Generalized q-Schur algebras and modular representation theory of finite groups with split (BN)-pairs

Richard Dipper and Jochen Gruber Mathematische Institut B Universität Stuttgart Postfach 80 11 40 70550 Stuttgart Deutschland

17th August 1998

Abstract

We introduce a generalized version of a q-Schur algebra (of parabolic type) for arbitrary Hecke algebras over extended Weyl groups. We describe how the decomposition matrix of a finite group with split BN-pair, with respect to a non-describing prime, can be partially described by the decomposition matrices of suitably chosen q-Schur algebras. We show that the investigated structures occur naturally in finite groups of Lie type.

Introduction

In a series of papers ([4, 5, 37, 14] Gordon James and the first named author used representations of Hecke- and of q-Schur algebras to derive a classification of the ℓ-modular irreducible representations of general linear groups G = GL_{n}(q) for primes ℓ which are coprime to q. It turned out that the ℓ-decomposition matrices of G are completely determined by decomposition matrices of certain q-Schur algebras by an algorithm involving only combinatorics as partitions and the Littlewood-Richardson rule (see [14]).

q-Schur algebras and finite groups with split (BN)-pairs 2

The connection between the group algebra of G on the one side, and Hecke- and q-Schur algebras on the other side is given by certain functors H, called quotients of Hom-functors and their preinverses, which were investigated in [6, 7]. First Hecke algebras come up as endomorphism rings of certain induced modules for G, and these functors connect their representations with those of G. This part of the theory was extented to arbitrary finite groups of Lie type in [10, 11]. For general linear groups q-Schur algebras turn up as endomorphism rings of certain Hecke algebra modules. In fact quotients of Hom-functors can be defined here as well, one obtains a q-analogue of the classical Schur functors. The functors H are then used to show that the q-Schur algebras are isomorphic to endomorphism rings of certain G-modules as well, and that one can define again quotients of Hom-functors S, now connecting representations of q-Schur algebras and of G. In [28] Hiss and the second named author extended these results to finite classical groups in the special case of linear primes ℓ (about a third of the primes dividing the group order).

E–mail: rdipper@mathematik.uni–stuttgart.de

jgruber@mathematik.uni-stuttgart.de A.M.S. subject classification (1991): 20C20, 20C33, 20G05, 20G40 This paper is a contribution to the DFG project on “Algorithmic number theory and algebra”. The authors acknowledge support from the Isaac Newton Institute, Cambridge

The main purpose of this paper is to exhibit how far these methods carry in the general case of finite groups G of Lie type for arbitrary primes ℓ different from the describing characteristic of the group. One of our main results is the extension of the second step, the q-Schur algebra approach in the situation of [10, 11]. In addition we shall show that the methods work in other cases too. Here a fundamental step has been done by Geck, Hiss and Malle, who showed that the endomorphism ring of Harish-Chandra induced cuspidal irreducible modules is always a Hecke algebra (see [24]), extending Howlett-Lehrer theory [32]. In addition they constructed corresponding quotients of Hom-functors.

However it should be pointed out that our results are not as nice and complete as in the case [14] of general linear or [28] of classical groups for linear primes: In general we can construct only a part of the irreducible representations of G and we get only partial information on the decomposition numbers in terms of generalized q-Schur algebras. It seems to be also clear, that the information provided by our results here is about everything what q-Schur algebras can give. To obtain information on the remaining irreducible representations of G and the corresponding decomposition numbers one probably needs fundamentally new methods.

We now describe briefly how we proceed in this paper. If G is a finite group with split BN-pair, one has the distribution of complex characters in so called Harish-Chandra series, HC-series for short. They are indexed by pairs whose first entry is a representative L of conjugacy classes of Levi subgroups of G, and the second entry is a cuspidal irreducible character of L. Attached to such an HC-series we have a Hecke algebra and one can define functors between the categories of G-modules and the module categories of the respective Hecke algebras.

In section one we first review the results on Hom-functors needed in the following. Then we define quotients of projective OG-lattices by factoring out constituents which do not belong to a fixed HC-series. These quotients are defined by a functor which is compatible with HC-induction.

In section two we introduce the central object of the paper, projective restriction systems associated with HC-series. It turns out that this axiomatic setting enables us to step up from using Hecke- to using q-Schur algebras to obtain information on representations belonging to the corresponding HC-series. We investigate how modules, which are contained in such a projective restriction system, behave under Harisch-Chandra restriction. We exhibit in particular a close relation between the poset structure of Levi subgroups of G and module structure of the modules in the projective restriction system. As a special case one can bear in mind the Steinberg lattices of finite groups of Lie type.

In section three we then connect our results of section two and those of section one to get our main result on decomposition numbers.

In section four we consider the most important groups with split BN-pairs, the finite groups of Lie type. We show that projective restriction systems appear regularly for these groups and apply our results to obtain at the end a new concrete result for the unipotent characters of finite unitary groups.

This paper has a long history: A preliminary version was out already at the end of 1996 and it was referred to in several papers which meanwhile appeared. In between there were several improvements and generalisations, which may explain the delay in part, (however the discovery of several gaps in proofs, which had to be closed, contributed subtantially as well). The main results were announced at several conferences, in [12] and in [27].

We wish to thank the organizers of the half year program on algebraic groups and related finite groups at the Isaac Newton Institute in Cambridge in the first half of 1997 and the University of Illinois at Chicago. During an extended visit at the Isaac Newton Institute by the first and an one year visit in Chicago in 1996/97 by the second author most of this article was written up.

1 Preliminaries

Throughout G denotes a finite group. We let O be a complete discrete valuation ring. The quotient field of O is denoted by K and its residue field by k. We assume that k is of characteristic ℓ for some prime ℓ dividing the order of G. Moreover we take (K, O, k) to be an ℓ-modular splitting system for G. Thus both fields, K and k are splitting fields for all subgroups of G.

For H ≤ G we denote the normalizer of H in G by N_{G}(H).

Modules are, if not stated otherwise, right modules, and homomorphisms act from the opposite side. Let R be a commutative domain. An R-order A is an R-algebra which is free and finitely generated as R-module, and an A-lattice is an A-module which is finitely generated and free as R-module. The category of A-lattices is denoted by 𝔏𝔞𝔱_{A} and of A-modules by 𝔐𝔬𝔡_{A}. Since R is commutative we may write the action of scalars on Amodules on the left assuming always that R acts centrally on every module.

If possible we omit tensor symbols. For instance, if B is some sublattice of A and M is a B-lattice we frequently write MA for the induced module M ⊗_{B} A. Or if R = O, and if M is an A-lattice, KA denotes the finite dimensional K-algebra K ⊗_{O} A, and KM the KA-module K ⊗_{O} M. The k-Algebra k ⊗_{O} A is denoted by kA or by Ā and the Ā-module k ⊗_{O} M by M̄. Similarly, if H is a group and χ is a KH-character, its associated Brauer character, that is the restriction of χ to ℓ-regular classes of H, is denoted by χ̄.

The radical of an A-lattice M is the intersection of its maximal sublattices and is denoted by Jac(M). The socle soc(M) of M is the maximal completely reducible submodule of M and the head hd(M) of M is the maximal completely reducible factor module of M. Note that hd(M) = M/Jac(M).

We need some results on quotients of Hom-functors from [6] and [7]. We begin with the basic setup there and supplement a few further results which will be needed later on.

Let T be a semiperfect R-algebra which is finitely generated as R-module. All occurring modules are, if not stated otherwise, finitely generated. Let M ∈ 𝔐𝔬𝔡_{T}, and let β : P → M be a projective presentation of M. Thus P ∈ 𝔐𝔬𝔡_{T} is projective, and β is an epimorphism. Let E = End_{T} (P). We take E_{β} = {ϕ ∈ E | ϕ(ker β) ⊆ ker β}. The endomorphism ring of M is denoted by H. Obviously J_{β} = {ψ ∈ E | im ψ ≤ ker β} is an ideal of E_{β} and E_{β}/J_{β} ≅ H as R-algebra canonically, (comp. [6, 2.1]). Our basic hypothesis is now the following:

Hypothesis 1.1 For M ∈ 𝔐𝔬𝔡_{T} we say that the projective presentation β : P → M of M satisfies 1.1, if E_{β} = E.

Then J_{β} is an ideal of E, and we identify H and E/J_{β} by the canonical isomorphism induced by β. We have now a functor

H = H^{β} = H_{M}^{β} : 𝔐𝔬𝔡_{T} → 𝔐𝔬𝔡_{H}

which takes the T -module V to the H-module Hom_{T} (P, V)/Hom_{T} (P, V)J_{β}. On maps H^{β} is defined in the obvious way.

Notation 1.2 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). The kernel kerP of P is the full subcategory of 𝔐𝔬𝔡_{T}, whose objects are T -modules V with Hom_{T} (P, V) = (0). So V ∈ kerP precisely if t_{P} (V) = V. 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). So A_{P} (V) is the maximal P -torsionless factor module of V. We define the functor Ĥ_{M} = Ĥ : 𝔐𝔬𝔡_{H} → 𝔐𝔬𝔡_{T} to be the composite functor A_{P} ∘ (− ⊗_{H} M). For V ∈ 𝔐𝔬𝔡_{T} the trace or combined image τ_{P} (V) of P in V is the T -submodule of V generated by the images of all homomorphisms ϕ : P → V.

The following lemma follows immediately from projectivity of P:

Lemma 1.3 Let P, V, U ∈ 𝔐𝔬𝔡_{T} and suppose that P is projective. Let f : V → U be T -linear. Then

f(τ_{P} (V)) ≤ τ_{P} (U) and f(t_{P} (V)) ≤ t_{P} (U).

Thus τ_{P} and t_{P} are really endofunctors of 𝔐𝔬𝔡_{T}.

It was shown in [6, 2.16] that Ĥ is a right inverse of the functor H. Thus in particular, if T is a finite dimensional algebra over some field, ℌ is a complete set of non-isomorphic irreducible H-modules, and 𝔗_{M} a complete set of non-isomorphic irreducible T -modules which are not taken to the zero module under the functor H, then H induces a bijection between 𝔗_{M} and ℌ [6, 2.28].

More precisely we have the following theorem (see [7, 1.2]):

Theorem 1.4 Suppose that β : P → M satisfies 1.1. Then

(i) If M is P -torsionless and X is a right ideal of H. Then

Ĥ(X) = XM ≅ A_{P} (X ⊗_{H} M).

(ii) If R is a field then 𝔗_{M} is a complete set of nonisomorphic irreducible constituents of hd(M). Then every indecomposable direct summand of M has simple head and factoring out the Jacobson radical induces a bijection between the isomorphism classes of indecomposable direct summands of M and the elements of 𝔗_{M}.

Let now T be an O-order in the semisimple K-algebra KT. We say V is an irreducible T -lattice if KV = K ⊗_{O} V is an irreducible KT -module.

Recall that an kT -module V is liftable, if V = k ⊗_{O} U for some T -lattice U. Moreover, if U, V are T -lattices, then

Hom_{KT} (KU, KV) ≅ K ⊗_{O} Hom_{T} (U, V) (1.5)

canonically. Similarly we have an homomorphism of k-spaces

(1.6) \overline{Hom_{T} (U, V)} = k ⊗_{O} Hom_{T} (U, V) → HomT̄(Ū, V̄)

which is injective but not necessarily surjective. The homomorphisms in the image of that map are called liftable, and similarly HomT̄(Ū, V̄) is said to be liftable, if this homomorphism is bijective.

We take M ∈ 𝔏𝔞𝔱_{T} and assume that the projective presentation β : P → M satisfies (1.1). By [6, 4.7] Hypothesis 1.1 (for R = O) is equivalent to ker(1_{K} ⊗_{O} β) having no irreducible constituent in common with KM. Moreover if this holds, the decomposition matrix of H is a submatrix of the decomposition matrix of T, [6, 4.10]. In [6, 4.7] it was shown that 1.1 holds then for 1_{R} ⊗_{O} β for all choices of R ∈ {K, O, k}. The three resulting Hom-functors and their right inverses are distinguished by a suffix R. We summarize:

Theorem 1.7 Let β : P → M be a projective presentation of the T -lattice M and suppose that K ker(β) and KM have no irreducible constituent in common. Then 1_{R} ⊗_{O} β satisfies Hypothesis 1.1 for all choices of R ∈ {K, O, k}, the endomorphism ring of M̄ is liftable and the decomposition matrix of H = End_{T} (M) is part of the decomposition matrix of T.

Lemma 1.8 Let β : P → M be a projective presentation of the T -lattice M satisfying Hypothesis 1.1. Then RM is RP -torsionless for R = K and O.

Recall that an algebra A over some field is called quasi Frobenius, if there exists a nondegenerate, associative bilinearform on A. This implies in particular, that the regular representation of A is injective, that is A is a self-injective algebra, and every homomorphism between two right ideals is given by left multiplication by some element of A.

If the form is in addition symmetric, we call A symmetric. For instance, group algebras over fields are symmetric algebras. Here is a generalisation to O-orders:

Notation 1.9 An O-order H is called integrally quasi Frobenius if there exists an associative bilinearform on H whose Gram determinant with respect to any basis of H is a unit in O.

A sublattice V of some O-lattice M is called pure if the factor module M/V is a torsion free (and hence free) O-module. If V ≤ M then the intersection of all pure sublattices of M which contain V is the unique minimal pure sublattice containing V and is denoted by \sqrt{V}. One checks easily that \sqrt{V} = KV ∩ M.

Note that the bilinearform on H induces nondegenerate associative bilinearforms on RH for R = K and R = k, such that these algebras are quasi Frobenius. In our applications later on we shall deal with Hecke algebras which are known to be integrally quasi Frobenius (indeed integrally symmetric).

Here is a result, which we will need later. For a proof see [7, 1.30 and 1.36]:

Theorem 1.10 Let β : P → M be a projective presentation of the T -lattice M satisfying Hypothesis 1.1. Let X and Y be pure right ideals of H = End_{T} (M). Suppose that H is integrally quasi Frobenius. Denote the associated Hom-functor by H = H^{β}. Then

(i) H(XM) = H(\sqrt{XM}) = X. (ii) Every homomorphism from \sqrt{XM} to \sqrt{Y M} maps XM into Y M. Restricting homor-

phisms induces an isomorphism

Hom_{RT} (R ⊗_{O} \sqrt{XM}, R ⊗_{O} \sqrt{Y M}) ⟶^{∼} Hom_{RT} (XM, Y M)

for R = O, K. (iii) The functor H induces an isomorphism

H_{R} : Hom_{RT} (R ⊗_{O} \sqrt{XM}, R ⊗_{O} \sqrt{Y M}) ⟶^{∼} Hom_{RH}(RX, RY), (1.11)

for R = O and K.

The following result is a refinement of [10, 3.2 and 3.3].

Lemma 1.12 Let G be a finite group and let (K, O, k) be a split ℓ-modular system for G for some prime ℓ. Let P be a projective OG-lattice and suppose that the irreducible KGmodule M occurs with multiplicity one as constituent of KP. Let X be an OG-lattice in M such that KX = M. Then

Hom_{OG}(P, X) ≅ O,

and for a generator φ of Hom_{OG}(P, X) we have the following: The projective cover of the image V = im (φ) is an indecomposable direct summand Q of P. The simple kG-module D = Q̄/JacQ̄ is the head of V̄, that is V̄ /JacV̄ ≅ D, and D occurs as composition factor of V̄ with multiplicity one. Moreover,

End_{kG}(V̄) ≅ k ⊗_{O} End_{OG}(V) ≅ k

and the image U of φ̄ = 1_{k} ⊗_{O} φ : P̄ → X̄ is an epimorphic image of V̄ and Q̄ is the projective cover of U. Thus U has simple head D occurring with multiplicity one as composition factor of U. In particular

End_{kG}(U) ≅ k.

Proof Let P = P_{1} ⊕ … ⊕ P_{m} be a decomposition of P into a direct sum of projective indecomposable OG-lattices. Then KP = KP_{1} ⊕ … ⊕ KP_{m} and KX occurs as irreducible direct summand of precisely one KP_{i}, say of KP_{1}, and we set Q = P_{1}. Thus restricting maps induces an isomorphism

Hom_{KG}(KP, KX) ≅ Hom_{KG}(KQ, KX),

where the left hand side is one dimensional. Since

Hom_{KG}(KP, KX) ≅ K ⊗_{O} Hom_{OG}(P, X) ≅ Hom_{KG}(KQ, KX)

we conclude that

Hom_{OG}(P, X) ≅ O ≅ Hom_{OG}(Q, X).

The generator φ of Hom_{OG}(P, X) is a generator of Hom_{OG}(Q, X) too, hence φ : Q → im (φ) = V is the minimal projective cover of V. Being indecomposable Q̄ has simple head hd(Q̄) = Q̄/Jac(Q̄) and this must be the head of V̄ as well. By Brauer’s reciprocity law the multiplicity of D as composition factor of X̄ equals the multiplicity of M = KX as irreducible constituent of KQ = KP_{1}, and hence is one. Thus D occurs only once as composition factor of V̄ as well. If ℘ denotes the generator of the unique maximal ideal (℘) of O, then k = O/(℘) and we have V̄ ≅ V/℘V, whereas the image U of φ̄ = 1_{k}⊗_{O}φ : P̄ → X̄ is given as U = V/(V ∩ ℘X). Since ℘V ⊆ ℘X, the module U is an epimorphic image of V̄ and the remaining claims of the lemma follow. □

Lemma 1.13 Let G and (K, O, k) be as in the previous lemma. Let X be an OG-lattice and suppose that its character χ decomposes χ = χ_{1} + χ_{2}, where χ_{2} is the character of a pure sublattice M of X and χ_{1} is an irreducible character afforded by S = X/M. Assume further that KS occurs in KP with multiplicity one, where P is the projective cover of S and let D = S̄/Jac(S̄). Then D is an irreducible kG-module and occurs with multiplicity one in S̄. Moreover

End_{kG}(S̄) ≅ k

and, if D does not occur as composition factor of M, then X = S ⊕ M, and P is indecomposable.

Proof Let β : P → S be an epimorphism. Then Lemma 1.12 implies immediately that P is indecomposable with head D = S̄/Jac(S̄) which is irreducible. Furthermore the multiplicity of D in S̄ is one and End_{kG}(S̄) ≅ k.

Consider the following diagram, where π : X → S is the canonical epimorphism:

ker β  ker ρ
[figure]
P
[figure: ∃ρ]
[figure: (0); M; X; π; S; β; (0)]

By projectivity of P we find ρ : P → X such that πρ = β. Let U = ker β. Now πρ(U) = β(U) = (0), hence ρ(U) ⊆ ker π. Tensoring over O by K yields

(1_{K} ⊗ ρ)(KU) ⊆ ker(1_{K} ⊗ π) = KM.

We conclude that either KU and KM have a composition factor in common, or KU ≤ ker(1_{K} ⊗ ρ), and hence U ≤ ker ρ. Since D is not a composition factor of M̄ we see, using Brauer reciprocity, that KM and KP do not have a composition factor in common, therefore the same holds for KU ≤ KP and KM.

Thus U ≤ ker ρ. We have an induced map ρ̂ from P/U ≅ S to X such that ρ = ρ̂α, where α : P → P/U is the natural epimorphism. Similarly β induces an isomorphism β̂ from P/ ker β to S. We have πρ̂ = β̂, and therefore πρ̂β̂^{−1} = 1_{S} and π splits, as desired.

□

Let G be a finite group with split BN-pair of characteristic p. We want to investigate representations of G in the non-describing characteristic case, hence we assume that the prime p is different from ℓ. The notion of a BN-pair means that we have given a Borel subgroup B and the monomial subgroup N of G. For the definition and details on groups with BN-pair we refer to [2]. Occasionally, G has to be viewed as a Levi subgroup of a larger finite group with split BN-pair G̃. We denote the BN-pair of G̃ by B̃ and Ñ and say that G̃ has a B̃Ñ-pair. In this situation we often assume that G is a standard Levi subgroup of G̃, that is, if B = UT and B̃ = T̃Ũ is the Levi decomposition of B and B̃ respectively, then T = T̃, B ⊆ B̃ and BŨ = B̃. As a consequence the monomial subgroup N of G, being the normalizer N_{G}(T), is contained in Ñ and W = N/T ≤ Ñ/T = W̃, indeed the Weyl group W of G is a standard parabolic subgroup of W̃ generated by a subset of the standard generators (simple reflections) of W̃.

The set of Levi subgroups of G will be denoted by L_{G} and for L ∈ L_{G}, we denote by L_{G,L} the set of Levi subgroups of G containing L. Note that every Levi subgroup of G is conjugate to a standard Levi subgroup.

For L ∈ L_{G} and R ∈ {K, O, F }, we define the functor Harish-Chandra induction R_{L}^{G} as follows: We choose a parabolic subgroup P of G such that L is a Levi complement of P. Let U be the Levi kernel, that is the unipotent radical, of P. Thus P is the semi direct product P = U ⋊ L. If M is an RL-lattice we may consider it as RP -lattice with trivial U-action. The corresponding functor is called inflation and is denoted by Inf_{L}^{P}. The functor R_{L}^{G} is now defined to be inflation followed by ordinary induction Ind_{P}^{G}. Thus for M ∈ 𝔏𝔞𝔱_{RL} we have:

R_{L}^{G}M = Ind_{P}^{G} ∘ Inf_{L}^{P}(M).

Dually we define Harish-Chandra restrictionT_{L}^{G} to be the restriction Res_{P}^{G} from RG-modules to RP -modules followed by Fix_{U}, which is the functor mapping the RP -module M to the RL-module Fix_{U}(M) = {m ∈ M | mu = m for all u ∈ U}. The fact that the order of U is invertible in R implies that R_{L}^{G} and T_{L}^{G} are adjoint functors, and the collection of functors R_{L}^{G} and their adjoints satisfy transitivity and a Mackey formula. They were investigated in a more general context in [10] and [8]. We point out that most of our general results in the following sections can be generalized to the situation in [8]. Morover, in this paper and independently in [33] it was shown that the functors R_{L}^{G} and their adjoints are independent of the choice of the parabolic subgroup P of which L is a Levi subgroup, provided the order of U is invertible in R. For fields R of characteristic 0 this result is known for a long time and was observed first by Deligne (comp. e.g. [42]). Of course these functors yield corresponding maps on characters and Brauer characters which again are denoted by R_{L}^{G} and T_{L}^{G}.

Harish-Chandra induction and restriction (or HC-induction and HC-restriction for short) provide the basic tools for the Harish-Chandra theory which subdivides the irreducible RG-modules for R = K or k into Harish-Chandra series (or HC-series for short). First an RL-module M is called cuspidal if T_{L′}^{L}(M) = (0) for every proper Levi subgroup L′ of L. The irreducible RG-module V belongs to the HC-series S(G/L, M)_{R} if V is a composition factor of the head (or alternatively of the socle) of R_{L}^{G}(M). If R = K we usually omit the index K. The subgroup L and the irreducible cuspidal RL-module M are unique up to conjugation in G respectively in the normalizer N_{G}(L) of L in G. Analogously we define the HC-series S(G/L, χ)_{R} for the irreducible cuspidal (Brauer-) character χ of L. This partition of the irreducible RG-modules and characters into HC-series is well known for fields R of chracteristic 0 (see e.g. [2]), and was introduced by Hiss in [29] for fields R of positive characteristic ℓ. If the RG-module X is in the HC-series S(G/L, M), the Levi subgroup L is called semisimple vertex and the irreducible cuspidal RL-module M semisimple source of X (see [8]). As remarked, those are unique up to conjugation in G and hence we may (and usually do) choose as representatives for semisimple vertices the unique standard Levi subgroups.

The following two lemmas are just formulated for the convenience of the reader. They can be easily derived from [29, 5.8] by Frobenius reciprocity and Mackey decomposition. Let now L ≤ M be Levi subgroups of G. We take an irreducible cuspidal kL-module X and an irreducible kG-module Y.

Lemma 1.14 Y lies in the HC-series S(G/L, X)_{k} if and only if we find Z, Z′ ∈ S(M/L, X)_{k} such that Y is isomorphic to some composition factor of hd(R_{M}^{G}Z) and to some composition factor of soc(R_{M}^{G}Z′).

Lemma 1.15 Y lies in the series S(G/L, X)_{k} if and only if there exist Z, Z′ ∈ S(M/L, X)_{k} such that Z is isomorphic to some composition factor of hd(T_{M}^{G}Y) and Z′ to some composition factor of soc(T_{M}^{G}Y).

Notation 1.16 Let 𝔛 be a set of irreducible characters of G and let Y be an OG-lattice affording the character χ. We may write uniquely χ = ψ + ψ′, where every irreducible constituent of ψ belongs to 𝔛 and no irreducible constituent of ψ′ belongs to 𝔛. We find pure sublattices V and V′ of Y such that V affords ψ and V′ affords ψ′. We define π_{𝔛}(Y) := Y/V′, the 𝔛-quotient of Y and ι_{𝔛}(Y) := V, the 𝔛-sublattice of Y. If 𝔛 consists of a single character ϕ say we write π_{ϕ}(Y) and ι_{ϕ}(Y) instead of π_{{ϕ}}(Y) and ι_{{ϕ}}(Y) respectively. Moreover, we set π_{𝔛}(χ) = ψ = ι_{𝔛}(χ). We define π_{𝔛}(Ȳ) := \overline{π_{𝔛}(Y/V′)} and ι_{𝔛}(Ȳ) := \overline{ι_{𝔛}(Y/V′)}. We note that the latter two depend on Y and not just on Ȳ. Obviously π_{ϕ}(Y) is zero or affords a multiple of the character ϕ. Let ℭ_{𝔛} be the full subcategory of the category of finitely generated OG-lattices whose objects are those Y ∈ 𝔏𝔞𝔱_{OG} satisfying ι_{𝔛}(Y) = Y.

The construction of π is precisely, what is needed to apply the results from the beginning of this section, as the following results demonstrates. It follows immediately from the definitions in 1.16:

Lemma 1.17 Let Y be a projective OG-lattice and 𝔛 a set of irreducible characters of G. Let M = π_{𝔛}(Y) and β : Y → M be the natural projection. Then β satisfies Hypothesis 1.1 and the requirements of Theorem 1.7. In particular the decomposition matrix of the O-order H = End_{OG}(M) is part of the ℓ-modular decomposition matrix of G.

By construction π and ι are functorial: If we have given Z ∈ 𝔏𝔞𝔱_{OG} with U = π_{𝔛}(Z) and U′ = ι_{𝔛}(Z) and if f : Y → Z, then f|_{V} : V → U and f|_{V′} : V′ → U′. Thus we have induced maps π_{𝔛}(f) : π_{𝔛}(Y) → π_{𝔛}(Z) and ι_{𝔛}(f) : ι_{𝔛}(Y) :→ ι_{𝔛}(Z). Thus π_{𝔛} and ι_{𝔛} are funtors from 𝔏𝔞𝔱_{OG} onto ℭ_{𝔛}, indeed both are left inverses of the embedding of ℭ_{𝔛} into 𝔏𝔞𝔱_{OG}. Obviously for 𝔛 = ∅ the functors π_{𝔛} and ι_{𝔛} are both the zero functor, taking every module to the zero module. If 𝔛 contains all irreducible characters of G we get the identity functors.

Lemma 1.18 Let Y, Z ∈ 𝔏𝔞𝔱_{OG}. Let 𝔛 be a set of irreducible characters of G. If either Y or Z is in ℭ_{𝔛} then the maps

π_{𝔛} : Hom_{OG}(Y, Z) → Hom_{OG}(π_{𝔛}(Y), π_{𝔛}(Z))

and

ι_{𝔛} : Hom_{OG}(Y, Z) → Hom_{OG}(ι_{𝔛}(Y), ι_{𝔛}(Z))

induced by the functors π_{𝔛} and ι_{𝔛} are isomorphisms.

Proof This is immediate by tensoring by K and comparing dimensions. □

Lemma 1.18 does not carry over to liftable kG-modules of the form Ȳ and Z̄ for Y, Z ∈ 𝔏𝔞𝔱_{OG} kG-modules, since the canonical embedding k ⊗_{O} Hom_{OG}(Y, Z) → Hom_{kG}(Ȳ , Z̄) is not surjective in general. However it is a bijection in case that either Y or Z is a projective OG-lattice (see e.g. [39, Theorem 14.7]).

Lemma 1.19 Let Y, Z ∈ 𝔏𝔞𝔱_{OG} such that Hom_{kG}(Ȳ , Z̄) is liftable. Let 𝔛 be a set of irreducible characters of G and let ℭ_{𝔛} be defined as in 1.16. Then we have:

(i) Let Z ∈ ℭ_{𝔛}. Then Hom_{kG}(Ȳ , Z̄) ≅ Hom_{kG}(π_{𝔛}(Ȳ), Z̄). (ii) Let Y ∈ ℭ_{𝔛}. Then Hom_{kG}(Ȳ , Z̄) ≅ Hom_{kG}(Ȳ , ι_{𝔛}(Z̄)).

Proof By Definition 1.16 we have π_{𝔛}(Ȳ) = Ȳ /V̄′ for V′ ∈ 𝔏𝔞𝔱_{OG} and ι_{𝔛}(V′) = (0). Denoting the projection from Ȳ onto π_{𝔛}(Ȳ) by π̄ we have an embedding

π̄^{∗} : Hom_{kG}(π_{𝔛}(Ȳ), Z̄) → Hom_{kG}(Ȳ , Z̄)

defined by π̄^{∗}(f) = f ∘ π̄. By assumption and Lemma 1.18 we have

Hom_{kG}(Ȳ , Z̄) = k ⊗_{O} Hom_{OG}(Y, Z) ≅ k ⊗_{O} Hom_{OG}(π_{𝔛}(Y), Z) ⊆ Hom_{kG}(π_{𝔛}(Ȳ), Z̄),

hence, comparing dimensions, π̄^{∗} is an isomorphism. ii) is proven analogously. □

We shall study now how the functors π and ι behave with respect to HC-induction and -restriction, where we take in 1.16 HC-series of characters of G as set 𝔛.

Notation 1.20 Let G be a group with split BN-pair which is a Levi subgroup of a group G̃ with split B̃Ñ-pair. Let M ⊂ L_{G} and let C be a set of cuspidal irreducible characters of elements of M. We say the pair (M, C) is closed under conjugation in G̃ if M ∈ M implies M^{x} ∈ M and χ ∈ C implies χ^{x} ∈ C for all x ∈ G̃ such that M^{x} ∈ L_{G}. If M ∈ L_{G} we set M_{M} := M ∩ L_{M}, and we define C_{M} to be the set of those KM′-characters χ with M′ ∈ M_{M} and χ ∈ C. Note that (M_{M}, C_{M}) is closed under conjugation in G̃ if (M, C) is. On the other hand if (M, C) is an arbitrary pair for G ∈ LG̃, then (M, C) is a pair for G̃. However, even if (M, C) is closed under conjugation by G, it is not necessarily closed under conjugation by G̃.

Now let (M, C) be fixed. We set for every M ∈ L_{G}

𝔛_{M} := ⋃_{{L∈M_{M} , \atop χ∈Irr(L)∩C_{M}}} S(M/L, χ).

Then we define π_{(M,C)}^{M} = π_{𝔛_{M}}. If no ambiguities can arise we omit the index (M, C) and write simply π^{M} = π_{𝔛_{M}}.

Remark 1.21 Obviously if M does not contain any L ∈ M then π_{𝔛_{M}} = π_{∅} is the zero functor. Moreover it should be pointed out that the restriction π^{M} of π to M ∈ L_{G} essentially depends upon the choice of (M, C), not on the functor π = π^{G} alone. For instance we choose x ∈ G such that L ≤ M and L^{x} ≤ M are not conjugate in M ∈ L_{G}, and we take an irreducible cuspidal character χ of L. Then χ^{x} is an irreducible cuspidal character of L^{x}. We set M = {L}, C = {χ}, M^{x} = {L^{x}} and C^{x} = {χ^{x}}. Then obviously

π_{(M,C)}^{G} = π_{(M^{x},C^{x})}^{G}

but

π_{(M,C)}^{M} ≅ π̸_{(M^{x},C^{x})}^{M}.

To apply Results 1.16 and 1.19 we need the following lemma which follows immediately from Mackey decomposition (compare [10, 1.14]) and from transitivity of HC-induction:

Lemma 1.22 Let ψ be a KG-character.

(i) Let M ∈ L_{G,L} and assume that all constituents of ψ belong to the HC-series S(G/L, χ), where χ is an irreducible cuspidal KL-character. Then every constituent of T_{M}^{G}ψ belongs to a series S(M/L^{x}, χ^{x}) for some x ∈ G such that L^{x} ≤ M.

(ii) Suppose G ∈ LG̃ and assume that every irreducible constituent of ψ belongs to some HC-series S(G/L^{x}, χ^{x}), where x ∈ G̃ such that L^{x} ≤ G. Then RG̃ ; Gψ belongs to S(G̃/L, χ).

Proof To prove (i) we may assume that ψ is irreducible. Then ψ is constituent of R_{L}^{G}χ by assumption and hence T_{M}^{G}ψ is a summand of

T_{M}^{G}R_{L}^{G}χ = ∑_{x} R_{L^{x}∩M}^{M}T_{L^{x}∩M}^{L^{x}}χ^{x},

where x runs through a set of double coset representatives of parabolic subgroups having M and L as Levi complements, by [10, 1.14]. But these double coset representatives can be chosen in N, (see [2, 2.1]). Moreover T_{L^{x}∩M}^{L^{x}}χ^{x} = (0) unless L^{x} ∩ M = L^{x}, that is L^{x} ≤ M by cuspidality of χ. Thus the result follows. Part (ii) is shown similarly, observing that for x ∈ N we have RG̃ ; L^{x} ≅ RG̃ ; L by [33] respectively [9, 5.2]. □

Note that in the lemma above we may take M and L to be standard Levi subgroups of G, and G to be a standard Levi subgroup of G̃ (by adjusting G, then M and then L by conjugating by an element of G̃ , G and M respectively). In this case we may take x to be an element of N or in (ii) of Ñ. Now Lemma 1.22 tells us that under HC-restriction we might get terms with semi simple HC-vertex in M^{x}, where x ∈ N such that L^{x} ≤ M. We therefore define

Definition 1.23 Let M ∈ L_{G,L}. Then we set

N_{L⊆M} = {x ∈ G|L^{x} ≤ M}.

The letter N stands here for the monomial group of G, since we can always assume that L is standard in M and M in G, and then choose x to be an element of N. Thus for instance

Ñ_{L⊆G} = {x ∈ G̃ | L^{x} ⊆ G}.

The following Corollary follows now easily from Lemma 1.22:

Corollary 1.24 Suppose that (M, C) is closed under conjugation in G and let M ∈ L_{G}. Then we have isomorphisms of functors

(i) For (M, C) as in 1.20 and M ∈ L_{G} we have

R_{M}^{G}π^{M} ≅ π^{G}R_{M}^{G}
R_{M}^{G}ι^{M} ≅ ι^{G}R_{M}^{G}.

(ii)

T_{M}^{G}π^{G} ≅ π^{M}T_{M}^{G}
T_{M}^{G}ι^{G} ≅ ι^{M}T_{M}^{G}.

Proof First note that it suffices to check that the functors agree on characters, since R_{M}^{G} and T_{M}^{G} preserve purity of sublattices. We show the second part of Corollary, the proof of the first being similar. So let ψ be an irreducible KG-character in S(G/L, χ), and L ∈ M. If ψ ∈ C then π^{G}(ψ) = ψ and π^{G}(ψ) = (0) otherwise. If L^{x} ≰ M for all x ∈ G, then T_{M}^{G}ψ = (0) by Mackey decomposition and then

0 = T_{M}^{G}π^{G}(ψ) = π^{M}(T_{M}^{G}ψ)

in all cases. Suppose L^{x} ≤ M. Since S(G/L, χ) = S(G/L^{x}, χ^{x}) we may assume x = 1, that is L ≤ M. Let χ ∈ C then χ^{x} ∈ C_{M} for all x ∈ N_{L⊆M} and hence

π^{M}(T_{M}^{G}ψ) = T_{M}^{G}ψ = T_{M}^{G}π^{G}(ψ).

If χ ∉ C then χ^{x} ∉ C for all x ∈ N_{L⊆M} and we get

0 = π^{M}(T_{M}^{G}ψ) = T_{M}^{G}(0) = T_{M}^{G}π^{G}(ψ).

The other isomorphisms of functors are shown similarly □

Corollary 1.25 Suppose that (M, C) is closed under conjugation in G and let M ∈ L_{G}. Suppose Y ∈ 𝔏𝔞𝔱_{M} is projective. Set X = π^{M}(Y) and consider

β : R_{M}^{G}Y → R_{M}^{G}X,

where β = R_{M}^{G}ψ setting ψ : Y → X to be the natural projection. Then β satisfies the requirements of Theorem 1.7. In particular the decomposition matrix of H = End_{OG}(R_{M}^{G}(X)) is part of the ℓ-modular decomposition matrix of G.

Proof By Corollary 1.24

π^{G}(R_{M}^{G}Y) = R_{M}^{G}π^{M}(Y) = R_{M}^{G}X.

Thus the result follows from 1.17 applied to the projective OG-module R_{M}^{G}Y. □

2 Projective restriction systems

We introduce now projective restriction systems for finite groups with split BN-pairs. First we describe the set up which we shall be dealing with:

Notation 2.1 Let G be a finite group with split BN-pair. We fix L ∈ L_{G} and an irreducible cuspidal KL-character χ_{L}. We assume that χ_{L} occurs with multiplicity one as an irreducible constituent of the character afforded by some projective OL-lattice Y_{L}. Thus we may apply Lemmas 1.12 and 1.13.

If G ∈ LG̃ for some group G̃ with B̃Ñ-pair such that B = B̃ ∩ G and N = Ñ ∩ G, we take in 1.20

M := {L^{x} | x ∈ Ñ_{L⊆G}} and C := {χ_{L}^{x} | x ∈ Ñ_{L⊆G}}.

For M ∈ L_{G} we write

π^{M} := π_{(M,C)}^{M}.

In particular for M = G we omit the superscript G and write π = π^{G}. A similar notation is used for ι.

Finally we set

X_{L} := π^{L}(Y_{L}), X′_{L} := ι^{L}(Y_{L}), and D_{L} := X̄_{L}/Jac(X̄_{L}).

Remark 2.2 By Lemma 1.13, D_{L} ≅ hd(X̄_{L}) is irreducible and is the head of the projective cover Ȳ_{1} of X̄_{L} which is isomorphic to a direct summand of Ȳ_{L}. If Y_{1} denotes its lift to an OL-sublattice of Y_{L}, which exists by projectivity of Y_{L}, then ι^{L}(Y_{L}) ⊆ Y_{1}. Since kL is a symmetric algebra, the head D_{L} of Ȳ_{1} and its socle are isomorphic. Thus the lattice X′_{L} is different from X_{L} unless X̄_{L} is irreducible, as D_{L} is the head of X̄_{L} and the socle of X̄′_{L}. Obviously we have

Hom_{kL}(X̄_{L}, X̄′_{L}) ≅ k.

Definition 2.3 Keep the notation introduced in 2.1. In view of Lemma 1.12 we may assume that Y_{L} is indecomposable and hence is the projective cover of X_{L}. Note that this implies that Ȳ_{L} is an principal indecomposable kL-module corresponding to the irreducible module D_{L}. In particular D_{L} is the head Ȳ_{L}/Jac(Ȳ_{L}) of Ȳ_{L}. Suppose in addition that χ_{G} is an irreducible constituent of multiplicity one in R_{L}^{G}χ_{L} such that the following holds:

(i) There exists a projective OG-lattice Y_{G} such that χ_{G} is the character of

π^{G}(Y_{G}) =: X_{G}.

Again we may (and do) assume that Y_{G} is indecomposable. (ii) For M ∈ L_{G,L} let T_{M}^{G}χ_{G} = ψ_{1} +…+ψ_{n} be a decomposition of T_{M}^{G}χ_{G} into irreducible

constituents. Let X_{i} be an OM-lattice affording ψ_{i} for i = 1, …, n. Then for i ≠ j, the kM-modules X̄_{i} and X̄_{j} have no composition factor in common.

We set X_{M} := π_{(L,χ_{L})}(T_{M}^{G}X_{G}) for every M ∈ L_{G,L} and X′_{M} := ι_{(M,χ_{M})}(R_{L}^{M}X_{L}). We denote the characters of X_{M} by χ_{M} and the projective cover of X_{M} by Y_{M}.

The projective restriction system to (L, χ_{L}) then consists of the data {X_{M}, Y_{M}, X′_{M} | M ∈ L_{G,L}} and is denoted by PR(X_{G}, Y_{L}).

Remark 2.4 Our notation is consistent, since part (i) of 2.3 implies

π(Y_{G}) = π_{(M,C)}^{G}(Y_{G}) = π_{(L,χ_{L})}(Y_{G}) = X_{G}.

Hence X_{G} = π_{(L,χ_{L})}(T_{G}^{G}X_{G}) since X_{G} is irreducible and χ_{G} ∈ S(G/L, χ_{L}) by assumption. Similarly Y_{L} is the projective cover of π_{(L,χ_{L})}T_{L}^{G}X_{G} = X_{L}, since

Hom_{OL}(T_{L}^{G}X_{G}, X_{L}) ≅ Hom_{OG}(X_{G}, R_{L}^{G}X_{L}) ≅ O

by Frobenius reciprocity. Moreover, if x ∈ N_{G}(L) is such that the conjugate character χ_{L}^{x} is different from χ_{L}, then χ_{L}^{x} cannot be a constituent of Y_{L}. With other words we have π^{L}(Y_{L}) = X_{L}. Otherwise the simple head of Ȳ_{L} is composition factor of X̄_{L} as well as of X̄_{L}^{x}, since Y_{L} is indecomposable. But X_{L}^{x} affords χ_{L}^{x}, and it is constituent of T_{L}^{G}X_{G}, hence X̄_{L}^{x} has by (ii) of 2.3 no composition factor with X̄_{L} in common. We also note that part (ii) is particularly satisfied if the summands of T_{M}^{G}χ_{G} are in pairwise different ℓ-modular blocks of OM. This will be the fact which we prove in concrete applications.

For the remainder of this section we fix a projective restriction system PR(X_{G}, Y_{L}). Throughout M ∈ L_{G,L}. The next lemma is trivial:

Lemma 2.5 Let x ∈ G̃, then the conjugate projective restriction system

PR^{x}(X_{G}, Y_{L}) = PR(X_{G}^{x}, Y_{L}^{x})

is the projective restriction system given as {X_{M}^{x}, Y_{M}^{x}, X_{M}^{x′} | M ∈ L_{G,L}}.

Observe that X_{M}^{x} is the conjugate module M^{x}-module (X_{M})^{x}. We have to distinguish this from certain other modules coming up in the conjugate restriction system:

Notation 2.6 Let x ∈ G̃ and M ∈ L_{G^{x},L^{x}}. Then the M-module

X_{M,x} = π_{(L^{x},χ_{L}^{x})}(T_{M}^{G}X_{G}^{x})

and its projective cover Y_{M,x} belong to the conjugate restriction system PR^{x}(X_{G}, Y_{L}). In particular, if M ∈ L_{G,L} and x ∈ G is such that L^{x} ⊆ M, then

PR^{x}(X_{G}, Y_{L}) = PR(X_{G}, Y_{L}^{x})

and

X_{M,x} = (X_{M^{x^{−1}}} )^{x} = π_{(L^{x},χ_{L}^{x})}(T_{M}^{G}X_{G}).

We denote the related terms similarly: For example χ_{M,x} denotes the character afforded by X_{M,x}.

In the following we shall use frequently the Mackey formula for HC-induction and restriction in connection with the property of X_{L} to be cuspidal. In the Mackey formula we deal with double coset representatives D_{LM} of parabolic subgroups containing L respectively M ∈ L_{G,L} as Levi complement. If L and M are standard Levi subgroups we may choose D_{LM} to be distinguished by general theory, that is it is contained in the monomial subgroup N of G as set of preimages of the distinguished double coset representatives of the corresponding standard parabolic subgroups of the Weyl group W = N/T of G, (for details see e.g. [2]). This choice then ensures, that the intersections M ∩ L^{x} with x ∈ D_{LM} are again standard Levi subgroups of G. Since X_{L} is cuspidal by assumption, all terms in the Mackey formula

T_{M}^{G}R_{L}^{G}X_{L} = ⨁_{x∈D_{LM}} R_{M∩L^{x}}^{M}T_{M∩L^{x}}^{L^{x}}X_{L}^{x} (2.7)

with L^{x} ≰ M are zero, that is we have to sum only over x ∈ N_{L⊆M} ∩ D_{LM}. But x ∈ G is contained in N_{L⊆M} precisely if LxM = xM. If in addition x ∈ D_{LM}, then L^{x} = L^{x} ∩ M ∈ L_{G}. Obviously M acts on N_{L⊆M} by right translation, where the orbit of x ∈ N_{L⊆M} under this action is the double coset xM = LxM.

Observe that N_{G}(L) ⊆ N_{L⊆M}, and that this group acts on N_{L⊆M} by left translation. We take N_{M,L}^{G} to be a set of representatives of N_{G}(X_{L}) × M-orbits on N_{L⊆M} (containing 1 ∈ N_{L⊆M}), where N_{G}(X_{L}) is the stabilizer of X_{L} in N_{G}(L). Note that each such orbit contains elements of D_{LM} and we choose therefore N_{M,L}^{G} to be a subset of our selected set of double coset representatives D_{LM}. If we need to emphasize the dependence of the chosen object X_{L} we write N_{M,L}^{G}(X_{L}), and we define similar representatives for other L-modules as for example N_{M,L}^{G}(D_{L}) etc.

It follows now that {L^{x} | x ∈ N_{M,L}^{G}} is a set of Levi subgroups of M which for x ≠ 1 are conjugate to L in G but not in M. Moreover L^{x} = L^{y} with x, y ∈ N_{M,L}^{G} and x ≠ y, implies X_{L}^{x} ≅ X̸_{L}^{y}.

Theorem 2.8 Let M ∈ L_{G,L} and x ∈ N_{M,L}^{G}. Then the OM-lattice X_{M,x} affords the character χ_{M,x} which is irreducible and occurs with multiplicity one as constituent of the character afforded by T_{M}^{G}(Y_{G}). Moreover

T_{M}^{G}X_{G} = ⨁_{x∈N_{M,L}^{G}} X_{M,x}.

Proof The character χ_{G} of X_{G} is contained in the HC-series S(G/L, χ_{L}). Hence by Lemma 1.22 every irreducible constitutent of T_{M}^{G}χ_{G} belongs to some S(M/L^{x}, χ_{L}^{x}) for some x ∈ N_{L⊆M}. We may assume that x ∈ N_{M,L}^{G}. By assumption χ_{G} occurs with multiplicity one in R_{L}^{G}χ_{L}, hence by Lemma 2.5 χ_{G} occurs with multiplicity one in R_{L^{x}}^{G}χ_{L}^{x} too. Therefore by Frobenius reciprocity and transitivity of HC-restriction

K ≅ Hom_{KL^{x}}(KX_{L}^{x}, T_{L^{x}}^{G}KX_{G}) ≅ Hom_{KM}(R_{L^{x}}^{M}KX_{L}^{x}, T_{M}^{G}KX_{G}),

and consequently T_{M}^{G}χ_{G} has precisely one constituent in S(M/L^{x}, χ_{L}^{x}) and this is by definition given as

π_{L^{x},χ_{L}^{x}}(T_{M}^{G}χ_{G}) = χ_{M,x}.

In particular χ_{M,x} is an irreducible character and, being in different HC-series of M, we have χ_{M,x} ≠ χ_{M,y} for x ≠ y in N_{M,L}^{G}. We have shown so far that

T_{M}^{G}χ_{G} = ∑_{x∈N_{M,L}^{G}} χ_{M,x}.

Being an OM-lattice in an irreducible KM-module, X_{M,x} is indecomposable. Note that X_{G} is an epimorphic image of Y_{G} and hence X_{M,x}, being an epimorphic image of T_{M}^{G}X_{G}, is an epimorphic image of the projective OM-lattice T_{M}^{G}Y_{G}. Hence the projective cover Y_{M,x} of X_{M,x} is a direct summand of T_{M}^{G}Y_{G}. In particular χ_{M,x} occurs as constituent of the character of T_{M}^{G}Y_{G}. Remark 2.4 in conjunction with Lemma 2.5 implies

X_{G} = π^{G}(Y_{G}) = π_{(L,χ_{L})}(Y_{G}) = π_{(L^{x},χ_{L}^{x})}(Y_{G}).

By Frobenius reciprocity, Corollary 1.24 and Lemma 1.18 we have

Hom_{OM}(T_{M}^{G}Y_{G}, X_{M,x}) ≅ Hom_{OG}(Y_{G}, R_{M}^{G}X_{M,x})
≅ Hom_{OG}(Y_{G}, R_{M}^{G}π^{M}X_{M,x})
≅ Hom_{OG}(Y_{G}, π^{G}R_{M}^{G}X_{M,x})
≅ Hom_{OG}(Y_{G}, R_{M}^{G}X_{M,x})
≅ Hom_{OG}(Y_{G}, π^{G}R_{M}^{G}(X_{M^{x^{−1}}} )^{x})
≅ Hom_{OG}(Y_{G}, π^{G}R_{M^{x^{−1}}}^{G} X_{M^{x^{−1}}} )
≅ Hom_{OG}(Y_{G}, π^{G}R_{M}^{G}X_{M})
≅ Hom_{OG}(π^{G}Y_{G}, π^{G}R_{M}^{G}X_{M})
≅ Hom_{OG}(X_{G}, π^{G}R_{M}^{G}X_{M})
≤ 1

We conclude that the multiplicity of χ_{M,x} in the character of Y_{M,x} is precisely one. But now part (ii) of Definiton 2.3 states exactly that the hypothesis of Lemma 1.13 are satisfied for T_{M}^{G}X_{G}, X_{M,x} and the projective cover Y_{M,x} of X_{M,x}. Thus X_{M,x} is a direct summand of T_{M}^{G}X_{G} for all x ∈ N_{M,L}^{G} and the theorem follows. □

Note that Theorem 2.8 states in particular that KX_{M} is irreducible. Result 1.12 now implies:

Corollary 2.9 Let M ∈ L_{G,L} and x ∈ N_{M,L}^{G}. Then D_{M,x} := X̄_{M,x}/Jac(X̄_{M,x}) is an irreducible kM-module occurring with multiplicity one as composition factor of X̄_{M,x}. Moreover

dim_{k}(End_{kM}(X̄_{M})) = 1,

and Ȳ_{M,x} is the projective cover of D_{M,x}. In particular Ȳ_{M,x} and hence Y_{M,x} is indecomposable.

Note that

D_{M,x} = hd(X̄_{M,x}) = hd((X̄_{M^{x^{−1}}} )^{x}) = (D_{M^{x^{−1}}} )^{x}. (2.10)

As a consequence of Theorem 2.8 and part (ii) of Definition 2.3 we get the following two results:

Corollary 2.11 Let M, M′ ∈ L_{G,L} with M′ ≤ M.

(i) Then π_{(L,χ_{L})}^{M′}(T_{M′}^{M}X_{M}) = X_{M′}. In fact

T_{M′}^{M}X_{M} = ⨁_{y∈N_{M′, L}^{M}} X_{M′,y}.

(ii) Let V be an indecomposable projective kM′-module such that V/Jac(V) is a composition factor of X̄_{M′}. Then

Hom_{kM′}(T_{M′}^{M}X̄_{M}, V) ≅ Hom_{kM′}(X̄_{M′}, V)

and

Hom_{kM′}(V, T_{M′}^{M}X̄_{M}) ≅ Hom_{kM′}(V, X̄_{M′}).

Corollary 2.12 Let M ∈ L_{G,L}. Then

PR_{M} := PR(X_{M}, Y_{L}) = {X_{M′}, Y_{M′}, X′_{M′} | M′ ∈ L_{M,L}}

is a projective restriction system to (L, χ_{L}).

We now investigate the projective cover Y_{G} of X_{G}:

Theorem 2.13 Let M ∈ L_{G,L}. Then

T_{M}^{G}Y_{G} = ⨁_{x∈N_{M,L}^{G}} Y_{M,x} ⊕ Z,

where Z is a projective OM-lattice with π^{M}(Z) = (0). Moreover,

π_{(L^{x},χ_{L}^{x})}(T_{M}^{G}Y_{G}) = X_{M,x},

and

π_{(L^{x},χ_{L}^{x})}(Y_{M,y}) = \begin{cases}π_{(L^{x},χ_{L}^{x})}(T_{M}^{G}Y_{G}) = X_{M,x} & if x = y \\ (0) & otherwise, \end{cases}

where x, y ∈ N_{M,L}^{G}. Thus in particular

π_{(M,C)}^{M}(Y_{M,x}) = π_{(L^{x},χ_{L}^{x})}(Y_{M,x}) = X_{M,x}.

Proof Since T_{M}^{G}X_{G} is an epimorphic image of T_{M}^{G}Y_{G}, the latter contains the projective cover of the former as a direct summand, hence by Theorem 2.8

T_{M}^{G}Y_{G} = ⨁_{x∈N_{M,L}^{G}} Y_{M,x} ⊕ Z,

for some projective OM-lattice Z. Now by Results 1.24 and 2.8

⨁_{x∈N_{M,L}^{G}} X_{M,x} ⊕ π^{M}(Z) = ⨁_{x∈N_{M,L}^{G}} π^{M}(Y_{M,x}) ⊕ π^{M}(Z)

= π^{M}(T_{M}^{G}Y_{G}) = T_{M}^{G}X_{G}

= ⨁_{x∈N_{M,L}^{G}} X_{M,x} = ⨁_{x∈N_{M,L}^{G}} π_{(L^{x},χ_{L}^{x})}(T_{M}^{G}X_{G}).

Hence π^{M}(Z) = (0) and

X_{M,x} = π_{(L^{x},χ_{L}^{x})}(T_{M}^{G}Y_{G}) = π_{(L^{x},χ_{L}^{x})}(Y_{M,x}).
□

Corollary 2.14 Let M ∈ L_{G,L}. Then

π_{(L^{x},χ_{L}^{x})}(RG̃ ; MY_{M}) = RG̃ ; MX_{M},

for every x ∈ G̃.

Proof We only need to observe that on OG̃-lattices πG̃ = πG̃ ; (M,C) = π_{(L^{x},χ_{L}^{x})} for every x ∈ G̃, since L^{x} and L are conjugate in G̃. Thus by Lemma 1.24

π_{(L,χ_{L})}(RG̃ ; MY_{M}) = πG̃(RG̃ ; MY_{M})

= RG̃ ; Mπ^{M}Y_{M}

= RG̃ ; MX_{M}

□

Notation 2.15 Let L ∈ L_{G,L}. In Theorem 2.8 we have seen that X̄_{M} has a unique maximal submodule, hence D_{M} := X̄_{M}/Jac(X̄_{M}) is irreducible. Now D_{M} can be cuspidal or not. We denote the subset of L_{G,L} of all M such that D_{M} is cuspidal by L^{c}(X_{G}, Y_{L}). Note that L ∈ L^{c}(X_{G}, Y_{L}) since χ_{L} and hence D_{L} = X̄_{L}/Jac(X̄_{L}) is cuspidal. Note too, that for x ∈ G the set L^{c}(X_{G}, Y_{L}^{x}) consists precisely of the subgroups M^{x} ∈ L_{G,L^{x}} with M ∈ L^{c}(X_{G}, Y_{L}). In particular, M ∈ L^{c}(X_{G}, Y_{L}) does not imply M ∈ L^{c}(X_{G}, Y_{L}^{x}) in general, hence if D_{M} is cuspidal and x ∈ N_{L⊆M} the kM-module D_{M,x} is not necessarily cuspidal. However the irreducible kM^{x}-module D_{M}^{x} = (D_{M})^{x} is cuspidal in this case.

We fix now M ∈ L^{c}(X_{G}, Y_{L}) and G′ ∈ L_{G,M}. Note that G′ ∈ L_{G,L}, and that N_{M⊆G′} ⊆ N_{L⊆G′}. Take x ∈ N_{M⊆G′}. The N_{G}(X_{L}) × G′-orbit N_{G}(X_{L})xG′ of x in N_{L⊆G′} is denoted by [x]_{G′,X_{L}} and we may assume that x ∈ N_{G′,L}^{G}. Thus X̄_{G′,x} is one of the indecomposable direct summands of T_{G′}^{G}X̄_{G} by Theorem 2.8.

Theorem 2.16 Keep the notation introduced above. Then there is precisely one composition factor D_{G′,M^{x}} of T_{G′}^{G}X̄_{G} in the HC-series S(G′/M^{x}, D_{M}^{x})_{k} and it is in fact a composition factor of the direct summand X̄_{G′,x} of T_{G′}^{G}X̄_{G}. Moreover, if V ≤ T_{G′}^{G}X̄_{G,x}, then D_{G′,M^{x}} is a composition factor of V if and only if D_{M}^{x} is a composition factor of T_{M^{x}}^{G′}V.

Proof Using Frobenius reciprocity, part (ii) of Definition 2.3, and Theorem 2.8, we have

Hom_{kG′}(T_{G′}^{G}X̄_{G}, R_{M^{x}}^{G′}D_{M}^{x}) = Hom_{kM^{x}}(T_{M^{x}}^{G}X̄_{G}^{x}, D_{M}^{x})

= ⨁_{y∈N_{M^{x},L^{x}}^{G}} Hom_{kM^{x}}(X̄_{M^{x},y}^{x}, D_{M}^{x})

= Hom_{kM^{x}}(X̄_{M}^{x}, D_{M}^{x}) ≅ k,

since in the last direct sum above the only nonzero term occurs for y = 1, again by part (ii) of Definition 2.3, (applied to the projective restriction system PR^{x}(X_{G}, Y_{L})). We conclude that T_{G′}^{G}X̄_{G} and the socle of R_{M^{x}}^{G′}D_{M}^{x} have a common composition factor. But this is a composition factor D_{G′,M^{x}} of T_{G′}^{G}X̄_{G} in S(G′/M^{x}, D_{M}^{x})_{k}.

Now let V ≤ T_{G′}^{G}X̄_{G}^{x}. Then the embedding V → T_{G′}^{G}X̄_{G}^{x} induces an embedding of Hom_{kG′}(R_{M^{x}}^{G′} Ȳ_{M}^{x}, V) into Hom_{kG′}(R_{M^{x}}^{G′} Ȳ_{M}^{x}, T_{G′}^{G}X̄_{G}^{x}), and we have by Theorem 2.8, using 2.3, Frobenius reciprocity, and in addition the fact that Y_{M} is projective:

Hom_{kM^{x}}(Ȳ_{M}^{x}, T_{M^{x}}^{G′}V) ≅ Hom_{kG′}(R_{M^{x}}^{G′} Ȳ_{M}^{x}, V)

≤ Hom_{kG′}(R_{M^{x}}^{G′} Ȳ_{M}^{x}, T_{G′}^{G}X̄_{G}^{x})

≅ Hom_{kM^{x}}(Ȳ_{M}^{x}, T_{M^{x}}^{G}X̄_{G}^{x})

≅ ⨁_{y∈N_{M^{x},L^{x}}^{G}} Hom_{kM^{x}}(Ȳ_{M^{x}}^{x}, X̄_{M^{x},y}^{x})

≅ Hom_{kM^{x}}(Ȳ_{M}^{x}, X̄_{M}^{x})

≅ k ⊗_{O} Hom_{OM^{x}}(Y_{M}^{x}, X_{M}^{x}) ≅ k ⊗_{O} O ≅ k.

(2.17)

Here we wrote X̄_{M^{x},y}^{x} for the construction 2.6 applied to the conjugate module X̄_{M}^{x}. The first isomorphism in Equation 2.17 implies that there is a non-trivial homomorphism

from R_{M^{x}}^{G′} Ȳ_{M}^{x} to V if and only if D_{M}^{x} is a composition factor of T_{M^{x}}^{G′}V. But in this case

Hom_{kG′}(R_{M^{x}}^{G′} Ȳ_{M}^{x}, V) ≅ Hom_{kG′}(R_{M^{x}}^{G′} Ȳ_{M}^{x}, T_{G′}^{G}X̄_{G}).

We observe that Ȳ_{M}^{x} is the projective cover of D_{M}^{x}, hence the projective cover of R_{M^{x}}^{G′}D_{M}^{x} is a direct summand of R_{M^{x}}^{G′} Ȳ_{M}^{x} which in turn contains the projective cover Ȳ_{G′,M^{x}} of D_{G′,M^{x}}, since this is contained in the HC-series S(G′/M^{x}, D_{M}^{x})_{k} and hence is an irreducible summand of hd(R_{M^{x}}^{G′}D_{M}^{x}). Thus a nonzero homomorphism from R_{M^{x}}^{G′} Ȳ_{M}^{x} into T_{G′}^{G}X̄_{G}^{x} (which exists and is unique up to scalar multiplies by 2.17), is nonzero on Ȳ_{G′,M^{x}}. From this the last assertion of the theorem follows easily.

It remains to show that D_{G′,M^{x}} is a composition factor of X̄_{G′,x}. To do this we apply Theorem 2.8 and the fact that homomorphisms from projective modules over k are liftable to O and obtain:

O ≅ Hom_{OG′}(R_{M^{x}}^{G′}Y_{M}^{x}, T_{G′}^{G}X_{G}) ≅ ⨁_{y∈N_{G′,L}^{G}} Hom_{OG′}(R_{M^{x}}^{G′}Y_{M}^{x}, X_{G′,y}),

hence there is precisely one y ∈ N_{G′,L}^{G} such that

Hom_{OG′}(R_{M^{x}}^{G′}Y_{M}^{x}, X_{G′,y}) ≠ (0). (2.18)

We tensor by K and use 1.20, Corollary 1.24, Lemma 1.18 and Frobenius reciprocity to get:

Hom_{KG′}(R_{M^{x}}^{G′}KY_{M}^{x}, KX_{G′,y}) = Hom_{KG′}(π^{G′}R_{M^{x}}^{G′}KY_{M}^{x}, KX_{G′,y})
= Hom_{KG′}(R_{M^{x}}^{G′}π^{M^{x}}KY_{M}^{x}, KX_{G′,y})
= Hom_{KG′}(R_{M^{x}}^{G′}KX_{M}^{x}, KX_{G′,y})
= Hom_{KM^{x}}(KX_{M}^{x}, T_{M^{x}}^{G′}KX_{G′,y})
≤ Hom_{KM^{x}}(R_{L^{x}}^{M^{x}}KX_{L}^{x}, T_{M^{x}}^{G′}KX_{G′,y})
= Hom_{KL^{x}}(KX_{L}^{x}, T_{L^{x}}^{G′}KX_{G′,y}).

(2.19)

Now π_{(L^{x},χ_{L}^{x})}(X_{G′,y}) ≠ (0) if and only if y = x, since X_{G′,y} = π_{(L^{x},χ_{L}^{x})}(T_{G′}^{G}X_{G}). Thus the first Hom-set in Formula 2.19 is zero for y ≠ x, and hence nonzero for x = y by Formula 2.18. We conclude that

Hom_{kM^{x}}(Ȳ_{M}^{x}, T_{M^{x}}^{G′} X̄_{G′,x}) = k ⊗_{O} Hom_{OG′}(R_{M^{x}}^{G′}Y_{M}^{x}, X_{G′,x}) ≠ (0).

and therefore D_{G′,M^{x}} is a composition factor of X̄_{G′,x} by the first part of the proof applied to V = X̄_{G′,x}. □

The previous theorem yields a unique composition factor D_{G′,M^{x}} of X̄_{G′,x} from the HC-series S(G′/M^{x}, D_{M^{x}}^{x}), for every x ∈ N_{G′,L}^{G} such that [x]_{G′,L} ∩ N_{M⊆G′} is not empty.

Let x, y ∈ N_{M⊆G′}. Then S(G′/M^{x}, D_{M^{x}}^{x}) = S(G′/M^{y}, D_{M^{y}}^{y}) if and only if there is a g ∈ G′ such that M^{xg} = M^{y} and D_{M^{xg}}^{xg} ≅ D_{M^{y}}^{y}. This means that x and y are in the same N (D_{M}) × G′-orbit of N_{M⊆G′}, where N (D_{M}) = stab_{N_{G}(M)}(D_{M}). Note that N (D_{M}) acts indeed on N_{M⊆G′} by left translation since N_{G}(M) does. On the other hand, Theorem 2.16 tells us that every direct summand of T_{G′}^{G}X̄_{G} contains at most one composition factor from one HC-series S(G′/M^{x}, D_{M}^{x}), and all such series occur, and we have shown:

Lemma 2.20 Keep the notation of Theorem 2.16. Let x ∈ N_{G′,L}^{G}. Then [x]_{G′,X_{L}} ∩ N_{M⊆G′} is either empty or a union of N (D_{M}) × G′-orbits in N_{M⊆G′}.

For general linear groups we have N (D_{M}) ≤ N (X_{L}) in the relevant cases, and the result is trivially true. We do not know if this holds in general. Theorem 2.16 implies now immediately:

Corollary 2.21 Suppose [x]_{G′,X_{L}} ∩ N_{M⊆G′} is empty. Then no composition factor of X_{G′,x} lies in an HC-series S(G′/M^{y}, D_{M^{y}}) for some y ∈ N_{M⊆G′}. If [x]_{G′,X_{L}} ∩ N_{M⊆G′} is the disjoint union of the N (D_{M}) × (G′ ∩ N)-orbits of y_{1}, …, y_{m} ∈ N_{M⊆G′},

[x]_{G′,X_{L}} ∩ N_{M⊆G′} = ⋃_{i=1}^{m}[y_{i}]_{G′,D_{M}} ,

then for every 1 ≤ i ≤ m there is precisely one composition factor D_{G′,M^{y_{i}}} of X_{G′,x} in the HC-series S(G′/M^{y_{i}}, D_{M}^{y_{i}}).

In the special case G = G′ we have obviously

N_{G,M}^{G}(D_{M}) = N_{G,L}^{G} = G, (2.22)

hence we have precisely one orbit

[1]_{G,X_{L}} = [1]_{G,D_{M}} = G. (2.23)

This implies:

Corollary 2.24 Let M ∈ L^{c}(X_{G}, Y_{L}). Then X_{G} has a unique composition factor D_{G,M} in S(G/M, D_{M}).

The next theorem provides information how the composition factor D_{G,M} behaves under HC-restriction:

Theorem 2.25 Let M ∈ L^{c}(X_{G}, Y_{L}) and M ≤ G′ ∈ L_{G,L}. Then

T_{G′}^{G}D_{G,M} = ⨁_{x∈N_{G′,M}^{G} (D_{M})} D_{G′,M^{x}}.

In particular, T_{G′}^{G}D_{G,M} is semisimple. Moreover, if C is any composition factor of X̄_{G} not isomorphic to some D_{G,M} then T_{G′}^{G}C has no composition factor equal to some D_{G′,M^{x}} for x ∈ N_{G′,M}^{G}(D_{M}).

Proof By definition we have D_{G,M} ∈ S(G/M, D_{M})_{k}. Hence, by Lemma 1.15, we find irreducible kG′-modules Z, Z′ ∈ S(G′/M, D_{M})_{k} such that Z is in the socle and Z′ is in the head of T_{G′}^{G}D_{G,M}.

By Theorem 2.16, T_{G′}^{G}X̄_{G} has a unique composition factor in S(G′/M, D_{M})_{k}, namely D_{G′,M}. Now D_{G,M} is a composition factor of X̄_{G} hence T_{G′}^{G}D_{G,M} is a subfactor of T_{G′}^{G}X̄_{G}. Thus every composition factor of T_{G′}^{G}D_{G,M} is a composition factor of T_{G′}^{G}X̄_{G}. We conclude that T_{G′}^{G}D_{G,M} has precisely one composition factor in S(G′/M, D_{M})_{k}, namely D_{G′,M} and we have shown Z = Z′ = D_{G′,M}. Being in the head and in the socle of T_{G′}^{G}D_{G,M}, the kG′-module D_{G′,M} splits off.

Applying Corollary 2.21 we get similarly that D_{G′,M^{x}} ∈ S(G′/M^{x}, D_{M}^{x}) splits of T_{G′}^{G}D_{G,M} for every x ∈ N_{G′,M}^{G}(D_{M}). Thus

T_{G′}^{G}D_{G,M} = ⨁_{x∈N_{G′,L}^{G}(D_{M})} D_{G′,M^{x}} ⊕ Y,

where Y is some kG′-module, whose composition factors do not belong to any of the series S(G′/M^{x}, D_{M}^{x}), x ∈ N_{M⊆G′}. But since D_{G,M} ∈ S(G/M, D_{M}), every irreducible summand of the socle and the head of T_{G′}^{G}D_{G,M}, hence of the socle and the head of Y, belongs to one of these HC-series. Therefore Y = (0).

We see in particular that the composition factors of T_{G′}^{G}X̄_{G} from these series are already composition factors of the subfactor T_{G′}^{G}D_{G,M} of T_{G′}^{G}X̄_{G} and the last assertion also follows.

□ Remark 2.26 Corollary 2.21 provides the recipe how the direct summands D_{G′,M^{y}} of T_{G′}^{G}D_{G,M} divide up into composition factors of the direct summands X_{G′,x} of T_{G′}^{G}X_{G}, where y runs through N_{G′,M}^{G}(D_{M}) and x through N_{M,L}^{G}.

Theorem 2.27 Let M, G′ ∈ L_{G,L} such that M ≤ G′. Then every homomorphism from R_{M}^{G′}Y_{M} to X_{G′} factors through R_{M}^{G′}X_{M} and

Hom_{OG′}(R_{M}^{G′}X_{M}, X_{G′}) ≅ O ≅ Hom_{OG′}(X_{G′}, R_{M}^{G′}X_{M}).

If ϕ = ϕ_{G′,M} in the first and ρ = ρ_{G′,M} in the second Hom-space are chosen such that every other homomorphism is a scalar multiple, then the following holds, writing 1_{k} ⊗_{O} ϕ = ϕ̄ and 1_{k} ⊗_{O} ρ = ρ̄:

(i) The kernel of ρ̄ has no composition factor isomorphic to D_{G′,M′} for M′ ∈ L^{c}(X_{G′}, Y_{L}). (ii) The image U of ϕ̄ has simple head and contains D_{G′,M′} as composition factor for

M′ ∈ L^{c}(X_{G′}, Y_{L}) if and only if M′ is conjugate in G′ to a subgroup of M. (iii) If in addition M ∈ L^{c}(X_{G′}, Y_{L}) then D_{G′,M} is the head U/Jac(U) of U = im (ϕ̄).

Moreover, if V ≤ X̄_{G′} and D_{G′,M} is a composition factor of V and M′ ≤ M, M′ ∈ L^{c}(X_{G′}, Y_{L}), then D_{G′,M′} is a composition factor of V.

Proof In view of Result 2.12 we may assume that G = G′. Now by Lemma 1.18 every homomorphism from R_{M}^{G}Y_{M} to X_{G} factors through π_{(L,χ_{L})}(R_{M}^{G}Y_{M}). But if M′ denotes the set of G-conjugates of L and C′ the set of G-conjugate characters of χ_{L}, then (M′, C′) is closed under conjugation in G. On the other hand on OG-lattices π_{(M′,C′)} = π_{(L,χ_{L})} and hence by Corollary 1.24 and Theorem 2.13

π_{(L,χ_{L})}(R_{M}^{G}Y_{M}) = π_{(M′,C′)}(R_{M}^{G}Y_{M}) = R_{M}^{G}π_{(M′,C′)}^{M}(Y_{M}) = R_{M}^{G}(π_{(L,χ_{L})}(Y_{M})) = R_{M}^{G}(X_{M}).

By Frobenius reciprocity and Theorem 2.8

(2.28) Hom_{OG}(R_{M}^{G}X_{M}, X_{G}) ≅ ⨁_{x∈N_{M,L}^{G}} Hom_{OM}(X_{M}, X_{M,x}) ≅ Hom_{OM}(X_{M}, X_{M}) ≅ O,

since KX_{M,x} ≅ K̸X_{M} for 1 ≠ x ∈ N_{M,L}^{G}. Similarly

Hom_{OG}(X_{G}, R_{M}^{G}X_{M}) ≅ O. (2.29)

We take now ϕ and ρ to correspond to 1 ∈ O in Equation 2.28 and 2.29 respectively. We first prove part (ii). By Frobenius reciprocity we have

k ≅ Hom_{kM}(Ȳ_{M}, T_{M}^{G}U) ≅ Hom_{kG}(R_{M}^{G} Ȳ_{M}, U)
≅ Hom_{kG}(R_{M}^{G} Ȳ_{M}, X̄_{G})
≅ Hom_{kG}(R_{M}^{G} X̄_{M}, X̄_{G})
≅ Hom_{kM}(Ȳ_{M}, X̄_{M})

(2.30)

which shows in particular that D_{M} is a composition factor of T_{M}^{G}U and hence X̄_{M} is a submodule of T_{M}^{G}U. Thus D_{M′} is a composition factor of T_{M′}^{G}U ≥ T_{M′}^{M}X̄_{M} for every M′ ∈ L^{c}(X_{G}, Y_{L}) with M′ ≤ M and hence D_{G,M′} is a composition factor of U by Theorem 2.16 (applied in the special case G = G′). Since all these composition factors have multiplicity one as composition factors of X̄_{G}, the same holds for the submodule U of X̄_{G}. Observe that the projective kG-module R_{M}^{G} Ȳ_{M} contains the projective cover Z of U as direct summand, and this is the projective cover of hd(U) as well. Since Hom_{kG}(R_{M}^{G} Ȳ_{M}, U) is one dimensional, the same is true for Hom_{kG}(Z, U) and hd(U) must be simple.

On the other hand let M′ ∈ L^{c}(X_{G}, Y_{L}) and assume that D_{G,M′} is a composition factor of U. Then it is a composition factor of R_{M}^{G} X̄_{M} since ϕ̄ maps this module onto U. Now D_{G,M′} is in the HC-series S(G/M′, D_{M′}) and therefore a composition factor of hd(R_{M′}^{G}D_{M′}). Thus R_{M′}^{G}Ȳ_{M′} contains the projective cover of D_{G,M′} and there is a nonzero homomorphism from R_{M′}^{G}Ȳ_{M′} to R_{M}^{G} X̄_{M}. We have:

(0) ≠ Hom_{kG}(R_{M′}^{G}Ȳ_{M′}, R_{M}^{G} X̄_{M})

≅ Hom_{kM′}(Ȳ_{M′}, T_{M′}^{G}R_{M}^{G} X̄_{M})

≅ ⨁_{z∈D_{M,M}} Hom_{kM′}(Ȳ_{M′}, R_{M′∩M^{z}}^{M′}T_{M′∩M^{z}}^{M^{z}}(X̄_{M^{z}}^{z})).

(2.31)

But the head of Ȳ_{M′} is D_{M′} and hence cuspidal by assumption. Thus such a homomorphism can exist only if M′ ≤ M^{z} for some z ∈ G or equivalently z^{−1} ∈ N_{M′⊆M}. Applying Frobenius reciprocity to 2.31 again and conjugating we conclude that

Hom_{kM^{′z^{−1}}} (Ȳ_{M^{′z^{−1}}} , T_{M^{′z^{−1}}}^{M} X̄_{M}) ≠ (0).

Setting M^{′z^{−1}} = M̃ we have by Corollary 2.8

Hom_{kM̃}(ȲM̃, TM ; M̃ X̄_{M}) = ⨁_{x∈N_{M̃,L}^{M}} Hom_{kM̃}(ȲM̃, X̄_{M̃,x})

= Hom_{kM̃}(ȲM̃, X̄M̃) = k.

Thus we have D_{G,M̃} ≅ D_{G,M′} since M̃ and M′ are conjugate in G and (ii) is shown.

To prove (iii) we observe that D_{G,M} is a composition factor of U by part (ii). Suppose D_{G,M} is not the head of U then there is a proper submodule V of U such that D_{G,M} is a composition factor of V. As in 2.30 we show that Hom_{kM}(Ȳ_{M}, T_{M}^{G}V) ≠ (0), thus im ϕ̄ ≤ V, a contradiction and therefore (iii) holds.

We now prove (i). First observe that

R_{M}^{G} X̄_{M} ≅ ⨁_{x} X̄_{M} ⊗ x,

where x runs through a set of coset repesentatives of a parabolic subgroup of G having Levi complement M. Let τ : R_{M}^{G} X̄_{M} → X̄_{M} be the projection onto the summand X̄_{M} = X̄_{M} ⊗1. By general theory the map ρ̂ defined to be the restriction of τ ∘ ρ̄ to T_{M}^{G} X̄ is KM-linear and is the map which corresponds to ρ under the isomorphism

(2.32) Hom_{kG}(X̄_{G}, R_{M}^{G} X̄_{M}) ≅ Hom_{kM}(T_{M}^{G} X̄_{G}, X̄_{M})

In particular ρ̂ is non-zero, indeed it is an epimorphism. By Theorem 2.8 and transitivity of HC-induction we have

T_{L}^{M}T_{M}^{G} X̄_{G} = T_{L}^{G}X̄_{G} = ⨁_{x∈N_{L,L}^{G}} X̄_{L,x},

in particular X̄_{L} occurs in T_{L}^{G}X̄_{G} and hence in T_{L}^{M}(T_{M}^{G} X̄_{G}) exactly once as a direct summand. Of course the same holds for X̄_{M}, that is X̄_{L} occurs in T_{L}^{M} X̄_{M} exactly once. Suppose that the unique subspace X̄_{L} of X̄_{G} is in the kernel of ρ̄ then it is also in the kernel of τ ∘ ρ̄ = ρ̂. But we may take ρ̂ to be the projection of

T_{M}^{G} X̄_{G} = ⨁_{x∈N_{M,L}^{G}} X̄_{M,x}

onto X̄_{M,1} ≅ X̄_{M} and therefore the restriction ρ̃ of ρ̂ to X̄_{M,1} is the identity map. Thus

T_{L}^{M} ρ̃: T_{L}^{M} X̄_{M,1} → X̄_{M}

maps the unique subspace X̄_{L} of T_{M}^{G} X̄_{G} into X̄_{L} ≤ T_{L}^{M} X̄_{M} ≤ X̄_{M}. Thus X̄_{L} can not be contained in the kernel of ρ̂ = τ ∘ ρ̄ and hence not in the kernel of ρ̄. Suppose D_{G,M} is a composition factor of ker(ρ̄) ≤ X_{G}. Then by (iii) D_{G,M′} is composition factor of ker(ρ̄) for every M′ ∈ L(X_{G}, Y_{L}) with M′ ≤ M. But L ∈ L^{c}(X_{G}, Y_{L}) by 2.15, hence D_{G,L} is a composition factor of ker(ρ̄). Thus we have a nonzero homomorphism from the projective cover of D_{G,L}, which is a direct summand of R_{L}^{G}Ȳ_{L} into ker(ρ̄), hence by Frobenius reciprocity from Ȳ_{L} into T_{L}^{G} ker(ρ̄) ≤ T_{L}^{G}X̄_{G}. But Hom_{kL}(Ȳ_{L}, T_{L}^{G}X̄_{G}) is one dimensional, and its image is the direct summand X̄_{L} of T_{L}^{G}X̄_{G}. We conclude that X̄_{L} is contained in T_{L}^{G} ker(ρ̄) and hence in ker(ρ̄), a contradiction. This proves (i) and the theorem is shown.

□

Corollary 2.33 Let G′ ∈ L_{G,L}. Then the head D_{G′} of X̄_{G′} is isomorphic to D_{G′,M} for some maximal M ∈ L^{c}(X_{G}, Y_{L}) with M ≤ G′.

Proof In view of Result 2.12 we may again assume that G = G′. If D_{G} is cuspidal we are done, since then G ∈ L^{c}(X_{G}, Y_{L}) is maximal, and D_{G} is the head of X̄_{G}.

Assume G ∉ L^{c}(X_{G}, Y_{L}) and let M ∈ L_{G} be the semisimple HC-vertex of D_{G}. We show that M ∈ L_{G,L}. By general assumption 2.3 χ_{G} occurs as constituent of R_{L}^{G}χ_{L}. Hence T_{M}^{G}χ_{G′} is a summand of

T_{M}^{G}R_{L}^{G}χ_{L} = ∑_{x∈D_{L,M}} R_{L^{x}∩M}^{M}T_{L^{x}∩M}^{L^{x}}χ_{L^{x}}^{x}.

But T_{L^{x}∩M}^{L^{x}}χ_{L^{x}}^{x} = (0) unless L^{x} ≤ M. This shows that for every composition factor S of X̄_{G} we have T_{M}^{G}S = (0) unless L^{x} ≤ M for some x ∈ G. As X_{G} and X_{G}^{x} are isomorphic for such an x, we may assume that the semisimple HC-vertex of D_{G} contains L.

The short exact sequence

0 → Jac(X̄_{G}) → X̄_{G} → D_{G} → 0

yields a short exact sequence

0 → T_{M}^{G}Jac(X̄_{G}) → ⨁_{y∈N_{M,L}^{G}} X̄_{M,y} → T_{M}^{G}D_{G} → 0

using Theorem 2.8. Consequently an irreducible direct summand S of the head of T_{M}^{G}D_{G} has to be the head of one of the summands of the middle term, and thus by induction it has to be of the form D_{M,y} for some y ∈ N_{M,L}^{G}. By Lemma 1.15, D_{M,y} is cuspidal and D_{G} is in the HC-series S(G/M, D_{M,y})_{k}. But D_{M,y} = (D_{M^{y^{−1}}} )^{y} by 2.10. Thus replacing M by M^{y−1} if necessary we may assume y = 1. By Theorem 2.16 the unique composition factor of X̄_{G} in the series S(G/M, D_{M}) is D_{G,M}. Theorem 2.27 part (iii) tells us now that M has to be maximal with M ∈ L^{c}(X_{G}, Y_{L}). □

Recall our definitions in 1.2. We apply Theorem 2.27 in the special situation M = L:

Corollary 2.34 Let G′ ∈ L_{G,L}. Then

ρ_{G′} = ρ_{G′,L} : X_{G′} → R_{L}^{G′}X_{L}

is injective and its image X_{ρ} = X_{ρ,G′} = ρ_{G′}(X_{G′}) is a sublattice of X′_{G′} ≤ R_{L}^{G′}X_{L} satisfying KX_{ρ} = KX′_{G′} and X′_{G′} = \sqrt{ρ(X_{G′})}. Morover every homomorphism from Y_{G′} into R_{L}^{G′}X_{L} factors through a multiple of ρ and is the combined image

X_{ρ} = τ_{Y_{G′}} (R_{L}^{G′}X_{L}) = τ_{Y_{G′}} (X′_{G′})

(which means X_{ρ} is the combined image of Y_{G′} in R_{L}^{G′}X_{L} and X_{G′} respectively). Finally the composition factors of ker ρ̄ and the cokernel V = cokerρ̄ = X̄′_{G′}/X̄_{ρ} coincide (multiplicities counted). In particular no composition factor of V has the form D_{G′,M′} for some M′ ∈ L^{c}(X_{G′}, Y_{L}).

Proof By Theorem 2.13 and Frobenius reciprocity

Hom_{OG′}(Y_{G′}, R_{L}^{G′}X_{L}) ≅ ⨁_{x∈N_{L,L}^{G}} Hom_{OL}(Y_{L,x}, X_{L}) ⊕ Hom_{OL}(Z, X_{L})
≅ Hom_{OL}(Y_{L}, X_{L})
≅ O,

since X_{L,x} ≅ X̸_{L} for 1 ≠ x ∈ N_{L,L}^{G} by Theorem 2.8, π_{(L,χ_{L})}(Y_{L}) = X_{L} and since X_{L} is cuspidal, by our basic assumption 2.1. Note that π_{(L,χ_{L})}(Z) = (0) by Theorem 2.13. Thus there is a unique homomorphism ρ̂ : Y_{G′} → R_{L}^{G′}X_{L} corresponding to 1 ∈ O. Again by Lemma 1.18

Hom_{OG′}(Y_{G′}, R_{L}^{G′}X_{L}) ≅ Hom_{OG′}(X_{G′}, R_{L}^{G′}X_{L}),

thus every homomorphism from Y_{G′} to R_{L}^{G′}X_{L} factors through X_{G′}, and we can choose ρ : X_{G′} → R_{L}^{G′}X_{L} such that ρ̂ is the composite of the natural projection of Y_{G′} onto X_{G′} = π^{G′}(Y_{G′}) and ρ. Thus im ρ is the combined image of Y_{G′} in R_{L}^{G′}X_{L} and in every submodule of R_{L}^{G′}X_{L} containing im ρ. Now 1_{K} ⊗_{O} ρ : KX_{G′} → R_{L}^{G′}KX_{L} has to be injective since KX_{G′} is irreducible by Theorem 2.8, hence ρ is injective, too. Thus 1_{K} ⊗_{O} ρ maps KX_{G′} isomorphically onto KX′_{G′} ≤ KR_{L}^{G′}X_{L}. In particular, since

X′_{G′} = \sqrt{X′_{G′}} = KX′_{G′} ∩ R_{L}^{G′}X_{L} = Kρ(X_{G′}) ∩ R_{L}^{G′}X_{L} = \sqrt{ρ(X_{G′})}

we have

ρ(X_{G′}) ≤ X′_{G′}.

Since KX_{G′} = KX′_{G′} the composition factors of X̄_{G′} and X̄′_{G′} coincide. Now by the first isomorphism theorem

X̄_{G′}/ ker ρ̄ ≅ im ρ̄ ≤ X̄′_{G′}.

Consequently ker ρ̄ and cokerρ̄ have the same composition factors. □

Lemma 2.35 Let G′ ∈ L_{G,L} and let S be a composition factor of X̄_{G′} with S ≠ D_{G′,M′} for all M′ ∈ L^{c}(X_{G′}, Y_{L}). Let M ∈ L_{G,L}. Then

Hom_{kG̃}(RG̃ ; M Ȳ_{M}, RG̃ ; G′S) = (0).

Thus no composition factor of RG̃ ; G′S is contained in the HC-series S(G̃/M, D_{M})_{k}.

Proof We have

Hom_{kG̃}(RG̃ ; M Ȳ_{M}, RG̃ ; G′S) = Hom_{kM}(Ȳ_{M}, T_{M}^{G}R_{G′}^{G}S)

= ⨁_{z∈D̃_{MG′}∩Ñ_{L⊆M}} Hom_{kM}(Ȳ_{M}, R_{M∩G′^{z}}^{M}T_{M∩G′^{z}}^{G′^{z}}S^{z})

= ⨁_{z∈D̃_{MG′}∩Ñ_{L⊆M}} Hom_{k(M∩G′^{z})}(T_{M∩G^{′z}}^{M} Ȳ_{M}, T_{M∩G^{′z}}^{G^{′z}}S^{z}),

(2.36)

where D̃_{MG′} denotes a suitable system of double coset representatives of parabolic subgroups of G̃ containing G′ and M as Levi complements. Now

(2.37) Hom_{k(M∩G^{′z})}(T_{M∩G^{′z}}^{M} Ȳ_{M}, T_{M∩G^{′z}}^{G^{′z}}S^{z})

= ⨁_{x∈N_{M∩G^{′z},L^{z}}^{M}} Hom_{k(M∩G^{′z})}(Ȳ_{M∩G^{′z},x}, T_{M∩G^{′z}}^{G^{′z}}S^{z}) ⊕ Hom_{k(M∩G^{′z})}(Z̄_{z}, T_{M∩G^{′z}}^{G^{′z}}S^{z}),

where Z_{z} is a projective OM ∩ G^{′z}-module with π^{M∩G^{′z}}(Z_{z}) = (0) by Theorem 2.13.

Now S^{z} is a subfactor of X̄_{G′}^{z}. By Corollary 2.11, applied to the projective restriction system PR^{z}(X_{G}, Y_{L}) = PR(X_{G}^{z}, Y_{L}^{z}), we have

T_{M∩G^{′z}}^{G^{′z}}X_{G′}^{z} = ⨁_{y∈N_{M∩G^{′z},L^{z}}^{G^{′z}}} X_{M∩G^{′z},y}^{(z)},

with X_{M∩G^{′z},y}^{(z)} = π_{(L^{zy},χ^{zy})}(T_{M∩G^{′z}}^{G^{′z}}X_{G′}^{z}) (see Notation 2.6). Similarly we denote the pro-

jective cover of X_{M∩G^{′z},y}^{(z)} by Y_{M∩G^{′z},y}^{(z)} and we write D_{M∩G^{′z},M^{′x}}^{(z)} for the composition factors of T_{M∩G^{′z}}^{G^{′z}}X_{G′}^{z} lying in series S((M ∩ G^{′z})/M^{′x}, D_{M^{′x}}) for M^{′x} ∈ L^{c}(X_{G′}^{z}, Y_{L}^{z}).

From Theorem 2.13 and Corollary 1.24 follows

T_{M∩G^{′z}}^{G^{′z}} X̄_{G′}^{z} = T_{M∩G^{′z}}^{G^{′z}}π^{M∩G^{′z}} Ȳ_{G′}^{z} = π^{G^{′z}}T_{M∩G^{′z}}^{G^{′z}} Ȳ_{G′}^{z},

and hence by Lemma 1.19

Hom_{k(M∩G^{′z})}(Z̄_{z}, T_{M∩G^{′z}}^{G^{′z}}S^{z}) ≤ Hom_{k(M∩G^{′z})}(Z̄_{z}, T_{M∩G^{′z}}^{G^{′z}} X̄_{G′}^{z})
≅ Hom_{k(M∩G^{′z})}(Z̄_{z}, π^{G^{′z}}T_{M∩G^{′z}}^{G^{′z}} Ȳ_{G′}^{z})
≅ Hom_{k(M∩G^{′z})}(π^{G^{′z}} Z̄_{z}, T_{M∩G^{′z}}^{G^{′z}} X̄_{G′}^{z})
≅ Hom_{k(M∩G^{′z})}((0), T_{M∩G^{′z}}^{G^{′z}} X̄_{G′}^{z})
= (0).

Now suppose that

Hom_{k(M∩G^{′z})}(Ȳ_{M∩G^{′z},x}, T_{M∩G^{′z}}^{G^{′z}}S^{z}) ≤ ⨁_{y∈N_{M∩G^{′z},L^{z}}^{G^{′z}}} Hom_{k(M∩G^{′z})}(Ȳ_{M∩G^{′z},x}, X̄_{M∩G^{′z},y}^{(z)}) ≠ (0),

say Hom_{k(M∩G^{′z})}(Ȳ_{M∩G^{′z},x}, X̄_{M∩G^{′z},y_{0}}^{(z)}) ≠ (0). Then using again Theorem 2.13 we see that

Ȳ_{M∩G^{′z},x} is the projective cover of X̄_{M∩G^{′z},y_{0}}^{(z)}. That is, we have Ȳ_{M∩G^{′z},x} ≅ Ȳ_{M∩G^{′z},y_{0}}^{(z)}, hence

its head is one of the modules D_{M∩G^{′z},M^{′x}}^{(z)} for some M^{′x} ∈ L_{G^{′z},L^{z}}^{c} by Corollary 2.33. Now Theorem 2.25 implies Hom_{k(M∩G^{′z})}(Ȳ_{M∩G^{′z},x}, T_{M∩G^{′z}}^{G^{′z}}S^{z}) = (0) and the lemma follows.

□

In the next result we extend Corollary 2.34 to HC-induced lattices. We define

Y = YG̃ = ⨁_{G′∈L_{G,L}} RG̃ ; G′Y_{G′}. (2.38)

Corollary 2.39 Let G′ ∈ L_{G,L} and let X_{ρ} ≤ X′_{G′} ≤ R_{L}^{G′} be defined as in Corollary 2.34. Then

RG̃ ; G′X_{ρ} = τ_{Y}(RG̃ ; G′X′_{G′}).

Proof By 2.34 X_{ρ} is image of Y_{G′} hence RG̃ ; G′X_{ρ} is epimorphic image of RG̃ ; G′Y_{G′}, which is a direct summand of Y. We conclude that

RG̃ ; G′X_{ρ} ≤ τ_{Y}(RG̃ ; G′X′_{G′}).

Let S be a composition factor of the cokernel of ρ̄. Then, by Lemma 2.34 S is not of the form D_{G′,M′} for some M′ ∈ L^{c}(X_{G′}, Y_{L}). Lemma 2.35 tells us

Hom_{kG̃}(RG̃ ; M Ȳ_{M}, RG̃ ; G′S) = (0).

Thus the image of every homorphism from Ȳ to RG̃ ; G′ X̄_{G′} has to be contained in im ρ̄. We lift this to O to get the desired result. □

Lemma 2.40 Suppose that every composition factor of R_{L}^{G}X̄_{L} is isomorphic to some D_{G,M} where M ∈ L^{c}(X_{G}, Y_{L}). Then we have

(i) The kL-module X̄_{L} is irreducible, that is X̄_{L} = D_{L}.

(ii) Let G′ ∈ L_{G,L}. Then every composition factor of R_{L}^{G′} X̄_{L} is of the form D_{G′,M} for some L^{c}(X_{G′}, Y_{L}) = {M ∈ L^{c}(X_{G}, Y_{L}) | M ≤ G′}.

(iii) There is an isomorphism between the poset of local submodules of X̄_{G′} and the poset of subgroups L^{c}(X_{G′}, Y_{L}) of G′.

(iv) RG̃ ; G′X_{G′} is a pure sublattice of RG̃ ; Lχ_{L} isomorphic to RG̃ ; G′X′_{G′} for all G′ ∈ L_{G,L}.

Proof Lemma 2.35 implies immediately part (i) and Theorem 2.27 part (iii) whence we have established part (ii) observing that all composition factors of X̄ are composition factors of R_{L}^{G}X̄_{L}, since KX_{G} is a constituent of R_{L}^{G′}X_{L} by Definition 2.3.

To show part (ii) let D be a composition factor of R_{L}^{G′} X̄_{L} and let D ∈ S(G′/M′, E)_{k}, where E is an irreducible cuspidal kM′-module and M′ ∈ L_{G′}. Since D is a composition factor of R_{L}^{G′} X̄_{L} and X ∈ 𝔏𝔞𝔱_{OL} is cuspidal, we may choose M′ such that L ≤ M′, that is M′ ∈ L_{G′,L}. Now R_{G′}^{G}(D) is a subfactor of R_{L}^{G}(X̄_{L}) and, by Lemma 1.14, every composition factor of the head of R_{G′}^{G}(D) is in S(G/M′, E)_{k}. Our general assumptions imply now that S(G/M′, E)_{k} = S(G/M, D_{M})_{k} for some M ∈ L^{c}(X_{G}, Y_{L}). Thus there exists x ∈ N such that M′ = M^{x} and D_{M}^{x} = E. Since M′ ∈ L_{G,L} we have M′ ∈ L^{c}(X_{G}, Y_{L}) and E = D_{M′}. But then D ∈ S(G′/M′, D_{M′}) as desired.

It remains to show (iv): In Theorem 2.27 we constructed a homomorphism 0 ≠ ρ_{G′,L} : X_{G′} → R_{L}^{G′}X_{L}, which is injective since KG′ is an irreducible KG′-module. Thus X_{G′} ⊂ R_{L}^{G′}X_{L}. But part ii) above and part i) of Theorem 2.27 imply that ρ̄_{G′,L} := 1_{k} ⊗_{O} ρ_{G′,L} : X̄_{G′} → R_{L}^{G′} X̄_{L} is injective, too, hence X_{G′} is a pure sublattice of R_{L}^{G′}X_{L}. By construction X′_{G′} is a pure sublattice of R_{L}^{G′}X_{L} affording the same character χ_{G′}. Since the multiplicity of χ_{G′} in R_{L}^{G′}χ_{L} is one, X′_{G′} has to be the image of ρ_{G′,L}. Since HC-induction preserves purity and isomorphisms part (iv) follows □

We point out that the assumption in Corollary 2.40 is frequently satisfied, as for example in the application for GL_{n}(q) and we shall see in the following section how the conclusion of Corollary 2.40 can be used to get information on decomposition numbers of G̃.

3 Hecke- and Schur algebras

We continue with the set up of the previous section. So let PR = PR(X_{G}, Y_{L}) be a projective restriction system, consisting of the data {X_{M}, Y_{M}, X′_{M}|M ∈ L_{G,L}}. We define Q = Q_{L} = RG̃ ; LY_{L} and βG̃ = β = RG̃ ; L(ψ), where ψ : Y_{L} → π^{L}(Y_{L}) = X_{L} is the natural projection.

Theorem 3.1 The epimorphism β : Q → RG̃ ; LX_{L} satisfies Hypothesis 1.1. Thus in particular the endomorphism ring of RG̃ ; LX̄_{L} is liftable and the decomposition matrix of the endomorphism ring End_{OG̃}(RG̃ ; LX_{L}) is part of the ℓ-modular decomposition matrix of G̃.

Proof By Corollary 1.24 and by 2.1

πG̃(Q) = πG̃(RG̃ ; LY_{L}) = RG̃ ; Lπ^{L}(Y_{L}) = RG̃ ; LX_{L},

hence the theorem follows from Result 1.25. □

Lemma 1.8 implies now immediately:

Corollary 3.2 The OG̃-lattice R_{L}^{G′}X_{L} is Q-torsionless.

Notation 3.3 We call HG̃ ; R = HG̃ ; R(PR) = End_{RG̃}(R ⊗_{O} RG̃ ; LX_{L}) the Hecke algebra over R associated with PR = PR(X_{G}, Y_{L}) on level G̃. If no ambiguities arise we omit super- and subscripts, for instance H_{R} is until further notice HG̃ ; R. Observe that Theorem 3.1 implies that H_{R} = RH_{O} = R⊗_{O} H_{O}. According to the general theory of quotients of Hom-functors as outlined in the first section, we have associated with H_{R} quotients of Hom-functors H_{R} given by

H_{R} : 𝔐𝔬𝔡_{RG̃} → 𝔐𝔬𝔡_{H_{R}} : V → Hom_{RG̃}(Q, V)/Hom_{RG̃}(Q, V)J_{β},

where J_{β} is the ideal of End_{RG̃}(Q) consisting of those endomorphisms of Q, whose image is contained in the kernel of β.

Let G′ ∈ L_{G,L}. We have seen in Lemma 2.34 that the irreducible character χ_{G′} of X_{G′} occurs exactly with multiplicity one in R_{L}^{G′}χ_{L}. By Fittings Lemma we find a one dimensional H_{K}^{G′}-module generated by σ = σ_{G′} such that KσR_{L}^{G′}X_{L} = KX_{G′}. We choose σ ∈ H_{O}^{G′} such that σH_{O}^{G′} is pure in H_{O}^{G′} and is a generator of this one dimensional O-space. Then σR_{L}^{G′}X_{L} is an O-lattice in KX_{G′}, but in general σR_{L}^{G′}X_{L} ≅ X̸_{G′} and σR_{L}^{G′}X_{L} ≅ X̸′_{G′} ≤ R_{L}^{G′}X_{L}. However X′_{G′} = ι_{(G′,χ_{G′})}(R_{L}^{G′}X_{L}) is pure in R_{L}^{G′}X_{L} by construction and affords χ_{G′}. This implies:

Lemma 3.4 Let U ≤ R_{L}^{G′}X_{L} with KU = KX′_{G′}. Then

U ≤ X′_{G′} = \sqrt{U}.

In particular

σR_{L}^{G′}X_{L} ≤ X′_{G′} and \sqrt{σR_{L}^{G′}X_{L}} = X′_{G′}.

Proof We have

\sqrt{U} = KU ∩ R_{L}^{G}X_{L} = KX′_{G} ∩ R_{L}^{G}X_{L} = X′_{G}.
□

The next result relates H_{O}^{G′} and HG̃ ; O:

Theorem 3.5 Let G′ ∈ L_{G̃,L}. Then the following holds for R ∈ {K, O, k}:

(i) H_{R}^{G′} is a subalgebra of HG̃ ; R and HG̃ ; R is free as left H_{R}^{G′}-module. (ii) If I is a right ideal of H_{O}^{G′} and Z = IR_{L}^{G′}X_{L} then

RG̃ ; G′Z = IHG̃ ; ORG̃ ; LX_{L} = IRG̃ ; LX_{L},

where IHG̃ ; O ≅ I ⊗_{H_{O}^{G′}} HG̃ ; O. In particular, if I is pure in H_{O}^{G′} then IHG̃ ; O is pure in HG̃ ; O.

In particular, the lattice σ_{G′}HG̃ ; O is a pure right ideal of HG̃ ; O.

Proof By Frobenius reciprocity

HG̃ ; O = Hom_{OG̃}(RG̃ ; LX_{L}, RG̃ ; LX_{L}) = Hom_{OG′}(R_{L}^{G′}X_{L}, TG̃ ; G′RG̃ ; LX_{L}).

Note that maps in the right hand side Hom-set correspond to maps in the left Hom-set simply by extending the map in the natural way to RG̃ ; LX_{L} = R_{L}^{G′}X_{L}OG̃, where R_{L}^{G′}X_{L} is considered as module for a parabolic subgroup whose Levi complement is G′. We choose a set D̃_{L,G′} of double coset representatives of parabolic subgroups of G̃ having L and G′ as Levi complements. Now by Mackey decomposition (see 2.7)

Hom_{OG′}(R_{L}^{G′}X_{L}, TG̃ ; G′RG̃ ; LX_{L}) ≅ ⨁_{x∈D̃_{L,G′}} Hom_{OG′}(R_{L}^{G′}X_{L}, R_{G′∩L^{x}}^{G′}T_{G′∩L^{x}}^{L^{x}}X_{L}^{x})

≅ ⨁_{x∈D̃_{L,G′}∩N_{L⊆G′}} Hom_{OG′}(R_{L}^{G′}X_{L}, R_{L^{x}}^{G′}X_{L}^{x}),

(3.6)

since X_{L} is cuspidal. But for x = 1 we get H_{O}^{G′′} = Hom_{OG′}(R_{L}^{G′}X_{L}, R_{L}^{G′}X_{L}) and for 1 ≠ x ∈ D̃_{L,G′}, L^{x} ≤ G′, Hom_{OG′}(R_{L}^{G′}X_{L}, R_{L^{x}}^{G′}X_{L}^{x}) is either (0), if L^{x} ≠_{G′} L, or L^{xy} = L for a y ∈ G′ but X_{L}^{xy} ≅ X̸_{L}. Or it is as left H_{O}^{G′}-module isomorphic to the regular module. A basis of this is then given for instance by choosing one isomorphism from R_{L}^{G′}X_{L} to R_{L^{x}}^{G′}X_{L}^{x} induced by conjugation certain y ∈ G′. For R = k, K we argue analogously. Part ii) follows now easily. □

Notation 3.7 Recall the definition of ρ_{G′} : X_{G′} → R_{L}^{G′}X_{L} in Corollary 2.34. The following OG̃-lattices will be considered:

X′ =X^{′G̃}= ⨁_{G′∈L_{G,L}} RG̃ ; G′X′_{G′}
X =XG̃ = ⨁_{G′∈L_{G,L}} RG̃ ; G′X_{G′}
X_{h}=XG̃ ; h = ⨁_{G′∈L_{G,L}} σ_{G′}RG̃ ; G′X_{L}
Y =YG̃ = ⨁_{G′∈L_{G,L}} Q_{G′},

setting QG̃ ; G′ = Q_{G′} = RG̃ ; G′Y_{G′}. Define ρ : X → X′ to be the homomorphism obtained by summing up the maps RG̃ ; G′ρ_{G′}, for G′ ∈ L_{G,L}. As in Corollary 2.34 we denote the image of ρ = ρ_{G′} in X′_{G′} by X_{ρ} = X_{ρ_{G′}}. Note that kX_{ρ} = X̄_{ρ} is not the image of ρ̄ = 1_{k} ⊗_{O} ρ in general, but the latter is always an epimorhic image of the former.

Here is our main result:

Theorem 3.8 Let the OG̃-lattices X = XG̃ and Y = YG̃ be defined as in 3.7. Then

πG̃(Y) = X.

Let β : Y → X denote the corresponding epimorphism. Then β satisfies Hypothesis 1.1 and the requirements of Theorem 1.7. In particular, the decomposition matrix of End_{OG̃}(X) is part of the ℓ-modular decomposition matrix of G̃.

Proof Corollary 2.14 implies

πG̃(Q_{M}) = πG̃(RG̃ ; MY_{M}) = RG̃ ; MX_{M}

for every M ∈ L_{G,L}. Since πG̃ obviously preserves direct sums the theorem follows using Corollary 1.25. □

Lemma 3.4 in conjunction with Theorem 1.10 imply that End_{OG̃}(X_{h}) = End_{OG̃}(X′) (see 3.12 below). Now Lemma 2.40 implies immediately:

Theorem 3.9 Suppose that every composition factor of R_{L}^{G}X̄_{L} is isomorphic to some D_{G,M} where M ∈ L^{c}(X_{G}, Y_{L}). Then X ≅ X_{ρ} = X′.

So under the assumption of the theorem we can immediately compute the decoposition of irreducible lattices occurring in X in terms of the Hecke algebra H_{O}.

Obviously we would now like to know End_{OG̃}(X) in general. We shall use the following result to show, that we can relate this endomorphism ring to the endomorphism rings of the other lattices defined in 3.7.

Theorem 3.10 Keep the notation introduced above. Let G′ ∈ L_{G,L}.

(i) We have σ_{G′}RG̃ ; LX_{L} ≤ RG̃ ; G′X_{ρ} ≤ RG̃ ; G′X′_{G′}, and therefore

X_{h} ≤ X_{ρ} ≤ X′,

where X_{ρ} = XG̃ ; ρ = ρ(X).

(ii) σ_{G′}RG̃ ; LX_{L} = τ_{Q_{L}}(RG̃ ; G′X_{ρ}) = τ_{Q_{L}}(RG̃ ; G′X′_{G′}) and τ_{Q_{G′}} (RG̃ ; G′X_{G′}) = RG̃ ; G′X_{ρ}, hence

τ_{Q_{L}}(X′) = X_{h} = τ_{Q_{L}}(X_{ρ}) and τ_{Y}(X′) = X_{ρ}.

Proof Note that Kσ_{G′}R_{L}^{G′}X_{L} = KX_{ρ} = KX′_{G′}. Thus Lemma 3.4 implies

σ_{G′}RG̃ ; LX_{L} ≤ RG̃ ; G′X_{G′} and RG̃ ; G′X_{ρ} ≤ RG̃ ; G′X′_{G′},

since HC-induction preserves injections.

Corollary 2.39 implies immediately

τ_{Q_{M}} (RG̃ ; G′X′_{G′}) ≤ RG̃ ; G′X_{ρ} = τ_{Y}(RG̃ ; G′X′_{G′}) (3.11)

for every M ∈ L_{G,L}, since Q_{M} is a direct summand of Y. Taking M = L we obtain

τ_{Q_{L}}(RG̃ ; G′X′_{G′}) ≤ RG̃ ; G′X_{ρ}.

Now σ_{G′}R_{L}^{G′}X_{L} is the image of the endomorphism σ of R_{L}^{G′}X_{L} and hence the image of the surjective composite map

R_{L}^{G′}Y_{L} → R_{L}^{G′}X_{L} →^{σ_{G′}} σ_{G′}R_{L}^{G′}X_{L},

hence σ_{G′}RG̃ ; LX_{L} is the image of the composite map

Q_{L} = RG̃ ; LY_{L} → RG̃ ; LX_{L} →^{σ̂} σ_{G′}RG̃ ; LX_{L}

setting σ̂ = RG̃ ; G′(σ_{G′}). We conclude that

σ_{G′}RG̃ ; LX_{L} ≤ τ_{Q_{L}}(RG̃ ; G′X′_{G′}) ≤ RG̃ ; G′X_{ρ}

and

σ_{G′}RG̃ ; LX_{L} = τ_{Q}(σ_{G′}RG̃ ; LX_{L}).

Summing up we get part i). By Theorem 3.5 σ_{G′}HG̃ ; O is pure in HG̃ ; O, hence by [6, 4.13] and Lemma 3.4 we have

σ_{G′}RG̃ ; LX_{L} = τ_{Q_{L}}(σ_{G′}RG̃ ; LX_{L}) = τ_{Q_{L}}(\sqrt{σ_{G′}RG̃ ; LX_{L}}) = τ_{Q_{L}}(RG̃ ; G′X′_{G′}).

Summing up we get

τ_{Q_{L}}(X′) = X_{h} = τ_{Q_{L}}(X_{ρ}).

Applying formula 3.11 for M = G′ we obtain

τ_{Q_{G′}} (RG̃ ; G′X_{G′}) ≤ RG̃ ; G′X_{ρ}

but obviously we have equality here, since RG̃ ; G′X_{ρ} is epimorphic image of Q_{G′} = RG̃ ; G′Y_{G′}. Summing up we get

τ_{Y}(X′) = X_{ρ}.
□

The additional hypothesis on HG̃ ; O (see 1.9) in the following result is satisfied in all our applications:

Theorem 3.12 Suppose the Hecke algebra HG̃ ; O is integrally quasi Frobenius. Let T := TG̃(PR) be the HG̃ ; O-module

T = ⨁_{G′∈L_{G,L}} σ_{G′}HG̃ ; O.

(i) Let the functor HG̃ ; O = H_{O} be defined as in 3.3. Then

H_{O}(X_{h}) = H_{O}(X) = H_{O}(X′).

(ii) Restricting endomorphisms to sublattices induces isomorphisms

End_{OG̃}(X_{h}) ≅ End_{OG̃}(X) ≅ End_{OG̃}(X′).

(iii) For every lattice Z = X_{h}, X, X′ the functor H_{O} induces an isomorphism

H_{O} : End_{HG̃ ; O}(T) → End_{OG̃}(Z).

Proof We may identify X with its image X_{ρ} under the inejection ρ. We apply Theorem 3.10. For any OG̃-lattice V we have always H_{O}(V) = H_{O}(τ_{Q_{L}}(V)) by [6, 2.17], hence i) follows.

By Lemma 1.3 the endomorphisms of X′ map the subspaces τ_{Q_{L}}X′ = X_{h} and τ_{Y}(X′) = X = X_{ρ} into itself, restriction defines so an homomorphism from End_{OG̃}X′ to End_{OG̃}(X_{h}) and to End_{OG̃}(X) repectively. Similarly we have an homomorphism from End_{OG̃}(X) to End_{OG̃}(X_{h}). By Theorem 1.10 part ii) restricting endomorphisms from X′ to X_{h} is an isomorphism between the corresponding endomorphism rings, therefore from X′ to X as well. This shows part ii). Part iii) follows now from the cooresponding part of 1.10. □

In the following we call the HG̃ ; O-lattice T the parabolic tensor space of HG̃ ; O. As usual k ⊗_{O} T = T̄. The endomorphism ring of T is denoted by S_{O} = SG̃ ; O(PR) and is called the parabolic q-Schur algebra of HG̃ ; O. The reason for these names is the following: We shall apply these results to finite reductive groups. Here HG̃ ; O is always a Hecke algebra associated with some reflection group W extended possibly by an abelian group. The one dimensional representation Oσ_{G′} of H_{O}^{G′} is a representation analogous to the alternating representation of symmetric groups. The space T is then isomorphic to the sum of modules analogous to the permutation representation of W on its parabolic subgroups (respectively to the sum over its parabolic subgroups corresponding to the Levi subgroups G′ ∈ L_{G,L}), which is again for symmetric groups W a space which is closely connected to tensor space. In general, endomorphism rings of such tensor spaces are now called q-Schur algebras. In type B for instance q-Schur algebras have been defined based not only on parabolic subgroups but on the larger set of reflection subgroups, ([15] and [16]). With this notation we have now the following main result for the projective restriction system PR using Theorems 1.7 and 3.12:

Corollary 3.13 Let PR be a projective restriction system such that the Hecke alhgebra HG̃ ; O is integrally Frobenius. Then the decomposition matrix of the associated parabolic q-Schur algebra SG̃ ; O(PR) is part of the ℓ-modular decomposition matrix of G.

Remark 3.14 We remark that in general the Hecke algebras H_{R}^{G′}(PR) and H_{R}^{G′′}(PR) for G′, G′′ ∈ L_{G,L} can be the same. This happens since HG̃ ; R(PR) is not a Hecke algebra defined over the Weyl group W̃ of G̃ but over some subquotient, namely over the group W (L, X_{L}) which is the stabilizer of X_{L} in NÑ(L)/L ≅ NW̃ (W_{L})/W_{L}, where W_{L} := (Ñ ∩ L)/T is the Weyl group of L. Note that in this case the HG̃ ; O-modules σ_{G′}HG̃ ; O and σ_{G′′}HG̃ ; O are isomorphic. Thus the resulting q-Schur algebras are Morita equivalent, and we could remove one of the corresponding summands in our summation

T_{O} = ⨁_{M∈L_{G,L}} σ_{M}HG̃ ; O.

We have seen that part of the ℓ-modular decomposition matrix of G̃ can be calculated by computing decomposition numbers of parabolic q-Schur algebras, which in turn are endomorphism rings of q-tensor space of Hecke algebras, provided the latter are integrally Frobenius. In the next section we shall construct projective restriction systems which satisfy this condition.

4 Finite groups of Lie-Type

4.1 Our main applications of the theory developed in the preceding sections are on finite groups of Lie type. For any such group, say G, bold face letter denotes its underlying algebraic group, that is, for example G will denote the underlying algebraic group for which G is the set of fixed points of some Frobenius morphism (the latter will always be assumed to be known). There exist finite groups of Lie type which can be defined via various non-isomorphic algebraic groups. However, which one is considered will always follow from the context. For finite groups of Lie type, the theory of Deligne and Lusztig plays an important role. There, the dual groups of finite groups of Lie type and the dual groups of connected algebraic groups are considered. We denote the dual group of some finite group of Lie type or some connected algebraic group with the same letter, added upper indices ’^{∗}’. For L ∈ L_{G} we view L^{∗} as Levi subgroup of G^{∗} in the usual way, after choosing an isomorphism between the root data of G and G^{∗} (see [2, 4.2]). We assume in the following that we have chosen once and for all such an isomorphism for the actually considered group G. Then the identification of L^{∗} with a Levi subgroup of G^{∗} becomes canonical.

The algebraic groups we consider are defined over algebraic closures of finite fields. Therefore let 𝔽_{q} denote the field of q elements for some power q of p and let 𝔽̄_{q} be its algebraic closure.

Finally, if s and z are elements of G, we write s ∼_{G} z if s is conjugate to z in G.

Let G be a connected reductive algebraic group over 𝔽̄_{q}. Let F be a Frobenius automorphism of G and let G be the set of F -fixed points. For simplicity of notation we assume that the center of G is connected, although most of the following could be formulated more generally. By general theory the center of every Levi subgroup of G is connected too.

First we want to summerize some well known facts.

4.2 By the theory of Deligne and Lusztig, the ordinary characters of G are distributed in pairwise disjoint series E(G, z), called (rational) Lusztig series, where z runs through a set of representatives of the conjugacy classes of semi simple elements of G^{∗}. For details see e.g. [2, chapter 7]. The Lusztig series are defined by the Deligne-Lusztig operator, which maps characters of generalized Levi subgroups of G to generalized characters of G. For our purpose it is enough to know that for L ∈ L_{G} this operator and HC-induction coincide ([2, 7.4.4]). In particular, the Lusztig series are unions of HC-series and R_{L}^{G}E(L, z) ⊂ E(G, z) for semisimple z ∈ L^{∗}. We have therefore, using Frobenius reciprocity:

Lemma 4.3 Two characters in different Lusztig series are contained in different HCseries. Moreover, if χ is contained in E(G, z) and ψ is a constituent of T_{L}^{G}χ, then ψ is in E(L, (z̃)) for some semisimple z̃ which is conjugate in G^{∗} to z.

There exists a canonical isomorphism between the Weyl groups W of G and W^{∗} of G^{∗} (by virtue of the fixed isomorphism between their root data, see e.g. [2, 4.2.3]). As the torus T of G is contained in every M ∈ L_{G}, the Weyl group W acts by conjugation on the set of Levi subgroups of G and on the set of pairs (M, χ), where M ∈ L_{G} and χ an irreducible character of M. We denote the stabilizer of M under this action by stab_{W} (M). On the other hand W^{∗}, and thus W (via the canonical isomorphism), acts on the set of Levi subgroups of G^{∗} and on the set of pairs (M^{∗}, (z)), where z ∈ M^{∗} is semisimple and (z) denotes its conjugacy class in M^{∗}. It follows immediately that stab_{W} (M) is the stabilizer of M^{∗} as well; (the identification of M^{∗} with a Levi subgroup of G^{∗} is just defined that way). Now let z ∈ M^{∗}. Then by the definition of the Deligne-Lusztig operator (see [2, 7.2]) we have

E(M, z)^{w} = E(M^{w}, z^{w}). (4.4)

4.5 It was shown by Broué and Michel in [1, 2.2] that for an ℓ-regular semisimple element z ∈ G^{∗}, the set

E_{ℓ}(G, z) = ⋃_{t∈C_{G^{∗}}(z)_{ℓ}} E(G, zt) (4.6)

forms a union of ℓ-modular blocks, where C_{G^{∗}}(z)_{ℓ} denotes the ℓ-elements of C_{G^{∗}}(z). Moreover, Geck and Hiss have shown in [21] that the characters in E(G, z) form a basic set of the blocks E_{ℓ}(G, z) if ℓ is a good prime for G (e. g. ℓ arbitrary if G is of type A and ℓ odd if G is of type B, C or D). That is their reduction modulo ℓ forms a ℤ-basis of the additive group of generalized Brauer characters in the blocks E_{ℓ}(G, z). Therefore, the decomposition matrix of the characters in E(G, z) determine uniquely the decomposition matrix of the whole blocks E_{ℓ}(G, z). Moreover, the latter can be derived from the former by decomposition of Deligne-Lusztig induced characters, as shown in [21]. An important property of G is that the Hecke algebra End_{KG}(R_{L}^{G}X) for some cuspidal irreducible KL-module X is always untwisted by [41, 4.23].

4.7 We want to exhibit a projective restriction system PR(X_{G}, Y_{L}) for (L, χ_{L}) and some L ∈ L_{G}. An important role for the representations of G plays a certain character, the Gelfand-Graev character Γ = Γ_{G} of G. For its definition and basic properties we refer to the standard literature, e.g. [2] and we list only a few facts on Γ which will be needed later. One of its main features is that many cuspidal irreducible characters are constituents of Γ and that all irreducible constituents of Γ occur with multiplicity one in it. These constituents are called regular characters. Each Lusztig series contains precisely one regular character. Another important property of Γ is that its HC-restriction T_{G′}^{G}(Γ_{G}) to the Levi subgroup G′ of G gives the Gelfand-Graev character Γ_{G′} of G′. Moreover Γ is induced from a linear character of the unipotent radical of a Borel subgroup which is in particular ℓ-regular. As a consequence there is a unique projective OG-lattice P_{G} affording Γ. Moreover,

T_{G′}^{G}P_{G} = P_{G′}. (4.8)

Let χ be an irreducible constituent of Γ_{G}. Then using the notation introduced in 1.16 we conclude that π_{χ}^{G}(P_{G}) is a lattice V affording the character χ. Now

Hom_{KG}(P_{G}, R_{L}^{G}KX_{L}) = Hom_{KL}(T_{L}^{G}Γ_{G}, KX_{L}) = Hom_{KL}(Γ_{L}, KX_{L}),

where X_{L} is a lattice affording χ_{L}. Thus the multiplicity of χ in R_{L}^{G}χ_{L} is one if χ ∈ E(G, z) and χ_{L} is regular, and zero otherwise.

4.9 For the following result we also need the fact, due to Hiss, ([30, 4.6.1]), that for a given block B of G the summand of P_{G} belonging to B is indecomposable. In fact every union 4.6 of blocks of G contains exactly one such indecomposable direct summand of P_{G}. Let M ∈ L_{G} and let B be a union 4.6 of blocks of M containing the Lusztig series E(M, y), where y ∈ M^{∗} is semi simple. Then we denote the unique indecomposable direct summand of P_{M} in B by Y_{M,(y)}. Since every Lusztig series contains a constituent of P_{M}, the summand Y_{M,(y)} is well defined, however the label y is only determined up to its ℓ-part by the indecomposable direct summand of P_{M}. In particular we have a bijection between direct indecomposable summands of P_{M} and M^{∗}-conjugacy classes of semi simple elements of order prime to ℓ.

Now consider the character ψ of T_{G′}^{G}Y_{G,(y)} for some semisimple ℓ-regular element y ∈ G^{∗}. Then 4.3 implies that ψ consists precisely of those summands of Γ_{G′} = T_{G′}^{G}Γ_{G} which are in some E(G′, x) where y is conjugate to x in G^{∗} (in particular T_{G′}^{G}Y_{M,(y)} = (0) in case that G^{′∗} contains no conjugate of y). Now we can refine Equation 4.8 to

T_{G′}^{G}Y_{G,(y)} = ⨁_{x∈C_{G^{′∗},(y)}} Y_{G′,(x)}, (4.10)

where C_{G^{′∗},y} is a set of representatives of the G^{′∗}-conjugacy classes making up the intersection of G^{′∗} with the G^{∗}-conjugacy class of y. In particular, the indecomposable direct summands of T_{G′}^{G}Y_{G,(y)} are contained in pairwise different blocks.

Notation 4.11 In the following let G be as above. Let L ∈ L_{G}. The underlying algebraic groups G and L have connected centers. We fix a semisimple element z ∈ L^{∗} and assume that we have a cuspidal element χ_{L} in E(L, z).

For M ∈ L_{G,L} such that M^{∗} contains the semisimple Element y ∈ G^{∗} we set X_{M,(y)} = π_{(L,χ_{L})}^{M}(Y_{M,(y)}).

Recall the action of W on the set of pairs (M^{∗}, (y)), where M^{∗} ∈ L_{G}, y ∈ M^{∗} is semi simple and (y) denotes its conjugacy class in M^{∗}. Then stab_{W} (L^{∗}) acts on the set of L^{∗}- conjugacy classes of semi simple elements. In the following stab_{W} (z) denotes the stabilizer of (z) under this action. We denote the ℓ′-part of any semisimple element y ∈ G^{∗} by y′.

Theorem 4.12 Suppose that χ_{L} is a regular character, and that stab_{W} (z) = stab_{W} (z′). Then PR(X_{G,(z)}, Y_{L,(z)}) is a projective restriction system for (L, χ_{L}) in G consisting of the data

PR(X_{G,(z)}, Y_{L,(z)}) = {X_{M}, Y_{M}, X′_{M} | M ∈ L_{G,L}},

where X_{M} = X_{M,(z)}, and Y_{M} and X′_{M} are defined as in 2.3.

Proof By 4.7 we only have to show part (ii) of Definiton 2.3.

As usual we denote the character of X_{G} = X_{G,(z)} by χ_{G}. Let M ∈ L_{G,L} and χ_{1} ≠ χ_{2} two summands of T_{M}^{G}χ_{G}. By 4.5 we are done if we can show that χ_{1} and χ_{2} are in two different unions of series defined in 4.6 and hence in different blocks. So assume that this is not the case. We may assume that both χ_{1} and χ_{2} are in the collection E_{ℓ}(M, z′) of blocks of M.

By Lemma 4.3 all summands of T_{L}^{G}χ_{G} and hence of T_{L}^{M}χ_{i} (i = 1, 2) are in series E(L, y) for y ∈ L^{∗} conjugate to z in G^{∗}. Moreover, by Equation 4.4 and general HC-theory, both T_{L}^{M}χ_{1} and T_{L}^{M}χ_{2} actually do have summands in in E_{ℓ}(L, z′) as the latter is the union of the series E(L, y) with y′ = z′. Thus T_{L}^{G}χ_{G} has at least two summands in E_{ℓ}(L, z′). However, by Mackey decomposition and cuspidality of χ_{L},

T_{L}^{G}χ_{G} ≤ R_{L}^{G}T_{L}^{G}χ_{L} = ⨁_{x∈stab_{W} (L)/(W ∩L)} χ_{L}^{x}.

By Frobenius reciprocity, χ_{L} has multiplicity one in T_{L}^{G}χ_{G}, hence it follows that there exists x ∈ stab_{W} (L) such that

χ_{L} ≠ χ_{L}^{x} ∈ E_{ℓ}(L, z′). (4.13)

Now we extend Equation 4.4 to the union of series E_{ℓ}(L, z′). As the two unions of series E_{ℓ}(L, z′) and E_{ℓ}(L, z^{′x}) are either equal or disjoint it follows

E_{ℓ}(L, z′) = E_{ℓ}(L, z′)^{x} = E_{ℓ}(L, z^{′x}),

hence x ∈ stab_{W} (z′). By assumption we then also have x ∈ stab_{W} (z). Thus

χ_{L}^{x} ∈ E(L, z)^{x} = E(L, z^{x}) = E(L, z).

However, by construction (see [2, chapter 8]), Γ_{L} is invariant under the action of stab_{W} (L), hence χ_{L}^{x} is regular. As there is only one regular character in E(L, z), namely χ_{L}, we conclude χ_{L}^{x} = χ_{L}, a contradiction to 4.13. □

Note that the assumption of the previuous theorem fits precisely the situation considered in [10, section 5]. There the Hecke-algebra part for R_{L}^{G}X_{L} was done, and our theorem here provides its extension to q-Schur algebras. For the general linear groups G = GL_{n}(q) this was done in [14]. In fact it was shown there, that we get a complete list of the irreducible ℓ-modular representations applying our main results 3.8,4.12 and Result 1.4 to the Lusztig series E(G, z), where z runs through a set of representatives of ℓ-regular conjugacy classes of G = GL_{n}(q). Thus we obtain from Results 2.40 and 3.9 a new proof for the following results from [4],[5],[37] and [14] (compare [5, 4.15, 6.11]):

Theorem 4.14 Let G = GL_{n}(q). Then every cuspidal irreducible KG-module remains irreducible if reduced modulo ℓ. On the other hand, every irreducible cuspidal kG-module is liftable to an irreducible KG-module. If L is a Levi subgroup of G, and X_{L} is an irreducible cuspidal OL-lattice such that R_{L}^{G}(X_{L}) is reduction stable then the combined image τ_{P_{G}}(R_{L}^{G}X_{L}) of the Gelfand-Graev representation in R_{L}^{G}X_{L} is pure in R_{L}^{G}X_{L}.

These results can be generalized to classical groups under the additional assumption that the prime ℓ is linear. For details we refer to [28]. Here is another situation, where our results apply:

Corollary 4.15 Suppose that χ_{L} is regular and χ̄_{L} is irreducible. Moreover, suppose that the stabilizers stab_{W} (χ_{L}) and stab_{W} (χ̄_{L}) of χ_{L} and χ̄_{L} under the action of stab_{W} (L) are equal. Then PR(X_{G,(z)}, Y_{L,(z)}) is a projective restriction system for (L, χ_{L}).

Proof χ_{L}^{x} for x ∈ stab_{W} (L) is regular (see the last proof), hence it is the unique regular character in E(L, z^{x}) and stab_{W} (z) ≤ stab_{W} (χ_{L}). As the converse follows from Equation 4.4, we have equality. Since there is a bijection between the set of irreducible Brauer characters of kL and representatives of isomorphism classes of indecomposable projective OL-lattices, we see that stab_{W} (χ̄_{L}) is equal to the stabilizer of Y_{L,(z′)} under the action of stab_{W} (L) (analogeously denoted by stab_{W} (Y_{L,(z′)})). As P_{L} is invariant under this action and has exactly one indecomposable direct summand in every union of series E_{ℓ}(L, z′), we get stab_{W} (Y_{L,(z′)}) = stab_{W} (z′) and the assertion follows from Theorem 4.12.

□

In the situation of Theorem 4.12 as well as of Corollary 4.15 we have exhibited projective restriction systems PR(X_{G}, Y_{L}), which allow us to apply our main results of the previous section. In both cases the Hecke algebra H = End_{OG}(R_{L}X_{L}) is a Hecke algebra associated with an extension of a reflection group and it is known that it admits an associative (in fact symmetric) bilinear form, whose determinant is a unit, hence is integrally Frobenius, ([10] in the first and [24] in the second case). Thus Theorem 3.13 implies:

Corollary 4.16 The decomposition matrix of the q-Schur algebra S_{O}^{G} = S_{O}^{G}(PR) is part of the ℓ-modular decomposition matrix of G.

We assume now that G is a Levi subgroup of some larger group of Lie type G̃. Again we assume that the center of G̃ is connected.

Theorem 4.17 Suppose that E(L, z) has only one cuspidal element χ_{L} and that z = z′. Let χ_{G} be a summand of R_{L}^{G}χ_{L} of multiplicity one. Suppose further that there exists some indecomposable projective OG-lattice Y_{G} such that the character of X_{G} = π_{(L,χ_{L})}^{G}(Y_{G}) is equal to χ_{G}. Then there exists an OL-lattice X_{L} with character χ_{L} and projective cover Y_{L} such that PR(X_{G}, Y_{L}) is a projective restriction system for (L, χ_{L}) in G̃.

Proof First we show part (i) of Definiton 2.3. To do so we show that if L^{g} = L with g ∈ G̃ but L^{g} ≁_{G} L then the character of Y_{G} has no summand in common with R_{L^{g}}^{G}χ_{L}^{g}. Recall that we can restrict attention to standard Levi subgroups, hence we can assume g ∈ Ñ, respectively we can take g = w in the Weyl group W of G̃. Then by 4.4

χ_{L}^{w} ∈ E(L, z)^{w} = E(L^{w}, z^{w})

(using the action of W on the conjugacy classes of G̃^{∗} via W ≅ W^{∗}) and hence

R_{L^{g}}^{G}χ_{L}^{g} ∈ E(G, z^{w}) ⊂ E_{ℓ}(G, z^{w}).

However, z = z′, hence if z^{w} is not conjugate to z in G then we have for the set of blocks E_{ℓ}(G, z^{w}) ∩ E_{ℓ}(G, z) = ∅. As Y_{G} is indecomposable, we are done.

For part (ii) of Definiton 2.3 we show that the summands of T_{M}^{G}χ_{G} for M ∈ L_{G,L} are in pairwise different blocks. So assume this is not the case, that is let χ_{1}, χ_{2} be summands of T_{M}^{G}χ_{G} in the same block. Again we may assume that one of them is χ_{M} ∈ E(M, z). By 4.2, every summand of T_{M}^{G}χ_{G} lies in some series E(M, y) ⊂ E_{ℓ}(M, y) for y ∈ M^{∗} conjugate to z in G^{∗}. As z = z′, it follows then by [1, 2.2] that both χ_{1} and χ_{2} are in E(M, z). Like in the proof of Theorem 4.12 we conclude that both of T_{L}^{M}χ_{i} has a summand in E(L, z). However, by general HC-theory, these summands must be cuspidal and thus by our assumption they must be equal to χ_{L}. Thus χ_{L} has multiplicity of at least two in T_{L}^{G}χ_{G}. By Frobenius reciprocity this gives a contradiction to our assumption. □

Remark 4.18 If L is a group of classical type, every Lusztig series has at most one cuspidal character (see [2, 13]).

The following corollary now follows immediately from the preceeding sections Corollary 4.19 Let the assumptions be as in Theorem 4.17. Then the q-Schur algebra SG̃ ; O(PR(X_{G}, Y_{L})) is defined and its decomposition matrix is a submatrix of the decomposition matrix of E(G̃, z).

An example of particular interest for the previous two Theorems are the regular characters in series E(L, y) for y = y′. We call such series ℓ-regular Lusztig series. Note that if some Lusztig series has just one element then this element is a regular character.

There are two main examples of the above, which have been investigated in previous papers by various authors. We keep the outline of the following examples sketchy, for further detailes the reader may refer to the cited references.

4.20 The first one is the well known theory for finite general linear groups. It was developed in several articles by G. James and the first named author (see [6], [4], [5], [14],

[37] and [38]). First one observes that all the Hecke algebras appearing as endomorphism rings of induced irreducible cuspidal KL-modules for L ∈ L_{G}, are defined over groups H isomorphic to direct products of symmetric groups. Secondly, any irreducible cuspidal KL-module for L ∈ L_{G} is regular. Therefore one can apply Theorem 4.12 to cuspidal characters in ℓ-regular Lusztig series. It follows that the decomposition matrix of the q-Schur algebra S_{O}(PR(X_{G,(z)}, Y_{L,(z)})) gives a part of the decomposition matrix of E(G, z), where z is ℓ-regular and L ∈ L_{G} is a minimal Levi subgroup containing z (for G a general linear group, we have G ≅ G^{∗} canonically). Moreover, it turns out that S_{O}(PR(X_{G,(z)}, Y_{L,(z)})) has the same number of isomorphism classes of projective indecomposable modules as S_{K}(PR(X_{G,(z)}, X_{L,(z)})) (see [14]). However, the latter are by definition in bijection to the elements S(G/L, χ_{L}), where χ_{L} is the unique (cuspidal) character in E(L, z). As HC-series and Lusztig series for finite general linear groups coincide, it follows that the decomposition numbers of the various q-Schur algebras defined over cuspidal irreducible regular characters in ℓ-regular Lusztig series gives us the complete decomposition matrix of all series E(G, z) for ℓ-regular elements z ∈ G^{∗} = G. The decomposition matrix of the other characters can now easily deduced by decomposition of Deligne Lusztig induced characters. However, the results in this article can be applied to an arbitrary group G such that G is of type A with connected center. Thus it follows that the decomposition numbers of the ℓ-regular Lusztig series of G are the same as the decomposition numbers of ℓ-regular Lusztig series of suitable general linear groups.

4.21 The second example concernes other finite classical groups G̃ such as symplectic, special orthogonal and general unitary groups. They were investigated in [28] by G. Hiss and the second named author. If one restricts attention to so called linear primes for these groups (see [28], Section 1 for definition, and note that linear primes are roughly half of all primes dividing the order of G̃), one can develop a theory similar to the one of finite general linear groups.

First one observes that the reduction modulo ℓ of every cuspidal character χ_{L} of a Levi subgroup L of G̃ is irreducible.

Secondly, let ϕ ∈ E(L, z), ψ ∈ E(M, s) for L, M ∈ LG̃ and z, s ∈ G̃^{∗} ℓ-regular, be two cuspidal irreducible characters not conjugate in N. Then two characters α ∈ S(G̃/L, ϕ) and β ∈ S(G̃/M, ψ) are lying in different blocks.

Thirdly, for every cuspidal irreducible character in an ℓ-regular Lusztig series E(L, z) for some L ∈ LG̃ one can define a projective restriction system and an associated q-Schur algebra of type B or D. Using the representation theory of Hecke algebras over classical extended Weyl groups for linear primes (see [13] and [28], Section 7) one again finds that the q-Schur algebra S_{O}(PR(X_{G}, Y_{L})) has the same number of isomorphism classes of projective indecomposable modules as the q-Schur algebra over K (see [28] Corollary 8.2). Thus we again have the complete decomposition matrix of all series E(G̃, z) for ℓ-regular elements z ∈ G̃^{∗}.

In these examples, Y_{L} is the unique indecomposable projective OL-lattice affording a character with some irreducible cuspidal constituent χ_{L} ∈ E(L, z), while X_{G} is a suitably chosen OG-lattice affording a character χ_{G} with multiplicity one in R_{L}^{G}χ_{L}. G ∈ LG̃ is chosen as follows: The Dynkin diagram of the Weyl group of L has at most one connected component I of type B or D (if it has no such component we set I = ∅). Then G is a maximal Levi subgroup of G̃ containing L such that its Weyl group has one connected component of type A and one equal to I.

We note that in both examples one can apply Corollary 2.40 to the projective restriction systems associated to irreducible cuspidal characters in ℓ-regular Lusztig series. We get a complete description of the module structure of the so called Steinberg lattice, that is the unique quotient lattice of the Gelfand-Graev lattice affording the Steinberg character. In case of general linear groups, the corresponding quotient of the Gelfand-Graev lattice affording other irrducible characters (even for those characters not lying in ℓ-regular Lusztig series) are described in [26]. There, also for the special linear groups corresponding results have been obtained.

4.22 Let G̃ be an arbitrary reductive algebraic group with connected center. For simplicity of notation we assume that the Dynkin diagram of G̃ is connected. However, the following can be formulated more generally. Let L ∈ LG̃ have a unique cuspidal character χ_{L} in the series E(L, z) for some z ∈ L^{∗}. Let X̂ be a KL-module with character χ_{L}. The Dynkin diagram of L has at most one connected component I which is not of type A. Let J be the union of the connected components of the Dynkin diagram of L of type A. Then there exists a maximal Levi subgroup G ∈ LG̃ whose Dynkin diagram has at most two connected components, one being of type A and having J as subset and one equal to I. Then stab_{W} (χ_{L}) is a direct product of symmetric groups. We take χ_{G} to be the character in S(G/L, χ_{L}) corresponding to the alternating character of stab_{W} (χ_{L}). In particular, π_{(L,χ_{L})}(T_{M}^{G}χ_{G}) = χ_{M} is irreducible for M ∈ L_{G,L} and π_{(L,χ_{L})}(T_{L}^{G}χ_{G}) = χ_{L} by [31], Theorem 5.9. Finally suppose that there exists an indecomposable projective OGlattice Y_{G} such that π_{(L,χ_{L})}(Y_{G}) = X_{G} has character χ_{G}. Setting X_{M} = π_{(L,χ_{L})}(T_{M}^{G}X_{G}) for M ∈ L_{G,L} and denoting the projctive cover of X_{M} by Y_{M}, we can apply Theorem 4.17 to get a projective restriction system PR(X_{G}, Y_{L}). Moreover, because of our assumptions, we can apply Corollary 4.19. Obviously, if L ≅ G′ × L_{1} and G ≅ G′ × L_{2}, where G′ is a finite group of Lie type with Dynkin diagram I (as it holds in the case of finite classical groups), it follows from the theory of linear groups that Y_{G} exists. Moreover, if L is of classical type, there always exists at most one cuspidal character in a Lusztig series of L. We can formulate the following

Corollary 4.23 Let the notation and assumptions be as in 4.22. Then the q-Schur algebra S_{O}(PR(X_{G}, Y_{L})) is defined and its decomposition matrix embeds into the decomposition matrix of E(G̃, z).

We want to discuss an application of our theory that has not been investigated in other articles, yet, the decomposition matrix of the unipotent characters of the general unitary groups for arbitrary odd primes. To do so we need some preliminary considerations about the general linear groups.

4.24 For the following Lemma the reader might recall the labeling of irreducible unipotent characters of the general linear group G = GL_{n}(q^{2}) by partitions α ⊢ n of n (see [2, 13.8]). Here, a unipotent Brauer character is a Brauer character appearing as summand of the reduction modulo ℓ of some unipotent character. Using the lexicographic order for partitions (see [37]), the Steinberg character has the lowest label. For any prime ℓ ≠ p, the decomposition matrix of the unipotent characters, arranged with respect to the lexicographic order along a vertical edge of the decomposition matrix, can be made unitriangular if the unipotent Brauer characters are arranged suitably on a horizontal edge of the decomposition matrix (see again [37]). Moreover, the decomposition matrix of the unipotent characters is a square matrix. Giving the i^{th} unipotent Brauer character the label of the i^{th} unipotent character, we get a labeling of the unipotent Brauer characters by the partitions of n. We remark that square unitriangularity of the decomposition matrix determines the labeling of the Brauer characters uniquely. This follows easily from the fact that a different labeling of the Brauer characters has the effect of multiplying the decomposition matrix with a permutation matrix and if the resulting matrix is again square unitriangular then the permutation matrix is necessarily the unit matrix.

Lemma 4.25 Let μ ≠ (1^{n}) be a partition of n. Then there exists an odd prime ℓ and some power q of p ≠ ℓ such that the unipotent Brauer character ϕ_{μ} of GL_{n}(q^{2}) with label μ appears in the reduction modulo ℓ of at least two different irreducible unipotent characters of G. Moreover, ℓ can be chosen to be linear for GU_{n}(q).

Proof The proof proceeds in two steps. First we prove the assertion for μ = (j, 1^{n−j}) for j ≥ 1. If n is odd, take ℓ to be a Zsigmondy prime of q^{n} − 1 (see [36, 8.3]) for some q an arbitrary power of some prime p. The order of q^{2} modulo ℓ is the same as the order of q modulo ℓ and ℓ is linear for GU_{n}(q). Now the assertion follows in this case immediately from [37, 6.5] by taking e = n in that theorem. For n = 2 the the assertion follows from [37, 6.5] using ℓ = 3 and q = 4. Now let n > 2 be even. Let q be an arbitrary power of some prime p (in case n = 6 take q ≠ 2). Let ℓ be a Zsigmondy prime of q^{2n} − 1. Then ℓ is again linear for GU_{n}(q) and the assertion follows in this case from [37, 6.5] by taking e = 2n. The second step is to proceed by induction. Let μ = (μ_{1}, …, μ_{r}). We set μ′ = (μ_{2}, …, μ_{r}) ⊢ n − μ_{1}. We can assume that (0) ≠ μ′ ≠ (1^{n−μ_{1}}). By induction hypothesis there exists ℓ ≠ p, a power q of p and a partition λ′ ≠ μ′ of n − μ_{1} such that the ℓ-modular unipotent Brauer character of GL_{n−μ_{1}}(q^{2}) with label μ′ is a constituent of the reduction modulo ℓ of the unipotent KGL_{n−μ_{1}}(q^{2})-character with label λ′ = (λ_{2}, …, λ_{s}) and ℓ is linear for GU_{n−μ_{1}}(q). Moreover, by unitriangularity of the decomposition matrix it follows that λ′ ≤ μ′ in the lexicographical order. Thus μ_{1} ≥ λ_{i} for 2 ≤ i ≤ s. By [37, 6.18] ϕ_{μ} is a constituent of the reduction modulo ℓ of the unipotent KGL_{n}(q)-character with label (μ_{1}, λ_{2}, …, λ_{s}). □

4.26 For the remainder of this section let G be the finite general unitary group over the field with q^{2} elements and let ℓ ≠ 2. We will investigate unipotent characters, thus we assume that χ_{L} is a cuspidal unipotent irreducible character for some L ∈ L_{G}. We remark however that the following can also be applied to some other ℓ-regular Lusztig series E(G, z) by 4.22, as the Levi subgroups of G are direct products of at most one unitary factor and general linear groups over 𝔽_{q^{2}}. The characters in E(G, z) are canonically labeled by the unipotent characters of C_{G}(z) and their decomposition matrices are the same (see [28]). By [2, 13.8] the irreducible unipotent KG-characters can be labeled by partitions of n. Similarly to the linear groups, it was shown in [23, 6.6] that the decomposition matrix of G is square lower unitriangular (for appropriate arrangements of the characters). We again get a labeling of the unipotent Brauer characters by partitions of n like in the case of GL_{n}(q). Now the unipotent characters of G (and thus the Brauer characters) can also be labeled by the bipartitions of {(n − r)/2}_{r}, where r runs through all positive integers less or equal to n such that n − r is even and the unitary group of degree r has a unipotent cuspidal character (see [28], for the relevant r see [2, 13.7]). Here a bipartition of n is an ordered pair of partitions α ⊢ a and β ⊢ b with a + b = n. A unipotent character has as label some bipartition of (n − r)/2 if and only if it lies in the HC-series of the cuspidal irreducible unipotent character χ_{L} of the Levi subgroup L ≅ GU_{r}(q) × GL_{1}(q^{2})^{(n−2)/2} of G (see [2, 13.8]).

Now consider S(G/L, χ_{L}). Then we have

Theorem 4.27 Let ψ be the character in S(G/L, χ_{L}) with least label in the lexicographical order. Then ψ appears with multiplicity one in R_{L}^{G}χ.

Proof We turn to the labeling of the characters in S(G/L, χ) by the bipartitions of s = (n − r)/2. Suppose that ψ has label (α, β) for partitions α ⊢ a and β ⊢ b = s − a. By the assumption on ψ, it follows from the lower unitriangularity of the decomposition matrices that there exists no odd prime ℓ′ and no power q′ of some prime p′ different to ℓ′ such that the ℓ′-modular irreducible unipotent Brauer character with the same label as ψ appears as constituent of the reduction modulo ℓ of some element in S(G′/L′, χ′) different to ψ. Here G′ denotes GU_{n}(q′), L′ the a standard Levi subgroup of G′ isomorphic to GU_{r}(q′) × GL_{1}(q′^{2})^{(n−2)/2} and χ′ the cuspidal irreducible unipotent character of G′. Recall that by general theory the multiplicities of summands ρ′ in R_{L′}^{G′}χ′ and ρ in R_{L}^{G}χ_{L} are the same, if ρ and ρ′ have the same label. Now assume that α ≠ (1^{a}). Then choose ℓ′ and q′ with Lemma 4.25 such that the ℓ′-modular irreducible unipotent Brauer character of GL_{a}(q′^{2}) with label α appears in the reduction modulo ℓ′ of some irreducible unipotent character with label γ ≠ α and ℓ′ is linear for GU_{a}(q′). Linearity of ℓ′ only depends on q′, thus ℓ′ is linear for GU_{n}(q′). Now it follows from [25, 3.36] that the ℓ-modular unipotent Brauer character of GU_{n}(q′) is a constituent of the reduction modulo ℓ of the character in S(G′/L′, χ′_{L}) with label (γ, δ), a contradiction to our statement at the beginning of the proof. Thus α = (1^{a}). Similarly one shows that β = (1^{b}).

Now recall the way how one passes from labeling of the characters in E(G, 1) by partitions of n to their labeling by bipartitions. This way is described for example in [18]. If r = 0, the partition (1^{n}) is the label of the Steinberg character and the least partition in the lexicographical order. Thus we may assume that r > 0. Assume that a > 0 and b > 0.

Then one can see immediately, that if λ is the partition labeling ψ, then λ must be of one of the following forms.

a) (m + 3 + b, m + 3 + (b − 1), …, m + 3, m, m − 1, …, 2, 1^{2a+1}),
b) (2 + b, 2 + (b − 1), …, 3, 1^{2a}),
c) (3 + b, 3 + (b − 1), …, 4, 1^{2a+1}),

for m ≥ 2. However, consider the partition δ = (m + b + 1, …, 2, 1^{2a+2b+1}) in case a), respectively δ = (b, …, 2, 1^{2a+2b+1}) in cases b) and c). Then δ is a partition of n and corresponds to a bipartition of the form ((0), (1^{s})) or ((1^{s}), (0)). Moreover, δ is lower in the lexicographical order than λ, a contradiction to our assumption on ψ. Thus it follows that a = 0 or b = 0. Now the assertion follows from the representation theory of the Weyl group of type B, see [2]. □

As a consequence of Proposition 4.27 we can apply Theorem 4.17 to get our last theorem. We use the following notation for it. Let χ_{L} be an irreducible cupidal unipotent character of L ∈ L_{G}. Let χ_{G} be the character of lowest label in S(G/L, χ_{L}). Let ϕ be the unipotent Brauer character of G with the same label as χ_{G} and let Y_{G} be an indecomposable projective OG-lattice such that Y_{G}/JacY_{G} has Brauer character ϕ. Let X_{M} = π_{(L,χ_{L})}(T_{M}^{G}Y_{G}) for M ∈ L_{G,L}, let χ_{M} be its character and let Y_{M} be the projective cover of X_{M}.

Theorem 4.28 The data {(χ_{M}, X_{M}, Y_{M}) | M ∈ L_{G,L}} define a projective restriction system PR(X_{G}, Y_{L}). In particular, the decomposition matrix of S_{O}(PR(X_{G}, Y_{L})) is a submatrix of the decomposition matrix of E(G, 1).

Proof By square unitriangularity of the decomposition matrix of E(G, 1) it follows that ϕ is not a constituent of the reduction modulo ℓ of any character in S(G/L, χ_{L}) different to χ_{L} and its multiplicity in χ̄_{L} is one. Thus π_{(L,χ_{L})}(Y_{G}) = X_{G} has character χ_{G}. Now the first statement follows from Theorem 4.17. The second statement follows from Corollary 4.19. □

4.29 We have so far only considered finite Lie groups whose underlying algebraic group has connected center. In case of classical groups it was shown in [28] that one can extend the results for linear primes to groups whose underlying algebraic group has non-connected center. In general this is done by embedding the group in a finite Lie groups whose underlying algebraic group has conncted center. In case of classical groups the center of the underlying algebraic group has two connected components. We get more problems in case the number of connected components increase. As a standard example for this to happen we can consider the finite special linear groups. These groups were treated in [26]. There it was shown how one can describe the decomposition matrix of a set of irreducible characters whose reduction modulo ℓ generate the group of generalized Brauer characters in terms of decomposition matrices of q-Schur algebras defined over extended Weyl groups of type A.

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