℓ}(L, λ).

(c) The block b_{ℓ}(L, λ) is isotypic to the principal ℓ-block of the normalizer of its defect group.

Broué and Michel [11, Théorème] have shown that part (c) actually holds under the weaker assumption that the Sylow ℓ-subgroup of G^{F} is abelian.

2.3 On ℓ-blocks of finite groups of Lie type

The results in the previous section have been generalized by Cabanes and Enguehard [13] in a program to determine all ℓ-blocks of finite groups of Lie type. In order to describe their results we first have to recall Lusztig’s classification of the irreducible complex characters of these groups. As before, let G be a connected reductive algebraic group, defined over a finite field 𝔽_{q}, with corresponding Frobenius morphism F : G → G. Let G^{∗} be a group in duality with G, with Frobenius morphism F^{∗}. We fix an isomorphism between the multiplicative group of the algebraic closure of 𝔽_{q} and the group (ℚ/ℤ)_{p′}, and an embedding of the latter into ℚ̄^{×}. Then the set of G^{F} -classes of pairs (T, θ), where T is an F -stable maximal torus of G and θ ∈ Irr(T^{F}), is in bijection with the set of G^{∗}^{F^{∗}}-classes of pairs (T^{∗}, s), where T^{∗} is an F^{∗}-stable maximal torus of G^{∗} and s ∈ T^{∗}^{F^{∗}}. We will write R_{T^{∗}}^{G}(s) for R_{T}^{G}(θ) if (T, θ) and (T^{∗}, s) correspond to each other in this way. This allows to introduce the so-called Lusztig series

E(G^{F} , s) := {χ ∈ Irr(G^{F}) | ⟨χ, R_{T^{∗}}^{G}(s)⟩ ≠ 0 for some T^{∗} ∋ s}

attached to the class of an F -stable semisimple element s of G^{∗}. Lusztig has shown that this defines a partition of the set of irreducible complex characters of G^{F}:

Irr(G^{F}) = ∐_{s} E(G^{F} , s) .

Here s runs over a system of representatives of conjugacy classes of semisimple elements in the dual group G^{∗}^{F^{∗}}. Furthermore, there is a bijection

E(G^{F} , s) ⟶^{∼} Uch(C_{G^{∗}}(s)^{F^{∗}})

of the Lusztig series E(G^{F} , s) with the set of unipotent characters of the centralizer C_{G^{∗}}(s) [93,94]. This is called the Jordan decomposition of characters.

Let ℓ be a prime dividing |G^{F} | but not dividing q. For a semisimple ℓ-regular element s ∈ G^{∗}^{F^{∗}} define

E_{ℓ}(G^{F} , s) := ⋃_{t} E(G^{F} , st)

where the union now is over semisimple ℓ-elements t in the centralizer C_{G^{∗}}(s)^{F^{∗}} of s. The following result of Broué and Michel [10, Théorème 2.2] shows a certain compatibility between Lusztig series and ℓ-blocks:

Theorem 2.19. (Broué/Michel) Each set E_{ℓ}(G^{F} , s) is a union of ℓ-blocks of G^{F}.

The description of arbitrary ℓ-blocks is analogous to that given for unipotent ℓ-blocks in the previous section, although at present some restrictions on the prime ℓ have to be made, and the result is phrased in terms of R_{L}^{G}, whose decomposition is not yet known in general. Let e be a positive integer. In extension of our above definitions we say that (L, λ) is e-cuspidal if L is e-split and λ ∈ Irr(L^{F}) is e-cuspidal. We let <<_{e} be the transitive closure of the relation ≤_{e} on e-split pairs introduced above. As an example for the type of results which can be obtained we cite:

Theorem 2.20. (Cabanes/Enguehard) Let (G, F) be as above, and ℓ a prime satisfying the following conditions: ℓ does not divide q, ℓ is good for G, ℓ ≠ 2, ℓ ≠ 3 if G^{F} has a component of type ^{3}D_{4}. Let e denote the order of q modulo ℓ.

(a) There is a natural bijection between the set of ℓ-blocks of G^{F} and the set of G^{F} -classes of e-cuspidal pairs (L, λ) with λ ∈ E(L^{F} , s) for some ℓ-regular s.

(b) Let b_{ℓ}(L, λ) be the block corresponding to the e-cuspidal pair (L, λ). Then

b_{ℓ}(L, λ) ∩ ⋃_{s} E(G^{F} , s) = {γ ∈ Irr(G^{F}) | (L, λ)<<_{e}(G, γ)}

where the union runs over ℓ-regular semisimple elements s.

(c) Let (L, λ) be e-cuspidal. Let G(s) be a group in duality with C_{G^{∗}}^{∘}(s), L(s) a Levi subgroup of G(s) in duality with C_{L^{∗}}^{∘}(s). By the Jordan decomposition of characters λ ∈ E(C_{L^{∗}}^{∘}(s)^{F} , 1) corresponds to an element of E(L(s)^{F} , 1), which we also call λ. Then (L(s), λ) is e-cuspidal. The defect group of b_{ℓ}(L, λ) is isomorphic to the defect group of the unipotent block b_{ℓ}(L(s), λ).

This was proved in [13, Theorem 3.3 and 3.4, and Rem. 3.6]. The ℓ-blocks of classical groups had previously been determined by Fong and Srinivasan [39], and Hiss [70] had obtained partial results in the case of exceptional groups.

Remark 2.21. It is not known whether the relations <<_{e} and ≤_{e} coincide. If so, we would obtain an e-Harish-Chandra theory for arbitrary Lusztig series, generalizing Theorem 2.13. Cabanes and Enguehard [13, (2.1.R)] give a technical condition on the decomposition of R_{L}^{G} which would imply the equality of <<_{e} and ≤_{e}.

Remark 2.22. Theorem 2.20 compares the defect groups of arbitrary ℓ-blocks to those of unipotent ℓ-blocks of (possibly) smaller groups. The defect groups of unipotent ℓ-blocks for large primes ℓ where described in Theorem 2.18(b). For small ℓ, when the Sylow ℓ-subgroup is no longer abelian, the structure of the defect group is explicitly given in [13, Theorem 3.5].

3 Hecke algebras

In Section 2.1 we described the endomorphism algebra of a Harish-Chandra induced cuspidal representation of a finite group G of Lie type. These endomorphism rings carry a lot of information on representations of G. We shall make this connection now more precise by defining certain functors which tie together representations of G and those of the Hecke algebras in Theorem 2.9. This motivates to study the representations of Hecke algebras abstractly. In subsequent sections we present some results on Iwahori-Hecke algebras associated with real (and in view of Section 2.2 with complex) reflection groups.

3.1 Hom-functors and Hecke algebras

We shall present the theory of Hom-functors only as far as it is needed for the representation theory of finite reductive groups and Hecke algebras. Proofs and details in a more general setting can be found in [19] and [20].

Throughout R is either a field or a complete discrete valuation ring. Ralgebras and modules of R-algebras are always assumed to be free of finite rank as R-modules. Let T be an R-algebra, M ∈ mod-T, and let β : P → M be a projective presentation of M. Assume henceforth that ker β is invariant under the endomorphism ring End_{T} (P) of P. Then J_{β} = {ϕ ∈ End_{T} (P) | βϕ = 0} is an ideal of End_{T} (P) and End_{T} (P)/J_{β} ≅ End_{T} (M) =: H canonically.

The functor H^{β} from mod-T to mod-H is defined on objects V ∈ mod-T by

H^{β} : V → Hom_{T} (P, V)/Hom_{T} (P, V)J_{β},

and on morphisms similarly. It is called Hom-functor associated with the projective presentation β : P → M of M.

The P -torsion submodule t_{P} (V) is the unique maximal submodule X of V ∈ mod-T with respect to the property that Hom_{T} (P, X) = {0}. We have a functor A_{P} : mod-T → mod-T taking V ∈ mod-T to V/t_{P} (V). We say that V is P -torsionless if t_{P} (V) = (0). We define the functor Ĥ^{β} : mod-H → mod-T to be the composite functor Ĥ^{β} = A_{P} ∘ (− ⊗_{H} M). It was shown in [19, 2.16] that Ĥ^{β} is a right inverse of the functor H^{β}.

We turn now to the calculation of decomposition numbers. For this let (K, O, k) be an ℓ-modular system. We let T_{O} be an O-order in some finite dimensional semisimple K-algebra T_{K}, where K is the quotient field of O. Let k be the residue field of O. Henceforth, tensoring is always tensoring over O if not stated otherwise, and objects tensored over O by K or k are denoted by an index K, k respectively. Thus in particular T_{k} = k ⊗ T_{O} is a finite dimensional k-algebra. Choosing T_{O}-lattices M_{O} in the irreducible T_{K}-modules M_{K} and recording the composition factors of M_{k} into a matrix (fixing an ordering of the irreducibles over K and k) yields the decomposition matrix of T_{O}.

We take a T_{O}-lattice M_{O} and let β_{O} : P_{O} → M_{O} be a projective presentation of M_{O}. Note that P_{O} is a T_{O}-lattice and that β_{k} = 1 ⊗ β_{O} : k ⊗ P_{O} → k ⊗ M_{O} is a projective presentation of M_{k} = k ⊗ M_{O}. Moreover, as in the case of group algebras we may lift idempotents, and homomorphism from projective T_{k}- modules are always liftable. The endomorphism ring of M_{R} for R ∈ {K, O, k} is denoted by H_{R}.

Theorem 3.1. (Dipper) Suppose that ker β_{O} is invariant under every endomorphism of P_{O}. Then ker β_{R} is invariant under End_{T_{R}}(P_{R}) for every choice of R ∈ {K, O, k}. Moreover End_{T_{R}}(M_{R}) = H_{R} ≅ R ⊗ End_{T_{O}}(M_{O}), and the decomposition matrix of H_{O} is the part of the decomposition matrix of T_{O} describing the composition multiplicities of the irreducible T_{k}-modules occurring in the head of M_{k} in the reductions from K to k of the irreducible constituents of the semisimple module M_{K}.

The theorem is an easy consequence of Fittings lemma and was shown first in [19, 4.7 and 4.10]. It is applied in the following situation. For a finite group G = G^{F} of Lie type we take a regular cuspidal representation C_{K} of some Levi subgroup L of G. The term regular means here, that C_{K} occurs as constituent of the Gelfand-Graev representation Γ_{K}^{L} of L. This is a special representation, which is multiplicity free and is induced from the unipotent radical of any Borel subgroup of L (see e.g. [15] for details). In particular Γ_{O}^{L} is a projective OLlattice, where the characteristic of K is now zero and that of k is a prime ℓ different from the describing characteristic p of G.

We recall the notion of Lusztig series introduced in Section 2.3. Thus corresponding to the character of C_{K} we have a representative s of a semisimple conjugacy class of the dual group L^{∗} of L. Since all irreducible direct summands of Γ_{K}^{L} occur with multiplicity one, there is up to isomorphisms precisely one OG-lattice C_{O}(s) affording this cuspidal representation such that C_{O}(s) is an epimorphic image of Γ_{O}^{L}. Among other things this implies that Γ_{O}^{L} contains a projective cover Q of C_{O}(s) as direct summand, the head of C_{k}(s) is simple occurring with multiplicity one as composition factor of C_{k}(s), and End_{kL}(C_{k}(s)) = k. We Harish-Chandra induce the epimorphism from Q onto C_{O}(s) to get a surjective map β_{O} : P_{O} → M_{O}(s), where P_{O} = R_{L}^{G}(Q) is a projective OG-lattice and M_{O}(s) = R_{L}^{G}(C_{O}(s)). As in Section 2.1, we set W (L) to be the relative Weyl group of L in G. Note that W (L) acts on the conjugacy classes of semisimple elements of L. For any t ∈ L^{∗}, denote the stabilizer of the L^{∗}-conjugacy class containing t in W (L^{∗}) by W (L, t). Note that, if R = K or if R = k and C_{k}(s) is irreducible, W (L, s) is isomorphic to the group W (L, C_{R}(s)) of Theorem 2.9. In [24, 4.10] it was shown that W (L, s) is a real reflection group provided that the center of G is connected.

Theorem 3.2. (Dipper/Fleischmann) Let s and β_{O} : P_{O} → M_{O}(s) be as above. Denote the ℓ-regular part of s by s′. Suppose that

W (L, s) = W (L, s′).

Then the kernel of β_{O} : P_{O} → M_{O}(s) is invariant under all endomorphisms of P_{O}. Moreover, the endomorphism ring of M_{O}(s) is the (extended) Iwahori-Hecke algebra H_{O}(W (L, s)) associated to W (L, s).

This theorem was proved in [24, 4.6] for connected groups with connected center and [24, Part II, Theorem 1] in full generality. As a consequence of Theorem 3.1 we obtain that the decomposition matrix of H_{O}(W (L, s)) is part of the ℓ-decomposition matrix of G.

3.2 Iwahori-Hecke algebras

The results of the previous section show that part of the decomposition numbers of finite groups of Lie type can be obtained from those of Iwahori-Hecke algebras. Therefore, we now briefly survey some features of the representation theory of Iwahori-Hecke algebras associated with reflection groups.

Let (W, S) be a finite Coxeter system, i.e., W is a finite group and S is a subset of W such that we have a presentation of W with

generators: s ∈ S,
relations: (st)^{m_{st}} = 1 for s, t ∈ S,

where (m_{st}) is a matrix of integers with m_{ss} = 1 and m_{ts} = m_{st} > 1 for s ≠ t. The matrices (m_{st}) such that the above presentation defines a finite group are classified. It turns out that the irreducible Coxeter groups fall into several infinite series and a number of exceptional types, which are conveniently described by the following diagrams:

[figure: (where m ≥ 5 in type I_{2}(m)).; E_{7}; 1; 3; 4; 2; 5; 6; 7; E_{8}; 1; 3; 4; 2; 5; 6; 7; 8; I_{2}(m); 1; m; 2; F_{4}; 1; 2; 4; 3; 4; H_{3}; 1; 5; 2; 3; H_{4}; 1; 5; 2; 3; 4; E_{6}; 1; 3; 4; 2; 5; 6; B_{n}; n≥2; 1; 4; 2; 3; n; A_{n}; n≥1; 1; 2; 3; n; D_{n}; n≥4; 1; 2; 3; 4; n]

The vertices are in bijection with S, and two vertices s ≠ t are joined by an edge if m_{st} > 2 (i.e., s, t do not commute with each other). An edge is labelled by the integer m_{st} unless we have m_{st} = 3 in which case we omit this label. For example, the diagram of type A_{n−1} defines the symmetric group on {1, …, n}, generated by the basic transpositions (1, 2), …, (n − 1, n).

An important tool in studying W is the length function l: W → ℕ_{0}: given w ∈ W we can write w = s_{1} ⋯ s_{n} with s_{i} ∈ S. If such an expression is of shortest possible length, we call it a reduced expression and let l(w) := n.

The Iwahori-Hecke algebra associated with (W, S) is defined as a deformation of the group algebra of W over a commutative ring R:

Definition 3.3. Let q_{s} ∈ R (s ∈ S) such that q_{s} = q_{t} whenever s, t ∈ S are conjugate in W. Then the Iwahori-Hecke algebra H = H_{R}(W; {q_{s} | s ∈ S}) over R is the associative algebra with 1 defined by the presentation with

generators: T_{w}, w ∈ W,
relations: T_{w} = T_{s_{1}} ⋯ T_{s_{n}} if w = s_{1} ⋯ s_{n} with s_{i} ∈ S is reduced,
T_{s}^{2} = q_{s}T_{1} + (q_{s} − 1)T_{s} for s ∈ S.

It is a non-trivial fact that H is free as an R-module, with a basis given by the elements {T_{w} | w ∈ W } (see [16, §68A]).

Now there is a universal choice for the pair (R, {q_{s} | s ∈ S}): we take the ring of Laurent polynomials A_{0} = ℤ[u_{s}^{±1} | s ∈ S] in indeterminates u_{s} such that u_{s} = u_{t} whenever s, t ∈ S are conjugate in W. The Iwahori-Hecke algebra defined with respect to the pair (A_{0}, {u_{s} | s ∈ S}) will be called the generic Iwahori-Hecke algebra associated with (W, S); we shall denote it by H. Given any commutative ring R and elements {q_{s} | s ∈ S} ⊆ R as above, we have a canonical isomorphism

H_{R} := R ⊗_{A_{0}} H ≅ H_{R}(W; {q_{s} | s ∈ S}),

where R is regarded as an A_{0}-module via the ring homomorphism A_{0} → R, u_{s} ↦ q_{s} (s ∈ S).

The reason why we took Laurent polynomials rather than just polynomials is the following: first of all, it implies that all basis elements T_{w} are invertible. Indeed, we have T_{w}^{−1} = T_{s_{n}}^{−1} ⋯ T_{s_{1}}^{−1} if w = s_{1} ⋯ s_{n} is a reduced expression and

T_{s}^{−1} = u_{s}^{−1}T_{s} + (u_{s}^{−1} − 1)T_{1} for all s ∈ S.

Secondly, it implies that the Gram matrix of the bilinear form H × H → A_{0} defined by

(T_{w}, T_{w′}) ↦ coefficient of T_{1} in T_{w}T_{w′},

is invertible over A_{0}, and hence H is a symmetric algebra.

The generic algebra and its character table. In view of our applications to groups of Lie type, we will now only consider the case where W is a finite Weyl group. Then it is known that ℚ is a splitting field for W. In order to obtain a similar statement for H, we consider the ring A = ℤ[u_{s}^{±1/2} | s ∈ S] ⊇ A_{0} and let K be its field of fractions. It follows from results of Benson/Curtis and Lusztig (see [4], [92] and also [47]) that the algebra H_{K} = K ⊗_{A_{0}} H is split semisimple. Hence, by Tits’ Deformation Theorem (see [16, §68A]), we have in fact H_{K} ≅ KW. In particular, there is a bijection between the irreducible characters of H_{K} and those of W. Explicitly, such a bijection is given as follows: since the structure constants of H_{K} with respect to the basis {T_{w} | w ∈ W } lie in A_{0}, a general argument shows that if χ is an irreducible character of H_{K} then χ(T_{w}) lies in the integral closure of A_{0} in K, i.e., we have

χ(T_{w}) ∈ ℤ[u_{s}^{1/2} | s ∈ S] for all w ∈ W .

Now consider the ring homomorphism θ: A → ℚ, u_{s}^{1/2} ↦ 1 (s ∈ S). Then Tits’ Deformation Theorem states more precisely that if χ is an irreducible character of H_{K} then the map w ↦ θ(χ(T_{w})) (w ∈ W) defines an irreducible character of W, and this gives the required bijection.

We now wish to define the character table of H_{K} as a square matrix of character values χ(T_{w}), where χ runs over the irreducible characters of H_{K} and w ∈ W runs over a set of representatives of the conjugacy classes of W. However, simple examples show that we may have χ(T_{w}) ≠ χ(T_{w′}) even if w, w′ ∈ W are conjugate. The question of a good choice of representatives is solved as follows.

Theorem 3.4. (Geck/Pfeiffer [58]) Let Cl(W) be the set of conjugacy classes of W. Let C ∈ Cl(W) and w, w′ ∈ C. Then, if w, w′ have minimal length in C, the basis elements T_{w} and T_{w′} are conjugate in H.

Thus, if we choose for each C ∈ Cl(W) a representative w_{C} ∈ C of minimal length, then the matrix of character values

(χ(T_{w_{C}}))_{χ,C} where χ ∈ Irr(H_{K}) and C ∈ Cl(W),

is independent of any choices, and it is called the character table of H.

Example 3.5. Let (W, S) be of type G_{2} = I_{2}(6), i.e., W is the dihedral group of order 12, generated by two involutions s, t such that st has order 6. There are 6 conjugacy classes, with representatives of minimal length as given in the following table.

T_{1}  T_{s}  T_{t}  T_{st}  T_{stst}  T_{ststst}
ind 1  u_{s}  u_{t}  u_{s}u_{t} (u_{s}u_{t})^{2}  (u_{s}u_{t})^{3}
ε_{1}  1  u_{s}  −1  −u_{s}  u_{s}^{2}  −u_{s}^{3}
ε_{2}  1  −1  u_{t}  −u_{t}  u_{t}^{2}  −u_{t}^{3}
ε  1  −1  −1  1  1  1
χ_{1} 2 u_{s} − 1 u_{t} − 1 \sqrt{u_{s}u_{t}} −u_{s}u_{t} −2\sqrt{u_{s}u_{t}}^{3}
χ_{2} 2 u_{s} − 1 u_{t} − 1 −\sqrt{u_{s}u_{t}} −u_{s}u_{t}  2\sqrt{u_{s}u_{t}}^{3}

The rows are indexed by the irreducible characters of H_{K}. There are 4 characters of degree 1 denoted by ind, ε_{1}, ε_{2}, ε and 2 characters of degree 2 denoted by χ_{1}, χ_{2}. We see from the above table that square roots of the parameters u_{s}, u_{t} do actually occur in character values.

The character tables of all Iwahori-Hecke algebras of all types are now known in the following sense: for classical types there exist recursive formulas in the spirit of the Murnaghan-Nakajama rule, for the exceptional types the tables have been computed explicitly.

– The table for type A_{n−1} was first computed in 1975 by A.J. Starkey (a Ph.D. student of J.A. Green in Warwick). The results are explained in [14].

– Murnaghan-Nakayama formulas for type A_{n} were found by A. Ram [110]; subsequently, different proofs of this result were found (see [107]) and generalizations were proved for types B_{n} and D_{n} (see [67] and [108], [109]).

– The tables for the exceptional types F_{4}, E_{6} and E_{7} were computed in [47] and [48], using CHEVIE [52] and GAP [114].

– The last remaining and biggest table, namely that of type E_{8}, was determined in Geck/Michel [57], again using CHEVIE and GAP.

The main tool for the computation of character tables of Iwahori-Hecke algebras of exceptional type is a general result about the eigenvalues of the basis elements T_{w} in the irreducible representations of H_{K}:

Theorem 3.6. (Geck/Michel [57]) Let ρ: H_{K} → K^{d×d} be an irreducible matrix representation and w ∈ W be an element which has minimal length in its conjugacy class. Then the eigenvalues of ρ(T_{w}) are of the form

ζ ∏_{s∈S} u_{s}^{f_{s}} where ζ is a root of unity and f_{s} ∈ ℚ for all s ∈ S.

Although Theorems 3.4 and 3.6 are formulated in general terms, their proofs depend on a case-by-case analysis. The first theorem depends on a strong form of conjugation in the underlying Weyl group, while the second theorem follows from a result about taking powers of suitable elements in the associated Artin-Tits braid groups. These results in turn were proved by inductive methods for the classical types A_{n}, B_{n}, D_{n} and were verified by explicit computations for the exceptional types using CHEVIE.

Remark 3.7. Theorems 3.4 and 3.6 remain correct for arbitrary finite Coxeter groups, i.e., also for non-Weyl groups, see [52,57]. The character tables of the Iwahori-Hecke algebras of the non-crystallographic types I_{2}(m), H_{3}, H_{4} are constructed from explicit matrix representations as computed by Kilmoyer/Solomon [16, §67C], Lusztig [92] and Alvis/Lusztig [1], respectively.

3.3 Hom-functors and q-Schur algebras

Having described Hecke algebras and exhibited their basic features we want to introduce a further type of finite dimensional algebras, q-Schur algebras, which are derived from Hecke algebras enlarging those, and whose representations carry also information on the representations for finite groups of Lie type. We first explain how this comes about.

Let us keep the notation of Section 3.1. Being an additive functor of abelian categories the Hom-functor H^{β} introduced there induces homomorphism on Hom-sets and endomorphism algebras. A special case is of particular interest (see [19, 2.21]):

Proposition 3.8. (Dipper) Let T be an R-algebra, M ∈ mod-T, and β : P → M be a projective presentation of M, such that ker β is invariant under all endomorphisms of P. Suppose that M is P -torsionless. Then H^{β} induces an isomorphism, again denoted by H^{β}:

H^{β} : Hom_{T} (XM, Y M) → Hom_{H}(X, Y),

where X, Y are right ideals of H.

This can again be applied to the representation theory of a finite group of Lie type G. The idea is roughly as follows: We take right ideals X_{i} of H_{O}, where H_{O} is one of the Hecke algebras coming up in Theorem 3.2, then we consider the OG-lattice

N_{O} = ⨁_{i} X_{i}M_{O}.

We construct a projective presentation γ_{O} : Q_{O} → N_{O} such that ker γ_{O} is invariant under all endomorphisms of Q_{O}. Then we apply Theorem 3.1 again to connect representations of RG with those of the endomorphism ring S_{R} = End_{RG}(N_{R}). On the other hand Proposition 3.8 yields that S_{R} is the endomorphism ring of the direct sum of the X_{i} as well and can hence be computed in terms of Hecke algebras again. However in 3.8 one needs often to replace X_{i}M_{O} by its purification in M_{O} which yields as well an isomorphism (for details see [19, 4.17] and [20, 1.30]) or even lattices in between ([25]).

The special case of the unipotent characters of G may serve as illustration: We take as Levi subgroup a maximally split torus T and let M_{R} = R_{T}^{G}(I_{R}), where I_{R} is the trivial RT -module. Thus M_{R} is the trivial B-module induced to G, where T ⊆ B is a Borel subgroup of G. The irreducible characters afforded by the constituents of M_{K} are unipotent characters. This situation was first investigated for R = K by Iwahori in [80] and is a special case of Theorem 2.9. The endomorphism ring End_{RG}(M_{R}) is the Iwahori-Hecke algebra H_{R} = H_{R}(W) associated with the Weyl group W of G, where the parameter values are determined by the field of definition of G. For instance if G is untwisted over the finite field with q elements, all parameters equal q. Note that this is a special case of the situation in Theorem 3.2. The corresponding Hom-functors are denoted by H_{R}, where as usual R ∈ {K, O, k}.

The Levi subgroups B ⊆ L ⊆ G are parametrized by the subsets J of the set S of basic reflections of the Weyl group. We denote by L_{J} the Levi subgroup of G corresponding to J ⊆ S. Similarly, W_{J} denotes the corresponding parabolic subgroup of W. For J ⊆ S we set

y_{J} = ∑_{w∈W_{J}}(−1)^{l(w)}ind(T_{w})^{−1}T_{w} ∈ H_{R},

where ind: H_{R} → R is the 1-dimensional representation such that ind(T_{s}) = q_{s} for s ∈ S. Moreover we define

Y_{R} = ⨁_{J⊆S} y_{J}H_{R}.

Note that Y_{R} ≅ R⊗Y_{O}, since y_{J}H_{O} is pure in H_{O}. Moreover St_{J,K} = y_{J}R_{T}^{L_{J}} (I_{K}) is a KL_{J}-submodule of M_{K} isomorphic to the Steinberg module of L_{J}, which occurs with multiplicity one in the Gelfand-Graev representation Γ_{J,K} of L_{J}. Consequently there is a unique OL_{J}-lattice St_{J,O} in St_{J,K} which is an epimorphic image of Γ_{J,O}, which hence contains a projective cover of St_{J,O}. We set N_{J,O} = R_{L_{J}}^{G} (St_{J,O}) and

N_{R} = R ⊗ ⨁_{J⊆S} N_{J,O},

for R ∈ {K, O, k}. Similarly we Harish-Chandra induce the projective cover of St_{J,O} and take the direct sum over all subsets J of S to obtain a projective presentation γ_{O} : Q_{O} → N_{O}. We have the following result:

Theorem 3.9. (Dipper/Gruber) The kernel of γ_{R} is invariant under all endomorphisms of Q_{R}. Moreover H_{R}(N_{R}) ≅ Y_{R} and H_{R} induces an isomorphism from End_{RG}(Y_{R}) onto End_{H_{R}}(Y_{R}).

As a consequence of Theorem 3.1 the decomposition matrix of End_{H_{O}}(Y_{O}) embeds into the decomposition matrix of G. For a proof (and a much more general statement, dealing with the situation of Theorem 3.2) see [25]. It should be noted too that for J ⊆ S the OG-sublattice N_{J,O} of M_{O} contains y_{J}M_{O} and is contained in the purification y_{J}M_{K} ∩ M_{O} of the latter, but is not equal to one of these in general.

The R-algebra End_{H_{R}}(Y_{R}) is called the q-Schur algebra associated with H_{R} (of parabolic type) and it is denoted by S_{R} = S_{R}(W) adding the parameters as additional indices if needed.

In [29] Dipper and James introduced q-Schur algebras associated with Weyl groups of type A and observed that their representations are closely related with those of finite general linear groups, (compare Section 4.3). Subsequently it was discovered by Dipper and Donkin that these come up as well as duals of the subcoalgebra of homogeneous parts in the quantum monoid of (n × n)-matrices, for details see [21].

In [51] and [66] Geck, Gruber and Hiss defined q-Schur algebras (somewhat smaller than the q-Schur algebras of parabolic type introduced above) for Weyl groups of types B and D, and used them to relate the decomposition matrices of classical groups with those of general linear groups (see Section 4.4).

Parshall and Wang showed in [106] that q-Schur algebras of type A are quasi hereditary. This is not always true for other types. In [36] Du, Parshall and Scott found that those satisfy the weaker condition to be stratified algebras. For further results see e.g. [62] and [112].

In [37] Du and Scott and independently in [32] Dipper, James, and Mathas constructed quasi hereditary q-Schur algebras of type B, called q-Schur^{2} algebras in [37] and (Q, q)-Schur algebras in [32] by using not only q-analogues of right ideals y_{J}H coming from parabolic subgroups of W, but from arbitrary reflection subgroups as well.

In [33] Dipper, James, and Mathas generalized this further by constructing cyclotomic q-Schur algebras which are associated with Ariki-Koike algebras. These are deformations of the group algebra of certain complex reflection groups, which are wreath products of a cyclic group of order r by the symmetric group 𝔖_{n}. For r = 1 one obtains the Iwahori-Hecke algebra of type A_{n} and for r = 2 of type B_{n}. Thus the previous results appear as special cases of this construction.

In [63] R. Green used similar ideas to construct q-Schur algebras associated with affine Hecke algebras of type A.

Using the quantum group approach Oehms defined a different generalized q-Schur algebra of type B, called symplectic q-Schur algebra in his dissertation [102]. The Brauer algebra acts on tensor space centralizing the action of symplectic and orthogonal groups. A q-deformation of the Brauer algebra is the Birman-Murakami-Wenzl algebra which acts on tensor space as well. Using a generalization of the FRT-construction the symplectic quantized monoid is defined as centralizer coalgebra of this action. This is a bialgebra graded by subcoalgebras of finite rank, whose duals are defined to be the symplectic q-Schur algebras.

3.4 Specializations and decomposition maps

Let H be a generic Iwahori-Hecke algebra or a suitable q-Schur algebra associated with a finite Weyl group W. Then any specialization θ: A → k into a field k gives rise to a specialized algebra H_{k} = k ⊗_{A} H. In Section 3.2 we have already encountered this situation in connection with Tits’ Deformation Theorem. Now we consider the case where the specialized algebra is not semisimple. Then we have a non-trivial decomposition map between the Grothendieck groups of representations of H_{K} and of H_{k}.

In this section, we consider only the case where all parameters u_{s} (s ∈ S) equal a single parameter u, say. In view of our applications to finite groups of Lie type, we are mainly interested in specializations of the following type: let q be a prime power, ℓ be a prime not dividing q, and

θ: A → 𝔽̄_{ℓ} such that θ(u) = q ∈ 𝔽_{ℓ}.

By the theory of Hom-functors (see Section 3.1), the associated decomposition matrix of H is a submatrix of the decomposition matrix of a corresponding finite group of Lie type over 𝔽_{q}. Now there are of course infinitely many choices for ℓ and q. But one can in fact control this situation by factorizing θ into two steps: one from A to a cyclotomic extension of ℚ and one from that cyclotomic extension to 𝔽̄_{ℓ}.

More precisely, let e ≥ 1 be minimal such that θ(u)^{e} = 1. Then we can factorize θ as follows. Let ζ_{e} ∈ ℂ be a primitive e-th root of unity and θ_{e}: A → ℂ be a ring homomorphism such that θ_{e}(u) = ζ_{e}. Then there exists a ring homomorphism θ_{e,ℓ}: ℚ[ζ_{e}^{1/2}] → 𝔽̄_{ℓ} such that θ = θ_{e,ℓ} ∘ θ_{e}.

Theorem 3.10. Let θ: A → 𝔽̄_{ℓ} be a specialization as above and D be the corresponding decomposition matrix between the irreducible representations of H_{K} and 𝔽̄_{ℓ} ⊗_{A_{0}} H. Then we have a factorization D = D_{e,ℓ} · D_{e}, where D_{e} and D_{e,ℓ} are the decomposition matrices induced by θ_{e} and θ_{e,ℓ}, respectively.

The article [59] provides a rather general framework in which a result as above can be proved. Thus, the decomposition matrix D_{e} can be viewed as a first approximation to D, and D itself is given by multiplying D_{e} with the “adjustment matrix” D_{e,ℓ} (a term introduced by James).

Example 3.11. Let H be an Iwahori-Hecke algebra of type G_{2}. Then D_{e} is not the identity matrix only for e ∈ {2, 3, 6}. In these cases, we have:

D_{e} e = 2  e = 3  e = 6
ind 1 . .  1 ….  1 ….
ε_{1} 1 . .  . 1 …  . 1 …
ε_{2} 1 . .  . . 1 . .  . . 1 . .
ε  1 . .  … 1 .  … 1 .
χ_{1} . 1 .  …. 1  1 . . 1 .
χ_{2} . . 1  1 . . 1 .  …. 1

If ℓ ∈ {2, 3} we only have to consider adjustment matrices D_{e,ℓ} for (e, ℓ) ∈ {(2, 2), (2, 3), (3, 3)}; they are given as follows:

( \begin{matrix}1 & . \\ . & 1 \\ . & 1 \end{matrix} ) ,  ( \begin{matrix}1 & . \\ 2 & . \\ . & 1 \end{matrix} ) ,  ( \begin{matrix}1 & . & . & . \\ . & 1 & . & . \\ . & . & 1 & . \\ . & . & . & 1 \\ . & . & 1 & 1 \end{matrix} ) ,  respectively.

If ℓ > 3, then D_{e,ℓ} is always the identity matrix of the appropriate size. These results are easily proved by explicit computations using the matrix representations given in [16, §67C].

The above example seems to indicate that most adjustment matrices are in fact identity matrices. A general argument (which is given for Iwahori-Hecke algebras in [44, Prop. 5.5]) shows that this is indeed the case for all sufficiently large primes ℓ. The term “sufficiently large” is specified in the following conjecture, which was first stated by James [84, §4] for Iwahori-Hecke algebras and q-Schur algebras of type A_{n−1} and then formulated for Iwahori-Hecke algebras of any type in [44, (5.6)].

Conjecture 3.12. If H is the generic Iwahori-Hecke algebra associated with a finite Weyl group, then the adjustment matrix D_{e,ℓ} is the identity matrix if ℓ does not divide the order of W.

Probably, in the case of q-Schur algebras and Iwahori-Hecke algebras of type A_{n−1}, this result will even be true under the strengthened condition that eℓ > n. The conjecture is established in [44] for blocks of defect 1, where the decomposition matrix is given in terms of a Brauer tree, similarly as for blocks of defect 1 in finite groups.

For explicit results on Iwahori-Hecke algebras of classical type, see Section 3.5. The known results for exceptional types are summarized as follows:

– For type G_{2} with equal parameters, see Example 3.11. (The case of unequal parameters is easily treated by similar methods.)

– The matrices D_{e} are completely determined for types F_{4}, E_{6} and E_{7}; see [55], [45], [48], and [98]. Partial results on the matrices D_{e} for type E_{8} are obtained by J. Müller [98]. The techniques used include R. Parker’s MEATAXE [105] and S. Linton’s VECTOR-ENUMERATOR [91]. In [99], the analogous matrices for the non crystallographic types I_{2}(m), H_{3}, H_{4} are determined.

– The above conjecture is established by explicit computation of all possible decomposition matrices for types F_{4} (with equal parameters) and E_{6}, see [55] and [45], respectively. K. Bremke [5] has considered those cases of unequal parameters in type F_{4} which actually occur in groups of Lie type.

In the case of Iwahori-Hecke algebras of exceptional type or of type A_{n}, B_{n}, D_{n} (n odd) the following result can be extracted from the above explicit results and the results in [26,34,104].

Theorem 3.13. (Geck [49]) If H is an Iwahori-Hecke algebra of a finite Weyl group (where all parameters are equal), then the matrix D_{e} always has a lower unitriangular shape, for a natural ordering of the rows and columns given in terms of the relation ≤_{LR} of Kazhdan-Lusztig [86].

3.5 Iwahori-Hecke and q-Schur algebras of classical type

We now turn to the Iwahori-Hecke algebras of classical type. We begin with the type A case. Thus W = 𝔖_{n} is the symmetric group, H = H_{R}(W; q), where R is a commutative ring and q is a unit of R. For fields R the irreducible H-modules were determined by Dipper and James in [26, 7.6] and it was shown that their decomposition matrices are lower unitriangular.

For a partition λ ⊢ n of n let 𝔖𝔗_{λ} denote the set of standard λ-tableaux. Murphy introduced in [100] and [101] a basis

{x_{𝔰𝔱} | 𝔰, 𝔱 ∈ 𝔖𝔗_{λ}, λ ⊢ n}

of H, called Murphy basis, which possesses remarkable properties. Indeed it is a cellular basis as defined by Graham and Lehrer in [60], and hence their general results on cellular algebras may be used to recover most of the results in [26] and some of [28]. We apply now the general theory of cellular algebras [60]. For a partition λ of n we define N^{λ} to be the R-module spanned by

{x_{𝔰𝔱} | 𝔰, 𝔱 ∈ 𝔖𝔗_{μ}, μ ⊢ n, μ ⊵ λ}

and N̄^{λ} to be the subspace spanned by those x_{𝔰𝔱} where μ ⊳ λ.

Theorem 3.14. Let λ be a partition of n. Then N^{λ} and N̄^{λ} are two sided ideals of H.

The Specht (or standard) module S^{λ} is now the submodule of H/N̄^{λ} generated by z_{λ} = x_{λ} + N̄^{λ}/N̄^{λ} ∈ H/N̄^{λ}, where x_{λ} := ∑_{w∈W_{λ}} T_{w}. On S^{λ} we have an H-invariant bilinear form whose radical is denoted by radS^{λ}. We set D^{λ} = S^{λ}/radS^{λ}.

Theorem 3.15. Suppose R is a field. Then the non-zero H-modules in

{D^{λ} | λ ⊢ n}

is a complete set of irreducible H-modules. If we choose a linear ordering of the partitions of n which is compatible with the dominance order downward then the matrix Δ describing the composition factors of Specht modules is lower unitriangular. If R is a field and H is semisimple then (0) ≠ S^{λ} = D^{λ} for every partition λ of n. Thus

{S^{λ} | λ ⊢ n}

is a complete set of non-isomorphic irreducible H-modules in this case.

It should be pointed out that the Kazhdan-Lusztig basis of H as defined in [86] is cellular in type A but not in general.

Theorem 3.15 states what can directly be concluded from the fact that H has a cellular basis. There are some further results which however do not follow directly from properties of a cellular basis, but require special proofs. In particular one knows which of the D^{λ} are nonzero in Theorem 3.15 by a result of Dipper and James [26, 7.6]. Thus let R be a field. It is known that H is not semisimple if and only if the parameter q ∈ R is a primitive eth root of unity for some 1 < e ≤ n or q = 1 and R is of characteristic ℓ ≤ n. We define e to be ℓ in the latter case. A partition λ is called e-regular, if there are at most e − 1 repetitions in λ. We then have:

Theorem 3.16. (Dipper/James) Let e be as above. Then D^{λ} ≠ (0) precisely if λ is e-regular.

The blocks of H are given by e-cores respectively p-cores in the group case by [28, 4.13] and [85, 4.29], and the primitive central idempotents were exhibited in [28, 5.4] to name a few further results.

In the special case of a field R of characteristic 0, where q is a primitive eth root of unity, an algorithm how to compute the decomposition matrices explicitly will be discussed in more detail in the next section.

We come now to other types of Iwahori-Hecke algebras and q-Schur algebras. In [97] Mathas produced a cellular basis of Ariki-Koike algebras generalizing results of Dipper, James and Murphy [34] on Iwahori-Hecke algebras of type B. Here the labels of the basis elements are standard multi-tableaux whose shapes are multi-partitions, which in turn label the standard and the irreducible modules. The decomposition matrix is again lower unitriangular. The question of actually classifying the irreducible modules is presently still open, although a lot of evidence towards a conjecture concerning this question has been obtained, (see [34, 8.13] for the case of Iwahori-Hecke algebras of type B and [97] for its generalization to Ariki-Koike algebras).

A similar picture evolves for q-Schur algebras and their generalizations. The case of type A was investigated first in [29] and [30] by Dipper and James. They defined q-Weyl modules, labelled by partitions of n and determined a semistandard basis of those, labelled by semistandard tableaux. The q-Weyl modules have unique maximal submodules for fields R and the unique irreducible quotients are precisely the irreducible modules for the q-Schur algebra. The decomposition matrix describing the composition factors of the q-Weyl modules is a square matrix and is unitriangular.

In [35] Du extended the Kazhdan-Lusztig basis for Iwahori-Hecke algebras to q-Schur algebras, and showed that it is cellular. He also found that it is dual to Lusztig’s canonical basis of quantum groups.

In his thesis [61] Richard Green extended the Murphy basis of the Iwahori-Hecke algebra of type A to q-Schur algebras using bideterminants and showed that it is cellular.

The cyclotomic q-Schur algebras (and hence the special cases for type A and B of Section 3.3) are cellular ([33]) as well as the symplectic q-Schur algebras of Oehms in [102]. Moreover, in both cases the bilinear form on standard modules are non-zero, and hence the decomposition matrices are square matrices and the algebras are quasi hereditary.

Under a special assumption there is a direct way to investigate Iwahori-Hecke and q-Schur algebras of type B, yielding more complete results. Thus let W = C_{2} ≀ 𝔖_{n} be the Weyl group of type B of rank n. The Iwahori-Hecke algebra H involves two parameter q and Q. Let

f_{n}(Q, q) = ∏_{i=−(n−1)}^{n−1}(Q + q^{i}).

Theorem 3.17. (Dipper/James) Let R be a commutative ring, and let q, Q be units of R. Suppose that f_{n}(Q, q) ≠ 0. Then H is Morita equivalent to

⨁_{a=0}^{n} H_{R}(𝔖_{a}; q) ⊗ H_{R}(𝔖_{n−a}; q).

For details we refer to [31]. As a consequence one obtains:

Theorem 3.18. (Gruber/Hiss) Let R, Q, q, H be as above and suppose again that f_{n}(Q, q) ≠ 0. Then the (Q, q)-Schur algebra S is Morita equivalent to the corresponding q-Schur algebra of parabolic type which in turn is Morita equivalent to the direct sum of q-Schur algebras associated with the Iwahori-Hecke algebras H_{R}(𝔖_{a}; q) ⊗ H_{R}(𝔖_{n−a}; q), where a runs from 0 to n.

This theorem was shown in [66] for those choices of parameters occurring in the representation theory of finite classical groups. The condition f_{n}(Q, q) ≠ 0 holds in roughly a third of all possible cases for these groups. Gruber and Hiss used their representations to establish Theorem 3.18 and to show even that using only parabolic subgroups of type A produces another Morita equivalent q-Schur algebra. However Theorem 3.18 holds in general as can be seen easily from [32]. Obviously Theorems 3.17 and 3.18 allow to carry results for the type A-case to those for the type B_{n}-case if f_{n}(Q, q) ≠ 0. In particular using 3.16 there is a complete description of the irreducible modules and decomposition matrix of H in terms of Iwahori-Hecke algebras of type A and that of the corresponding q-Schur algebra of type B in terms of q-Schur algebras of type A. In Section 4.4 this will be applied to representations of classical groups.

Iwahori-Hecke algebras of type D seem far less accessible. In particular there is no cellular basis known, and only q-Schur algebras of parabolic type have been considered so far. Pallikaros generalized Theorem 3.17 to Iwahori-Hecke algebras of type D in [104]. However, his results are not as complete as in the type B case. In particular, he does not get a complete classification of the irreducible modules in the case of even n. Note, however, that the triangular shape for the decomposition matrix is established in [49], using a completely different approach. There are some more results in special cases, due to Gruber and Hiss in [66], using the fact that Iwahori-Hecke algebras of type D are contained as subalgebras in Iwahori-Hecke algebras of type B for a special choice of the second parameter.

There are many numerical results on decomposition matrices of Iwahori-Hecke and q-Schur algebras, of which we shall mention only two, since they are connected with Conjecture 3.12. Mathas wrote the share package SPECHT in GAP [114] to compute decomposition numbers of Iwahori-Hecke algebras of type A at roots of unity in characteristic zero. In his thesis which is under preparation, Lipp used an implementation in GAP of combinatorial rules [84] for the computation of decomposition numbers of q-Schur algebras of type A to verify the James conjecture in many special cases (compare Section 3.4). In general he shows that the conjecture holds for fields R of small characteristic and based on results of Richards [111] proved a block wise version of James conjecture for blocks of weight ≤ 3.

3.6 The theory of Lascoux-Leclerc-Thibon

Let H_{n} be the generic Iwahori-Hecke algebra associated with the symmetric group 𝔖_{n}. Consider a specialization θ: A → ℂ which takes the parameter u to a primitive e-th root of unity (for some e ≥ 1), and let D_{e} be the corresponding decomposition matrix. Lascoux, Leclerc and Thibon [89] conjectured that the problem of computing D_{e} can be translated to that of computing the canonical basis (in the sense of Kashiwara and Lusztig) of a certain highest weight module for an affine Kac-Moody algebra. This has now been proved by Ariki [3]. (In October 1995, Grojnowski also announced a proof, following the ideas in [64].) Here, we shall only describe the purely algorithmic part of the solution.

For any n ≥ 0, let F_{n} be the ℚ(u)-vector space with a basis consisting of the partitions λ ⊢ n; by convention, F_{0} has as basis the empty partition, ∅. Let

F := ⨁_{n≥0} F_{n}, the so-called “Fock space”.

As usual, a partition λ ⊢ n is represented by a Young diagram, which is a collection of n boxes arranged in left-justified rows, where the number of boxes in each row is given by the parts of λ.

The e-residue of a box which occupies the j-th position in the i-th row say, is the unique r ∈ {0, …, e − 1} such that r ≡ j − i (mod e).

Next, if we can remove the right-most box in a row and the result is still a Young tableau (for a partition of n − 1) we call that box a removable box. (For example, the right-most box in the last row is always removable.)

Finally, if we can add a new box to some row and the result is again a Young tableau (for a partition of n + 1), we call that row an indent row. (For example, we can always add a box to the first row; note that it is allowed to add a new row consisting of one box to the bottom of the diagram.) The e-residue of such a row is the e-residue of the new box that we can add.

We shall now define linear maps f_{r}: F → F (0 ≤ r ≤ e − 1) such that f_{r}(F_{n}) ⊆ F_{n+1} for all n, r. Let λ ⊢ n and 0 ≤ r ≤ e − 1. Suppose that the i-th row is an indent row of λ whose e-residue is r. Then we let N(λ, r, i) be the number of indent rows strictly above i whose e-residue is r, minus the number of removable boxes in rows above i whose e-residue is r. We define

f_{r}(λ) = ∑_{i}u^{N(λ,r,i)} λ_{[i]},

where the sum is over all indent rows i of λ with e-residue r, and λ_{[i]} denotes the partition of n + 1 obtained by adding a new box to the i-th row. For any k ≥ 1, we let [k]_{u} := (u^{k} − u^{−k})/(u − u^{−1}), and define operators f_{r}^{(k)} by

f_{r}^{(1)} := f_{r} and f_{r}^{(k)} = f_{r}^{(k−1)}/[k]_{u} (k ≥ 2).

Let K_{n} ⊆ F_{n} be the subspace generated by all f_{ι}^{κ}(∅) := f_{i_{n}}^{(k_{n})} ⋯ f_{i_{1}}^{(k_{1})}(∅), where ι = (i_{1}, …, i_{n}) and κ = (k_{1}, …, k_{n}) are any sequences of integers with 0 ≤ i_{1}, …, i_{n} ≤ e − 1 and k_{1}, …, k_{n} ≥ 1.

Lemma 3.19. For any e-regular partition ν ⊢ n, there exist ι, κ as above such that

K_{ν} := f_{ι}^{κ}(∅) = ν + ℤ[u^{±1}]-linear combination of various λ ⊢ n, λ ⊲ ν.

Here, ⊲ denotes the usual dominance order. Note that this element depends on the choice of ι, κ, but for any choice, the elements {K_{ν}} form a basis of K_{n}.

The proof in [89, §6] also yields an algorithm for finding suitable sequences ι, κ for each ν. Now, using an interpretation of F as a module for an affine Kac-Moody algebra and applying the theory of canonical bases, it can be shown that there exist unique elements {G_{μ}} in K_{n} (for e-regular μ ⊢ n) such that

G_{μ} = μ + ∑_{λ⊢n,λ⊲μ} d_{λμ}(u)λ where d_{λμ}(u) ∈ ℤ[u] and d_{λμ}(0) = 0.

Lemma 3.20. Each G_{μ} is uniquely a linear combination G_{μ} = ∑_{ν} γ_{νμ}(u)K_{ν} where γ_{νμ}(u) ∈ ℤ[u, u^{−1}] are such that γ_{νμ}(u^{−1}) = γ_{νμ}(u).

The proof in [89, §6] shows that the γ_{νμ}(u) can be computed by a “triangular algorithm”. All this is implemented in Mathas’ GAP package SPECHT [114], and in Veigneau’s package ACE [116]. Here is an example, for e = 2 and n = 5:

\begin{aligned}& K_{(5)} & K_{(41)} & K_{(32)} \\ \begin{matrix}κ : \\ ι : \end{matrix} & {11111 \atop 01010} & {1211 \atop 0101} & {122 \atop 010} \\ (5) & 1 & . & . \\ (41) & . & 1 & . \\ (32) & 1 & . & 1 \\ (311) & 2u & . & u \\ (221) & u^{2} & . & u^{2} \\ (2111) & . & u & . \\ (11111) & u^{2} & . & . \end{aligned}  \begin{aligned}G_{(32)} & = K_{(32)} \\ G_{(41)} & = K_{(41)} \\ G_{(5)} & = K_{(5)} − K_{(32)} \end{aligned}

Now let us finally come to D_{e}, the decomposition matrix that we are trying to compute. By [26], we know already that D_{e} has rows labelled by all partitions λ ⊢ n and columns labelled by the e-regular partitions μ ⊢ n. The following result, which is Conjecture 6.9(ii) in [89], shows that D_{e} is determined by G_{μ}:

Theorem 3.21. (Ariki [3, Theorem 4.4]) For any e-regular partition μ ⊢ n, let us write G_{μ} = ∑_{λ⊢n} d_{λμ}(u) λ where d_{λμ}(u) ∈ ℤ[u]. Then we have D_{e} = (d_{λμ}(1)).

We remark that Ariki proves in fact a more general result, which also applies to Iwahori-Hecke algebras of type B_{n} and even to the cyclotomic Hecke algebras associated with the wreath product of 𝔖_{n} and a finite cyclic group. Moreover, Leclerc and Thibon [90] have proposed a generalization to the q-Schur algebra.

4 Decomposition matrices

The concept of the decomposition matrix provides a link between the “ordinary” complex representations of a finite group and the “modular” representations over a field of prime characteristic. In the case of finite groups of Lie type, the ordinary character tables are relatively well understood, due to the work of Lusztig and Shoji (see [93,115] and the references there). The situation for modular representations in the defining characteristic is settled by Lusztig’s conjecture, if that characteristic is not too small (see [2] and the references there). In this section we consider the case of non-defining characteristic, and give a survey of the present state of knowledge. Most of the results were first proved for the groups GL_{n}(q) (see [18,29]). The precise formulation of possible generalizations to other types of groups was in many cases only possible after extensive calculations in exceptional groups of low rank (see [70]).

4.1 Basic sets

The notion of a basic set (as defined below) was originally conceived by R. Parker as a computational tool for the explicit determination of decomposition numbers of sporadic simple groups. Since then, it has also proved useful as a conceptual tool in the modular representation theory of finite groups of Lie type.

First we give the general definitions. Let G be a finite group, ℓ a prime, and B a union of ℓ-blocks of G. We take the point of view that we know the ordinary irreducible characters of G belonging to B. Let D be the decomposition matrix of B. One of the most basic theoretical results that we have is that the elementary divisors of D are all 1. This leads to the following:

Definition 4.1. A subset of the set of ordinary irreducible characters in B is called an ordinary basic set for B if the submatrix of D with rows labelled by the elements in that subset has determinant ±1.

More generally, a set of Brauer characters is called a basic set for B if it is linearly independent and if every irreducible Brauer character belonging to B is an integral linear combination of the elements of that set. Thus, if we have an ordinary basic set for B, the restrictions of its members to the ℓ′-elements of G yield a basic set for B. In the literature, one sometimes finds a definition where also generalized Brauer characters are allowed to be members of a basic set, but we do insist on having actual characters.

Note that basic sets always exist (e.g., take the irreducible Brauer characters belonging to B), but it is an open question whether or not ordinary basic sets always exist. (K. Lux has checked that ordinary basic sets exist, for example, for all blocks of sporadic simple groups.)

The importance of the existence of an ordinary basic set lies in the following facts. First of all, its cardinality is the number of irreducible Brauer characters in B. Moreover, any ordinary irreducible character in B outside the basic set can be written—on the ℓ′-elements of G—as an integral linear combination of the elements in the basic set. Thus, in order to know the whole decomposition matrix of B, it is sufficient to determine that part of it which has rows labelled by the elements in the basic set. Note that this is a square matrix of non-negative integers with determinant ±1.

It has turned out that, under some mild conditions, blocks of finite groups of Lie type always possess ordinary basic sets which have a simple description in terms of Lusztig series. To state these results, we fix the following notation. Let G be a connected reductive group defined over the finite field 𝔽_{q}, with corresponding Frobenius map F: G → G. Let G^{∗} be the dual group over 𝔽_{q}, with Frobenius map F^{∗}. Recall that we have a partition of the irreducible characters of the finite group G^{F} into Lusztig series, E(G^{F} , s), one for each F^{∗}-stable class of semisimple elements s ∈ G^{∗}.

Let ℓ be a prime not dividing q. For each F -stable semisimple element s ∈ G^{∗} of order prime to ℓ, we have a corresponding union of ℓ-blocks E_{ℓ}(G^{F} , s); see Theorem 2.19. We have the following general result:

Theorem 4.2. (Geck/Hiss [50,46]) Assume that ℓ is a good prime for each simple component of G, i.e., ℓ ≠ 2 in types B_{n}, C_{n}, D_{n}, ℓ ≠ 2, 3 in types G_{2}, F_{4}, E_{6}, E_{7}, and ℓ ≠ 2, 3, 5 in type E_{8}. Assume, moreover, that ℓ does not divide the order of the largest quotient of the group Z(G)/Z^{∘}(G) on which F acts trivially. Then, for any F^{∗}-stable semisimple element s ∈ G^{∗} of order prime to ℓ, the characters in E(G^{F} , s) form an ordinary basic set for E_{ℓ}(G^{F} , s).

If the assumption on ℓ or on the center of G is not satisfied, the assertion of the above theorem does not necessarily hold; for example, take G^{F} = SL_{3}(q) where ℓ = 3 | q − 1 or G^{F} = G_{2}(q) and ℓ = 2. Nevertheless, we believe that ordinary basic sets for finite groups of Lie type in non-defining characteristic will always exist. Some partial results in this direction are obtained in Part III of [46], where 2-modular Brauer characters of groups of classical type and with a connected center are considered. In particular, it is shown that in this case the number of irreducible Brauer characters in E_{2}(G^{F} , s) equals the number of unipotent classes of C_{G^{∗}}(s)^{∗}^{F^{∗}}.

In [51], we have found the number of irreducible Brauer characters in B = E_{ℓ}(G^{F} , 1) for exceptional groups G with a connected center, under some mild restrictions on q. We take this opportunity to correct an error in the tables in [51, (5.5)(c) and (6.6)] which was found by F. Lübeck. The error is caused by a preceding error in [115, (5.1)], where the supporting sets for cuspidal character sheaves in type E_{8} are stated incorrectly. (Details of the corrections will appear elsewhere.) The correct numbers of irreducible Brauer characters in E_{ℓ}(G^{F} , 1), for bad primes ℓ, are given as follows:

Type ℓ = 2 ℓ = 3 ℓ = 5
G_{2}  9  8
F_{4}  28  35
E_{6}  27  28
E_{7}  64  72
E_{8}  131  150  162

(Thus it is not necessary to consider the case of type E_{8} with q a power of 3 separately.) We hope to be able to settle the question of the existence of ordinary basic sets in these cases by explicitly computing the necessary parts of the character tables in the framework of CHEVIE [52].

4.2 Triangular shape

Consider again a union B of ℓ-blocks for some finite group G, and let D be the corresponding decomposition matrix. Let IBr_{ℓ}(B) be the set of irreducible Brauer characters in B. Assume that there exists an ordinary basic set S for B; see Definition 4.1. Let D_{S} be the submatrix of D with rows labelled by the elements in S.

Definition 4.3. We say that D has a triangular shape with respect to the ordinary basic set S, if it is possible to order the sets S and IBr_{ℓ}(B) in such a way that the matrix D_{S} has a lower triangular shape with 1 along the diagonal.

There are examples where this is not possible; e.g., for G = SL_{2}(8) and ℓ = 2. But there are no counter-examples known for finite groups of Lie type in nondefining characteristic! The advantage of having a triangular shape as above is that we then have a canonical labelling of the irreducible Brauer characters in B by the ordinary characters in S. Indeed, we have:

Lemma 4.4. Assume that D has a triangular shape with respect to S. Choose labellings S = {χ_{1}, …, χ_{m}} and IBr_{ℓ}(B) = {θ_{1}, …, θ_{m}} such that the corresponding matrix D_{S} is lower triangular with 1 along the diagonal. Then the bijection

S ↔ IBr_{ℓ}(B), χ_{i} ↔ θ_{i} (1 ≤ i ≤ m),

does not depend on the chosen labellings.

Now consider again a finite group of Lie type G^{F} over 𝔽_{q} as in the previous section, and let ℓ be a prime not dividing q.

As in the case of ordinary characters, the union of unipotent blocks E_{ℓ}(G^{F} , 1) plays a distinguished role in the modular representation theory of the group G^{F}. A reduction to this case may be possible by an—as yet conjectural—Jordan decomposition of blocks and decomposition numbers. More precisely, assume that G has a connected center and that s ∈ G^{∗} is an F^{∗}-stable semisimple element of order prime to ℓ.

Conjecture 4.5. (Broué [6, Section 2A]/Hiss [70, Section 4.4]) If the ordinary irreducible characters in E(G^{F} , s) and in E(C_{G^{∗}}(s)^{∗}^{F^{∗}}, 1) are matched via the Jordan decomposition of characters (see Section 2.3), then the decomposition matrix of E(G^{F} , s) equals that of E(C_{G^{∗}}(s)^{∗}^{F^{∗}}, 1).

In the case where C_{G^{∗}}(s) is a Levi L^{∗} subgroup of G^{∗}, the Jordan decomposition is given by the operator R_{L}^{G} and Broué has in fact conjectured that this should define a Morita equivalence between the corresponding unions of blocks. We will come back to these questions below in the discussion of general linear groups and the linear prime case for classical groups.

By Theorem 4.2, the unipotent characters of G^{F} form an ordinary basic set for E_{ℓ}(G^{F} , 1) provided that ℓ is a good prime for G and the center of G is connected. Suppose we are in this situation. Then it is known that the corresponding decomposition matrix has a triangular shape in the following cases:

– G^{F} = GL_{n}(q), any ℓ; see Section 4.3 below. In this case, we also have a Jordan decomposition of blocks and decomposition numbers.

– G^{F} is a finite classical group, ℓ is a linear prime; see Section 4.4. There is also a Jordan decomposition of blocks and decomposition numbers.

– G^{F} = GU_{n}(q), any ℓ; see [42].

– G^{F} is an exceptional group of low rank (possibly with further conditions on q or ℓ); see [70] for G_{2}(q), [43] for ^{3}D_{4}(q), and [120] for F_{4}(q). A discussion of a special case in E_{6}(q) is contained in [51, §7].

Some partial results are known in the bad prime case. It is shown in [46, Part III] that if G^{F} is a finite classical group, q is large and ℓ = 2, then the decomposition matrix of E_{2}(G^{F} , 1) also has a triangular shape but only with respect to ordinary basic sets which do not necessarily consist of unipotent characters.

Among all irreducible characters, the cuspidal ones play a special role from the viewpoint of Harish-Chandra theory. It has been shown by Dipper and James [18] that, if G^{F} = GL_{n}(q), then the ℓ-modular reduction of an ordinary cuspidal irreducible character remains irreducible as a Brauer character, and all cuspidal irreducible Brauer characters arise in this way. This need not be the case for other types of groups: there are examples of cuspidal ordinary characters which do not remain irreducible under ℓ-modular reduction; and there are examples of cuspidal Brauer characters which do not occur in the ℓ-modular reduction of any ordinary cuspidal irreducible character. However, the second named author has formulated in his Ph.D. thesis (1990) the following:

Conjecture 4.6. (Geck) The ℓ-modular reduction of a cuspidal unipotent character remains irreducible as a Brauer character.

If this were true, we would then have a connection between the theory of Harish-Chandra series of ordinary characters and of ℓ-modular Brauer characters. The conjecture is known to be true in the following cases:

– G^{F} = GU_{n}(q), any ℓ; see [53].

– G^{F} is a finite classical group, q is large and ℓ = 2; see [56].

– G^{F} is an exceptional group of low rank (possibly with further conditions on q or ℓ); see [70] for G_{2}(q), [43] for type ^{3}D_{4}(q), and [51, §7] for E_{6}(q).

The proofs of these results use the fact that the decomposition matrix is known to have a triangular shape in these cases, and a kind of Harish-Chandra theory for generalized Gelfand-Graev representations in order to produce ”sufficiently many zeroes” in the decomposition matrix (see [56]).

4.3 Decomposition matrices of general linear groups

For G = G^{F} = GL_{n}(q) we have a complete classification of the ℓ-modular irreducible representations in terms of q-Schur algebras. The ordinary irreducible characters in geometric conjugacy classes belonging to semisimple ℓ-regular conjugacy classes of G constitute an ordinary basic set for G and the ℓ-modular decomposition matrix of G is unitriangular. Indeed the computation of the ℓmodular decomposition matrix of G can be completely reduced to the computation of decomposition matrices of q-Schur algebras of type A using the approach of Section 3.3. How this is done will be described briefly below.

Thus let G = GL_{n}(q) for some prime power q, and let ℓ be a prime not dividing q. Here G^{∗} = G hence the geometric conjugacy classes are labelled by conjugacy classes of semisimple elements of G. We choose a set C of representatives of these classes, and for s ∈ C we denote the corresponding geometric conjugacy class of G by E(G, s). For general linear groups geometric conjugacy classes coincide with Harish-Chandra series.

We let (K, O, k) be a split ℓ-modular system for G and fix s ∈ C. We choose a minimal Levi subgroup L = L_{s} containing s. Then the corresponding cuspidal OL-lattice C_{O}(s) is regular, that is it occurs in the Gelfand-Graev representation of L (see Section 3.1).

We concentrate now on the special case where the ℓ-regular part s′ of s is the identity 1 of G. This is justified, since the union E_{ℓ}(G, s) of all ℓ-blocks of G with semisimple part s (compare Theorem 2.19) has the same decomposition matrix as the union E_{ℓ}(C_{G}(s), 1) of unipotent ℓ-blocks of C_{G}(s), i.e., Conjecture 4.5 holds for G. There is, however, so far no independent proof of this fact, it follows a posteriori by showing that the decomposition matrices of the same q-Schur algebras determine these decomposition matrices with the same algorithms. Everything what is said below can be generalized to the case of arbitrary s ∈ C.

Thus s is an ℓ-element of G. Then W (L_{s}, 1) in Theorem 3.2 is the normalizer W (L_{s}) of L_{s} in the Weyl group W of G. We have:

Theorem 4.7. (Dipper) C_{k}(s) = k ⊗ C_{O}(s) is an irreducible cuspidal kLmodule. There exists an ℓ-element t ∈ C such that L = L_{t} = L_{s} and the subgroup W (L, t) of W (L) equals W (L). Moreover C_{k}(t) ≅ C_{k}(s) and W (L, s) is a parabolic subgroup of W (L) = W (L, t).

This result can be used to reduce the computation of the decomposition numbers of the geometric conjugacy class E(G, s) completely to those of E(G, t). For details and a proof we refer to [20, Section 4] and [29]. With other words it suffices to calculate decomposition numbers of Harish-Chandra series E(G, s) satisfying the hypothesis of Theorem 3.2. In particular we may assume that H_{R}(W (L, s)) = R⊗_{O}H(W (L, s)) for R ∈ {K, O, k}, and write H_{R}(s) := H_{R}(W (L, s)). Similarly, we write S_{R}(s) := S_{R}(W (L, s)) for the corresponding q-Schur algebras.

Let e = ℓ if ℓ divides q − 1 and e be the multiplicative order of q modulo ℓ otherwise. The standard and the irreducible H_{R}(s)-modules are now labelled by multi-partitions respectively regular multi-partitions corresponding to the tensor factors of H_{R}(s), where R = K or k. Here regular means e_{i}-regular in the ith part were e_{i} = e if d_{i} = 1 and e_{i} = ℓ otherwise. We now apply Theorems 3.1 and 3.9 (in their general form [25]) to obtain:

Theorem 4.8. (Dipper/James) The decomposition matrix Δ_{s} of the q-Schur algebra S_{O}(s) is part of the decomposition matrix of E(G, s). The columns corresponding to regular multi-partitions are the columns of the decomposition matrix of H_{O}(s). If L_{s} = GL_{d}(q)^{m} for natural numbers m, d with md = n, then Δ_{s} is the complete ℓ-modular decomposition matrix of E(G, s).

This was proved in [29] up to the last part, which is a result of James in [83]. Consider now the special case s = 1. Then L can be taken as the Borel subgroup of all lower triangular matrices in G and H_{R}(s) = H_{R}(𝔖_{n}; q). The rows and columns of the decomposition matrix Δ_{1} of the corresponding q-Schur algebra are labelled by partitions of n. An easy counting argument (applied to all semisimple ℓ-regular elements s ∈ C) shows that Δ_{1} is the complete decomposition matrix of E(G, 1), that is, we have a complete list of the irreducible kG-modules in the union of the unipotent blocks. We have thus:

Theorem 4.9. (Dipper/James) There is a bijection between the irreducible kGmodules and the irreducible S_{O}(s)-modules, where s runs through a set of representatives of semisimple ℓ-regular conjugacy classes of G. Moreover the ℓmodular decomposition matrix of G is lower unitriangular.

The irreducible kG-modules were determined by Dipper in [18] in terms of Hecke algebras and it was shown there that the decomposition matrix of G is lower unitriangular. James derived in [81], [82], and [83] a classification by labels coming from q-Schur algebras (without defining these), as in the theorem. In [27] Dipper and James exhibited their connection: For s ∈ C as above and a multipartition λ occurring as label for some irreducible S_{k}(s)-module let D(s, λ) be the corresponding kG-module. Taking a multi-partition λ of n which is not eregular there is corresponding to the e-ℓ-adic decomposition λ̃ of λ (which is a multi partition) a semisimple ℓ-element s such that λ̃ occurs as a label of an irreducible H_{k}(s)-module. We have (for a proof see [27, 5.1]):

Theorem 4.10. (Dipper/James) Let λ, s and λ̃ be as above. Then

D(1, λ) = D(s, λ̃).

The theorem looks like Steinberg’s tensor product theorem. Using quantum groups, Dipper and Du showed in [23] that indeed there is a tensor product theorem behind this coming from a tensor product theorem for q-Schur algebras. Moreover, the classification of the irreducible kG-modules in terms of Hecke algebras gives the Harish-Chandra series of kG. With other words the irreducible kG-module D(1, λ) has semisimple vertex L_{s} and semisimple source C_{s,k}, which is the cuspidal kL_{s}-module corresponding to s ∈ L_{s} (see Definition 2.5. For a proof see [23, 4.3.12]).

Theorem 4.10 connects column labels of Δ_{s} with those of Δ_{1}, where s ∈ C is an ℓ-element. However there may be entries in the decomposition matrix of E(G, s) which are in a column not matching any column in Δ_{s}. As can be seen from Theorem 4.8 this happens only if L_{s} is not of the form GL_{d}(q)^{m}. In this case there is another combinatorial algorithm involving only inverting matrices Δ_{s} and the Littlewood-Richardson rule to calculate those entries of the decomposition matrix of G. For proofs and details we refer to [29, Section 7] or [27]. We have:

Theorem 4.11. (Dipper/James) The ℓ-modular decomposition matrix of finite general linear groups for non-describing characteristic ℓ can be computed from the decomposition matrices of q-Schur algebras by an algorithm which involves only the Littlewood-Richardson rule and the inversion of triangular matrices.

The decomposition numbers of q-Schur algebras for integral parameters q and hence of GL_{n}(q) are known up to n = 10 by work of James, [84].

4.4 Decomposition matrices of classical groups

In this section we survey the present knowledge on decomposition numbers of classical groups. To simplify the exposition, we shall assume here that our groups arise from algebraic groups with connected center. This excludes, e.g., the symplectic groups or the special orthogonal groups in even dimensions. Many of the results to follow are also valid for such groups. We exclude the general linear groups, since these have been dealt with in the previous section.

The classical groups. Thus a classical group in this section is one of the groups of the following list. Let 𝔽_{q} denote the field with q elements, where q is a power of a prime p, and let n be a positive integer. We write G_{n}(q) for any one of the following groups.

(1) The unitary group GU_{n}(q) of an n-dimensional vector space over 𝔽_{q^{2}} with a non-degenerate unitary form.

(2) The special orthogonal group SO_{n}(q) of an n-dimensional vector space over 𝔽_{q} with a quadratic form of maximal Witt index, where n is odd.

(3) The conformal symplectic group CSp_{n}(q) of an n-dimensional vector space over 𝔽_{q} with a non-degenerate symplectic form (such that n is even).

(4) The conformal special orthogonal group CSO_{n}^{+}(q) of an n-dimensional vector space over 𝔽_{q} with a quadratic form of maximal Witt index, where n is even.

(5) The conformal special orthogonal group CSO_{n}^{−}(q) of an n-dimensional vector space over 𝔽_{q} with a quadratic form of Witt index n/2 − 1, where n is even.

Linear and very large primes. Given a non-zero integer a and a prime ℓ not dividing a we write d := d(a, ℓ) for the multiplicative order of a modulo ℓ, i.e., d is the smallest positive integer such that ℓ divides a^{d} − 1.

Definition 4.12. Let G = G_{n}(q) be one of the groups in the list above, and let ℓ be a prime not dividing q. Put

e := e(G, ℓ) := \{ \begin{matrix}d(−q, ℓ), & if G is a unitary group, \\ d(q, ℓ), & otherwise. \end{matrix}

If G is unitary, ℓ is called linear for G, if e(G, ℓ) is even. In the other cases, ℓ is called linear for G, if ℓ and e(G, ℓ) are odd. We call ℓ very large for G, if e(G, ℓ) > n/2.

Notice that a linear prime for a unitary group is odd. Also, if ℓ is very large for G, a Sylow ℓ subgroup of G is cyclic.

Decomposition matrices. There is a great amount of information on the ℓdecomposition matrices of G if ℓ is very large or linear. First of all, the hypotheses of Theorem 4.2 on basic sets of Brauer characters are satisfied in these cases. Thus in order to compute the irreducible Brauer characters of G in a union of blocks E_{ℓ}(G, s) (see Section 2.3 for the notation), it suffices to find the decomposition numbers of the characters in E(G, s). The matrix of these decomposition numbers is a square matrix, but we can say a lot more about its entries in case of linear or very large primes. For example, if ℓ is linear, the decomposition matrix of E(G, s) is equal to the decomposition matrix of a generalized q-Schur algebra (see Section 3.3).

The results of Fong and Srinivasan [40, (9B)] and of Gruber and Hiss [66, Theorem 9.3] allow a reduction to the case of unipotent characters. Namely, let G = G^{F} be one of the classical groups above, ℓ a linear or a very large prime for G, and s a non-isolated semisimple ℓ′-element in the dual group G^{∗}, i.e., C_{G^{∗}}(s) is contained in a proper Levi subgroup of G^{∗}. Then the decomposition matrices of E(G, s) and of E(C_{G^{∗}}(s), 1) are the same, i.e., Conjecture 4.5 is true in these cases.

Unfortunately, no a priori proof of this reduction is known. For very large primes it follows from the fact that, for the two cases, corresponding arguments can be given to establish the links on the Brauer trees. For linear primes it follows from the fact that the two q-Schur algebras which determine the decomposition numbers of E(G, s) and E(C_{G^{∗}}(s), 1) are isomorphic to each other.

We may thus, for the purpose of this survey, restrict attention to the decomposition numbers for unipotent characters. It turns out that in case of linear and very large primes, the distribution of the unipotent characters into Harish-Chandra series (of ordinary characters) plays an important role.

Theorem 4.13. (Fong/Srinivasan [40], Gruber/Hiss [66, Theorem 9.3]) Let G be a classical group from the list above and let ℓ be a linear or a very large prime for G. Then the decomposition matrix of the unipotent characters is a block diagonal matrix

diag(Δ_{1}, Δ_{2}, …, Δ_{r}),

where the Δ_{i} are square matrices corresponding to the Harish-Chandra series of ordinary characters, i.e., each Δ_{i} gives the decomposition into irreducible Brauer characters of the restricted ordinary characters of a particular Harish-Chandra series of unipotent characters.

This theorem no longer holds in general. An example is provided by the group GU_{3}(q) and a prime ℓ ≥ 5 dividing q + 1.

We are now going to describe the matrices Δ_{i} which occur in the preceding theorem. We define the a-invariant of a unipotent character χ to be the largest r ≥ 0 such x^{r} divides the generic degree of χ (see the paragraph following Definition 2.10).

Theorem 4.14. (Fong/Srinivasan [40]) Let G be a classical group from the list above and let ℓ be a very large prime for G. Consider a Harish-Chandra series S of unipotent characters of G. Suppose that the elements of S fall into the ℓ-blocks B_{1}, …, B_{s} of G. Then, for 1 ≤ i ≤ s, the a-invariants of the unipotent characters in S ∩ B_{i} are all different. Moreover, arranging these characters according to increasing a-invariants, the decomposition matrix of S ∩ B_{i} equals J(b_{i}) (where b_{i} = |S ∩ B_{i}| and J(b_{i}) denotes the lower unitriangular Jordan (b_{i} × b_{i})-matrix). With this ordering of the characters in S, the decomposition matrix Δ of S is the block diagonal matrix diag(J(b_{1}), J(b_{2}), …, J(b_{s})).

We finally turn to the case of linear primes. For this purpose we have to describe, in some more detail, the parameterization of the characters inside a Harish-Chandra series of unipotent characters. This is the same for all classical groups and all Harish-Chandra series except for the principal series of the conformal group CSO_{n}^{+}(q). For reasons of simplicity we therefore omit the latter from the following considerations. Then the characters of a Harish-Chandra series of unipotent characters are parameterized by bi-partitions of some positive integer m ≤ n/2. This is due to the fact that the Hecke algebra corresponding to the Harish-Chandra series (according to Definition 2.5) is the Iwahori-Hecke algebra of type B_{m}. In the following theorem, δ equals 2 if G is a unitary group, and 1, otherwise. Also, if λ is a partition, we write |λ| for the sum of its parts.

Theorem 4.15. (Gruber/Hiss [66, Corollary 8.11]) Let G be a classical group from the list above and let ℓ be a linear prime for G. Let Δ denote the decomposition matrix of a Harish-Chandra series S of unipotent characters of G (where S is not the principal series if G = CSO_{n}^{+}(q)).

For a non-negative integer j let Λ_{j} denote the decomposition matrix of the unipotent characters of GL_{j}(q^{δ}). Then Δ is a block diagonal matrix of the form

Δ = [ Λ_{0} ⊗ Λ_{m}  0  …  0 ; 0 Λ_{1} ⊗ Λ_{m−1} ; ...  ⋮ ; ⋮ ; Λ_{m−1} ⊗ Λ_{1} 0 ; 0  …  0  Λ_{m} ⊗ Λ_{0} ] .

Here, ⊗ denotes the Kronecker product of the matrices.

To be more precise, the elements of S are labelled as χ = χ^{(μ_{1},μ_{2})} by bipartitions (μ_{1}, μ_{2}) of m via ordinary Harish-Chandra theory. The Brauer characters occurring as constituents of the elements in S also have a labeling as φ = φ^{(ν_{1},ν_{2})} by bi-partitions (ν_{1}, ν_{2}) of m. The multiplicity of φ^{(ν_{1},ν_{2})} in χ^{(μ_{1},μ_{2})} is 0, if |μ_{1}| ≠ |ν_{1}|, and it equals d_{μ_{1}ν_{1}}(q^{δ})d_{μ_{2}ν_{2}}(q^{δ}), otherwise. Here, d_{λ_{1}λ_{2}}(q^{δ}) is the ℓ-decomposition number of GL_{|λ_{1}|}(q^{δ}) which corresponds to the unipotent character parameterized by the partition λ_{1} and the Brauer character parameterized by λ_{2}.

Let us state one corollary of the results described in this section.

Corollary 4.16. Let G be a classical group from the list above and let ℓ be a linear or a very large prime for G. Then the ordinary irreducible characters of G can be arranged in such a way that the decomposition matrix is lower unitriangular.

This follows from the theorems given in this section (except for the conformal orthogonal groups CSO_{n}^{+}(q), where it is also true) and from the corresponding fact for the general linear groups, see Theorem 4.10.

We also have an important result on the size of the decomposition numbers.

Theorem 4.17. (Gruber/Hiss [66, Corollary 8.11]) Let G = G_{n}(q) be a classical group from the list above and let ℓ be a linear prime for G. Then the ℓ-decomposition numbers of G are bounded above by the order of the Weyl group of G.

Note that this bound on the decomposition numbers is independent of q and ℓ.

4.5 Decomposition matrices of Sp_{4}(q)

The determination of decomposition matrices of classical groups for non-linear, not very large primes appears to be a very hard problem. The answer is not even completely known for one of the “smallest” series of classical groups, the 3-dimensional unitary groups GU_{3}(q). Geck has determined their decomposition numbers in [41], with one exception in case ℓ divides q + 1.

A similar problem, again for the case ℓ divides q + 1, has been left open by White, who investigated the decomposition numbers for the symplectic groups Sp_{4}(q) [119]. In this case, however, Okuyama and Waki [103] were able to solve White’s problem. It should be noted that here, again, the decomposition numbers of the unipotent characters are bounded independently of q and ℓ. In fact, the largest decomposition number of a unipotent character is 2.

In contrast to the case of linear primes the ℓ-modular decomposition numbers do not only depend on the integer e(G, ℓ) (see Definition 4.12) but also on the highest power of ℓ dividing q + 1.

4.6 Decomposition matrices of exceptional groups

Let us shortly summarize the knowledge on the decomposition matrices (in nondefining characteristic) for the exceptional groups of Lie type. Let G be an exceptional group of Lie type.

Complete decomposition matrices. Complete decomposition matrices are known only if G is a Suzuki group ^{2}B_{2}(q) [96] or a Ree group ^{2}G_{2}(q) [87,71]. Almost complete results are available for Steinberg’s triality groups ^{3}D_{4}(q) [43] and for the Chevalley groups G_{2}(q) [69]. For example, all ℓ-modular decomposition numbers are known for G_{2}(q) in case ℓ is greater than 3 and does not divide q + 1. If ℓ > 3 divides q + 1, the results of [69] allow to compute all but two of the ℓ-modular irreducible characters. The remaining problems in these two series are very similar to those already encountered in the 3-dimensional unitary groups and the 4-dimensional symplectic groups (see Section 4.5). We hope that they can be solved with methods similar to those introduced by Okuyama and Waki in [103].

Brauer trees. If we consider primes ℓ with the property that the Sylow ℓsubgroups of G are cyclic, the situation is much better. In this case the decomposition matrices are described by Brauer trees. These are known for all exceptional groups of Lie type except for E_{7}(q) and E_{8}(q) (and some exceptions for ^{2}E_{6}(q)) [75,74].

4.7 Harish-Chandra series of Brauer characters

In order to describe the Harish-Chandra series of Brauer characters, we have to describe the cuspidal irreducible Brauer characters first. For reasons of simplicity, we only consider unipotent characters. As always we let G = G^{F} be a finite group of Lie type of characteristic p, and let ℓ be a prime different from p.

The Gelfand-Graev representation of G is projective and has a unique direct summand containing the Steinberg representation as a constituent. We denote by φ_{St} the corresponding irreducible ℓ-modular character of G.

General linear groups. Let G = GL_{n}(q), and let e be as in 4.3, i.e., e = ℓ, if ℓ | q − 1, and e = d(q, ℓ) is the order of q modulo ℓ, otherwise.

Theorem 4.18. (Dipper/James) If G has a cuspidal unipotent Brauer character φ, then φ = φ_{St}. Moreover, φ_{St} is cuspidal if and only if n = 1 or n = eℓ^{j} for some j ≥ 0.

If λ = (1^{a_{1}}, 2^{a_{2}}, …) is a partition of n, we write

L_{λ} := GL_{1}(q)^{a_{1}} × GL_{2}(q)^{a_{2}} × ⋯  (1)

for the corresponding standard Levi subgroup of G. If φ is a cuspidal unipotent Brauer character of L_{λ}, we must have

λ = (1^{m_{−1}}, e^{m_{0}}, (eℓ)^{m_{1}}, (eℓ^{2})^{m_{2}}, …),  (2)

and

φ = φ_{λ,St} := φ_{St}^{m_{−1}} ⊗ φ_{St}^{m_{0}} ⊗ φ_{St}^{m_{1}} ⊗ ⋯

by Theorem 4.18. Thus the Harish-Chandra series of unipotent Brauer characters of G are in bijection with the partitions of n of the form (2). The question of which unipotent Brauer characters lie in the Harish-Chandra series corresponding to such a partition is answered in the following theorem. By the results of Section 4.3 we have a natural labelling of the unipotent Brauer characters of G by partitions of n. Let φ_{μ} denote the unipotent Brauer character of G corresponding to the partition μ of n.

Theorem 4.19. (Dipper/Du [23, 4.3.12]) Let μ be a partition of n with e-ℓ-adic expansion

μ = μ^{(−1)} + eμ^{(0)} + eℓμ^{(1)} + · · · .

Then φ_{μ} lies in the Harish-Chandra series corresponding to the partition λ of the form (2) with m_{i} := |μ_{i}| for i = −1, 0, 1, … Classical groups. We now let G = G_{n}(q) be a classical group as in 4.4, and we let ℓ be a linear prime for G. In this case the cuspidal unipotent Brauer characters are easy to describe.

Theorem 4.20. (Gruber/Hiss [66, Corollary 8.8]) The group G has a cuspidal unipotent Brauer character φ if and only if it has an ordinary cuspidal unipotent character χ. In this case φ is liftable to χ. Moreover, χ is an ℓ-defect 0 character of G/Z(G).

It follows in particular, that G has at most one cuspidal unipotent Brauer character. The Levi subgroups of G are of the form

L_{a,λ} = G_{a}(q) × L_{λ},

where a ≤ n is an integer such that n − a is even, G_{a}(q) is a group of the same type as G, λ is a partition of (n − a)/2, and L_{λ} is the Levi subgroup of GL_{(n−a)/2}(q^{δ}) introduced in (1) with δ as in Theorem 4.15. It is well known that L_{a,λ} has an ordinary cuspidal unipotent character only for special values of a and then only for λ = (1^{m}).

Let 0 ≤ a_{1} < a_{2} < … < a_{r} ≤ n denote the integers such that n − a_{i} is even and such that G_{a_{i}}(q) has a cuspidal unipotent character χ_{i}. The ordinary unipotent characters in the Harish-Chandra series corresponding to χ_{i} are labelled by bi-partitions of n_{i} = (n − a_{i})/2. (Again we exclude the case of the principal series of CSO_{n}^{+}(q).) Thus the unipotent characters are labelled by the triples (a_{i}; λ_{1}, λ_{2}) where (λ_{1}, λ_{2}) is a bi-partition of n_{i} and i runs from 1 to r. (It is not difficult to translate this labelling of the unipotent characters to the more common one by symbols or partitions.) In turn, by Corollary 4.16, the unipotent Brauer characters of G are labelled by such triples.

Theorem 4.21. Let (a_{i}; μ_{1}, μ_{2}) be the label of a unipotent Brauer character φ of G, and let μ_{1}^{(j)} and μ_{2}^{(j)}, j = −1, 0, 1, …, denote the parts of the e-ℓ-adic expansion of μ_{1} and μ_{2}, respectively. (Here, e is defined as in 4.3 with respect to q^{δ}.) Then φ lies in the Harish-Chandra series corresponding to L_{a_{i},λ}, where λ is the partition of (n − a_{i})/2 of the form (2) with m_{j} = |μ_{1}^{(j)}| + |μ_{2}^{(j)}|, j = −1, 0, 1, …

This theorem was conjectured for the unitary groups in [53, Conjecture 9.2], and proved in the case that φ lies in an ℓ-block with a cyclic defect group [53, Proposition 9.3]. In the general case it follows from the results of [66, Section 8].

The unitary groups. We now give a result which indicates the existence of many cuspidal unipotent Brauer characters which are not liftable to ordinary characters. Let G = GU_{n}(q) and let ℓ be an odd prime dividing q + 1. The unipotent Brauer characters of G have a natural parameterization by partitions of n (see Section 4.2). Let φ_{μ} denote the unipotent Brauer character of G corresponding to the partition μ of n.

Theorem 4.22. (Geck/Hiss/Malle, Gruber) The unipotent Brauer character φ_{μ} is cuspidal if and only if two consecutive parts of μ differ by at most 1. (This is the same as saying that μ′, the partition conjugate to μ, is 2-regular).

We can also describe the Harish-Chandra series in this case.

Theorem 4.23. (Geck/Hiss/Malle, Gruber) Let μ be a partition of n. Write μ = μ_{1} + 2μ_{2}, where μ_{1} and μ_{2} are uniquely determined by the fact that μ′_{1} is 2-regular. Define m = |μ_{1}| and m_{j} = |μ_{2}^{(j)}|, where the μ_{2}^{(j)} are the parts of the ℓ-adic expansion of μ_{2}, j = 0, 1, 2, . . ..

Then φ_{μ} lies in the Harish-Chandra series of φ_{μ_{1}} ⊗ φ_{St}^{m_{0}} ⊗ φ_{St}^{m_{1}} ⊗ ⋯, where φ_{St}^{m_{j}} is the ℓ-modular Steinberg character of GL_{ℓ^{j}}(q^{2})^{m_{j}}, j = 0, 1, 2, …

These two results were conjectured in [53, Conjecture 9.6], and proved to be true in [54, Theorem 4.12] for ℓ > n, and in general by Gruber in [65].

Exceptional groups. The modular Harish-Chandra series and the endomorphism algebras of Harish-Chandra induced cuspidal modules occurring in exceptional groups G(q) were determined by Geck/Hiss/Malle in [54] in all cases where the order d = d(q, ℓ) of q modulo ℓ is odd and ℓ does not divide the order of the Weyl group of G, except for d ∈ {3, 5} for G = E_{8}(q). In order to indicate the nature of the results to expect we reproduce from Table 5.4 in loc. cit. the results pertaining to Harish-Chandra series of G = E_{7}(q) in the form of a table.

Its first column contains the number d = d(q, ℓ), the order of q modulo ℓ. The second column of the table gives the pairs (L, X), where L is a Levi

subgroup of G and X is a cuspidal simple kL-module. The Levi subgroups are denoted by their types. The trivial kL-module (which is cuspidal only if L is the maximally split torus), is denoted by 1, the Steinberg module, if it is cuspidal, by φ_{St}. Further cuspidal modules are denoted by c or c_{i}, i = 1, …, 6.

The third and fourth columns then give the groups R = R(L, X) and C = C(L, X), and the sixth column the endomorphism algebra H = H(L, X) = End_{kG}(R_{L}^{G}(X)) (see Theorem 2.9). The parameters of these all lie in the finite field 𝔽_{ℓ} ⊆ k, i.e., q also denotes its image in 𝔽_{ℓ}. There are some parameters p_{i} ∈ 𝔽_{ℓ}^{∗} \ {−1}, i = 1, …, 6, which are not yet known explicitly.

The last column of the table gives the number of simple kG-modules in the Harish-Chandra series corresponding to (L, X), which is equal to the number of simple H-modules by Corollary 2.4.

Table. Modular Harish-Chandra series for E_{7}(q)

d  L  X  R  C H  #irr.
3  1  1  E_{7}  1 H(E_{7}; q)  32
A_{2}  φ_{St}  A_{5}  2 H(A_{5}; q) : 2  14
A_{2} × A_{2} φ_{St} × φ_{St} G_{2} × A_{1} 1 H(G_{2}; q, 1) ⊗ H(A_{1}; 1) 8
D_{4}  c  C_{3}  1 H(C_{3}; q, q)  4
E_{6}  c_{1}, c_{2} A_{1}  1 H(A_{1}; 1)  2 + 2
E_{6}  c_{3}, …, c_{6} A_{1}  1 H(A_{1}; p_{i}), i = 1, …, 4 8

+ 6 cuspidals

5  1  1  E_{7}  1 H(E_{7}; q)  54
A_{4}  φ_{St}  A_{2}  2 H(A_{2}; q) : 2  6
D_{4}  c  C_{3}  1 H(C_{3}; q, q^{4})  10
E_{6}  c_{1}, c_{2} A_{1}  1 H(A_{1}; q^{4})  2 + 2

+ 2 cuspidals

7  1  1  E_{7}  1 H(E_{7}; q)  58
D_{4}  c  C_{3}  1 H(C_{3}; q, q^{4})  10
A_{6}  φ_{St}  A_{1}  1 H(A_{1}; p_{5})  2
E_{6}  c_{1}, c_{2} A_{1}  1 H(A_{1}; q^{2})  2 + 2

+ 2 cuspidals

9  1  1  E_{7}  1 H(E_{7}; q)  58
D_{4}  c  C_{3}  1 H(C_{3}; q, q^{4})  10
E_{6}  c_{1}, c_{2} A_{1}  1 H(A_{1}; 1)  2 + 2
E_{6}  φ_{St}  A_{1}  1 H(A_{1}; p_{6})  2

+ 2 cuspidals

References

1. D. Alvis and G. Lusztig, The representations and generic degrees of the Hecke algebra of type H_{4}, J. reine angew. Math. 336 (1982), 201–212; Correction: ibid. 449 (1994), 217–218.

2. H.H. Anderson, J.C. Jantzen, and W. Soergel, Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p, Astérisque 220 (1884).

3. S. Ariki, On the decomposition numbers of the Hecke algebra of G(m, 1, n), J. of Math. Kyoto Univ. 36 (1996), 789–808.

4. C.T. Benson and C.W. Curtis, On the degrees and rationality of certain characters of finite Chevalley groups, Trans. Amer. Math. Soc. 165 (1972), 251–273; Corrections and additions, ibid. 202 (1975), 405–406.

5. K. Bremke, The decomposition numbers of Hecke algebras of type F_{4} with unequal parameters, Manuscripta Math. 83 (1994), 331–346.

6. M. Broué, Isométries parfaites, types de blocs, catégories dérivées, in Représentations Linéaires de Groupes Finis, Astérisque 181–182 (1990), 61–92.

7. M. Broué and G. Malle, Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis, Math. Ann. 292 (1992), 241–262.

8. M. Broué and G. Malle, Zyklotomische Heckealgebren, in: Représentations unipotentes génériques et blocs des groupes réductifs finis, Astérisque 212 (1993), 119–189.

9. M. Broué, G. Malle, and J. Michel, Generic blocks of finite reductive groups, in: Représentations unipotentes génériques et blocs des groupes réductifs finis, Astérisque 212 (1993), 7–92.

10. M. Broué and J. Michel, Blocs et séries de Lusztig dans un groupe réductif fini, J. reine angew. Math. 395 (1989), 56–67.

11. M. Broué and J. Michel, Blocs à groupes de défaut abéliens des groupes réductifs finis, in: Représentations unipotentes génériques et blocs des groupes réductifs finis, Astérisque 212 (1993), 93–118.

12. M. Cabanes, A criterion for complete reducibility and some applications, in: Représentations linéaires de groupes finis, Luminy, 16-21 mai 1988, Astérisque 181–182 (1990), 93–112.

13. M. Cabanes and M. Enguehard, Local methods for blocks of reductive groups over a finite field, in: Finite reductive groups: Related structures and representations (ed. M. Cabanes), pp. 141–163. Birkhäuser, Basel, 1997.

14. R.W. Carter, Representation theory of the 0-Hecke algebra, J. Algebra 104 (1986), 89–103.

15. R.W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, Chichester, 1985.

16. C. W. Curtis and I. Reiner, Methods of representation theory, vols. 1, 2, Wiley, Chichester, 1981, 1987.

17. P. Deligne and G. Lusztig, Duality for representations of a reductive group over a finite field II, J. Algebra 81 (1983), 540–545.

18. R. Dipper, On the decomposition numbers of the finite general linear groups, Trans. Amer. Math. Soc. 290 (1985), 315–343; II, 292 (1985), 123–133.

19. R. Dipper, On quotients of Hom-functors and representations of finite general linear groups I, J. Algebra 130 (1990), 235–259.

20. R. Dipper, On quotients of Hom-functors and representations of finite general linear groups II, Preprint, Stuttgart 1997.

21. R. Dipper and S. Donkin, Quantum GL_{n}, Proc. London Math. Soc. 63 (1991), 165–211.

22. R. Dipper and J. Du, Harish-Chandra vertices, J. reine angew. Math. 437 (1993), 101–130.

23. R. Dipper and J. Du, Harish-Chandra vertices and Steinberg’s tensor product theorem for general linear groups in non-describing characteristic, Proc. London Math. Soc., to appear.

24. R. Dipper and P. Fleischmann, Modular Harish-Chandra theory I, Math. Z. 211 (1992), 49–71; II, Arch. Math. 62 (1994), 26–32.

25. R. Dipper and J. Gruber, Generalized q-Schur algebras and modular representation theory of finite groups with split (B, N)-pairs, Preprint, Stuttgart 1997.

26. R. Dipper and G. D. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. 52 (1986), 20–52.

27. R. Dipper and G. D. James, Identification of the irreducible modular representations of GL_{n}(q), J. Algebra 104 (1986), 266–288.

28. R. Dipper and G. D. James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. 54 (1989), 57–82.

29. R. Dipper and G. D. James, The q-Schur algebra, Proc. London Math. Soc. 59 (1989), 23–50.

30. R. Dipper and G. D. James, q-tensor space and q-Weyl modules, Proc. Amer. Math. Soc. 327 (1991), 251–282.

31. R. Dipper and G.D. James, Representations of Hecke algebras of type B_{n}, J. Algebra 146 (1992), 454–481.

32. R. Dipper, G.D. James, and A. Mathas, The (Q, q)-Schur algebra, Proc. London Math. Soc., to appear.

33. R. Dipper, G.D. James, and A. Mathas, Cyclotomic q-Schur algebras, Math. Z., to appear.

34. R. Dipper, G.D. James, and G.E. Murphy, Hecke algebras of type B_{n} at roots of unity, Proc. London Math. Soc. 70 (1995), 505–528.

35. J. Du, Kazhdan-Lusztig bases and isomorphism theorems for q-Schur algebras, Contemp. Math. 139 (1992), 121–140.

36. J. Du, B. Parshall, and L. Scott, Stratifying endomorphism algebras associated to Hecke algebras, Preprint, Sydney 1996.

37. J. Du and L. Scott, The q-Schur^{2} algebra, Preprint, Charlottesville 1997.

38. P. Fong and B. Srinivasan, Generalized Harish-Chandra theory for unipotent characters of finite classical groups, J. Algebra 104 (1986), 301–309.

39. P. Fong and B. Srinivasan, The blocks of finite classical groups, J. reine angew. Math. 396 (1989), 122–191.

40. P. Fong and B. Srinivasan, Brauer trees in classical groups, J. Algebra 131 (1990), 179–225.

41. M. Geck, Irreducible Brauer characters of the 3-dimensional special unitary groups in non-defining characteristic, Comm. Algebra 18 (1990), 563–584.

42. M. Geck, On the decomposition numbers of the finite unitary groups in nondefining characteristic, Math. Z. 207 (1991), 83–89.

43. M. Geck, Generalized Gelfand-Graev characters for Steinberg’s triality groups and their applications, Comm. Algebra 19 (1991), 3249–3269.

44. M. Geck, Brauer trees of Hecke algebras, Comm. Algebra 20 (1992), 2937–2973.

45. M. Geck, The decomposition numbers of the Hecke algebra of type E_{6}, Math. Comp. 61 (1993), 889–899.

46. M. Geck, Basic sets of Brauer characters of finite groups of Lie type, II, J. London Math. Soc. 47 (1993), 255–268; III, Manuscripta Math. 85 (1994), 195–216.

47. M. Geck, On the character values of Iwahori-Hecke algebras of exceptional type, Proc. London Math. Soc. 68 (1994), 51–76.

48. M. Geck, Beiträge zur Darstellungstheorie von Iwahori-Hecke-Algebren, Habilitationsschrift, Aachen 1994, Aachener Beiträge zur Mathematik, Band 11, Aachen, 1995.

49. M. Geck, Kazhdan-Lusztig cells and decomposition numbers, Institut de Mathématiques de Jussieu, Paris, Preprint (1997).

50. M. Geck and G. Hiss, Basic sets of Brauer characters of finite groups of Lie type, J. reine angew. Math. 418 (1991), 173–188.

51. M. Geck and G. Hiss, Modular representations of finite groups of Lie type in non-defining characteristic, in Finite reductive groups: Related structures and representations (ed. M. Cabanes), pp. 195–249. Birkhäuser, Basel, 1997.

52. M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer, CHEVIE — A system for computing and processing generic character tables, AAECC 7 (1996), 175–210.

53. M. Geck, G. Hiss, and G. Malle, Cuspidal unipotent Brauer characters, J. Algebra 168 (1994), 182–220.

54. M. Geck, G. Hiss, and G. Malle, Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type, Math. Z. 221 (1996), 353–386.

55. M. Geck and K. Lux, The decomposition numbers of the Hecke algebra of type F_{4}, Manuscripta Math. 70 (1991), 285–306.

56. M. Geck and G. Malle, Cuspidal unipotent classes and cuspidal Brauer characters, J. London Math. Soc. 53 (1996), 63–78.

57. M. Geck and J. Michel, Good elements in finite Coxeter groups and representations of Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 74 (1997), 275–305.

58. M. Geck and G. Pfeiffer, On the irreducible characters of Hecke algebras, Adv. in Math. 102 (1993), 79–94.

59. M. Geck and R. Rouquier, Centers and simple modules for Iwahori-Hecke algebras, in Finite reductive groups: Related structures and representations (ed. M. Cabanes), pp. 251–272. Birkhäuser, Basel, 1997.

60. J.J. Graham and G.I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), 1–34.

61. R.M. Green, q-Schur algebras and quantized enveloping algebras, Thesis, University of Warwick, 1995.

62. R.M. Green, Hyperoctahedral Schur algebras, J. Algebra, to appear.

63. R.M. Green, The affine q-Schur algebra, Preprint, Oxford 1997.

64. I. Grojnowski, Representations of affine Hecke algebras (and quantum GL_{n}) at roots of unity, Internat. Math. Research Notices 5 (1994), 215–217.

65. J. Gruber, Green vertex theory, Green correspondence and Harish-Chandra induction, J. Algebra 186 (1997), 476–521.

66. J. Gruber and G. Hiss, Decomposition numbers of finite classical groups for linear primes, J. reine angew. Math. 485 (1997), 55–91.

67. T. Halverson and A. Ram, Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of classical type, Trans. Amer. Math. Soc. 348 (1996), 3967–3995.

68. Harish-Chandra, Eisenstein series over finite fields, in Functional analysis and related fields, pp. 76–88. Springer, New York 1970.

69. G. Hiss, On the decomposition numbers of G_{2}(q), J. Algebra 120 (1989), 339–360.

70. G. Hiss, Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nicht-definierender Charakteristik, Habilitationsschrift, RWTH Aachen, 1990.

71. G. Hiss, The Brauer trees of the Ree groups, Comm. Algebra 19 (1991), 871–888.

72. G. Hiss, Harish-Chandra series of Brauer characters in a finite group with a split BN-pair, J. London Math. Soc. 48 (1993), 219–228.

73. G. Hiss, Supercuspidal representations of finite reductive groups, J. Algebra 184 (1996), 839–851.

74. G. Hiss and F. Lübeck, The Brauer trees of the exceptional Chevalley groups of types F_{4} and ^{2}E_{6}, Arch. Math., to appear.

75. G. Hiss, F. Lübeck, and G. Malle, The Brauer trees of the exceptional Chevalley groups of type E_{6}, Manuscripta Math. 87 (1995), 131–144.

76. G. Hiss and J. Shamash, 3-Blocks and 3-decomposition numbers of G_{2}(q), J. Algebra 131 (1990), 371–387.

77. G. Hiss and J. Shamash, 2-Blocks and 2-decomposition numbers of the Chevalley groups G_{2}(q), Math. Comp. 59 (1992), 645–672.

78. R. B. Howlett and G. I. Lehrer, Induced cuspidal representations and generalized Hecke rings, Invent. Math. 58 (1980), 37–64.

79. R. B. Howlett and G. I. Lehrer, On Harish-Chandra induction for modules of Levi subgroups, J. Algebra 165 (1994), 172–183.

80. N. Iwahori, On the structure of the Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 10 (1964), 215–236.

81. G.D. James, Unipotent representations of the finite general linear groups, J. Algebra 74 (1982), 443–465.

82. G.D. James, Representations of general linear groups, Cambridge Univ. Press, Cambridge, 1984.

83. G.D. James, The irreducible representations of the finite general linear groups, Proc. London Math. Soc. 52 (1986), 236–268.

84. G.D. James, The decomposition matrices of GL_{n}(q) for n ≤ 10, Proc. London Math. Soc. 60 (1990), 225–265.

85. G.D. James and A. Mathas, A q-analogue of the Jantzen-Schaper theorem, Proc. London Math. Soc. 74 (1997), 241–274.

86. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 40 (1979), 165–184.

87. P. Landrock and G. O. Michler, Principal 2-blocks of the simple groups of Ree type, Trans. Amer. Math. Soc. 260 (1980), 83–111.

88. A. Lascoux, B. Leclerc, and J.-Y. Thibon, Une conjecture pour le calcul des matrices de décomposition des algèbres de Hecke du type A aux racines de l’unité, C. R. Acad. Sci. Paris Sér. I 321 (1995), 511–516.

89. A. Lascoux, B. Leclerc, and J.-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Physics 181 (1996), 205–263.

90. B. Leclerc and J.-Y. Thibon, Canonical bases of q-deformed Fock spaces, Internat. Math. Research Notices 9 (1996), 447–456.

91. S.A. Linton, On vector enumeration, Linear Algebra Appl. 192 (1993), 235–248.

92. G. Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981), 490–498.

93. G. Lusztig, Characters of reductive groups over a finite field, Ann. Math. Studies 107, Princeton U. Press, Princeton, 1984.

94. G. Lusztig, On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 157–166.

95. G. Lusztig and N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. 19 (1979), 41–52.

96. R. P. Martineau, On representations of the Suzuki groups over fields of odd characteristics, J. London Math. Soc. 6 (1972), 153–160.

97. A. Mathas, Canonical bases and the decomposition numbers of Ariki-Koike algebras, Preprint, London 1996.

98. J. Müller, Zerlegungszahlen für generische Iwahori-Hecke-Algebren von exzeptionellem Typ, Dissertation, RWTH Aachen, 1995.

99. J. Müller, Decomposition numbers for generic Iwahori-Hecke algebras of non crystallographic type, J. Algebra 189 (1997), 125–149.

100. G.E. Murphy, On the representation theory of the symmetric groups and associated Hecke algebras, J. Algebra 152 (1992) 492–513.

101. G.E. Murphy, The representations of Hecke algebras of type A_{n}, J. Algebra 173 (1995), 97–121.

102. S. Oehms, Symplektische q-Schur-Algebren, Dissertation, Stuttgart, 1997.

103. T. Okuyama and K. Waki, Decomposition numbers of Sp(4, q), preprint, 1997.

104. C. Pallikaros, Representations of Hecke algebras of type D_{n}, J. Algebra 169 (1994), 20–48.

105. R. A. Parker, The computer calculation of modular characters (the Meat-Axe), in Computational group theory (ed. M.D. Atkinson). Academic Press, London, 1984.

106. B. Parshall and J.P. Wang, Quantum linear groups, Memoirs Amer. Math. Soc. 439 (1991).

107. G. Pfeiffer, Young characters on Coxeter basis elements of Iwahori-Hecke algebras and a Murnaghan-Nakayama formula, J. Algebra 168 (1994), 525–535.

108. G. Pfeiffer, Charakterwerte von Iwahori-Hecke-Algebren von klassischem Typ, Dissertation, Aachener Beiträge zur Mathematik, Band 14, Aachen, 1995.

109. G. Pfeiffer, Character values of Iwahori-Hecke algebras of type B, in: Finite reductive groups: Related structures and representations (ed. M. Cabanes), pp. 195– 249. Birkhäuser, Basel, 1997.

110. A. Ram, A Frobenius formula for the characters of the Hecke algebras, Invent. Math. 106 (1991), 461–488.

111. M.J. Richards, Some decomposition numbers for Hecke algebras of general linear groups, Proc. Camb. Phil. Soc. 119 (1996).

112. H. Rui, On endomorphism algebras arising from multiparameter Hecke algebras, J. Algebra, to appear.

113. K. -D. Schewe, Blöcke exzeptioneller Chevalley-Gruppen, Dissertation, Bonner Mathematische Schriften, nr 165, Bonn, 1985.

114. M. Schönert et al., GAP — Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, RWTH Aachen, Germany, fourth ed., 1994.

115. T. Shoji, Character sheaves and almost characters of reductive groups I, II, Adv. in Math. 111 (1995), 244-313 and 314–354.

116. S. Veigneau, ACE — An algebraic combinatorics environment for the computer algebra system MAPLE, User’s reference manual, Version 2.0, Institut Gaspard

Monge (1996), http://weyl.univ-mlv.fr/∼sv/ACE

117. M.-F. Vigneras, Sur la conjecture locale de Langlands pour GL(n, F) sur F̄_{l}, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), 905–908.

118. K. Waki, A note on the decomposition numbers of Sp(4, q), J. Algebra 185 (1996), 105–112.

119. D. White, Decomposition numbers of Sp(4, q) for primes dividing q±1, J. Algebra 132 (1990), 488–500.

120. E. Wings, Über die unipotenten Charaktere der Chevalley-Gruppen vom Typ F_{4} in guter Charakteristik, Dissertation, RWTH Aachen, 1995.