On quotients of Hom-functors and representations of finite general linear groups II

Richard Dipper Mathematisches Institut B Universität Stuttgart Postfach 80 11 40 70550 Stuttgart Deutschland

23rd March 1998

Abstract

This is a second paper on quotients of Hom-functors and their applications to the representation theory of finite general linear groups in non-describing characteristic. After some general result on quotients of Hom-functors and their connection to Harish-Chandra theory these contructions are used to obtain a full classification of the ℓmodular irreducible representations of GL_{n}(q) for some prime power q which is not divisible by the prime ℓ and to explain some facts on their Harish-Chandra series and decomposition numbers.

0 Introduction

This is a second paper dealing with representations of finite general linear groups G in nondescribing characteristic and quotients of Hom-functors in fulfilment of an anouncement in [9], promising new proofs for the classification of the irreducible representations of G.

In the first three sections the general theory of those functors is further developed. In particular some errors of [9] are corrected. The relation between the functors for different projective resolutions is investigated as well as the image of the different right inverses of those.

Our main applications of these functors concern representations of finite reductive groups G in nondescribing characteristic, (they have however been successfully applied to p-adic groups as well, see [46]). Quotients of Hom-functors arise here especially in connection with Harish-Chandra theory, which is used to subdivide the irreducible G-modules into Harish-Chandra series in terms of Hecke algebras associated with reflection groups. Partial results in this direction were obtained first in [7], [8] (for general linear groups) and in [15],

[16] for arbitrary finite reductive groups. In [28] Geck, Hiss and Malle established then the general case: They showed that Harish-Chandra induced irreducible cuspidal modules in finite reductive groups in nondescribing characteristic always satisfy Hypothesis 1.1 and the results of [9] apply. In particular the Harish-Chandra series can be described in terms of the irreducible representations of endomorphism rings of these modules. In section two we give details on these results in the general setting of Hom-functors and Harish-Chandra theory as developed in [13]. Geck, Hiss and Malle showed too that those endomorphism rings are Hecke algebras associated with certain (abelian extensions) of reflection groups extending Howlett-Lehrer theory [36]. However, since there is no complete classification of the irreducible cuspidal representations of finite reductive groups G, the Harish-Chandra series of G are still not determined.

E-mail: rdipper@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”.

About twenty five years ago Green and Sawada utilized different functors, connecting representations of G-modules with those of the endomorphism ring of a certain induced Gmodule, to obtain a classification of the irreducible G-modules in the describing characteristic case. These methods have been reinvestigated recently by Cabanes in [3]. Subsequently Geck and Hiss showed in [27], that they can be applied in the nondescribing characteristic case as well and they produced an alternate method to obtain the Harish-Chandra series in terms of Hecke algebras. Our approach here and the Green-Sawada-Cabanes approach use different assumptions: We need Hypothesis 1.1, whereas there the assumption is needed, that the endomorphism ring in question is a selfinjective algebra. The proofs in both approaches are very similar. In addition, if both conditions hold, the corresponding functors coincide on a large class of modules including those irreducible G-modules, which are not taken to the zero module by the functors. More details on these functors will be given in an upcoming paper by Volker Schubert.

There is a further question besides the classification of the irreducible G-modules, which can be studied using quotients of Hom-functors, namely the calculation of decomposition numbers. To do this, we need an integral theory describing the transition from characteristic zero to characteristic ℓ representations of G using quotients of Hom-functors between module categories of O-orders over integral domains O. Results on these from [9] are revised (and corrected in part) in section one. The key theorems here are [9, 4.10] and Theorem 1.30 of this paper. The applications are twofold: First we are able to describe part of the ℓ-modular decomposition matrix of G in terms of decomposition matrices of Hecke algebras. Secondly we use 1.30 to enlarge Hecke algebras to q-Schur algebras (which are not selfinjective in general even over fields) to apply the theory of quotients of Hom-functors a second time.

All this works especially well, when G is a finite general linear group. Thus the second part of the paper, starting with section four, is devoted to the representation theory of GL_{n}(q). As a consequence we get a classification of the irreducible ℓ-modular representations of G = GL_{n}(q) for primes ℓ not dividing q and information on decomposition numbers in terms of Hecke- and q-Schur algebras. Most of this was done before in [7],[8],[38], [39] and [40], however the method and the proofs here are new and more conceptual.

At the end of section six we recover and generalize a result of Takeuchi [45] who constructed q-tensor space directly as subquotient of the group algebra of G and showed, that it satisfies Schur-Weyl duality. The result follows from our investigation in section three, where we demonstrate, that this is a general result on quotients of Hom-functors with respect to selfinjective endomorphism rings.

The representation theory of finite general linear groups in nondescribing characteristic is an especially well behaved application of our general set-up. Hiss and Gruber used similar methods to classify the irreducible ℓ-modular representations of classical groups in [31] for so called linear primes ℓ (roughly a third of all primes) and reduce the computation of the ℓ-modular decomposition matrix of these groups to those of q-Schur algebras.

In [18] Gruber and the author apply the general theory of this paper to representations of arbitrary finite groups G with split BN-pair. Generalised q-Schur algebras are constructed and used, to produce many irreducible ℓ-modular G-modules for nondescribing primes ℓ, and it is shown that their decomposition matrices are part of the ℓ-modular decomposition matrix of G. The results however are not as complete in general as in the case of general linear or classical groups with respect to linear primes. We neither get a classification of all irreducible G-modules nor do we obtain a complete description of the decomposition matrix of G in terms of generalised q-Schur algebras.

For an application of quotients of Hom-functors to p-adic groups and some general results on the functors we refer to a recent preprint of M. F. Vigneras [46].

I wish to thank the Australian Research Council which supported a visit at the University of New South Wales in Sydney in summer 1994, when a first version of this paper was drafted. I also wish to thank the Isaac Newton Institute in Cambridge and the organizers of a half year program in the first half of 1997 on representation theory of algebraic groups and related finite groups. During an extended visit at the Institute in this program, this paper was drafted in its present form, and was written up to a large extend.

I wish also to thank Volker Schubert for many helpful discussions and the referee for several valuable comments.

1 Quotients of Hom-functors

In general we shall take over notation and main definitions from [9]. However there will be exceptions.

We begin with the basic setup of [9] and supplement a few further results which will be needed later on. Thus R is a commutative noetherian ring, and T is a semiperfect R-algebra which is finitely generated as R-module. All occurring modules are, if not stated otherwise, finitely generated. The category of finitely generated right T -modules is denoted by mod_{T}, of finitely generated left T -modules by _{T} mod. 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. [9, 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.

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 τ_{P} (V) of P in V is the T -submodule of V generated by the images of all homomorphisms ϕ : P → V.

It was shown in [9, 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 [9, 2.28].

For an arbitrary module V the Jacobson radical Jac(V) of V is the intersection of all maximal submodules of V, the head hd(V) is the factor module V/Jac(V). Thus hd(V) is the largest semisimple factor module of V. Similarly the socle soc(V) of V is the largest semisimple submodule of V. For a ring S the Jacobson radical Jac(S) of S is defined to be the radical of the regular representation S_{S} of S.

More precisely Results [9, 2.28,2.32, and 2.20ii] say:

Theorem 1.2 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}.

It was remarked in [9, 2.4] that the functor H = H_{M} is independent of the choice of the projective presentation β : P → M. This however is not correct. Here are some precise statements about what happens, if we change the projective presentation of M to another one, for which Hypothesis 1.1 as well holds. Obviously it is enough to consider the minimal projective cover β_{1} : P_{1} → M of M. Note that by general theory P_{1} splits from the kernel of β, that is ker β = ker β_{1} ⊕ P_{2}, P = P_{1} ⊕ P_{2} and β = β_{1} ⊕ 0 in the following short exact sequence:

[figure: 0; ker β_{1} ⊕ P_{2}; P_{1} ⊕ P_{2}; β_{1} ⊕ 0; M; 0.] (1.3)

We can write down elements of P as column vectors with two components, the first one from P_{1} and the second one from P_{2}, and accordingly the endomorphisms of P by 2 × 2-matrices with entries in the appropriate Hom-spaces:

E_{β} = (\begin{matrix}E_{1} & Hom_{T} (P_{2}, P_{1}) \\ Hom_{T} (P_{1}, P_{2}) & E_{2} \end{matrix}) , (1.4)

setting E_{i} = E_{β_{i}} for i = 1, 2. For a ring S and S-modules V_{1}, V_{2} and for U ≤ V_{1} we define Hom_{S}(V_{1}, V_{2})U to be the submodule of V_{2} spanned by all images im ϕ for restrictions ϕ to U of homomorphisms from V_{1} to V_{2}. We calculate:

(\begin{matrix}E_{1} & Hom_{T} (P_{2}, P_{1}) \\ Hom_{T} (P_{1}, P_{2}) & E_{2} \end{matrix}) (\begin{matrix}ker β_{1} \\ P_{2} \end{matrix})

= (\begin{matrix}E_{1} ker β_{1} + Hom_{T} (P_{2}, P_{1})P_{2} \\ Hom_{T} (P_{1}, P_{2}) ker β_{1} + E_{2}P_{2} \end{matrix})

⊆ (\begin{matrix}ker β_{1} + τ_{P_{2}}(P_{1}) \\ P_{2} \end{matrix}) .

Thus β satisfies 1.1 if and only if τ_{P_{2}}(P_{1}) ⊆ ker β_{1} or equivalently

Hom_{T} (P_{2}, P_{1}) = Hom_{T} (P_{2}, ker β_{1}).

Assume 1.1 holds. Then

J_{β} = (\begin{matrix}J_{1} & Hom_{T} (P_{2}, ker β_{1}) \\ Hom_{T} (P_{1}, P_{2}) & E_{2} \end{matrix}) .

We denote the Hom-set Hom_{T} (P_{i}, P_{j}) by E_{i,j} for i, j = 1, 2 and i ≠ j. Thus in particular Hom_{T} (P_{2}, ker β_{1}) = E_{2,1}. Homomorphisms from P into a T -module V are written as row vectors α = (α_{1}, α_{2}), where α_{i} ∈ Hom_{T} (P_{i}, V) for i = 1, 2. We have:

Hom_{T} (P, V)J_{β}

= ^{(Hom_{T} (P_{1}, V), Hom_{T} (P_{2}, V))} (\begin{matrix}J_{1} & E_{2,1} \\ E_{1,2} & E_{2} \end{matrix})

= (Hom_{T} (P_{1}, V)J_{1} + Hom_{T} (P_{2}, V)E_{1,2}, Hom_{T} (P_{2}, V)) ,

setting J_{1} = J_{β_{1}}.

We have shown:

Lemma 1.5 Let β = (β_{1}, 0) : P_{1} ⊕ P_{2} → M be given as above. Then β satisfies 1.1 if and only if

τ_{P_{2}}(P_{1}) ⊆ ker β_{1}.

Moreover for V ∈ mod_{T} we have

H^{β}(V) ≅ H^{β_{1}}(V)/(Hom_{T} (P_{2}, V)E_{1,2}).

Thus H^{β} = H^{β_{1}} if and only if every homomorphism from P_{1} to P_{2} factors through a linear combination of endomorphisms of P_{1} whose image is contained in ker β_{1}, that is H^{β_{1}}(P_{2}) = (0). If this condition does not hold, then H^{β} is a proper quotient of H^{β_{1}}.

Note that τ_{P_{2}}(P_{1}) ⊆ ker β_{1} is equivalent to Hom_{T} (P_{2}, M) = (0) or, if R is a field, that no composition factor of the head of P_{2} occurs as composition factor of M. Thus we have:

Lemma 1.6 Let β = (β_{1}, 0) as above. Then β satisfies 1.1 if and only if Hom_{T} (P_{2}, M) = (0). In this case M is P -torsionless if and only if it is P_{1}-torsionless.

Of course the last condition in 1.5 holds in particular if Hom_{T} (P_{1}, P_{2}) = (0) and hence we have:

Corollary 1.7 Suppose that P_{2} is a projective T -module and assume that Hom_{T} (P_{1}, P_{2}) = (0) and that τ_{P_{2}}(P_{1}) ⊆ ker β_{1}. Then β = β_{1} ⊕0 : P_{1} ⊕P_{2} → M satisfies 1.1, and H^{β} = H^{β_{1}}.

We are now ready to sharpen results [9, 2.28 and 2.32] in special cases. A ring H is called selfinjective if the regular module H_{H} is injective.

Theorem 1.8 Suppose β : P → M satisfies 1.1. Assume that R is a field, and that M is P -torsionless. If H is selfinjective, then every element of ℌ is isomorphic to some minimal right ideal X of H and the corresponding element Ĥ(X) of 𝔗_{M} is isomorphic to XM. Moreover 𝔗_{M} is up to isomorphism a complete set of the irreducible constituents of soc(M) as well as of hd(M), every indecomposable direct summand of M has simple socle and simple head, and taking socles respectively heads induce bijections between the isomorphism classes of indecomposable direct summands of M and the elements of 𝔗_{M}. Socle and head of the indecomposable direct summands of M are isomorphic if in addition H is a symmetric algebra.

Proof The first part of the theorem follows immediately from 1.2. By the same result 𝔗_{M} is up to isomorphism precisely the set of irreducible constituents of hd(M), hence of the head of the projective cover P_{1} of M. Since M is P -torsionless by assumption it is P_{1}-torsionless by 1.6. Thus every irreducible submodule of M is isomorphic to an irreducible constituent of hd(P_{1}), hence to an element of 𝔗_{M}. Let S be an element of 𝔗_{M}. By [9, 2.27 and 2.28] S = Ĥ(X) for some irreducible H-module X which we may assume to be a minimal right ideal of H since H is selfinjective by assumption. By Theorem 1.2 we have S = XM and is therefore contained in the socle of M and therefore 𝔗_{M} is up to isomorphisms a complete set of irreducible constituents of soc(M). The last assertion of the theorem follows now easily from the fact, that the head and the socle of an indecomposable projective H-modules are isomorphic if H is a symmetric algebra. □

As remarked in the introduction, using the Green-Sawada-Cabanes approach one can obtain a similar theorem replacing our Hypothesis 1.1 by the assumption that H is a symmetric finite dimensional algebra over some field and that every composition factor of the head of M is also a composition factor of the socle and vice versa. For details see [27, 2.8-2.9].

We turn now to the right inverse Ĥ = Ĥ^{β} = Ĥ_{M}^{β} of H^{β}. This as well depends on the choice of P in general since A_{P} does. Here are some results relating Ĥ_{M}^{β} and Ĥ_{M}^{β_{1}} for the minimal projective cover P_{1} of M. Being the composite functors of − ⊗_{H} M with A_{P} respectively A_{P_{1}} they are distinguished only by the latter functors. Obviously for V ∈ mod_{T} the module A_{P_{1}}(V) is an epimorphic image of A_{P} (V), since kerP ⊆ kerP_{1} and therefore t_{P} (V) ≤ t_{P_{1}}(V). We have shown:

Lemma 1.9 Let X ∈ mod_{H}. Then there is a natural epimorphism from Ĥ_{M}^{(P)}(X) onto Ĥ_{M}^{(P_{1})}(X).

The kernel of the map in 1.9 is obviously A_{P_{1}}(Ĥ_{M}^{(P)}(X)).

We shall prove now some auxiliary results for later use. First we want to deduce that the ideal J_{β_{1}} of the endomorphism ring E_{1} = End_{T} (P_{1}) is contained in the Jacobson radical Jac(E_{1}) of E_{1}. For this it suffices for example to assume that T is noetherian. In all our applications T has this property.

Lemma 1.10 Suppose that T is noetherian. Then

J_{1} ⊆ Jac(E_{1}).

Proof Since T is noetherian and P_{1} is finitely generated every surjective endomorphism of P_{1} is actually an isomorphism, see e.g. [6, 5.8]. From this one concludes immediately that {ϕ ∈ E_{1} | im ϕ ⊆ V } is a maximal right ideal of E_{1} for a maximal submodule V of P_{1} and one gets every maximal right ideal of E_{1} in this way. As a consequence

{ϕ ∈ E_{1} | im ϕ ⊆ Jac(P_{1})} = Jac(E_{1}).

Since β_{1} : P_{1} → M is a minimal projective cover and T is semiperfect,

ker β_{1} ⊆ P_{1}Jac(T) ⊆ Jac(P_{1}),

by general theory, (see e.g. [6, 6.21 and 5.6]). The claim follows now immediately from the definition of J_{1}. □

We assume from now on that T is noetherian. Lemma 1.10 implies the following useful criterion to determine if a given T -module is mapped to the zero module or not. We keep the notation of 1.5 and denote the class of all T -modules which are taken to isomorphic H-modules by the functors H^{β} and H^{β_{1}} by 𝔐^{β} and the class of all T -modules which are taken to the same H-module under all Hom-functors H^{β} by 𝔐.

Corollary 1.11 Let V ∈ 𝔐^{β}. Then

H^{β}(V) = (0) ⇔ Hom_{T} (P_{1}, V) = (0).

Proof This follows immediately from 1.10 and Nakayamas Lemma ([5, 5.7]). □

Remark 1.12 Lemma 1.5 and Corollary 1.7 provide criterions for 𝔐^{β} to be mod_{T}. In the applications which we are mostly interested in, we have often that H^{β} = H^{β_{1}} or we apply 1.11 to T -modules of which we know by some other reason that it belongs to 𝔐. Later we shall determine under special assumptions large classes of modules in 𝔐 (see 1.25).

Having a left inverse the functors Ĥ_{M}^{β} and Ĥ_{M}^{β_{1}} are injective on objects. In particular they take only the zero object to zero. Moreover, H-modules which are decomposable are taken to decomposable T -modules by these functors. However they do not preserve indecomposability in general.

Lemma 1.13 Let X ∈ mod_{H} be indecomposable, and let V be one of the T -modules X ⊗_{H} M, X ⊗_{E} P, Ĥ_{M}^{β}(X) or A_{P} (X ⊗_{E} P). Let V = V_{1} ⊕ V_{2} ⊕ ⋯ ⊕ V_{k} be a decomposition of V into a direct sum of indecomposable T -modules. Then there is an index i ∈ {1, …, k} such that the following holds (viewing X if needed as E-module via the epimorphism E → H):

(i) H_{M}(V_{i}) = X and H_{M}(V_{j}) = (0) for i ≠ j ∈ {1, …, k}. (ii) V_{j} ≅ V̸_{i} for i ≠ j ∈ {1, …, k}. (iii) Hom_{T} (P_{1}, V_{i}) ≠ (0) and Hom_{T} (P_{1}, V_{j}) = (0) for i ≠ j ∈ {1, …, k}.

Proof Since H_{M} is a left inverse of each of the four functors which appear as a choice in the definition of V from X by [9, 2.16], the H-module H_{M}(V) ≅ X is indecomposable by assumption. Since H_{M} preserves direct sums there is precisely one direct summand V_{i} which is taken by H_{M} to X, and all other direct summands V_{j}, j ≠ i are taken to the zero module. This proves part one. The second part of the lemma follows immediately and the third one using Result 1.11 for the first two choices of V, since they are taken back to X under the functor H^{β_{1}}.

For the remaining two choices for V we observe that the functor A_{P} preserves direct sums and the result follows easily using the projectivity of P_{1}. □

For the next lemma we assume that T is a finite dimensional algebra over some field (or more generally an artin algebra). Since P_{1} is the minimal projective cover of M, these two modules have the same head. The previous lemma implies now immediately:

Corollary 1.14 In the situation of Lemma 1.13 suppose that H^{β} = H^{β_{1}}, where T is a finite dimensional algebra over some field. Then no composition factor of the head of P_{1} hence of M occurs as composition factor of V_{j} for i ≠ j ∈ {1, …, k}.

There is a further application of [9, 2.21] in connection with Result 1.13:

Corollary 1.15 Keep the notation of Lemma 1.13 and suppose that M is P -torsionless. Let X be an indecomposable right ideal of H. Then

Ĥ_{M}^{β}(X) = XM

is indecomposable.

Proof By [9, 2.20] we have that Ĥ_{M}^{β}(X) = XM, and by [9, 2.21] H_{M}^{β} induces an isomorphism between the endomorphism rings of X and XM. Since X is indecomposable, its endomorphism ring is local, hence the endomorphism ring of XM is local as well and XM is indecomposable as desired. □

In the applications we have in mind, H is a selfinjective finite dimensional algebra over some field, and hence every irreducible H-module is isomorphic to a minimal right ideal of H. Note too that this result applies in particular to principal projective indecomposable H-modules of H. These are principal right ideals X = xH generated by an idempotent, and for those one has obviously

XM = xM ≅ x ⊗_{H} M = X ⊗_{H} M, (1.16)

such that in this case one gets Ĥ_{M}^{β}(X) already as tensor product X ⊗_{H} M provided, M is P -torsionless.

The following lemma is a refinement of Corollary 1.15:

Lemma 1.17 Let X be an indecomposable direct summand of an ideal of H and suppose that M is P -torsionless. Then

Ĥ_{M}^{β}(X) = XM = A_{P} (X ⊗_{H} M)

is indecomposable.

Proof The claim follows as in 1.15 observing that − ⊗_{H} M and t_{P} preserve direct sums. □

We now turn to the more specialized situation described in [9, section4]. So let O be a discrete complete valuation ring with quotient field K and residue field F. Let T_{K} be a semisimple K-algebra and let T_{O} be an O-order in T_{K}. We say V_{O} is an irreducible T_{O}-lattice if V_{K} = K ⊗_{O} V_{O} is an irreducible T_{K}-module.

Recall that an T_{F} -module V is liftable, if V = F ⊗_{O} U for some T_{O}-lattice U. Moreover, if U_{O}, V_{O} are T_{O}-lattices, then

Hom_{T_{K}}(U_{K}, V_{K}) ≅ K ⊗_{O} Hom_{T_{O}}(U_{O}, V_{O}) (1.18)

canonically. Similarly we have an homomorphism of F -spaces

F ⊗_{O} Hom_{T_{O}}(U_{O}, V_{O}) → Hom_{T_{F}} (U_{F} , V_{F}) (1.19)

which is injective but not necessarily surjective. The homomorphisms in the image of that map are called liftable, and similarly Hom_{T_{F}} (U_{F} , V_{F}) is said to be liftable, if this homomorphism is bijective.

We take M_{O} ∈ mod_{T_{O}} to be O-free and assume that the projective presentation β : P_{O} → M_{O} satisfies (1.1). We let H_{O} be End_{T_{O}}(M_{O}). Similarly we indicate for other objects that they are defined over O, by adding a subscript O. For R ∈ {K, O, F } and an object X_{O} we abbreviate R ⊗_{O} X_{O} by X_{R}. By [9, 4.7] 1.1 (for R = O) is equivalent to ker β_{K} having no irreducible constituent with M_{K} in common. Moreover if this holds, the decomposition matrix of H_{O} is a submatrix of the decomposition matrix of T_{O}, [9, 4.10].

In [9, 4.7] it was also shown that 1.1 holds for β_{R} = 1_{R} ⊗_{O} β for all choices of R ∈ {K, O, F }. The three resulting Hom-functors and their right inverses are distinguished by a suffix R, but usually we drop the suffix M_{R}, if no ambiguities arise.

We adopt the notation introduced at the beginning of the section but add a further index R, where R is as usual one of the rings F, O or K. In particular β_{1,R} : P_{1,R} → M_{R} is a projective cover of M_{R} for R = O or F, and β_{R} = β_{1,R} ⊕ 0 for all choices of R.

Lemma 1.20 Let β_{O} : P_{O} → M_{O} be a projective presentation of the T_{O}-lattice M_{O}. Then M_{R} is P_{R}-torsionless for R = K and O.

Proof This follows immediately from the fact that β_{K} is an epimorphism and that Hom_{T_{K}}(P_{K}, M_{K}) ≅ K ⊗_{O} Hom_{T_{O}}(P_{O}, M_{O}) . □

Remark 1.21 Note that the Hom-functor H_{O}^{β} does not take T_{O}-lattices to H_{O}-lattices in general, since Hom_{T_{O}}(P_{O}, V_{O})J_{β_{O}} may be a sublattice of Hom_{T_{O}}(P_{O}, V_{O}) such that the corresponding factor module has O-torsion, that is Hom_{T_{O}}(P_{O}, V_{O})J_{β_{O}} is not pure in Hom_{T_{O}}(P_{O}, V_{O}). As a consequence, whereas H_{R}^{β}(V_{R}) = R ⊗_{O} H_{O}^{β}(V_{O}) is still true for R = K and R = F, the dimension of H_{F}^{β}(V_{F}) may be larger as that of H_{K}^{β}(V_{K}), since the O-torsion part of H_{R}^{β}(V_{R}) is killed by tensoring with K but survives, if we tensor with F. In the next theorem we investigate what happens, if V_{O} is taken to a lattice by the Hom-functor H^{β_{1}}.

The next theorem tells us that for many applications we may neglect the fact, that different projective presentations produce different Hom-functors.

Theorem 1.22 Suppose that β_{1,O} : P_{1,O} → M_{O} satisfies 1.1. Let P_{2,O} be a projective T_{O}-lattice. Then

β_{R} = β_{1,R} ⊕ 0 : P_{1,R} ⊕ P_{2,R} → M_{R}

satisfies 1.1 for all choices of R if and only if the image of any homomorphism from P_{2,R′} to P_{1,R′} is in ker β_{R′}, for just one ring R′ in {K, O, F }. Morover, if 1.1 is satisfied by β_{R} and X_{O} = H_{O}^{β_{1}}(V) is an H_{O}-lattice, where V is a T_{O}-lattice, then

H_{R}^{β}(V) ≅ H_{R}^{β_{1}}(V) ≅ R ⊗_{O} X

for all choices of R.

Proof By [9, 4.7] and 1.5 it suffices to show for the first part of the theorem that the condition for R′ = F implies the condition for R′ = O. But if there is a homomorphism from P_{2,O} to P_{1,O}, whose image is not contained in ker β_{1,O} we get a nonzero homomorphism from P_{2,O} to M_{O}, and hence from P_{2,F} to M_{F} as well. But this lifts to an endomorphism of P_{F} = P_{1,F} ⊕ P_{2,F} which does not leave ker β_{F} invariant, a contradiction.

To prove the second part of the theorem we observe that H_{K}^{β} = H_{K}^{β_{1}}, since

P_{K} = ker(β_{K} ⊕ M_{K}) = P_{1,K} ⊕ P_{2,K},

and M_{K} has no irreducible composition factor with P_{2,K} in common, since P_{2,K} is contained in ker β_{K}. Thus every homomorphism P_{1,K} → P_{2,K} factors through J_{1,K} = J_{β_{1,K}} and we are done by 1.5.

Thus we assume that X = H_{O}^{β_{1}}(V) is free as O-module. As we have seen in 1.5, the functor H_{O}^{β} takes V to an epimorphic image X̂ of X. Obviously KX ≅ H_{K}^{β_{1}}(KV) and similarly KX̂ ≅ H_{K}^{β}(KV), hence by our argument above KX ≅ KX̂. Since tensoring by K is exact, we conclude that the kernel of the epimorphism τ from X onto X̂, that is τ is an isomorphism.

If σ_{F} : P_{1,F} → P_{2,F} and ρ_{F} : P_{2,F} → V_{F} = F ⊗_{O} V are given, we can lift them to get maps σ_{O} and ρ_{O} for the corresponding T_{O}-lattices. But by the above and 1.5 we can express ρ_{O}σ_{O} as a linear combination of the composite of maps ϕ_{O} ∈ J_{1,O} and ψ_{O} : P_{1,O} → V and we have the following commutative diagram (indicating by the sum symbol that we have to take linear combinations of the composite maps):

[figure: P_{1,O}; σ_{O}; P_{2,O}; ker_{1,O}; P_{1,O}; ∑ ϕ_{O}; ∑ ψ_{O}; V,; ρ_{O}] (1.23)

hence ρ_{F} σ_{F} is a linear combination of the corresponding composite maps ψ_{F} ϕ_{F}. Using 1.5 one checks now easily that

H_{F}^{β}(V_{F}) ≅ H_{F}^{β_{1}}(V_{F}) ≅ X_{F} .
□

For the right inverses of the Hom-functors we have a similar result:

Theorem 1.24 Assume that β_{O} : P_{O} → M_{O} satisfies 1.1 and let β_{1,O} : P_{1,O} → M_{O} be the projective cover of M_{O}. Let X_{O} be an H_{O}-lattice and V_{O} = X_{O} ⊗_{H_{O}} M_{O}. Then the following holds:

(i) There is a natural isomorphism between K ⊗_{O} V_{O} and X_{K} ⊗_{H_{K}} M_{K} = V_{K}. (ii) V_{K} is P_{1,K} und P_{K}-torsionless, and

Hom_{T_{K}}(P_{1,K}, V_{K}J_{1,K}) = (0) = Hom_{T_{K}}(P_{K}, V_{K}J_{1,K}).

(iii) Hom_{T_{O}}(P_{1,O}, V_{O})J_{1,O} = (0) = Hom_{T_{O}}(P_{O}, V_{O})J_{1,O}. (iv) The O-torsion submodule of V_{O} is precisely the P_{1,O}-torsion submodule t_{P_{1,O}}(V_{O}) of

V_{O} and coincides with t_{P_{O}}(V_{O}).

(v) Ĥ_{R}^{β}(X_{R}) = Ĥ_{R}^{β_{1}}(X_{R}) = V_{R}/t_{P_{1,R}}(V_{R}) for all choices of R.

Proof Part one of the theorem is easy and left to the reader. To prove part two we may assume that X_{K} is an irreducible H_{K}-module, since H_{K} is semisimple. But then X_{K} = eH_{K} for some primitive idempotent e of H_{K} and V_{K} ≅ eM_{K} by [9, 2.20 (ii)]. Now part two follows from 1.20 and the fact that there are no homomorphisms from ker β_{1,K} respectively from ker β_{K} to M_{K}. Part three follows now immediately.

To prove part four let U be the O-torsion submodule of V_{O}. Then obviously Y = Hom_{T_{O}}(P_{1,O)}, U) is an O-torsion module as well. Suppose Y ≠ (0). Then by 1.11 the proper epimorphic image H_{O}^{β_{1}}(U) of Y is O-torsion and a submodule of

H_{O}^{β_{1}}(V) = H_{O}^{β_{1}} Ĥ_{O}^{β_{1}}(X) = X,

which in turn is O-torsionfree by assumption. But this is a contradiction. Thus Y = (0).

K ⊗_{O} (V_{O}/U) ≅ K ⊗_{O} V_{O} = V_{K}

which is P_{1,K}- and P_{K}-torsionless by part three, hence V_{O}/U is P_{1,O}- and P_{O}-torsionless, as desired. Now part four follows from [1, section 5]. Part five for R = O or K is an obvious consequence. For R = F we observe first that homomorphisms from projective modules are always liftable, and hence F ⊗_{O} U is P_{1,F} - and P_{F} -torsionless (but not necessarily equal to t_{P_{1,F}} (F ⊗_{O} V_{O}) or to t_{P_{F}} (F ⊗_{O} V_{O})). However, let P_{F} = P_{1,F} ⊕ P_{2,F} as in the beginning of this section. But now P_{2,K} and V_{K} do not have a composition factor in common, since P_{2,K} belongs to ker β_{K}. Thus V_{O}/U is in ker(P_{2,O}), and the same holds if we tensor by F (note that U splits from V_{O} as O-space). From this, part five follows for F = R easily. □

We remark that in part five of the theorem Ĥ_{F}^{β}(X_{F}) = (F ⊗_{O} V)/t_{P_{F}} (F ⊗_{O} V) is not in general equal to F ⊗_{O} Ĥ_{O}^{β}(X_{O}), since the latter might still have a submodule which is in ker(P_{F}). All what we claim here is, that the functors for β and β_{1} send also on the F -level X_{F} to the same T_{F} -module.

As remarked in 1.21 we cannot expect that the functor H_{O}^{β} takes lattices to lattices. But there are large classes of lattices where this happens, and where we can hence apply Theorem 1.22.

Remark 1.25 Let R as above. Take the class ℑ_{M}(Ĥ)_{R} of all T_{R}-modules V_{R} which arise in the following way: Let Ĥ_{O} be any of the four right inverses of the Hom-functor H_{O}^{β} for any projective presentation β_{O} : P_{O} → M_{O}, satisfying 1.1, and take V_{R} = R ⊗_{O} U_{O}, where U_{O} is any sublattice or O-free epimorphic image of Ĥ_{O}(X_{O}), and X_{O} is any H_{O}-lattice. Then all Hom-functors derived from projective presentations of M_{O} satisfying 1.1 coincide on ℑ_{M}(Ĥ)_{R}. Thus ℑ_{M}(Ĥ)_{R} ⊆ 𝔐_{R}. (Recall that 𝔐_{R} consists of all T_{R}-modules which are taken to the same H_{R}-modules by all Hom-functors H^{β}).

Proof We observe that by part three of the Theorem 1.24

Hom_{T_{O}}(P_{1,O}, X_{O} ⊗ M_{O})J_{1,O} = (0), (1.26)

hence the same holds, if X_{O} ⊗ M_{O} is replaced by a sublattice or epimorphic image (regardless, if it is O-free or not). Since all T_{O}-lattices Y = Ĥ_{O}(X_{O}) arise in this way too, we have the same result if we replace X_{O} ⊗ M_{O} by sublattices or epimorphic images of Y in 1.26, in particular for all members of ℑ_{M}(Ĥ)_{O}. By [9, 4.12] H^{β_{1}}O(V_{O}) is free as O-module for V_{O} ∈ ℑ_{M}(Ĥ)_{O} and the result follows from 1.22. □

Remark 1.27 Note that indeed [9, 4.12] describes an even wider class of T_{O}-lattices to which we may apply Theorem 1.22. Note too, that in particular all sublattices V_{O} of M_{O} belong to this class by similar arguments, since M_{O} is P_{O} torsionless by 1.20 and hence there can be no non trivial homomorphisms from ker β_{R} to V_{R} for R = K and hence for R = O.

In [9, 4.17] there is another mistake which I would like to straighten out here. Let X, Y be pure right ideals of H_{O}, thus they have O-complements in H_{O}. The purification of a sublattice U of M_{O} is defined to be \sqrt{U} = KU ∩ M_{O}. We know by 1.20 that M_{O} is P_{O}-torsionless. By [9, 2.20 and 4.14]

H_{R}(XM) ≅ H_{R}(R ⊗_{O} \sqrt{X_{O}M_{O}}) ≅ X_{R}, (1.28)

where R is one of the rings F, O, or K, and a similar statement holds for Y_{R}. Note that the lattices XM_{R} and Y M_{R} belong to 𝔐_{R} by Remark 1.25, hence we have not to specify the projective presentation of M_{O} which we use.

In [9, 2.21] it was shown that the functor H_{R} = H_{M_{R}} induces an isomorphism, also denoted by H_{R}:

H_{R} : Hom_{T_{R}}(X_{R}M_{R}, Y_{R}M_{R}) ⟶^{∼} Hom_{H_{R}}(X_{R}, Y_{R}). (1.29)

In [9, 4.17] this result was extended to the purification of the corresponding sublattices of M_{O}. However the result needs an additional assumption, and the hypotheses in [9, 4.17], that M_{O} is P_{O}-torsionless can be dropped in view of Result 1.20:

Theorem 1.30 Suppose that β_{O} : P_{O} → M_{O} satisfies 1.1. Let X = X_{O} and Y = Y_{O} be pure right ideals of H_{O} = End_{T_{O}}(M_{O}). Suppose every homomorphism from X_{O} to H_{O} is given as λ_{h} for some h ∈ H_{O}, where λ_{h} : X_{O} → H_{O} is left multiplication by h. Then the associated Hom-functor H_{R} induces an isomorphism

H_{R} : Hom_{T_{R}}(R ⊗_{O} \sqrt{X_{O}M_{O}}, R ⊗_{O} \sqrt{Y_{O}M_{O}}) ⟶^{∼} Hom_{H_{R}}(X_{R}, Y_{R}), (1.31)

for R = O and K. If in addition M_{F} is P_{F} -torsionfree and Hom_{H_{F}} (X_{F} , Y_{F}) is liftable then Formula 1.31 holds for R = F as well.

Proof By [9, 2.22 ii)] the homomorphism

(1.32) H_{O} : Hom_{T_{O}}(\sqrt{X_{O}M_{O}}, \sqrt{Y_{O}M_{O}})

→ Hom_{H_{O}}(H_{O}(\sqrt{X_{O}M_{O}}), H_{O}(\sqrt{Y_{O}M_{O}}))
= Hom_{H_{O}}(X_{O}, Y_{O})

is injective. Let ϕ : X_{O} → Y_{O} be H_{O}-linear. We use the natural embedding i_{O} : Y_{O} → H_{O} to find h ∈ H_{O} such that i_{O}ϕ = λ_{h}, where λ_{h} is left multiplication by h. By [9, 2.21] ϕ = λ_{h} = H_{O}(ψ) for some T_{O}-linear map ψ : X_{O}M_{O} → Y_{O}M_{O}. We denote by ψ_{K} an extension of 1 ⊗_{O} ψ : X_{K}M_{K} → Y_{K}M_{K} to an endomorphism of M_{K}. Obviously we can choose ψ_{K} to be the left multiplication λ_{h} by h ∈ H_{O} ⊆ H_{K}. Let m ∈ \sqrt{X_{O}M_{O}}, that is m ∈ M_{O} and ℘^{k}m ∈ X_{O}M_{O} for some natural number k, where ℘ is a generator of the unique maximal ideal of the complete discrete valuation ring O. So

℘^{k}(ψ_{K}(m)) = ψ_{K}(℘^{k}m) = h℘^{k}m = ℘^{k}hm

is an element of Y_{O}M_{O}. On the other hand

ψ_{K}(m) = λ_{h}(m) = hm ∈ M_{O},

and hence ψ_{K}(m) ∈ M_{O} ∩ Y_{K}M_{K} = \sqrt{Y_{O}M_{O}}. We have shown that the restriction of ψ_{K} to \sqrt{X_{O}M_{O}} extends the T_{O}-linear map ψ : X_{O}M_{O} → Y_{O}M_{O} to a T_{O}-linear map from \sqrt{X_{O}M_{O}} to \sqrt{Y_{O}M_{O}}. This shows that H_{O} in 1.32 is surjective, and hence an isomorphism. Thus our claim holds for R = O and obviously for R = K as well.

Now assume that M_{F} is P_{F} -torsionless, and that Hom_{H_{F}} (X_{F} , Y_{F}) is liftable. Let \overline{ϕ} : X_{F} → Y_{F} be H_{F} -linear. By assumption \overline{ϕ} = 1_{F} ⊗_{O} λ_{h} = λ\overline{h} for some h ∈ H_{O} setting \overline{h} = 1_{F} ⊗_{O} h ∈ H_{F}. We set ψ = H_{O}^{−1}(λ_{h}) and conclude that H_{F} (1_{F} ⊗_{O} ψ) = \overline{ϕ}. Thus

(1.33) H_{F} : Hom_{T_{F}} (F ⊗_{O} \sqrt{X_{O}M_{O}}, F ⊗_{O} \sqrt{Y_{O}M_{O}})

→ Hom_{H_{F}} (H_{F} (F ⊗_{O} \sqrt{X_{O}M_{O}}), H_{F} (F ⊗_{O} \sqrt{Y_{O}M_{O}}))
= Hom_{H_{F}} (X_{F} , Y_{F})

is surjective by [9, 4.12, ii]. By [9, 2.22 ii] this map has to be injective too, finishing the proof of the theorem. □

Remark 1.34 In the situation of the theorem we have indeed shown: The functor H_{F} induces an epimorphism

H_{F} : Hom_{T_{F}} (F ⊗_{O} \sqrt{X_{O}M_{O}}, F ⊗_{O} \sqrt{Y_{O}M_{O}}) → Hom_{H_{F}} (X_{F} , Y_{F}),

provided Hom_{H_{F}} (X_{F} , Y_{F}) is liftable. We used the hypothesis, that M_{F} is P_{F} -torsionless only to ensure, that this map is injective too. There are however other sufficient conditions for this to hold. For instance, our proof shows also that the restriction of H_{F} to the subspace F ⊗_{O} Hom_{T_{O}}(\sqrt{X_{O}M_{O}}, \sqrt{Y_{O}M_{O}}) of Hom_{T_{F}} (F ⊗_{O} \sqrt{X_{O}M_{O}}, F ⊗_{O} \sqrt{Y_{O}M_{O}}) is injective. Thus, if Hom_{T_{F}} (F ⊗_{O} \sqrt{X_{O}M_{O}}, F ⊗_{O} \sqrt{Y_{O}M_{O}}) is liftable, H_{F} is an isomorphism.

Recall that an arbitrary ring S is selfinjective if the regular representation S_{S} is an injective S-module. If so every homomorphism from a right ideal X into S can be extended to an endomorphism of S and hence is left multiplication by some element of S. In general H_{O} is not selfinjective, but in all our applications H_{F} is not only selfinjective, but even stronger a symmetric algebra. That means that there is a symmetric nondegenerate bilinear form < , > on H_{F} which is associative that is < xh , y >=< x , hy > for all h, x, y ∈ H_{F}. In our applications this form can be realized over O: We have an associative bilinear form < , >: H_{O} × H_{O} → O such that its Gram determinant with respect to an O-basis of H_{O} is a unit in O which induces the form on H_{F}. If such a form on H_{R} is given, we define the subset X^{⊥} orthogonal to the subset X of H_{R} to be {y ∈ H_{R} |< y , x >= 0 ∀x ∈ X}, (R ∈ {K, O, F }). We shall show now, that the existence of such a form on H_{O} is sufficient for the assumption in Theorem 1.30 to hold. We first need an auxiliary result on left annihilators which generalizes well known facts on bilinear forms on algebras over fields. For convenience we include a proof.

For a ring S and a subset X of S we denote the left annihilator of X in S by l(X). Similarly we define the right annihilator r(X) of X.

Lemma 1.35 Let

< , >: H_{O} × H_{O} → O : (x, y) ↦< x , y >∈ O

be an associative bilinear form whose Gram determinant is a unit in O. Let X be a right ideal of H_{O}. Then the left annihilator l(X) of X is a pure left ideal of H_{O} of O-rank rk(H_{O}) − rk(X). Moreover l(X) = l(\sqrt{X_{O}}) = X^{⊥}. If in addition X is pure in H_{O}, then

l(X_{F}) = F ⊗_{O} l(X_{O}).

In particular dim_{F} (l(X_{F})) = dim_{F} (H_{F}) − dim_{F} (X_{F}).

Proof It is easy to check that l(X) = l(\sqrt{X_{O}}), and that l(X) is a pure left ideal of H_{O}.

Let X be pure in H_{O}. Then X_{R} ≤ H_{R} for R = F or K. We have obviously induced forms on H_{R} which again are denoted by < , >. Since the Gram determinant of < , > over O is a unit these forms are nondegenerate. Obviously we have also H_{O}^{⊥} = (0). Now for R ∈ {K, O, F } and y ∈ H_{R} we have

yX_{R} = (0) ⇔< yX_{R} , H_{R} >= (0) ⇔< y , X_{R}H_{R} >= (0) ⇔< y , X_{R} >= (0) ⇔ y ∈ X_{R}^{⊥},

and hence l(X_{R}) = X_{R}^{⊥}.

The form < , > induces an homomorphism from H_{R} into the dual space H_{R}^{∗} = Hom_{R}(H_{R}, R) sending h ∈ H_{R} to

f_{h} : H_{R} → R : x ↦< x , h >∈ R,

and one checks easily that this map is H_{R}-linear on the left. Moreover it is an isomorphism, its matrix with respect to a dual pair of bases of H_{R} and H_{R}^{∗} given as Gram matrix of < , > with respect to the given basis of H_{R}. Thus it is an isomorphism of left H_{R}-modules. The embedding of X_{R} = R ⊗_{R} X into H_{R} induces an left H_{R}- module epimorphism from H_{R}^{∗} onto X_{R}^{∗} = Hom_{R}(X_{R}, R), given by restricting linear maps from H_{r} to X_{R}. One checks easily that the kernel of the composite map

H_{R} → H_{R}^{∗} → X_{R}^{∗}

is X_{R}^{⊥} = l(X_{R}). Observe that R ⊗_{O} l(X) ⊆ l(X_{R}). The lemma now follows comparing dimensions. □

Since the induced form on H_{R} for R = K or F is nondegenerate, H_{R} is a Frobenius algebra. Such algebras are in particular selfinjective, (see e.g. [6, 9.9]), and hence every homomorphism from X_{R} to H_{R} is given by left multiplication by elements of H_{R}. In the situation of the previous lemma we can indeed lift this property to the O-level:

Theorem 1.36 Let < , > be an associative bilinear form on H_{O} whose Gram determinant is a unit. Let X be a pure right ideal of H_{O}. Then

Hom_{H_{F}} (X_{F} , H_{F}) ≅ F ⊗_{O} Hom_{H_{O}}(X_{O}, H_{O})

and every homorphism from X to H_{O} is given by left multiplication by some element of H_{O}.

Proof Let ψ : X_{F} → H_{F} be H_{F} -linear. Since H_{F} is a Frobenius algebra we find h̄ ∈ H_{F} such that ψ = λh̄. Let h ∈ H_{O} be such that h̄ = 1_{F} ⊗_{O} h. Then ψ = λh̄ = 1_{F} ⊗_{O} λ_{h}, and hence every homomorphism from X_{F} into H_{F} is liftable.

Let ϕ : X → H_{O} be H_{O}-linear, and let ϕ̄ = 1_{F} ⊗_{O} ϕ : X_{F} → H_{F}. The extension of ϕ to a homomorphism from X_{K} to H_{K} is again denoted by ϕ. Since H_{K} is a Frobenius algebra we find a ∈ H_{K} such that ϕ = λ_{a}. Since a ∈ H_{K} we find 0 ≤ k ∈ ℤ such that ℘^{k}a = b ∈ H_{O}, where again ℘ is a generator of the unique maximal ideal of the valuation ring O. We may choose a and b such that k is minimal such that ϕ = λ_{a} and ℘^{k}a = b ∈ H_{O}.

Suppose k ≠ 0. Note that λ_{b} = λ_{℘^{k}a} = ℘^{k}λ_{a} = ℘^{k}ϕ. We have

bx = ℘^{k}ax = ℘^{k}λ_{a}x = ℘^{k}ϕx ∈ ℘^{k}H_{O} ∀x ∈ X,

and hence

b̄ = 1_{F} ⊗_{O} b ∈ l(X_{F}) = F ⊗_{O} l(X),

by lemma 1.35. Thus we find c ∈ l(X) with b̄ = 1_{F} ⊗_{O} c, and hence d ∈ H_{O} such that b = c + ℘d. Since b and ℘d differ only by something in the annihilator of X (namely c), we have λ_{b} = λ_{℘d} on X, hence

℘^{k}ϕ = λ_{b} = λ_{℘d} = ℘λ_{d}

on X, and therefore

ϕ = ℘^{1−k}λ_{d}

a contradiction to the minimality of k. Thus k = 0. But then a = b ∈ H_{O} and ϕ = λ_{a} = λ_{b} as desired. □

In the situation of Theorem 1.30 we need later some additional information in the special case, where Y_{R} = H_{R}. Here all the Hom-sets in Formulas 1.29 and 1.31 inherit left H_{R}-module structures by viewing M_{R} and H as left H-modules.

Lemma 1.37 In the situation of 1.30 let Y_{R} be the regular representation H_{R} and suppose that H_{R} is an isomorphism for all choices of R. Then H_{R} in 1.29 and 1.30 are isomorphisms of left H_{R}-modules.

Proof Obviously we may assume R = O and omit the index R. Let V be XM or \sqrt{XM}, and let H = H^{β}. Since V is contained in M, the ideal J_{β} annihilates Hom_{T} (P, V) and (compare 1.27). Thus (embedding mod_{H} into mod_{E} by the inflation functor) Hom_{T} (P, M) = H(M) = H and Hom_{T} (P, V) = H(V).

Let f : V → M be T -linear, α ∈ Hom_{T} (P, V), h ∈ H, and p ∈ P. Then

(H(hf))(α) : p → (hf)(α(p)) = h(f ∘ α(p)),

which is as well the image of p under (h(H(f)))(α), hence

H(hf) = hH(f),

as desired, since the canonical isomorphism Hom_{T} (P, M) → H is an H-E-bimodule map. □

The question arises if the situation of 1.1 comes up often. The following remark, which was pointed out to me by Gerhard Hiss, shows that we can produce situation 1.1 in the general setting described in [9, section 4].

Remark 1.38 Let P_{O} be a projective T_{O}-lattice. Let M be a subset of irreducible T_{K}- modules occurring as constituent of P_{K}, and suppose that not every composition factor of P_{K} is isomorphic to an element of M. Let U be the sum of all submodules of P_{K} whose constituents belong to M. Thus no constituent of a complement of U in P_{K} belongs to M. Take U_{O} = U ∩ P_{O}. Then U_{O} is a pure sublattice of P_{O} and U = U_{K}. Moreover, setting M_{O} = P_{O}/U_{O} and β : P_{O} → M_{O} to be the natural epimorphism, we have constructed a projective presentation of M_{O} satisfying 1.1. Obviously every situation as described in [9, section 4] and in Results 1.20 through 1.37 arises in this way.

2 Hom-functors and Harish-Chandra theory

We shall give now a first application which is important for finite reductive groups but can be formulated in a more general context.

Let G be a finite group and let F be a field. For a subfactor L of G, given by a pair (P, U) of subgroups of G with U ⊴ P and L = P/U, we denote Harish-Chandra induction (HC-induction for short) from mod_{F L} to mod_{F G} by R_{L}^{G} and its right adjoint functor, Harish-Chandra truncation (HC-restriction for short), by T_{L}^{G}. Thus R_{L}^{G} lifts an F L-module to an F P -module by letting U act trivially and then induces it from P to G, and dually T_{L}^{G} restricts a F G-module to P and takes U-fixpoints to yield an F L-module ([15]). If in addition the order of U is invertible in the field F, a condition which we shall usually assume in the following, T_{L}^{G} is adjoint on both sides to R_{L}^{G}. In [13] those functors were used to generalize Greens theory of vertices and sources from subgroups of G to subfactors.

We mostly adopt the notation introduced in [13] and [10] and refer the reader to those papers for details. For two subfactors L, K of G, given by the pairs (P, U) and (Q, V) of subgroups of G, we define the shift intersection to be given by the pair

L ∩^{∼} K = ((P ∩ Q)V, (U ∩ Q)V).

This is a relation which is neither commutative nor associative, however L ∩^{∼} K ≅ K ∩^{∼} L naturally. How to conjugate subfactors is obvious. We have a Mackey decomposition theorem [13, 1.4]:

Theorem 2.1 Let L, K be a subfactors of G given by the pairs (P, U), Q, V respectively. Suppose that the orders of U and V are invertible in F. Let M be an F L-module. Then

T_{K}^{G} ∘ R_{L}^{G}(M) ≅ ⨁_{x∈P \G/Q} R_{L^{x}∩^{∼}K}^{K} ∘ T_{K∩^{∼}L^{x}}^{L^{x}}(M^{x})

where M^{x} denotes the conjugate module for the conjugate subfactor L^{x} = x^{−1}Lx of G, and P \G/Q is a set of P -Q-double coset representatives in G.

We remark that in Theorem 2.1 all the intersections are again given by pairs (P′, U′) of subgroups of G (with U′ ⊴ P) such that the order of U′ is invertible in F.

A system M of subfactors of G is called a Mackey system, if it is closed under shift intersection and conjugation and if it contains G. We say that M is p-modular, p a prime, if all the subgroups U determining the elements of M are p-regular, that is their order is not divisible by p. Thus if F has characteristic p and M is p-modular, we may apply 2.1 to subfactors in M. For L ∈ M the set

M_{L} = {H ∩^{∼} L | H ∈ M}

defines a Mackey system in L, which is p-modular, if M is so. Note that M_{L} ⊆ M.

Thus fix now M to be a Mackey system of subfactors of G and assume that it is pmodular, if F has characteristic p. An F G-module M is called cuspidal (with respect to M), if T_{L}^{G}(M) = (0) for all subfactors L ≠ G in M. Observe that T_{L}^{G}(M) = (0), if L is actually a subgroup of G, given as pair (L, (1)). Thus, if M contains a proper subgroup, then F G has no cuspidal modules. If L ∈ M and M is an F L-module, we say M is cuspidal (with respect to M), if it is cuspidal with respect to M_{L}. Of course, even if M contains a proper subgroup and hence does not have cuspidal representations, the same might be not true for M_{L}. We point out that the definition of “cuspidal” in [13] and [10] is slightly different excluding subfactors in M which are actually subgroups of G, hence correspond to a pair (L, (1)). The adjusted definition seems to be more consistent and practical. Indeed [13, 4.6] is false under the old definition but holds assuming ours here.

The following theorem relates Harish-Chandra theory and the results of [9]:

Lemma 2.2 Let M be a Mackey system for G which is p-modular, if the characteristic of the field F is p > 0, and let L be a subfactor of G contained in M. Let M̃ be an irreducible cuspidal F L-module, β̃ : X̃ → M̃ a minimal projective cover of M̃. Then

R_{L}^{G}(β̃) : R_{L}^{G}(X̃) → R_{L}^{G}(M̃)

satisfies Hypothesis 1.1.

Proof The lemma generalizes a result of M. Geck, G. Hiss and G. Malle in [28, 2.2] on finite reductive groups G, where M is given as the set of Levi subgroups L considered as subfactors (P, U), where P = LU is a parabolic subgroup of G with unipotent radical U and Levi complement L. The proof there (based on [15, 2.7]) carries easily over to the general case. For convenience we shall give details here.

We set β = R_{L}^{G}(β̃), X = R_{L}^{G}(X̃) and M = R_{L}^{G}(M̃). We adopt the notation of section one (replacing P there by our module X). Thus E = End_{F G}(X), H = End_{F G}(M), E_{β} = {ϕ ∈ E | ϕ(ker β) ≤ ker β} and J_{β} = {ψ ∈ E | im ψ ≤ ker β}. We have a map

β^{∗} : E → Hom_{F G}(X, M) : ϕ ↦ β ∘ ϕ,

which is surjective by projectivity of X, and which has kernel J_{β}. Thus

dim_{F} Hom_{F G}(X, M) = dim_{F} E − dim_{F} J_{β}.

On the other hand

dim_{F} H = dim_{F} E_{β} − dim_{F} J_{β},

since H ≅ E_{β}/J_{β} canonically. This implies now that E_{β} = E if and only if

dim_{F} Hom_{F G}(X, M) = dim_{F} Hom_{F G}(M, M).

We now apply adjointness of the functors R_{L}^{G} and T_{L}^{G} and Mackey decomposition 2.1 to get:

(2.3)  Hom_{F G}(M, M) ≅ ⨁_{x∈P \G/P} Hom_{F L}(M̃, R_{L^{x}∩^{∼}L}^{L} ∘ T_{L∩^{∼}L^{x}}^{L^{x}}(M̃^{x})

= ⨁_{x∈N_{G}(P,U)∩P \G/P} Hom_{F L}(M̃, M̃^{x}),

where L is given by the pair (P, U) and N_{G}(P, U) is defined to be the set {x ∈ G | L^{x} ∩^{∼} L = L} of elements of G, which “shift” normalize the pair (P, U). The second equation follows from the fact, that M̃ is cuspidal, hence truncation to every proper subfactor of L in M_{L} is zero. Similarly we see:

Hom_{F G}(P, M) ≅ ⨁_{x∈N_{G}(P,U)∩P \G/P} Hom_{F L}(P̃, M̃^{x}). (2.4)

Since M̃ is irreducible, dim_{F} Hom_{F G}(M̃, M̃^{x}) is one, if M̃ ≅ M̃, and zero otherwise. Since P̃ is the minimal projective cover of M̃, we have similarly dim_{F} Hom_{F G}(P̃, M̃^{x}) is one, if M̃ ≅ M̃, and zero otherwise. The lemma follows. □

In the situation of the theorem we have now the functors H = H_{M} : mod_{F G} → mod_{H} and Ĥ = Ĥ_{M} : mod_{H} → mod_{F G} described in section one [9, 2.28] implies the result [28, 2.4] of Geck, Hiss and Malle (again in slightly generalized form):

Corollary 2.5 The functors H and Ĥ provide a bijection between the isomorphism classes of the irreducible F G-modules occurring in the head of M = R_{L}^{G}(M̃) and the isomorphism classes of irreducible H-modules.

Suppose now we have found all irreducible H-modules. Application of the functor Ĥ will then produce the corresponding irreducible F G- modules. Another way is to exhibit primitive idempotents e in H. Then eM is an indecomposable direct summand of M and has simple head. The irreducible F G-module eM/Jac(eM) corresponds to the irreducible head of the projective indecomposable H-module eH. All this was shown in [9, 2.32].

We relate this to the notion of semisimple vertices and sources, which was introduced in [13] based on work of Hiss [34]. An irreducible F G-module M is weakly L-projective (L ∈ M), if T_{L}^{G}(M) ≠ (0), and L is called a semisimple vertex of M, if it is minimal with this property. By [13, 4.9] there is then an irreducible cuspidal F L-module (unique up to conjugation by certain group elements) such that M is a direct summand of the head of R_{L}^{G}(N). N is called a semisimple source of M. If the Mackey system M is in particular nice ([13, 4.13]), then the isomorphism classes of irreducible F G-modules are partitioned into subsets labelled by semisimple vertices and sources. In this case we get a full set of nonisomorphic irreducible F G-modules with semisimple vertex L and semisimple source N precisely as set of nonisomorphic components of hd(M) = hd(R_{L}^{G}(N)), and hence by exhibiting a full set of nonisomorphic irreducible H-modules.

3 Double centralizer theorems

We continue with our general discussion. We shall see later, that the main result of this section provides the general background for a paper of Takeuchi [45] . We begin with an auxiliary lemma. Thus for the moment let A be a finite dimensional semisimple K-algebra and assume that K is a splitting field for A. For M ∈ mod_{A} and an idempotent e of A we consider the space Me as eAe-module, where the eAe-action is given by restriction. Then the following result holds:

Lemma 3.1 Assume that Se ≠ (0) for every composition factor S of M. Then the restriction of A-endomorphisms of M to the subset Me of M defines an isomorphism

End_{A}(M) ≅ End_{eAe}(Me).

Proof Obviously the restriction ϕ ↦ ϕ|_{Me} is an algebra homomorphism from End_{A}(M) into End_{eAe}(Me). The condition on e implies, that the A-module eA is a generator for M, that is M is an epimorphic image of a direct sum of copies of eA. From this we conclude immediately, that MeA = M, and therefore the restriction map above is injective. Moreover acting by primitive central idempotents one sees easily, that we may assume that A is simple. Thus M = S ⊕ S ⊕ ⋯ ⊕ S, where S is the unique simple A-module. Let the number of direct summands in this sum be m, and let e = e_{1} + ⋯ + e_{k} be a decomposition of e into a sum of primitive orthogonal idempotents. Using Schur’s lemma we conclude that End_{A}(M) is the matrix algebra M_{m}(K). Note that by the Wedderburn theorems k ≤ dim_{K}(S). Moreover by Fitting’s theorem (see e.g. [41, 1.4]) we have

Me ≅ Hom_{A}(eA, M) = Hom_{A}(eA, S) ⊕ ⋯ ⊕ Hom_{A}(eA, S) ≅ Se ⊕ ⋯ ⊕ Se,

as K-space. On the other hand Se is a k-dimensional K-subspace of S, and computing dim_{K}(eAe) to be k^{2} one concludes that Se is an irreducible eAe-module. Therefore Me is a direct sum of m copies of the irreducible eAe-module Se, and hence

dim_{K}(End_{eAe}(Me)) = m^{2} = dim_{K}(End_{A}(M)).

The lemma follows. □

We return now to the situation of section one and keep the notation introduced there. As usual R is one of the rings K, O, F. In particular, if not stated otherwise we assume that modules for an O-order (e.g. T_{O}, or H_{O}) are free and finitely generated over O, (thus are lattices). We remind the reader of the notion of liftable modules and homomorphisms (see 1.19).

Given a T_{R}-module V the annihilator ann_{T_{R}}(V) of V in T_{R} consists of all elements t ∈ T_{R} which annihilate V, that is which satisfy vt = 0 for all v ∈ V. Note that ann_{T_{R}}(V) is an ideal, and that we have always a canonical injective algebra homomorphism from T/ann_{T_{R}}(V) into the bicentralizer End_{E}(V), where E = End_{T_{R}}(V). The module V is said to be balanced or has the double centralizer property, if this map is surjective too, that is if End_{E}(V) ≅ T_{R}/ann_{T_{R}}(V). The ring T is called balanced, if every right T -module is balanced. It is shown in [24, III.3.2], that this is equivalent to every left T -module being balanced. Note that finite dimensional semisimple algebras over fields are always balanced. This can be seen easily, using Wedderburn’s structural theory and Fitting’s lemma. It is also a special case of a result of Nesbitt and Thrall [42], which states that uniserial rings are balanced. For details see [24]. As an consequence we have the following auxiliary result:

Lemma 3.2 Let M = M_{O} be a T_{O}-lattice, and write M_{R} = R ⊗_{O} M_{O} for R = F, K. Set H_{O} = End_{T_{O}}(M_{O}) and H_{R} = R ⊗_{O} H_{O}. Suppose that End_{T_{F}} (M_{F}) is liftable, that is H_{F} = End_{T_{F}} (M_{F}). Moreover assume that End_{H_{F}} (M_{F}) is liftable as well. Then M_{R} is balanced for every choice of R ∈ {K, O, F } if and only if ann_{T_{F}} (M_{F}) = F ⊗_{O} ann_{T_{O}}(M_{O}).

Proof We have

dim_{F} (End_{T_{F}} (M_{F})) = dim_{K}(End_{T_{K}}(M_{K}))
dim_{F} (End_{H_{F}} (M_{F})) = dim_{K}(End_{H_{K}}(M_{K}))

by assumption. The annihilator condition in the lemma is equivalent to

dim_{F} (T_{F} /ann_{T_{F}} (M_{F})) = dim_{K}(T_{K}/ann_{T_{K}}(M_{K})),

hence the result for R = F follows immediately comparing dimensions: We need only to observe that T_{F} /ann_{T_{F} (M_{F})} is contained in End_{H_{F}} (M_{F}), and that End_{H_{K}}(M_{K}) is isomorphic to T_{K}/ann_{T_{K}}(M_{K}), since T_{K} is semisimple and hence balanced.

It remains to show the result for R = O. Obviously the assumption that the endomorphism rings over F are liftable implies, that M_{R} is balanced for R = F if it is balanced for F = O, hence the annihilator condition holds in this case.

Thus conversely suppose that the annihilator condition holds. We have seen that M_{F} is balanced. Let f : M → M be E-linear, E = E_{O}. Let 𝔭 be a generator of the unique maximal ideal of O. Since 1 ⊗ f : K ⊗_{O} M → K ⊗_{O} M is an element of End_{E_{K}}(M_{K}) we find t ∈ T_{O} and 0 ≤ k ∈ ℤ such that 1 ⊗ f is right multiplication by 𝔭^{−k}t. Suppose k ≠ 0. Then multiplication by t = 𝔭^{k}f maps M_{O} into 𝔭M_{O} and consequently t̄ = t + 𝔭T_{O} ∈ T_{F} annihilates M_{F}. Our annihilator condition says that we find r ∈ ann_{T_{O}}(M_{O}) satisfying r̄ = t̄, hence t_{1} ∈ T_{O} such that t = r + 𝔭t_{1}. Since r annihilates M_{O}, t operates precisely as 𝔭t_{1} on M_{O}. Thus t_{1} operates as 𝔭^{k−1}f on M_{O}. Continuing in this fashion one gets t_{k} ∈ T_{O} such that the action of f on M_{O} is multiplication by t_{k}. This shows that M_{O} is balanced as desired. □

We take now an T_{O}-lattice M = M_{O} and let β_{O} : P_{O} → M_{O} be a projective presentation of M_{O} satisfying Hypothesis 1.1. We assume that M is cyclic, that is M = mT for an m ∈ M, and that P_{O} is direct summand of T_{O}. Thus P_{O} = eT_{O} for an idempotent e ∈ T_{O}, and we may assume that β takes e to m. Thus the endomorphism ring E_{R} = End_{T_{R}}(eT_{R}) = eT_{R}e, where we denote the image of e in T_{R} again by e. Write the T -endomorphism ring of M_{R} as H_{R} for R ∈ {K, O, F }. Note that by [9, 4.8] End_{T_{F}} (M_{F}) is liftable, that is H_{F} = F ⊗_{O} H_{O}, so our notation is consistent. Recall that by [9, 4.7] β_{R} satisfies 1.1 for all choices of R, and that ker β_{K} and M_{K} have no constituent in common. Finally we denote ker β_{R} by U_{R}, and remark that indeed U_{R} = R ⊗_{O} U_{O} by [9, 4.7].

We note that in this situation the functor H_{R} takes V ∈ mod_{T_{R}} to V e/V eJ_{β_{R}}e by [9, 3.6]. Moreover [9, 2.8 and 3.4] imply the following lemma:

Lemma 3.3 Let the T_{O}-lattice M_{O} be as above. Then for R ∈ {K, O, F } we have

M_{R}e ≅ H_{R}

as H_{R}-eT_{R}e-bimodule, where the right action by eT_{R}e is given by inflating the right H_{R}- action to E_{R} = eT_{R}e along the epimorphism eT_{R}e → eT_{R}e/J_{β_{R}} ≅ H_{R}. Thus in particular M_{R}e is a balanced eT_{R}e-module.

We apply our lemmas now in the following special situation:

Situation 3.4 We assume that β_{O} : P_{O} → M_{O} satisfies Hypothesis 1.1. Note that M_{O} is P_{O}-torsionless by Lemma 1.20. Suppose that P_{R} = eT_{R} for an idempotent e of T_{O}.

We take pure right ideals X_{1,O}, …, X_{k,O} of H_{O}. The T_{R}-module N_{i,R} is defined as

N_{i,R} = R ⊗_{O} \sqrt{X_{i,O}M_{O}}.

Set

X_{R} = ⨁_{i=1}^{k} X_{i,R},

and

N_{R} = ⨁_{i=1}^{k} N_{i,R}.

We denote the endomorphism ring End_{H_{O}}(X_{O}) by S = S_{O}, and we assume that End_{H_{F}} (X_{F}) is liftable, hence we may apply [9, 4.17] by 1.30. In particular defining the functor H_{R} = H_{R}^{β} as in section one we have by Equation 1.28 that H_{R}(N_{R}) ≅ X_{R}, and H_{R} induces by 1.30 an isomorphism

H_{R} : End_{T_{R}}(N_{R}) ⟶^{∼} End_{H_{R}}(X_{R}) = S_{R} = R ⊗_{O} S.

Thus End_{T_{F}} (N_{F}) is liftable as well. Assume that there are pairwise orthogonal idempotents e = e_{1}, e_{2}, …, e_{k} and projective presentations β_{i,R} : e_{i}T_{R} → N_{i,R}, which satisfy 1.1. Assume in addition that e_{1} = e, thus N_{1,O} = M_{O} and X_{1,O} = H_{O}.

Suppose that N_{i,O} = M_{O}e_{i}T_{O}, and note that this condition means, that N_{i,R} is the e_{i}T_{R}-trace τ_{i,R}(M_{R}) in M_{R}, writing τ_{i,R} instead of τ_{P_{R}}, for P_{R} = e_{i}T_{R}. This is obvious for R = O and K, and follows for R = F, since homomorphisms from projective modules are always liftable and N_{i,O} is pure in M_{O}. We define the idempotent f = f_{R} of T_{R} to be the sum e_{1} + … + e_{k}, (where we denote idempotents in T_{R} for the various rings R by the same symbols which is justified by uniqueness of lifting idempotents).

We define γ_{R} = β_{1,R} + … + β_{k,R}. The last condition N_{i,O} = τ_{i,O}(M_{O}) in 3.4 implies in particular that ker γ_{K} and N_{K} have no irreducible constituent in common and hence γ_{R} : fT_{R} → N_{R} also satisfies 1.1. To see this we take an irreducible constituent S from ker γ_{K}. Then S is an irreducible constituent of ker β_{i,K} for some i. Suppose S is constituent of N_{j,K}. Then i ≠ j. But then there is a nontrivial T_{K}-linear map from e_{i}T_{K} into N_{j,K}, whose image contains S. Since τ_{i,K}(M_{K}) = N_{i,K}, this implies that S is contained in N_{i,K}, a contradiction to our assumption 3.4.

The situation described in 3.4 seems to be very special, but it is precisely what we need for our applications and for the promised generalization of Takeuchi’s result [45]. Note that all modules involved satisfy the requirements of Remark 1.25 and hence belong to ℑ_{M}(Ĥ)_{R}. Thus the T_{R}-modules N_{i,R} and N_{R} depend not really on the chosen projective presentation β_{O} : P_{O} → M_{O}.

For a T_{R}-module V define

V^{#} = Hom_{T_{R}}(V, T_{R}). (3.5)

Note that V^{#} is canonically a left T_{R}-module by the left T_{R}-module structure of T_{R}. We extend this notation in the obvious way to other modules, for example H_{R}-modules. We observe that −^{#} is a contravariant functor from mod_{T} to _{T} mod, which acts on homomorphisms by composition. More precisely if α : U → V is a homomorphism in mod_{T} then α^{#} takes f : V → T to f^{α^{#}} = f ∘ α. Writing α^{#} to the right as superscript has the advantage that now

(α ∘ β)^{#} = α^{#} ∘ β^{#}.

This is consistent with our convention to write homomorphisms opposite to the side of the ring action. Since # is involutive it induces bijections on Hom-sets. Therefore we have:

Lemma 3.6 Let V ∈ mod_{T}. Then End_{T} (V) ≅ End_{T} (V^{#}) canonically.

We need another auxiliary result:

Lemma 3.7 Let V ∈ mod_{T}, and let f be an idempotent element of T. Then

ann_{fT f}(V f) = f(ann_{T} (V))f.

Proof Obviously f(ann_{T} (V))f ⊆ ann_{fT f}(V f). Let x ∈ ann_{fT f}(V f), and let v ∈ V. Then x = fxf and vx = (vf + v(1 − f))fxf = 0. The result follows. □

Theorem 3.8 Keep the notation introduced in 3.4. Then the following holds for every choice of R ∈ {K, O, F }:

(i) The left H_{R}-module M_{R}f is isomorphic to X_{R}^{#}, and End_{H_{R}}(M_{R}f) ≅ S_{R} canonically. (ii) The ring fT f acts on M_{R}f by right multiplication as H_{R}-endomorphisms such that

the canonical map fT_{R}f → End_{H_{R}}(M_{R}f) is surjective. (iii) The endomorphism ring of the fT_{R}f-module M_{R}f is End_{fT_{R}f}(M_{R}f) = H_{R} by re-

striction. (iv) The fT_{R}f-module M_{R}f is balanced.

Proof We may omit indices R for the moment. Obviously Mf is a submodule of the left H-module M. Since the e_{i}, i = 1, …, k are pairwise orthogonal idempotents,

Mf = ⨁_{i=1}^{k} Me_{i},

as H-module. Thus to show i) it suffices to show that Me_{i} ≅ X_{i}^{#} for all i ∈ {1, …, k}. It is well known that for an idempotent e of the ring T and V ∈ mod_{T} the abelian group V e is canonically isomorphic to Hom_{T} (eT, V). This isomorphism is natural in V, hence it is an isomorphism of left S-modules if V is an S-T -bimodule for some ring S. Thus for V = M we have an isomorphism of left H-modules

Me_{i} ≅ Hom_{T} (e_{i}T, M). (3.9)

Let ϕ : e_{i}T → M be T -linear. Now 3.4 states that im ϕ ⊆ N_{i} ⊆ M, hence we may view ϕ as element of Hom_{T} (e_{i}T, N_{i}), and that β_{i} : e_{i}T → N_{i} satisfies 1.1. Thus by [9, 2.8] ker β_{i} ≤ ker ϕ, hence ϕ factorizes through N_{i}, and we have:

Hom_{T} (e_{i}T, M) ≅ Hom_{T} (N_{i}, M). (3.10)

and this isomorphism respects the left H-action on these Hom-sets coming from the left module structure of M too. We apply the functor H and get from 3.4 and 1.37 an isomorphism of left H-modules:

Hom_{T} (N_{i}, M) ≅ Hom_{H}(X_{i}, H), (3.11)

Equations 3.9, 3.10, 3.11 and 3.5 together with Lemma 3.6 give part i).

Next we prove ii): Note that N_{1} = M by assumption 3.4. This shows that

ann_{T} (N) ⊆ ann_{T} (M). (3.12)

But all the direct summands N_{i} of N are contained in M, hence every element of T which annihilates M, annihilates N_{i} for all i ∈ {1, …, k}, and therefore N, and we have equality in Equation 3.12.

The image of fT f in End_{H}(Mf) is fT f/ann_{fT f}(Nf) and by Lemma 3.7 we have

(3.13) ann_{fT f}(Mf) = f(ann_{T} (M))f = f(ann_{T} (N))f = ann_{fT f}(Nf).

We observe that Lemma 3.3 applies to the T -module N. Thus taking E = fT f = End(fT) we see that N is a balanced E-module, and is isomorphic to S as S-E-bimodule. We conclude that

fT_{R}f/ann_{fT_{R}f}(M_{R}f) = fT_{R}f/ann_{fT_{R}f}(N_{R}f) ≅ S_{R} (3.14)

for every choice of R. Now ii) follows from i).

To prove iii) for R = K we may use 3.1, observing that an irreducible constituent of M_{K} cannot be annihilated by f, since we have an epimorphism from fT_{K} onto M_{K} = N_{1,K}. An alternate proof which works for all choices of R goes as follows: First as left H-module Mf decomposes into the direct sum

Mf = ⨁_{i=1}^{k} Me_{i}.

But Me_{1} = Me ≅ H as left H-modules, hence Mf contains the left regular H-module as direct summand. By [5, 59.4] the left H-module Mf is balanced. By ii) and i) its endomorphism ring is

End_{H}(Mf) ≅ S ≅ fT f/ann_{fT f}(Mf), (3.15)

and therefore

(3.16) End_{fT f}(Mf) = End_{fT f/ann_{fT f} (Mf)}(Mf) = H

by the double centralizer property. We check now that all hypotheses of Lemma 3.2 are satisfied for M_{O}f and part iv) follows. □

We note that part ii) of Theorem 3.8 says in particular, that the endomorphism ring S_{R} of X_{R} (and of N_{R}) is a subfactor of T_{R}.

In the applications which we have in mind, the algebra H_{F} is often a Frobenius algebra. Note that H_{K} is always Frobenius, since it is semisimple, and note too that a Frobenius algebra is in particular selfinjective. So assume that H_{F} is selfinjective, and let H = H_{R} for R = F or K. Let I be a right ideal of H. Then, for h ∈ H, left multiplication by h defines an H-linear λ_{h} : I → H. Since H is selfinjective, every homomorphism from I to H_{H} lifts to an endomorphism of the right regular module H_{H} and hence is left multiplication by an element of H. Thus

I^{#} = Hom_{H}(I, H) ≅ H/l(I), (3.17)

where l(I) denotes the left annihilator of I in H.

We apply this to the case, where I = xH is a principal right ideal and conclude that

Hx → Hom_{H}(xH, H) : hx ↦ λ_{h} (3.18)

is a well defined isomorphism.

Lemma 3.19 Let H_{F} be self injective, and let x ∈ H_{O} be such that H_{O}x is pure in the left regular H_{O}-module _{H_{O}}H_{O}. Then the map

H_{R}x → (xH_{R})^{#} ≅ Hom_{H_{R}}(xH_{R}, H_{R}) : hx ↦ λ_{h}

is a well defined isomorphism for all choices of R ∈ {K, O, F }. So in particular, if in addition xH_{O} is pure in the right regular module H_{O H_{O}}, then Hom_{H_{F}} (xH_{F} , H_{F}) is liftable.

Proof The only case not yet shown is R = O. Let ϕ : xH_{O} → H_{O} be an H_{O}-linear map. Formula 3.18 applied to the case R = K implies that ϕ(x) ∈ H_{K}x ∩ H_{O} which in turn equals H_{O}x, since the latter is pure by assumption. Thus ϕ(x) = hx for some h ∈ H_{O}. Clearly now ϕ = λ_{h}.

xH_{O} being pure in H_{O} means that xH_{F} = F ⊗_{O} xH_{O} (writing again x for 1 ⊗ x ∈ F ⊗_{O} H_{O} ≅ H_{F}). The second assertion follows. □

We observe that taking # commutes with taking direct sums. Thus Lemma 3.19 applied to the Situation 3.4 gives immediately the following reformulation of the part i) of Theorem 3.8:

Corollary 3.20 Keep the notation of 3.4 and assume in addition that H_{F} is selfinjective. Suppose that X_{i} is a cyclic module X_{i} = x_{i}H_{O} for some x_{i} ∈ H_{O} such that both, H_{O}x_{i} and x_{i}H_{O} are pure in the left respectively right regular H_{O}-module for 1 ≤ i ≤ k. Then

M_{R}f ≅ X_{R}^{#} ≅ ⨁_{i=1}^{k} H_{R}x_{i}

for all choices of R ∈ {K, O, F }.

4 Hecke functors for general linear groups

For the moment let G be an arbitrary finite reductive group. Suppose the field of definition of G is the finite field with q elements for some power q of some prime p. We denote the Weyl group of G by W. The system M of Levi subgroups of G considered as subfactors (P, U) by choosing a parabolic subgroup P of G with unipotent radical U such that L = P/U satisfies the requirements of a Mackey system as defined in the second section. Henceforth the terms as HC-induction, restriction, cuspidal, vertices are always meant with respect to this special Mackey system. Note that for a Levi subgroup L of G the Mackey system M_{L} is the Mackey system of Levi subgroups of L. For details we refer to [4]. Although HC-induction and restriction involve the parabolic subgroup P which the Levi subgroup L is a Levi complement of. It is a classical result over fields of characteristic zero (see e.g. [4]), that HC-induction and restriction are independent of the choice of P and the same was shown by Howlett and Lehrer in [37] and independently by Du and the author in [13, 5.2] for arbitrary commutative rings R, in which q1_{R} is invertible. We take ℓ to be a prime number different from p and a split ℓ-modular system (K, O, F) for G. Note that q is invertible in any ring R ∈ {K, O, F } and may be identified with the invertible element q1_{R} of R. As a consequence the functors corresponding to conjugate Levi subgroups are naturally equivalent. Thus the irreducible RG-modules (R = K or F) are partitioned according to their semisimple vertices and sources into Harish Chandra series, (HC-series for short), (compare [4] for the classical and [34] for the ℓ-modular case).

By [28, 2.4] (compare 2.5) the task to find the irreducible RG-modules R = F, K subdivides into the following two parts:

Procedure 4.1 To determine the irreducible RG-modules we have to proceed as follows:

(i) For any Levi subgroup L find the irreducible cuspidal RL-modules. (ii) For a Levi subgroup L and a cuspidal irreducible RL-module C find the irreducible

H-modules, where H = End_{RG}(R_{L}^{G}(C)).

For R = K the structure of the endomorphism ring H was determined by Howlett and Lehrer in [36]. In the special case of general linear groups their result extended (in view of [21, 3.3]) to the case R = F in [7, 3.6] (details will be provided later in this section in Theorem 4.32). The general case was done in [28].

We turn now to finite general linear groups. Thus let G = GL_{n}(q), where q is a power of some prime p. Let g ∈ G. In [21] the conjugacy class g^{G} of g was labelled by an (n, ∞)-index:

g^{G} ↔ I(g) = ( \begin{matrix}d_{1} & d_{2} & … & n_{1} & n_{2} & … \\ s_{1} & s_{2} & … & λ^{(1)} & λ^{(2)} & … \end{matrix} ) , (4.2)

where the meaning of the parts of this index is explained below.

Let g = su be the Jordan decomposition of g, that is the decomposition of g into its semisimple part s and the unipotent part u ∈ C_{G}(s). Note that this is precisely the decomposition of g into its p-regular part s and p-part u. We may assume that s is given in its canonical rational form. Thus

s = (s_{1})^{×n_{1}} × (s_{2})^{×n_{2}} × · · · · · · , (4.3)

where s_{1}, s_{2}, … are pairwise different elements of some large extension field of GF (q) and (s_{i}) is the companion matrix of the minimum polynomial of s_{i} over GF (q). In order to make things uniquely defined we shall usually order the s_{i} according to their degrees d_{i} downward and within a fixed degree d according to an arbitrary but fixed ordering of the corresponding elements of GF (q^{d}) on the orbits under the Galois group. For a given i the label d_{i} in 4.2 is now the degree of the minimum polynomial of s_{i} over GF (q), that is the rank of the matrix (s_{i}). Note that n = n_{1}d_{1} + n_{2}d_{2} + . . . . . . and

C_{G}(s) ≅ GL_{n_{1}}(q^{d_{1}}) × GL_{n_{2}}(q^{d_{2}}) × · · · · · · , (4.4)

where × means again matrix direct sum (which is isomorphic to the direct product) and the right hand side of 4.4 is suitably imbedded into G. We set

n_{s} = n = (n_{1}, n_{2}, . . . . . .). (4.5)

The G-conjugacy classes of the unipotent part u in C_{G}(s) is given by the canonical rational form, which is of course the same as the Jordan form. Hence it is described by the distribution of ones above the diagonal, that is a multipartition

λ = (λ^{(1)}, λ^{(2)}, . . . . . .), (4.6)

where λ^{(i)} = (λ_{1}^{(i)}, …) is a partition of n_{i}. We abbreviate this by λ ⊢ n and and we use for multicompositions μ of n similarly the notation μ ⊧ n. It is obvious that the left half of the index in 4.2 describes the semisimple and the right half the unipotent part of g, and hence those parts of the index are called the semisimple and unipotent part of I(g) respectively.

For short we may label the conjugacy classes of G by the pair (s^{G}, λ).

The ℓ-regular classes of G are now precisely given by pairs (s^{G}, λ) where s ∈ G is an ℓ-regular semisimple element, since unipotent elements are p-elements, and therefore especially ℓ-regular. The corresponding index in [21] looks precisely as in 4.2, but is called an (n, ℓ)-head index to indicate that its semisimple part is ℓ-regular.

We want to describe the irreducible KG-modules. Their classification was done (in terms of character theory) by J. Green in 1955, [29].

First we use Deligne-Lusztig theory to determine the irreducible cuspidal KG-modules (for details see e.g. [43]). Take a semisimple element s ∈ G whose minimum polynomial has degree n. It generates together with the center of G a maximal (non split) torus T which is minisotropic, that is it is not contained in any proper Levi subgroup of G. The Deligne-Lusztig operator taking characters of T to generalized characters of G is denoted by R_{T}^{G}. Now T being abelian is isomorphic to its character group, and we may choose such an isomorphism; (we have to do this in a consistent manner, that means at the same time for G and all its standard Levi subgroups). Denote the image of s by ŝ. Then R_{T}^{G}(ŝ) is up to sign the character of an irreducible cuspidal KG-module which we shall denote by C_{s}. We may label these module by the pair (s^{G}, (1)), where (1) is the only partition of the number 1 or following [21] the head (n, ∞) index

C_{s} ↔ I(C_{s}) = ( \begin{matrix}n & 1 \\ s & (1) \end{matrix} ) . (4.7)

We remark that the G-conjugacy classes of parabolic subgroups are parametrized by compositions of n. A set of representatives of these consists of the set of all parabolic subgroups containing one fixed Borel subgroup B of G. We choose B to be the set of all upper triangular matrices in G. Then the standard parabolic subgroup corresponding to the composition μ of n (written as μ ⊧ n) is denoted by P_{μ} and its Levi subgroup by L_{μ}. Moreover, if μ = (μ_{1}, μ_{2}, …)

L_{μ} = GL_{μ_{1}}(q) × GL_{μ_{2}} × ⋯,

diagonally embedded into G.

Now let s ∈ G be an arbitrary semisimple element. We assume it is given in its canonical rational form as in Formula 4.3. Let the composition μ = μ_{s} of n be given by

(4.8) μ_{s} = (\underbrace{d_{1}, d_{1}, …, d_{1}}_{n_{1}-times}, \underbrace{d_{2}, …, d_{2}}_{n_{2}-times}, . . . . . .),

where the n_{i} and the d_{i} are defined in 4.4. Together with the center of G the element s generates a minisotropic maximal torus T_{s} of L_{s} = L_{μ}. Consequently, since L_{s} is a direct product of general linear groups, the generalized character R_{T_{s}}^{L_{S}}(ŝ) is up to sign the character afforded by an irreducible cuspidal KL_{s}-module C_{s}. Indeed we can decompose C_{s} into an outer tensor product:

C_{s} ≅ C_{s_{1}}^{⊗n_{1}} ⊗ C_{s_{2}}^{⊗n_{2}} ⊗ · · · · · · (4.9)

All modules considered are semisimple, hence in order to find the irreducible components of M_{s} = R_{L_{s}}^{G}(C_{s}), we have to inspect the endomorphism ring H_{s} = End_{KG}(M_{s}) and apply Fitting’s Lemma or equivalently 2.5 (considering the projective presentation β = id : M_{s} → M_{s} of M_{s}).

Let ℭ_{ss} be a set of representatives of conjugate semisimple elements of G e.g. given in the form 4.3. Then the set of irreducible components of M_{s} is precisely the HC-series with semisimple vertex L_{s} = L_{μ} and semisimple source C_{s} (and coincides with the Deligne-Lusztig series associated with (T_{s}, ŝ) which is defined to be the set of irreducible constituents of R_{T_{s}}^{G}(ŝ)). For general linear groups this series is as well a geometric conjugacy classes of G and is denoted by E(s^{G}) or simply by E(s). (This is special for general linear groups. For arbitrary finite groups of Lie type geometric conjugacy classes in general are unions of Deligne-Lusztig series).

The structure of H_{s} has been determined by Howlett and Lehrer in [36] to be the Hecke algebra associated with the ramification group W_{s} = W (L_{s}, C_{s}), that is the stabilizer of C_{s} in W factored by the Weyl group of L_{s}. In the case of general linear groups it is again a Weyl group of type A (comp. [7]), indeed W_{s} is isomorphic to the Weyl group of C_{G}(s) (see 4.4). As subgroup of W it is it is given as follows:

W_{s} = 𝔖_{\{\overline{1},... ,\overline{n_{1}}\}} × 𝔖_{\{\overline{n_{1}+1},... ,\overline{n_{1}+n_{2}}\}} × . . . . . . , (4.10)

where \overline{i} denotes the i-th subset in the set partition of \{1, …, n\} given by μ ⊧ n. Thus, if \overline{i} = \{b + 1, …, b + j\} and \overline{i + 1} = \{b + j + 1, …, b + 2j\} belong to the same subset above, (that is both, i and i + 1 are between n_{1} + n_{2} + ⋯ n_{a} and n_{1} + ⋯ n_{a} + n_{a+1} for some a), then the corresponding generating transposition of W_{s} is the permutation in W = 𝔖_{n} which swaps b + x and b + x + j for 1 ≤ x ≤ j and keeps all other numbers fixed.

For a Coxeter group W generated by basic reflections v_{1}, …, v_{r}, a commutative ring R and a unit q ∈ R the Hecke algebra H_{R,q}(W) over R with respect to q associated with W has R-basis {T_{w} | w ∈ W } and multiplication is defined by the following rule:

(4.11)  T_{v}T_{w} = \{ \begin{matrix}T_{vw} & if l(vw) = l(w) + 1 \\ qT_{vw} + (q − 1)T_{w} & otherwise \end{matrix} ,

for w ∈ W and v ∈ {v_{1}, …, v_{r}}. Here l : W → ℕ denotes the length function which is defined to be the least number k for w ∈ W such that one can w write as word with k factors in the generators. It can be shown that these relations indeed determine an associative R-algebra.

Theorem 4.12 (R.Howlett, G.Lehrer) Let s ∈ G be semisimple. Then

H_{s} = End_{KG}(M_{s}) ≅ H_{K,q^{d_{1}}} (𝔖_{n_{1}}) ⊗ H_{K,q^{d_{2}}} (𝔖_{n_{2}}) ⊗ · · · · · · ,

Explicit representations for H_{K,q}(W) were constructed by P.N. Hoefsmit in [35] for Weyl groups of classical type. For Weyl groups of type A, that is symmetric groups, the representations of the associated Hecke algebras H_{R,q}(𝔖_{r}) were investigated by G. James and the author in [19] for arbitrary R and q and we shall use the notation developed there. Thus the irreducible H_{s}-modules are labelled by multipartitions λ ⊢ n, where n = n_{s} = (n_{1}, n_{2}, …) and λ are given in 4.5 and 4.6. The H_{s}-module corresponding to λ ⊢ n is the Specht module S^{λ} and may be realized as right ideal in H_{s}. Thus

S(s, λ) = S^{λ}M_{s}

is an irreducible KG-submodule of M_{s} by Fitting’s Lemma. We have:

Theorem 4.13 Let G = Gl_{n}(q) and K a splitting field for G of characteristic zero. Then

{S(s, λ) | s ∈ ℭ_{ss}, λ ⊢ n_{s}}

is a complete set of nonisomorphic irreducible KG-modules. Here C_{ss} denotes a set of representatives of semisimple conjugacy classes in G.

By the discussion at the beginning of this section, we may parametrize the irreducible KG-modules as well by (n, ∞)-indices.

Remark 4.14 In 4.13 we made implicitly some choices and it is a priori not clear that all of our constructions are independent of those. As an example we consider G = GL_{2}(q) and s = 1 ∈ G. Thus the Levi subgroup L_{s} is the maximal split torus T of invertible diagonal matrices and we have two parabolic subgroups containing T as Levi subgroup, namely the groups B_{1} of invertible upper and B_{2} of lower triangular matrices. The corresponding Hecke algebra H = End_{KG}(R_{T}^{G}(K_{T})) is the Hecke algebra H = H_{K,q}(𝔖_{2}) (which in turn is isomorphic to the group algebra K𝔖_{2}), where K_{T} denotes the trivial KT -module. Now H has two one dimensional representations, namely the trivial and the alternating one corresponding to the partitions λ = (2) and μ = (1^{2}) respectively. By [13, 37] the HCinduction R_{T ⊂B_{i}}^{G} = Ind_{B_{i}}^{G}Inf_{T}^{B_{i}} for i = 1, 2 does not depend on the choice of the embedding T as Levi subgroup into a parabolic subgroup of G. In particular we have an isomorphism ϕ : R_{T ⊂B_{1}}^{G}(K_{T}) → R_{T ⊂B_{2}}^{G}(K_{T}), hence their endomorphism rings are isomorphic too. Those, however, are given by the Howlett-Lehrer construction [36]. It is not immediate that both constructions give precisely the same answer, the trivial and the alternating representation of H could be interchanged by the isomorphism ϕ. In this special case this cannot happen, because the corresponding irreducible direct summands of R_{T ⊂B_{i}}^{G}(K_{T}) are the trivial and the Steinberg module, which are distinguishable by their degrees. In general one checks easily that there is only one critical choice involved, namely the order of the factors in the representation 4.3 of the canonical rational form of s or equivalently the precise choice of the canonical rational form of s and hence the Levi subgroup L_{s} among the conjugate standard Levi subgroups. Thus one has to show the following: Given standard Levi subgroups L_{i} of G with i = 1, 2 which are conjugate in G (and hence by an element d of W) then the natural transformation B_{d} from R_{L_{1}}^{G} to R_{L_{2}}^{G} induces an H_{s}-KG-bimodule isomorphism from R_{L_{1}}^{G}(C_{s}) onto R_{L_{2}}^{G}(C_{s}^{d}), where C_{s} is defined in 4.9 and C_{s}^{d} denotes the conjugate module. But this can be checked directly from the construction of the natural transformation B_{d} in [13, 5.8]. Note that this fact then justifies our convention in 4.2 fixing for the corresponding (n, ∞)-indices the order of the terms and the order of the parts of s.

For later use we remark that an analogous statement holds, if K is replaced by F (or O) and (n, ∞)-indices by (n, ℓ)-indices (compare [21, 2.5]).

It should be pointed out here that we know a lot more about the irreducible KG-modules S(s, λ), for example their dimensions. G. James even constructed a “standard basis” for them. For details we refer to his paper [40], and to [38, 39].

We now turn to the ℓ-modular representation theory of G. For this we have to consider suitable reductions modulo ℓ of the irreducible KG-modules constructed in Theorem 4.13. We adopt the convention to drop sub- and superscripts whenever there is no danger of ambiguities. Moreover, the ring R of scalars we are working with is often indicated by a subscript R. More specifically, whenever an object is defined over the order O, then an index R means that we tensor the object over O with R. Usually R is F, O, or K. Finally the symbol “=_{G}” means “equal up to conjugation in G”, and we adopt a similar notation for further symbols, as “⊆_{G}” etc.

Here are some results on cuspidal representations:

4.15 Let C_{K} be a cuspidal irreducible KG-module. Then the following holds:

(i) Any reduction modulo ℓ of C_{K} is cuspidal. The converse holds too: The lift to KG of a liftable cuspidal F G-module is cuspidal.

(ii) Let C_{F} be a reduction modulo ℓ of C_{K}. Then C_{F} is irreducible. Thus C_{K} has an up to isomorphism a uniquely determined OG-lattice C_{O}.

The first part follows immediately from the definition of cuspidal modules. Part 2 was first proved in [7, 8]. However the proof there requires the full classification of the irreducible F G-modules and is therefore not useful for our treatment here. There is a direct proof in [40] involving the Gelfand-Graev representation of G and none of our forthcoming results here. The reader should therefore consult [40] for details. It should be pointed out that this part does not hold in general for finite groups of Lie type. At the end of section five in 5.23 we shall prove that the converse of part 2 above holds for G = GL_{n}(q) as well: Every cuspidal irreducible F G-module is liftable and hence lifts to an irreducible cuspidal KG-module. This as well does not hold for other types of finite reductive groups. For example, if G is a finite reductive group with Dynkin diagram G_{2} defined over GF (q) such that 5 divides q +1 but 25 does not, then G has a 5-modular irreducible cuspidal representation which does not occur in any reduction modulo 5 of any complex irreducible cuspidal representation of G, and hence is in particular not liftable. Moreover G has plenty complex cuspidal irreducible representations which do not remain irreducible upon reduction modulo 5. For details see Appendix C of [32].

For g ∈ G we denote the ℓ-regular part of g by g′. Thus g = g′v = vg′ for some ℓ-element v in C_{G}(g). The set of ℓ-elements of a subset U of G is denoted by U_{ℓ}.

The following theorem can be derived from [25] respectively from [21]. Its generalization to arbitrary finite groups of Lie type is due to M. Broué an J. Michel [2]:

Theorem 4.16 Let s be a semisimple ℓ-regular element of G. Then

E_{ℓ}(s) = ⋃_{y∈C_{G}(s)_{ℓ}} E(sy),

is the set of irreducible KG-modules in a union of ℓ-blocks of G.

The following results concern induced cuspidal representations. Note that HC-induction commutes with base change, that is for s ∈ G semisimple

M_{F,s} = F ⊗_{O} M_{O,s} ≅ R_{L_{s}}^{G}(C_{F,s})

canonically, justifying our notation.

Lemma 4.17 Let s, t ∈ G be semisimple.

(i) Assume that L_{s} =_{G} L_{t}. Suppose s′ =_{G} t′. Then C_{F,s} ≅ C_{F,t}, and hence M_{F,s} ≅ M_{F,t}. (ii) Let s, t be semisimple elements of G such that s′ ≠_{G} t′. Then the geometric conjugacy

classes E_{s} and E_{t} belong to disjoint ℓ-blocks of G.

Proof To see the first part one checks that the characters of C_{K,s} and C_{K,t} coincide on ℓ-regular conjugacy classes of G if and only if s and t have the same ℓ-regular parts, and the claim follows. For details see [21, 3.3] and for a generalization to finite groups of Lie type [33]. The second part of the lemma follows immediately from 4.16, but can also seen by inspecting the linear characters of the center of KG, (comp. [21]). □

Next we need the Gelfand-Graev representation of G and of Levi subgroups. Thus, for μ ⊧ n let U_{μ} the unipotent radical of the standard Borel subgroup of L_{μ}. Thus U_{μ} consists of all unipotent upper triangular matrices in L_{μ}. We take a linear character χ of U_{μ} whose restriction to any root subgroup X_{i} is nontrivial for any simple root α_{i}. More precisely, X_{i} consists of those matrices which have ones on the diagonal and the only nonzero entry occurs in row i and column i + 1, where 1 ≤ i ≤ n − 1 is chosen such that X_{i} is contained in L_{μ}. If L = L_{μ} we have an idempotent

e_{L} = e_{μ} = \frac{1}{|U_{μ}|} ∑_{u∈U_{μ}} χ(u)u, (4.18)

which is contained in OL, since the order of U_{μ} is invertible in O. By general theory the OL-lattice Γ_{L,O} = e_{L}OL is projective and hence up to isomorphisms the only OL-lattice in the KL-module e_{μ}KL = Γ_{L,K}. For R as usual the RL-lattice Γ_{L,R} is the Gelfand-Graev module of RL. If L = L_{μ} we also write Γ_{L,R} = Γ_{μ,R}.

Here are some properties of the Gelfand-Graev representation Γ_{R} = Γ_{G,R}:

4.19 Let s be a semisimple element of G, and let n_{s} be defined as in 4.5. Then the following holds:

(i) The geometric conjugacy class E(s) of G has precisely one irreducible KG-module with Γ_{K} in common, namely the KG-module S_{K}(s, λ) of Theorem 4.13, where λ ⊢ n is given as λ = (λ^{(1)}, λ^{(2)}, …) = ((1^{n_{1}}), (1^{n_{2}}), …).

(ii) The Gelfand-Graev representation Γ_{K} is multiplicity free. (iii) Every cuspidal irreducible KG-module occurs in Γ_{K} precisely once as composition

factor.

(iv) The Hom-set Hom_{RG}(Γ_{R}, M_{R,s}) is free of rank one as R-module for all choices of R ∈ {K, O, F }.

Proof The first two parts are well known (see e.g. [4]), and the remaining parts are straightforward consequences hereof. □

Again parts three and four of 4.19 hold for general linear groups only. Part one is true for finite reductive groups with connected center, and part two in general. We define the multipartition ω_{s} ⊢ n_{s} to be

ω_{s} = ((1^{n_{1}}), (1^{n_{2}}), . . . . . .). (4.20)

We remark that everything what has been done so far for G applies as well to Levi subgroups L of G. We denote the corresponding objects for L by an additional subscript. So for a semisimple element s of G which is (up to conjugation in G) contained in L we have an associated geometric conjugacy class E_{L}(s) (taking s to be an element of L). Immediately from the definition of geometric conjugacy classes and transitivity of HCinduction follows that the modules in E_{G}(s) are the irreducible constituents of the modules R_{L}^{G}(N) for N ∈ E_{L}(s). Thus

R_{L}^{G}(E_{L}(s)) = E_{G}(s). (4.21)

But note that the semisimple element t in L may be not conjugate in L to our given s but in G, and hence E_{L}(s) ∩ E_{L}(t) = ∅ but E_{G}(s) = R_{L}^{G}(E_{L}(s)) = R_{L}^{G}(E_{L}(t)) = E_{G}(t). We can however describe precisely when this happens: Both s and t are given in its canonical rational form, but t is derived from s by permuting and mixing parts of s in equation (4.1) of equal degree. Here is an example:

\begin{aligned}s & = ⋯ × (s_{i}) × ⋯ × (s_{i})× & (s_{i}) & × & (s_{i+1}) & ×(s_{i+1}) × ⋯ \\ t & = ⋯ × (s_{i}) × ⋯ × (s_{i})× & (s_{i+1}) & × & (s_{i}) & ×(s_{i+1}) × ⋯, \end{aligned} (4.22)

where in the notation of Equation 4.3 d_{i} = d_{i+1}.

Fix s to be a semisimple element of G given as in 4.3. Set again M_{R,s} = R ⊗_{O} M_{O,s} with M_{O,s} = R_{L_{s}}^{G}(C_{O,s}) and let L = L_{s}.

Lemma 4.23 There exist a projective presentation

β_{O,s} = β_{O} : P_{O,s} → M_{O,s}

with the following property: Every composition factor of the kernel ker β_{K} of β_{K} = β_{K,s} has the form S(s′y, λ) for some λ ⊢ n_{s′y} with 1 ≠ y ∈ C_{L}(s′)_{ℓ} such that s ≠_{L} s′y.

Proof Let L = L_{s}, and denote the Gelfand-Graev representation of OL by Γ_{O,s}. We apply Result 4.19: There exists a nonzero homomorphism ψ_{O} from an indecomposable direct summand P_{1} onto C_{O,s} and the kernel of ψ_{K} contains only summands of the form S_{L}(t, ω_{t}). Being composition factor of the same indecomposable projective OG-module tensored by K as S_{L}(s, (1)) = C_{K,s} both modules must be in the same ℓ-block of L. Result 4.17 implies that s′ =_{L} t′. We apply 4.19 once more to conclude that s and t cannot be conjugate in L. Thus t = s′y has the required form. Now we take P_{O,s} = R_{L}^{G}(P_{1}) and β_{O} = R_{L}^{G}(ψ_{O}) then Equation 4.21 yields the desired result. □

Of course we would like to show that we are in situation [9, 4.7], by showing that ker β_{K} has no constituent in common with M_{K,s}. For this we need in Lemma 4.23 that s ≠_{G} s′y holds. Unfortunately it can happen that these elements are conjugate in G as Equation 4.22 demonstrates.

Lemma 4.24 Let β_{O} be defined as above. Let S(s′y, λ) be an irreducible composition factor of ker β_{K}, such that s and s′y of L are conjugate in G. Then L_{s} = L_{s′y} and s = s′ỹ for some y =_{G} ỹ ∈ C_{L}(s′)_{ℓ}. Moreover if s′ is given in its canonical rational form as in 4.3

s′ = (s′_{1})^{×n_{1}} × ⋯ × (s′_{k})^{×n_{k}},

then there exists an index 1 ≤ a ≤ k and ℓ-elements {y_{1}, y_{2}, … } in some extension field GF (q^{d_{a}m}) of GF (q^{d_{a}}) with the following properties, (where d_{a} is the degree of the minimum polynomial s_{a}):

(i) The minimum polynomial of y_{i} over GF (q^{d_{a}}) has degree m for all i. (ii) The elements of {y_{1}, y_{2}…. } have pairwise different minimum polynomials. (iii) In C_{G}(s′) = ⋯ × GL_{n_{a}}(q^{d_{a}}) × ⋯ the companion matrix (y_{i}) of the minimum polyno-

mials of the y_{i} occurs with the same positive multiplicity in y and ỹ, for i = 1, 2, …, but in different order.

Conversely if s = sỹ and t = s′y are given as above, then s =_{G} t and M_{K,s} ≅ M_{K,t}. In this case every irreducible composition factor of the image M_{K,s} under β_{K,s} is composition factor of ker β_{K,s} as well.

Proof Immediately from Equation 4.22 and Lemma 4.23 follows that y satisfies the (i), (ii) and (iii). To see the converse we observe that it is enough to show that the KL_{s}- module C_{K,t} is contained in the kernel of ψ_{K,s} which maps P_{1,K,s} onto C_{K,s}, (defined in the proof of Lemma 4.23 and now given appropriate indices). Note that C_{K,s} and C_{K,t} are not isomorphic, since s and t are not conjugate in L_{s} = L_{t}. On the other hand by part (i) of Lemma 4.17 we have C_{F,s} ≅ C_{F,t}, since s and t have equal ℓ-regular parts. Consequently we have an epimorphism from P_{1,F,s} onto C_{F,t} and by Brauers reciprocity law C_{K,t} is composition factor of P_{1,K,s} as well. Being different from C_{K,s} it has to be in the kernel of ψ_{K,s}. Thus the lemma holds. □

We may visualize the situation in the following example:

s′ = ⋯ × \underbrace{((s′_{a}) × ⋯ × (s′_{a}))}_{m-factors} × ⋯ × \underbrace{((s′_{a}) × ⋯ × (s′_{a}))}_{m-factors} × ⋯
ỹ = ⋯ ×  (y_{1})  × ⋯ ×  (y_{2})  × ⋯
y = ⋯ ×  (y_{2})  × ⋯ ×  (y_{1})  × ⋯
s′ỹ = ⋯ ×  (s′_{a}y_{1})  × ⋯ ×  (s′_{a}y_{2})  × ⋯
s′y = ⋯ ×  (s′_{a}y_{2})  × ⋯ ×  (s′_{a}y_{1})  × ⋯  .

(4.25)

Note that indeed the minimum polynomial of s′_{a}y_{i}, i = 1, 2 over GF (q) has degree md_{a}. The canonical rational form of s′, s′y, s′ỹ in 4.25 has to be taken in G, that of y, ỹ in C_{L}(s′).

Definition 4.26 Let s ∈ G be semisimple:

s = ⋯ × (s_{a})^{n_{a}} × ⋯ × (s_{b})^{n_{b}} × · · · .

Let s_{i} = s′_{i}y_{i} be the decomposition of s_{i} into its ℓ-regular and ℓ-singular parts and let d_{i} be the degree of the minimum polynomial of s_{i}. Then we say that s is ℓ-stable when it satisfies the following condition: Whenever (s′_{a}) = (s′_{b}) and d_{a} = d_{b} then y_{a} and y_{b} have the same minimum polynomial over GF (q^{d′_{a}}), where d′_{a} is the degree of the minimum polynomial of s′_{a} over GF (q).

We remark that our definition of ℓ-stable here is precisely the hypothesis in [15, 4.6] for the special case of general linear groups (which are known to coincide with their dual groups). Lemmas 4.23 and 4.24 (in connection with [9, 4.7]) imply now immediately:

Theorem 4.27 Let s ∈ G be semisimple. Then the projective presentation

β_{R,s} : P_{R,s} → M_{R,s}

satisfies Hypothesis 1.1 for all choices R ∈ {K, O, F } if and only if s is ℓ-stable.

Immediate from the definition we have:

Lemma 4.28 Let s ∈ G be semisimple and ℓ-regular. Then s is ℓ-stable.

It is obvious how to turn an arbitrary semisimple element into a ℓ-stable one. If for example s = s′ỹ is given as in Lemma 4.24 we may change the direct factor (s′_{a}y_{2}) into (s′_{a}y_{1}) and do this for all y_{i} ∈ {y_{2}, … } and for all indices a to get a semisimple element s̃ which is ℓ-stable and satisfies s′ = s̃′. Part (i) of Result 4.17 says, that over F our constructions depend only on the ℓ-regular part s′ of s and the “shape” of its ℓ-singular part y. We call two semisimple elements s and t of G ℓ-equivalent, and we write s ∼_{ℓ} t, if s′ =_{G} t′ and L_{s} =_{G} L_{t}. In this case

M_{F,s} ≅ M_{F,s̃} ≅ M_{F,t̃} ≅ M_{F,t},

where the ℓ-stable elements s̃ and t̃are constructed as above. Obviously ∼_{ℓ} is an equivalence relation, and each equivalence class contains ℓ-stable elements, since s ∼_{ℓ} s̃.

Now Result 4.17 implies:

Lemma 4.29 Let s, t ∈ G be semisimple, and let t ∼_{ℓ} s. Then

C_{F,s} ≅ C_{F,t}
P_{F,s} ≅ P_{F,t}
M_{F,s} ≅ M_{F,t}

Thus replacing β_{F,s} : P_{F,s} → M_{F,s} by β_{F,s̃} we obtain a projective presentation of M_{F,s} which satisfies Hypothesis 1.1.

We apply [9] to attach to each geometric conjugacy class E(s) for which s ∈ G is ℓ-stable, three pairs of functors:

H_{s} = H_{R,s} : mod_{RG} → mod_{H_{R,s}}
Ĥ_{s} = Ĥ_{R,s} : mod_{H_{R,s}} → mod_{RG}, (4.30)

where R ∈ {K, O, F }. If s is not ℓ-stable, we have to replace in case of R = F the element s by s̃ and get the functors H_{F,s̃}, Ĥ_{F,s̃}.

Recall that these functors are defined as follows:

\begin{aligned}H_{s} & = & Hom_{RG}(P_{s}, −)/Hom_{RG}(P_{s}, −)J_{s} \\ Ĥ_{s} & = & A_{P_{s}} ∘ (− ⊗_{H_{s}} M_{R,s}). \end{aligned} (4.31)

Here H_{s} = H_{R,s} = R ⊗_{O} H_{O,s} is the endomorphism ring End_{RG}M_{R,s}, if s is ℓ-stable or R = O or K. Otherwise we again replace s by s̃ and apply Lemma 4.29. Moreover J_{s} = J_{R,s} is defined as in section 1.

The next result generalizes Theorem 4.12 and was shown in [7, 3.6]:

Theorem 4.32 Let s ∈ G be semisimple and ℓ-stable. Then

End_{RG}(M_{r,s}) = H_{R,s} ≅ H_{R,q}(W_{s}) = H_{R,q^{d_{1}}} (𝔖_{n_{1}}) ⊗ H_{R,q^{d_{2}}} (𝔖_{n_{2}}) ⊗ ⋯,

for R ∈ {K, O, F }, where the d_{i} are given as degrees of the irreducible factors of the minimum polynomial and the n_{i} as multiplicities of these factors in the characteristic polynomial of s.

Note that by construction

W (L_{s}, C_{s}) ≤ W (Ls̃, Cs̃). (4.33)

Indeed W (L_{s}, C_{s}) is a standard Young subgroup of W (Ls̃, Cs̃). Thus

F ⊗_{O} End_{OG}(M_{O,s}) ≤ End_{F,s}(M_{F,s}) = F ⊗_{O} End_{OG}(M_{O,s̃}) ≅ H_{F,s̃},

and the inequality is proper, if s is not ℓ-stable.

We shall now construct “standard” ℓ-stable elements. For this we need to know how to find ℓ-elements in finite extensions of GF (q^{i}). The following results follow easily from elementary facts in finite field theory.

For 0 ≤ i ∈ ℤ define e(i) to be the smallest natural number k such that

1 + q^{i} + (q^{i})^{2} + ⋯ + (q^{i})^{k−1} ≡ 0 mod ℓ. (4.34)

Note that e(i) is the multiplicative order of q^{i} in the finite field GF (l), unless q^{i} ≡ 1 mod ℓ, in which case e(i) = ℓ. In any case e(i) ≤ ℓ. Let e = e(1). If q ≡ 1 mod ℓ, then q^{i} ≡ 1 mod ℓ for all 1 ≤ i ∈ ℤ. Otherwise q^{i} ≡ 1 mod ℓ if and only if i = el^{m} for some 0 ≤ m ∈ ℤ. Replacing e by e(i) we obtain

e(e(i)ℓ^{m}) = l, (4.35)

for all nonnegative integers m. Moreover GF (q^{ij}) contains an ℓ-element which is not contained in any proper subfield if and only if j = e(i)ℓ^{m} for some 0 ≤ m ∈ ℤ.

Definition 4.36 We choose now for each field GF (q^{i}) and each number e(i)ℓ^{m} for every nonnegative integer m a fixed ℓ-element

y_{i}(m) ∈ GF (q^{ie(i)l^{m}}),

which is not contained in any proper subfield. Thus its minimum polynomial over GF (q^{i}) has degree e(i)ℓ^{m}, and its companion matrix is denoted by (y_{i}(m)) (compare 4.3). If s_{i} is a ℓ-regular element of GF (q^{i}) whose minimum polynomial over GF (q) has degree i, then s_{i}y_{i}(m) generates GF (q^{ie(i)ℓ^{m}}) over GF (q), hence its companion matrix (s_{i}y_{i}(m)) is an ie(i)ℓ^{m} × ie(i)ℓ^{m}-matrix.

A composition μ of k is called e(i)-ℓ-adic, if it has the form:

μ = (1, 1, …, 1, e(i), …, e(i), e(i)ℓ, …, e(i)ℓ, e(i)ℓ^{2}, …).

Let s in G be a fixed semisimple ℓ-regular element, and let n = n_{s} be defined as in Equation 4.5. A multi composition

μ = (μ^{(1)}, μ^{(2)}, …, μ^{(i)}, …) ⊧ n

is called e-ℓ-adic, if μ^{(i)} ⊧ n_{i} is e(d_{i})-ℓ-adic for all i, where

e = e_{s} = (e(d_{1}), e(d_{2}), …).

The corresponding parabolic subgroup W_{μ} of W_{s} is called e-ℓ-parabolic as well as the corresponding parabolic subgroup P_{μ}, Levi subgroup L_{μ} etc. of C_{G}(s). For multipartitions we use a similar notation. The set of e-ℓ-adic multipartitions of n_{s} is denoted by 𝔓_{s}^{e,ℓ}

For λ ∈ 𝔓_{s}^{e,ℓ} we define the element y_{λ} ∈ C_{G}(s)_{ℓ} as follows: Given

C_{G}(s) = ⋯ × GL_{n_{i}}(q^{d_{i}}) × ⋯

and correspondingly

λ = (…, λ^{(i)}, …)

with

λ^{(i)} = (…, \underbrace{e(d_{i})ℓ^{m}, …, e(d_{i})ℓ^{m}}_{j-times} …) ⊢ n_{i},

we define y_{λ} = ⋯ × y_{λ^{(i)}} × ⋯ with

y_{λ^{(i)}} = ⋯ × \underbrace{(y_{d_{i}}(m)) × ⋯ × (y_{d_{i}}(m))}_{j-times} × · · · .

The element

s_{λ} = sy_{λ}

is obviously an ℓ-stable semisimple element of G.

Note that two semisimple conjugate elements of G are ℓ-equivalent. The next result can be seen easily:

Lemma 4.37 Let s be a semisimple ℓ-regular element of G. Then

{s_{λ}| λ ∈ 𝔓_{s}^{e,ℓ}}

is a complete set of representatives of ℓ-equivalence classes of semisimple elements in G with semisimple part conjugate to s.

So every semisimple s element of G is ℓ-equivalent to precisely one standard ℓ-stable element defined above. We shall denote it by s̃, (refining our previous definition). Thus

s̃ = s′_{λ} (4.38)

for an λ ∈ 𝔓_{s′}^{e,ℓ}.

We define ℭ_{ss}^{ℓ} to be the subset of the representatives ℭ_{ss} of G-classes of semisimple elements consisting of the ℓ-regular elements in ℭ_{ss}, and we define

𝔖𝔗_{ℓ} = {s_{λ} ∈ G | s ∈ ℭ_{ss}^{ℓ}, λ ∈ 𝔓_{s}^{e,ℓ}}, (4.39)

thus 𝔖𝔗_{ℓ} is a set of representatives of ℓ-equivalence classes of elements of G consisting of ℓ-stable elements. Moreover we may assume that 𝔖𝔗_{ℓ} ⊆ ℭ_{ss}.

Theorems [9, 2.28 and 4.10] imply immediately:

Lemma 4.40 To each geometric conjugacy class E(s^{G}) of G we have attached six functors (choosing s as representative in s^{G}):

H_{R,s} : mod_{RG} → mod_{H_{R,s}}
H_{F,s̃} : mod_{F G} → mod_{H_{F,s̃}}
Ĥ_{R,s} : mod_{H_{R,s}} → mod_{RG}
Ĥ_{F,s̃} : mod_{H_{F,s̃}} → mod_{F G}  ,

where R = O or K.

Those functors induce bijections between

• The set of irreducibles in the geometric conjugacy class E(s) and the set of nonisomorphic irreducible H_{K,s}-modules.

• The set of nonisomorphic irreducible composition factors in the head of M_{F,s} and the nonisomorphic irreducible H_{F,s̃}- modules.

If in addition s is ℓ-stable, then those functors provide an embedding of the decomposition matrix of the triple

F ⊗_{O} H_{O,s} ≅ H_{F,s} ← H_{O,s} → H_{K,s} ≅ K ⊗_{O} H_{O,s},

and the decomposition numbers of OG describing the composition multiplicities of the irreducibleconstituents of hd(M_{F,s}) in reductions modulo ℓ of the elements of E(s).

In the next section we shall use results in representation theory of Hecke algebras to derive a first classification of the irreducible F G-modules. As pointed out in section 2 this will give us as well the subdivision of those irreducibles according to their semisimple vertices and sources. However, besides the labelling of the irreducible F G-modules we shall obtain more detailed information as for example about decomposition numbers. For the classification of the irreducibles alone and their subdivision in to Harish-Chandra series one needs only Result 4.16, a result of Fong and Srinivasan in [25] saying that the Brauer characters of KG-modules of the form S_{K}(s, λ), where λ ⊢ n_{s} and s ∈ ℭ_{ss} is ℓ-regular, are a basis of the space of Brauer characters of G, and the fact that the decomposition matrix of G is unitriangular (Corollary 6.44). All but the last part is known to hold for arbitrary finite reductive groups analogously, (for details and an extensive bibliography we refer to [17]). The unitriangularity of the decomposition matrix is conjectured. For general linear and unitary groups it can be achieved by HC-inducing ordinary Gelfand-Graev characters of all Levi subgroups and cutting by blocks (see [26] for details).

5 The irreducible F G-modules

In the last section we constructed the Hecke functors H_{s} and Ĥ_{s} attached to the geometric conjugacy class indexed by the semisimple element s. Over F we only need to consider representatives s of ℓ-equivalence classes. Our goal in this section is to construct a complete set of irreducible F G-modules. We know from [9], that we may construct irreducible F Gmodules by applying the functor Ĥ_{s} to irreducible H_{s}-modules.

We begin by describing briefly some results on representations of H = H_{R,s}(𝔖_{n}), for details we refer to [19]. Here for the moment R is an arbitrary commutative domain and q is a unit in R. For a composition μ = (μ_{1}, μ_{2}, …) ⊧ n we let W_{μ} = 𝔖_{{1,2,... ,μ_{1}}} × 𝔖_{{μ_{1}+1,... ,μ_{1}+μ_{2}}} × ⋯ be the corresponding standard parabolic subgroup of W = 𝔖_{n}. In section 4 a basis of H was presented together with the defining relations for the multiplication (see Equation 4.11). Obviously the R-space spanned by {T_{w} | w ∈ W_{μ}} is closed under multiplication and hence a subalgebra H_{μ} of H which is of course the Hecke algebra associated with W_{μ}. We define two special elements of H_{μ} ≤ H:

𝔵_{μ} = ∑_{w∈W_{μ}} T_{w}  𝔶_{μ} = ∑_{w∈W_{μ}}(−q)^{l(w)}T_{w}  . (5.1)

Both elements generate one dimensional representations of H_{μ}, and we denote the induced representations 𝔵_{μ}H and 𝔶_{μ}H by M^{μ} and M̃^{μ} respectively. Let λ be a composition of n and denote its dual partition by λ′. There is a unique element of shortest length (a distinguished double coset representative) w_{λ} of W such that W_{λ}^{w_{λ}} ∩ W_{λ′} = 1. We define 𝔷_{λ} = 𝔵_{λ}T_{w_{λ}}𝔶_{λ′} and the H_{s}-module S^{λ} = S_{R}^{λ} = 𝔷_{λ}H_{R,s}. Recall that a partition λ = (λ_{1}, λ_{2}, …) is e - restricted for the positive integer e, if λ_{i+1} − λ_{i} < e for all indices i, and it is e-regular if the dual partition λ′ is e-restricted. We summarize:

5.2 For λ ⊢ n we call S^{λ} Specht module.

• There is a symmetric associative bilinear form < , > on H whose Gram determinant with respect to any basis of H is a unit. In particular, for fields R the Hecke algebra H is a symmetric algebra and consequently every irreducible H-modules is isomorphic to a minimal right ideal of H.

• The Specht modules are free as R-modules and have a basis independent of the choice of R and q. If R is a field and H is semisimple, then {S^{λ} | λ ⊢ n} is a complete set of nonisomorphic irreducible H-modules.

• If R is a field then H is not semisimple precisely if q is an e-th root of unity in R for some e ≤ n.

There is an antiautomorphism ∗ of H induced by T_{w}^{∗} = T_{w^{−1}}. Thus we can define dual modules, and H-invariant bilinear forms.

• For λ ⊢ n there is an H-invariant symmetric bilinear form on S^{λ}, universally defined for all choices of R and q. The radical S^{λ⊥} of that form is either S^{λ} or is the unique maximal submodule of S^{λ}.

Let now R be a field of characteristic ℓ ≥ 0, and let q be a primitive e-th root of unity or, if q = 1, let e = ℓ for ℓ > 0 and e = ∞ otherwise.

• S^{λ⊥} ≠ S^{λ} precisely if λ is e- regular.

Set D^{λ} = S^{λ}/S^{λ⊥} if the partition λ of n is e-regular.

• {D^{λ} | λ ⊢ n, λ is ℓ-regular} is a complete set of nonisomorphic irreducible H-modules.

• The composition factors of S^{λ} are of the form D^{μ} for some e-regular partition μ of n satisfying λ ⊴ μ, and D^{λ} has composition multiplicity 1 in S^{λ} provided λ is e-regular.

Here ⊴ denotes the usual dominance ordering on the set Λ(n) of compositions of n. Thus if λ = (λ_{1}, λ_{2}, …) and μ = (μ_{1}, μ_{2}, …) are compositions of n we write λ ⊴ μ if ∑_{i=1}^{j} λ_{i} ≤ ∑_{i=1}^{j} μ_{i} for all j.

The last part of 5.2 asserts in particular, that the decomposition matrix Δ = Δ_{H} of H is unitriangular, where the partitions respectively the e-regular partitions of n are ordered compatibly with the dominance ordering downward.

Of course all these results carry over to tensor products of Hecke algebras associated with symmetric groups. In particular it carries over to our Hecke algebras H_{R,s}(W_{s}) = H_{s} of Theorem 4.32. We have to adjust however notation. Partitions have to be replaced by multipartitions λ ⊢ n_{s}, where again s is a semisimple element of G. The dominance ordering of partitions can be extended to multipartitions too: We define that λ ⊴ μ for λ, μ ⊢ n_{s} if λ^{(i)} ⊴ μ^{(i)} for all i. Note that q^{i} is a primitive e-th root of unity in F if and only if e = e(i), provided q^{i} ≢ 1 mod ℓ, where e(i) is defined in 4.34.

Thus in the notation of the previous section, setting n = n_{s} and e = e_{s} (compare 4.36) the set {S_{K}^{λ} | λ ⊢ n} is a complete set of irreducible nonisomorphic H_{K,s}-modules and of course these modules are precisely the Specht modules used in Theorem 4.13. We say that the multipartition λ = (λ^{(1)}, λ^{(2)}, …) of n is e-regular, if λ^{(i)} is e(d_{i})-regular for all i, and we denote the set of multipartitions of n_{s} by 𝔓_{s} and the subset of e-regular multipartitions of n_{s} by 𝔓_{s}^{reg}. Thus, if s is ℓ-stable, then {D^{μ}| μ ∈ 𝔓_{s}^{reg})} is a complete set of nonisomorphic irreducible H_{F,s}-modules and the decomposition matrix Δ_{s} = Δ_{H_{s}} of H_{O,s} is unitriangular.

We continue to use the notation introduced in the previous section. Thus henceforth C_{R,s} denotes the cuspidal RL_{s}-module as defined in 4.7,R ∈ {K, O, F }, where the Levi subgroup L_{s} was introduced in 4.8. Moreover M_{R,s} = R_{L_{s}}^{G}(C_{R,s}).

The next result tells us that we may often ignore the functor A_{P_{R,s}} in our definition of Ĥ_{R,s}, simplifying calculations:

Theorem 5.3 Let s ∈ G be semisimple, and let the projective presentation β = β_{R,s} : P_{R,s} → M_{R,s} be defined as in Lemma 4.23 and Theorem 4.27 for R ∈ {K, O, F }. Then M_{R,s} is P_{R,s}-torsionless for R = K and R = O. If in addition s is ℓ-stable, then M_{F,s} is P_{F,s}-torsionless as well.

Proof For R = K and R = O the theorem is Lemma 1.20. Thus let s be ℓ-stable. We present a proof which works for all choices of R.

We drop indices, when possible. We have to show that

Hom_{RG}(P, X) ≠ (0)

for all nonzero submodules X of M_{s}. Recall that L_{s} is the smallest standard Levi subgroup of G containing s. Let P_{s} = L_{s}U_{s} be the corresponding parabolic subgroup. Let Û_{s} = |U_{s}|^{−1} ∑_{u∈U_{s}} u. Recall the definition of the idempotent e_{μ_{s}} = e_{L_{s}} (Formulas 4.18 and 4.8), which generates the Gelfand-Graev representation of L_{s}, and remember our construction of P_{1} and P = R_{L_{s}}^{G}(P_{1}) in Lemma 4.23: Being a principal indecomposable projective L_{s}-module, P_{1} = e_{s}RL_{s} for some primitive idempotent e_{s} of RL_{s} which is one of the summands in a decomposition of e_{L_{s}} into a sum of orthogonal primitive idempotents. Thus P = R_{L_{s}}^{G}(P_{1}) = e_{s}Û_{s}RG.

Adjointness of HC-induction and HC-restriction (see e.g. [15, 1.7]) implies:

(5.4) Hom_{RG}(P, X) = Hom_{RG}(R_{L_{s}}^{G}(P_{1}), X) = Hom_{RL_{s}}(P_{1}, T_{L_{s}}^{G}(X)).

For a subgroup H of G we denote the normalizer of H by N_{G}(H). Now we apply Mackey decomposition ([13, 1.4]) to get

T_{L_{s}}^{G}(X) ≤ T_{L_{s}}^{G}(M_{s}) = ⨁_{d∈D_{μ,μ}} R_{L_{s}^{d}∩L_{s}}^{L_{s}}T_{L_{s}^{d}∩L_{s}}^{L_{s}^{w}}(C_{s}^{d}) = ⨁_{d∈N_{G}(L_{s})∩D_{μ,μ}} C_{s}^{d}. (5.5)

where Dμ denotes the set of distinguished right coset representatives W_{μ} of W, that is, d ∈ D_{μ} is the unique shortest element in W_{μ}d. Moreover for λ, μ ⊧ n let D_{λ,μ} = D_{λ}^{−1} ∩ D_{μ}. Note that the elements of D_{λ,μ} are distinguished W_{λ}-W_{μ}-double coset representatives, that is consist of the shortest elements in each double coset. We also use in Equation 5.5 that D_{λ,μ} is as well a set of P_{λ}-P_{μ}-double coset representatives in G, and for the second part, that C_{s} is cuspidal and hence its HC-restriction of any of its conjugates to a proper Levi subgroup is zero.

We let 𝔟_{s} be the block of OL_{s} containing C_{s}, and let ε_{s} be the corresponding central idempotent (considered as element of F L_{s} or OL_{s} ⊆ KL_{s}). Let d ∈ N_{G}(L_{s}) ∩ D_{μ,μ} and assume that C_{s}^{d} belongs to 𝔟_{s}. Note that C_{s}^{d} = C_{s^{d}}. By Theorem 4.16 the ℓ-regular parts of s and s^{d} are conjugate in L_{s}. Since s is ℓ-stable by assumption, s =_{L_{S}} s^{d}, that is d ∈ W_{s}, where W_{s} ≤ W is defined as in Formula 4.10. Note that W_{s} ⊆ D_{μ,μ}. We conclude that

⨁_{d∈N_{G}(L_{s})∩D_{μ,μ}} C_{s}^{d}ε_{s} = ⨁_{d∈W_{s}} C_{s}, (5.6)

Since P_{1} = e_{s}RL_{s}, we have by general theory that

Hom_{RL_{s}}(P_{1}, T_{L_{s}}^{G}(X)) ≅ XÛ_{s}e_{s}.

Let 0 ≠ x ∈ X. Let U be the unipotent radical of the fixed Borel subgroup B of G consisting of upper triangular matrices. Thus U is the subgroup of all upper unitriangular matrices in G. Denote the opposite group of lower unitriangular matrices in G by V, and for w ∈ W define U_{w}^{−} to be the subgroup {u ∈ U | wu ∈ V } of U. Since

M_{s} = R_{L_{s}}^{G}(C_{s}) = ⨁_{w∈D_{μ},u∈U_{w}^{−}} C_{s} ⊗ wu

and X ≤ M_{s}, we may write

x = m_{∘} ⊗ 1 + ∑_{1≠w∈D_{μ},u∈U_{w}^{−}} m_{w,u} ⊗ wu, (5.7)

for certain m_{w,u}, m_{∘} ∈ C_{s}. We may assume that m_{∘} ≠ 0. To see this note, that we can always multiply by a suitable element of G (of the form (wu)^{−1}) to get a nonzero coefficient m_{∘}. For R = O we may assume in addition that m_{∘} is not contained in 𝔭C_{O,s}, where 𝔭 as in the proof of Lemma 3.2 is the generating element of the maximal ideal of O. Otherwise we may replace x by 𝔭^{−k}x for some nonnegative integer k to get a suitable element of X_{O}. Set x_{1} = xÛ_{s}ε_{s}. Then Equations 5.5 and 5.6 imply

0 ≠ x_{1} = m_{∘} ⊗ 1 + ∑_{1≠w∈W_{s}} m_{w} ⊗ wÛ_{s} ∈ X, (5.8)

for certain m_{w} ∈ C_{s}, since U_{w}^{−} ≤ U_{s} for w ∈ W_{s}. To see this one has to use that w is distinguished. Choose 0 ≠ ψ_{R} ∈ Hom_{RL_{s}}(P_{1}, C_{s}), and let m = ψ_{R}(e_{s}). Then me_{s} = m ≠ 0 and we find an element a ∈ RL_{s} such that m_{∘}a = m. We have

0 ≠ x_{1}ae_{s} = m ⊗ 1 + ∑_{1≠w∈W_{s}} m_{w}(ae_{s})^{w^{−1}} ⊗ wÛ_{s}. (5.9)

We have shown that XÛ_{s}e_{s} = T_{L_{s}}^{G}(X)e_{s} ≠ (0), and the theorem follows from Formula 5.4. □

We remark that in [40, 4.7] a similar but weaker result is shown. Moreover the theorem can be applied to the case where s is not necessarily ℓ-stable. For R = K and O it holds as stated, for R = F we have to replace s by s̃, (compare to Lemma 4.29).

Recall that 𝔖𝔗_{ℓ} denotes a set of representatives of ℓ-equivalence classes of semisimple elements of G consisting of ℓ-stable elements (see 4.39). We fix s ∈ 𝔖𝔗_{ℓ}. By Theorem 4.27 the projective presentation β_{O,s} : P_{O,s} → M_{O,s} of M_{O,s} satisfies Hypothesis 1.1 and by 4.40 the functors H_{F,s} and Ĥ_{F,s} produce a bijection between a complete set of nonisomorphic irreducible H_{F,s}-modules and the nonisomorphic irreducible F G-modules occurring as composition factors in hd(M_{F,s}). Thus with n = n_{s} (see 4.5) and e = e_{s} (see 4.36) we define for λ ⊢ n using 5.2

S_{R}(s, λ) = R ⊗_{O} \sqrt{Ĥ_{O,s}(S_{O}^{λ})} (5.10)

for R ∈ {K, O, F }. If in addition λ ∈ 𝔓_{s}^{reg} we get

D(s, λ) = Ĥ_{F,s}(D^{λ}). (5.11)

It is well known that H_{R} = H_{R,s} is a symmetric R-algebra for fields R (see e.g. [19, 2.3]). Thus by Theorems 5.3 and 1.2 :

S_{R}(s, λ) = R ⊗_{O} \sqrt{S_{O}^{λ}M_{O,s}} = R ⊗_{O} \sqrt{z_{λ}M_{O,s}}, (5.12)

where z_{λ} ∈ H_{O,s} is defined as in the beginning of this section. For λ ∈ 𝔓_{s}^{reg} we have:

D(s, λ) = D^{λ}M_{F,s}. (5.13)

for μ ⊢ n and λ ∈ 𝔓_{s}^{reg} we denote the multiplicity of D^{λ} as composition factor of S_{F}^{μ} by d_{μλ}. The matrix

Δ = Δ_{s} = (d_{μ,λ})

is the decomposition matrix of H_{O,s}. We denote the set {D^{λ} | λ ∈ 𝔓_{s}^{reg}} of irreducible H_{F,s}-modules by ℌ_{s}.

Theorem 5.14 Let s ∈ 𝔖𝔗_{ℓ}. Then

𝔗_{s} = {D(s, λ) | λ ∈ 𝔓_{s}^{reg}} (5.15)

is precisely the Harish-Chandra series of irreducible F G-moduls with semisimple HC-vertex L_{s} and semisimple HC-source C_{F,s}. If s ≠ t ∈ 𝔖𝔗_{ℓ} then 𝔗_{s} and 𝔗_{t} are disjoint. If μ ⊢ n, then the multiplicity of D(s, λ) as composition factor of S_{F} (s, μ) equals the decomposition number d_{μλ} of H_{O,s}. Thus Δ_{s} is part of the ℓ-decomposition matrix of G.

Proof By 4.40 the functors H_{F,s} and Ĥ_{F,s} produce a bijection between 𝔗_{S} and ℌ_{s}, and 𝔗_{s} is the set of composition factors of hd(M_{F,s}). By Corollary 2.5 (compare 4.1) 𝔗_{s} is precisely the set of irreducible F G-modules having semisimple HC-vertex L_{s} and semisimple source C_{F,a}, since M_{F,s} = R_{L_{s}}^{G}(C_{F,s}). If s ≠ t ∈ 𝔖𝔗_{ℓ} then especially s =_{ℓ} t by 4.39 , and hence C_{F,s} and C_{F,t} cannot be conjugate in G by the definition of ℓ-equivalence in Lemma 4.29. By [13, 4.16] 𝔗_{s} and 𝔗_{t} are disjoint. The theorem follows using [9, 4.10], (compare 4.40). □

So far the set

𝔗 = ⋃_{s∈𝔖𝔗_{ℓ}} 𝔗_{s} (5.16)

is a set of nonisomporphic irreducible F G-modules, which is partitioned into HC-series 𝔗_{s}. We show now, that 𝔗 is indeed a complete set of non-isomorphic irreducible F G-modules.

We modify the (n, ∞)-index I(s) attached to the pair (s, λ) for λ ⊢ n_{s} in 4.2, by replacing the elements s_{1}, s_{2}, … in the lower lefthand part of I(s) belonging to some extension field of GF (q) by their ℓ-regular parts s′_{1}, s′_{2}, … to get an (n, ℓ)-index, that is we have a map

(s, λ) ↦ I_{ℓ}(s, λ) = ( \begin{matrix}d_{1} & d_{2} & …, & n_{1} & n_{2} & …, \\ s′_{1} & s′_{2} & …, & λ^{(1)} & λ^{(2)} & …, \end{matrix} ) (5.17)

For convenience we give the details of the definition of (n, ℓ)-indices, (see [21, 2.4]):

Definition 5.18 An (n, ℓ)-index is an array

I = ( \begin{matrix}d_{1} & d_{2} & …, & d_{N} & n_{1} & n_{2} & …, & n_{N} \\ s_{1} & s_{2} & …, & s_{N} & λ^{(1)} & λ^{(2)} & …, & λ^{(N)} \end{matrix} ) ,

such that

(i) the s_{i} are representatives of orbits under the Galois group of the extension field GF (q^{d_{i}}) of GF (q) as in 4.3. In addition s_{i} is ℓ-regular and there exists an ℓ-element y_{i} ∈ GF (q^{d_{i}}) with s_{i}y_{i} of degree d_{i} over GF (q).

(ii) The integers n_{i} are positive for all i and λ^{(i)} ⊢ n_{i}. (iii) n_{1}d_{1} + ⋯ + n_{N}d_{N} = n

Moreover in I the entries are unordered with respect to 1, …, N, that is, if π is a permutation of (1, 2, …, N) the (n, ℓ)-indices

( \begin{matrix}d_{1} & d_{2} & …, & d_{N} & n_{1} & n_{2} & …, & n_{N} \\ s_{1} & s_{2} & …, & s_{N} & λ^{(1)} & λ^{(2)} & …, & λ^{(N)} \end{matrix} )

and

( \begin{matrix}d_{1π} & d_{2π} & …, & d_{Nπ} & n_{1π} & n_{2π} & …, & n_{Nπ} \\ s_{1π} & s_{2π} & …, & s_{Nπ} & λ^{(1π)} & λ^{(2π)} & …, & λ^{(Nπ)} \end{matrix} )

are identified. An (n, ℓ)-index is called

(i) a head, if the s_{i} are pairwise different, and s_{i} has degree d_{i} over GF (q) for all i, (ii) a foot, if λ^{(i)} is e(d_{i})-regular for all i, where e(d_{i}) is defined in 4.34, and (iii) special, if s_{i} = s_{j} implies either i = j or d_{i} ≠ d_{j}.

We remark that all head indices are special. The indices above are defined for all primes ℓ and l = ∞. The latter case produces (n, ∞)-indices, and the head (n, ∞)-indices parametrize the conjugacy classes of G, (compare 4.2). Similarly, for our prime ℓ, the ℓ-regular conjugacy classes of G are parametrized by head (n, ℓ)-indices by [21, 2.8].

Lemma 5.19 Let s ∈ 𝔖𝔗_{ℓ} and λ ∈ 𝔓_{s}^{reg}. Then I_{s}(s, λ), defined in 5.17 is special foot, and the map

(s, λ) ↦ I_{ℓ}(s, λ)

is a bijection between {(s, λ) | s ∈ 𝔖𝔗_{ℓ}, λ ∈ 𝔓_{s}^{reg}} and the set of special foot (n, ℓ)-indices.

Proof Obviously I_{ℓ}(s, λ) is an (n, ℓ)-index, and it is a foot, since λ is e-regular, by Definition of e-regularity in the beginning of this section. Let i ≠ j and suppose s′_{i} = s′_{j}. Then s_{i} ≠ s_{j}, hence d_{i} ≠ d_{j} by the construction 4.36 of standard ℓ-stable semisimple elements of G. Thus I_{ℓ}(s, λ) is a special foot. Conversely 4.36 shows too that every special foot (n, ℓ)-index corresponds to exactly one pair (s, λ) with s ∈ 𝔖𝔗_{ℓ} and λ ∈ 𝔓_{s}^{reg}. □

Remark 5.20 Permuting the parts of s corresponds to the analogous permutation in the associated (n, ℓ)-index. As remarked in 4.14 our constructions are invariant under such operations justifying [21, 2.5] and our convention to fix the order of the parts of s in Formula 4.3.

It was shown in [21, 2.13] (compare [7, 6.4] and [8, Section 2]) that there is a bijection between the set of special foot and the set of head (n, ℓ)-indices, which in turn label the ℓ-regular conjugacy classes of G as remarked above. Thus we have shown:

Theorem 5.21 (The irreducible FG-modules) The set

𝔗 = {D(s, λ) | s ∈ 𝔖𝔗_{ℓ}, λ ∈ 𝔓_{s}^{reg}}

is a complete set of nonisomorphic irreducible F G-modules. Moreover

𝔗 = ⋃_{s∈𝔖𝔗_{ℓ}} 𝔗_{s}

is the decomposition of 𝔗 into Harish-Chandra series.

Remark 5.22 The bijection [21, 2.13] between head and special foot (n, ℓ)-indices is given by an e-ℓ-adic decomposition of the involved multipartitions. The classification in Theorem 5.21 was derived in [7, 6.16] and [8, 3.7].

We come now to the promised proof that all irreducible cuspidal F G-modules are liftabel. In 5.21 we have found all irreducible F G-modules in the set 𝔗 and therefore we have to determine which of those are cuspidal. By construction this happens precisely for such D(s, λ which satisfy L_{s} = G. But then s is ℓ-regular, the multipartition λ consists precisely of one part (1) and D(s, λ) = D(s, (1)) = C_{F,s}. Thus we have:

Corollary 5.23 Let ℭ_{ss}^{n} be the subset of all elements s of ℭ_{ss} whose minimum polynomial has degree n. Then

{D(s, (1)) | s ∈ ℭ_{ss}^{n}}

is a complete set of nonisomorphic cuspidal irreducible F GL_{n}(q)-modules. In particular every irreducible cuspidal F GL_{n}(q)-module is liftable.

6 q-Schur algebras

In this section we shall derive a classification of the irreducible F G-modules in terms of head (n, ℓ)-indices. This was first done by James in [40] without employing q-Schur algebras. Here we shall recover this classification using our general set up in the first section.

Let s ∈ ℭ_{ss}. Thus by 4.3

s = (s_{1})^{×n_{1}} × (s_{2})^{×n_{2}} × ⋯,

and n = n_{1}d_{1}+n_{2}d_{2}+⋯, where d_{i} is the degree of the minimum polynomial of s_{i} ∈ GF (q^{d_{i}}) over GF (q). By 4.8 L_{s} = L_{μ}, where μ = μ_{s} is given as

μ_{s} = (\underbrace{d_{1}, …, d_{1}}_{n_{1}-times}, \underbrace{d_{2}, …, d_{2}}_{n_{2}-times}, . . . . . .).

We set

L̂_{s} = GL_{n_{1}d_{1}} × GL_{n_{2}d_{2}} × · · · · · · . (6.1)

To λ = (λ^{(1)}, λ^{(2)}, …) ⊧ n_{s} we define the composition λ̃ of n by

λ̃ = (d_{1}λ_{1}^{(1)}, d_{1}λ_{2}^{(1)}, …, d_{1}λ_{k}^{(1)}, …, d_{2}λ_{1}^{(2)}, …, d_{i}λ_{j}^{(i)}, . . . . . .). (6.2)

The next lemma is obvious:

Lemma 6.3 Let s ∈ ℭ_{ss}. Then L_{s} ≤ L̂_{s}. The map λ ↦ λ̃ ⊧ n for λ ⊧ n_{s} induces a bijection W_{λ} ↦ Wλ̃ between the set of parabolic subgroups W_{λ} of the ramification group W_{s}, defined in 4.10 and the set of all parabolic subgroups of the Weylgroup W = 𝔖_{n} of G containing the Weyl group W (L_{s}) of L_{s} and being contained in the Weyl group W (L̂_{s}) of L̂_{s}. Thus

{Lλ̃ | λ ⊧ n_{s}} and {Pλ̃ | λ ⊧ n_{s}}

is precisely the set of standard Levi respectively parabolic subgroups placed between L_{s} and L̂_{s}, and P_{s} and P̂_{s} respectively, (extending the hat-notation to parabolic subgroups in the obvious way).

We have seen in 4.10 that the ramification group W_{s} may be identified with a subgroup of W. Moreover the standard parabolic subgroups of W_{s} are parametrized by {λ | λ ⊧ n_{s}}. Obviously we have W_{s} ≤ W (L̂_{s}) ≤ L̂_{s}.

Theorem 4.32 implies:

Lemma 6.4 Let λ ⊧ n_{s}, and let R ∈ {K, O, F }. Suppose that s is ℓ-stable if R = F. Then

End_{RLλ̃}(R_{L_{s}}^{Lλ̃}C_{R,s}) ≅ H_{R,s}(W_{λ}),

where W_{λ} is the parabolic subgroup of W_{s} corresponding to λ ⊧ n_{s}, and

H_{R,s}(W_{λ}) = H_{R,q^{d_{1}}} (W_{λ^{(1)}}) ⊗ H_{R,q^{d_{2}}} (W_{λ^{(2)}}) ⊗ · · · · · · .

Here W_{λ^{(i)}} is the parabolic subgroup of 𝔖_{n_{i}} coresponding to λ^{(i)} ⊧ n_{i}.

Note that

W_{λ} = W_{s} ∩ W (Lλ̃) = W_{s} ∩ Lλ̃ = W_{s} ∩ Pλ̃. (6.5)

If R = F and s is not ℓ-stable, we have to replace s by s̃ by 4.29. The next two lemmas are easy to prove using transitivity of HC-induction (see [15, 1.12]):

Lemma 6.6 Let A ⊆ H_{R,s}(W_{λ}) ⊆ H_{R,s}, and let L̃ = Lλ̃, L = L_{s}. Then

A(R_{L}^{G}C_{R,s}) = RG ; L̃(ARL̃ ; LC_{R,s}),

as RG-modules.

Lemma 6.7 Let L̃ be a standard Levi subgroup of G containing L = L_{s}. Let X be a pure sublattice of RL̃ ; LC_{O,s}. Then RG ; L̃X is a pure sublattice of R_{L}^{G}C_{O,s}.

We fix now s ∈ 𝔖𝔗_{ℓ} and hence may omit indices s. Thus L = L_{s}, L̂ = L̂_{s} and n = n_{s} etc. We fix as well λ ⊧ n and set L̃ = Lλ̃. Note that W_{λ} = W_{λ^{(1)}} × W_{λ^{(2)}} × ⋯, where λ = (λ^{(1)}, λ^{(2)}, …) and λ^{(i)} ⊧ n_{i}. We set

𝔶_{λ} = 𝔶_{λ^{(1)}}𝔶_{λ^{(2)}} ⋯ ∈ H(W_{λ}) ⊆ H, (6.8)

where 𝔶_{λ^{(i)}} ∈ H_{R,q^{d_{i}}} is defined as in Formula 5.1, replacing q by q^{d_{i}}. We define 𝔵_{λ} ∈ H(W_{λ}) analogously.

Observe that R𝔶_{λ} is an one dimensional H(W_{λ})-lattice. Consequently

𝔶_{λ}RL̃ ; LC_{K,s} = St_{λ,K}

is an irreducible KL̃-module. Parts (i) and (iv) of 4.19 applied to L̃ say:

Lemma 6.9 St_{λ,K} is the unique irreducible constituent of the Gelfand-Graev representation Γ_{λ,K} of L̃ which is contained in the geometric conjugacy class of L̃ associated with s ∈ L̃ ⊆ G. Moreover this component occurs with multiplicity one as composition factor of

RL̃ ; LC_{K}.

The idempotent eL̃ ∈ OL̃, generating the Gelfand-Graev representation Γ_{λ,O} of OL̃ was introduced in Formula 4.18. By general arguments the RL̃-module

Y_{λ,R} = RL̃ ; LC_{R}eL̃RL̃, (6.10)

for R ∈ {K, O, F }, is the RL̃-submodule of RL̃ ; LC_{R} generated by the images of all homomorphisms Φ : Γ_{λ,R} → RL̃ ; LC_{R}. Thus in the language of [9, 2.15] (compare section 1), Y_{λ,R} is the Γ-trace τ_{Γ}(RL̃ ; LC_{R}), setting momentarily Γ = Γ_{λ,R}. James showed in [40, 7.22] that the dimensions of Y_{λ,F} and Y_{λ,K} coincide. Consequently Y_{λ,O} is a pure sublattice of RL̃ ; LC_{O}. Lemma 6.9 implies that Y_{λ,K} = St_{λ,K} = 𝔶_{λ}RL̃ ; LC_{K}. We conclude that Y_{λ,O} = 𝔶_{λ}RL̃ ; LC_{K} ∩ RL̃ ; LC_{O} and hence contains 𝔶_{λ}RL̃ ; LC_{O}. We have therefore:

Theorem 6.11 (G. James) Let s ∈ 𝔖𝔗_{ℓ} and λ ⊧ n_{s}. Then the OL̃-lattice Y_{λ,O} is the

purification \sqrt{𝔶_{λ}RL̃ ; LC_{O}} of 𝔶_{λ}RL̃ ; LC_{O} in RL̃ ; LC_{O}.

Remark 6.12 Most of what we have done so far can in this section carries over in slightly adapted form to arbitrary finite groups with (BN)-pair. However Theorem 6.11 does not. How this happens and what can be done in this more general situation will be investigated in a forthcoming paper of the author and J. Gruber. It also will contain a direct much shorter proof of 6.11.

By Part (iv) of 4.19 there exists an epimorphism Ψ_{λ,O} : Γ_{λ,O} → Y_{λ,O}, thus Γ_{λ,O} contains the minimal projective cover β_{λ,O} : P_{λ,O} → Y_{λ,O} of Y_{λ,O}. As in 4.23 we see that P_{λ,O} is indecomposable, hence P_{λ,O} = f_{λ,s}OL̃ for some primitive idempotent f_{λ,s} ∈ OL̃. Recall the construction of the composition λ̃ of n associated with λ ⊧ n in Formula 6.2. The idempotent f̂_{λ,s} ∈ OPλ̃ is obtained by multiplying f_{λ,s} by the idempotent Û λ̃ affording the trivial representation of the unipotent radical U λ̃ of the parabolic subgroup Pλ̃ of G. Similarly êλ̃ = eL̃Û λ̃. Note that by construction in Formula 4.18

{êλ̃ | λ ⊢ n_{s}}

is a set of orthogonal primitive idempotents of OU of the group U of unipotent upper triangular matrices, affording pairwise different linear characters of U.

We set

P_{R,s} = ⨁_{λ⊢n_{s}} RG ; L̃P_{λ,R}, N_{λ,R} = RG ; L̃Y_{λ,R}, and N_{R,s} = ⨁_{λ⊢n_{s}} N_{λ,R}. (6.13)

A projective presentation of N_{R,s} is given as

β_{R,s} = ⨁_{λ⊢n_{s}} R_{Lλ̃}^{G}β_{λ,R} : P_{R,s} → N_{R,s}, (6.14)

where the subscript R indicates as usual, that we have tensored the corresponding O-object by R ∈ {K, O, F }. Note that HC-induction commutes with tensoring by R, thus all these objects are well defined.

Obviously

RG ; L̃P_{λ,R} = f̂_{λ,s}RG

and we may decompose the idempotents êλ̃ ∈ OPλ̃ into a sum of orthogonal primitive idempotents and choose f̂_{λ,s} among those such that

{f̂_{λ,s} | λ ⊢ n_{s}}

is a set of pairwise orthogonal idempotents of OG.

We set

f_{s} = ∑_{λ⊢n_{s}} f̂_{λ,s} ∈ OG, (6.15)

and we conclude that

P_{R,s} = f_{s}RG.

Note that for ω = ((1^{n_{1}}), (1^{n_{2}}), …) ⊢ n_{s} we have 𝔶_{ω} = 1 ∈ H_{R,s}, hence

f̂_{ω,s}RG ≅ P_{R,s} (6.16)

where P_{R,s} is defined as in 4.23. In particular P_{O,s} is direct summand of P_{O,s}. Moreover, for λ ⊢ n_{s} we have

N_{λ,R} ≤ M_{R,s}. (6.17)

Consequently, for fields R, every simple submodule of N_{R,s} is a submodule of M_{R,s} too. Therefore it occurs in hd(P_{R,s}) by 5.3, and hence in hd(P_{R,s}). From this we conclude immediately:

Theorem 6.18 Let s ∈ 𝔖𝔗_{ℓ}. Then N_{R,s} is P_{R,s}-torsionless for R ∈ {K, O, F }.

Theorem 6.19 Let s ∈ ℭ_{ss} be ℓ-stable. Then

β_{R,s} : P_{R,s} → N_{R,s}

satisfies Hypothesis 1.1 for any choice of R ∈ {K, O, F }.

Proof Let λ ⊢ n_{s}. All irreducible components of N_{λ,K} are contained in the geometric conjugacy class E(s) associated with s ∈ ℭ_{ss}: This is immediate from the construction using Formula 4.21. The composition factors of the kernel of β_{λ,K} belong to geometric conjugacy classes of the form E(s′y) for some y ∈ C_{G}(s′)_{ℓ} such that s ≠_{Lλ̃} s′y, by 4.19. Since s is ℓ-stable we have s ≠_{G} s′y by 4.24. This holds for all λ ⊢ n_{s}, and we conclude, that the irreducible constituents of ker β_{K,s} and N_{K,s} belong to different geometric conjugacy classe. In particular these modules have no composition factor in common and the theorem foolows from [9, 4.7]. □

We apply [9, 2.2 and 2.15] to construct functors connecting representations of G with those of End_{RG}(N_{R,s}). We use now 1.30 to investigate this endomorphism ring in terms of H_{R,s}.

Lemma 6.20 Let s ∈ ℭ_{ss} and λ ⊧ n_{s}. Then 𝔶_{λ}H_{s,O} is a pure sublattice of H_{s,O}.

Proof The lemma is easily reduced to the case, where s = 1 ∈ G, hence H_{R,s} = H_{R,q}(𝔖_{n}), and λ = (n) ⊢ n. Now the lemma follows easily from defintion 5.1. □

We define the q-Schur algebra S_{R,s} to be

S_{R,s} = End_{H_{R,s}}( ⨁_{λ⊢n_{s}} 𝔶_{λ}H_{R,s}). (6.21)

Note that S_{R,s} ≅ R ⊗_{O} S_{s,O} by 6.20. We set

E_{R,s} = ⨁_{λ⊢n_{s}} 𝔶_{λ}H_{R,s}. (6.22)

Note that by 5.2 H_{O,s} admits an associative bilinear form whose Gram determinant is a unit. By Theorem 1.36 the assumptions of 1.30 are satisfied in view of Result 5.3. Using Equation 1.28, functoriality of the Hom-functor H_{R,s} of 4.30, and Theorem 1.30 we get:

Theorem 6.23 Let s ∈ ℭ_{ss} be ℓ-stable. Then

H_{R,s}(N_{R,s}) = E_{R,s} (6.24)

and the functor H_{R,s} induces an R-algebra isomorphism also denoted by H_{R,s}:

H_{R,s} : End_{RG}(N_{R,s}) → End_{H_{R,s}}(E_{R,s}) = S_{R,s}. (6.25)

Thus End_{RG}(N_{R,s}) is the q-Schur algebra S_{R,s}. In [22] q-Schur algebras were introduced, related to representations of finite general linear groups, and their representations were investigated in [22] and [23]. In [11] it was shown that they (or rather Morita equivalent algebras) are building blocks of quantum coordinate rings of general linear groups. Indeed here q-Schur were defined as endomorphism ring of a certain action of the Hecke algebra H = H_{q,R}(𝔖_{r}) on tensor space E^{⊗r}, with E = R^{n}, such that

E^{⊗r} ≅ ⨁_{λ∈Λ(n,r)} 𝔵_{λ}H

as H-module. Here Λ(n, r) is the set of compositions of r into n parts. For n = r one gets an algebra Morita equivalent to ours here, observing that 𝔵_{λ}H and 𝔶_{λ}H are conjugate H-modules under an outer automorphism of H.

We describe next some facts about q-Schur algebras using results [23],[11], and [12], and we refer for details and proof to these papers. As in the beginning of section 5 we define for λ ⊧ n the element w_{λ} ∈ 𝔖_{n} to be the unique distinguished W_{λ}-W_{λ′}-double coset representative with W_{λ}^{w_{λ}} ∩ W_{λ′} = (1) and extend this notation to multi compositions:

w_{λ} = w_{λ^{(1)}}w_{λ^{(2)}} ⋯ ∈ W_{s} (6.26)

for λ = (λ^{(1)}, λ^{(2)}, …) ⊧ n_{s}. Moreover we define

𝔷_{λ} = 𝔷_{λ^{(1)}}𝔷_{λ^{(2)}} ⋯ ∈ H_{R,s}. (6.27)

Let λ ⊢ n_{s}. We may premultiply 𝔶_{λ}H_{R,s} by 𝔵_{λ}′T_{w_{_{λ}′}}, where λ′ = (λ^{(1)′}, λ^{(2)′}, …). We get so an epimorphism from 𝔶_{λ}H_{R,s} onto the Specht module S_{R}^{λ′} to the dual weight λ′ of λ, (see 5.2). We apply [19, 4.12] and [22, 2.1 and 2.4] to the special case of semisimple Hecke algebras and see that S_{K}^{λ′} occurs exactly once as composition factor of 𝔶_{λ}H_{K,s} and if S_{K}^{μ′} occurs as well with μ ⊢ n_{s}, then μ′ ⊵ λ. Consequently there is for any commutative ring R precisely one indecomposable direct summand Y_{λ′,R} of 𝔶_{λ}H_{R,s} such that 𝔵_{λ}′T_{w_{_{λ}′}} Y_{λ′,R} ≠ (0) and then

𝔵_{λ}′T_{w_{_{λ}′}} Y_{λ′,R} ≅ S_{R}^{λ′} (6.28)

is an epimorphic image of Y_{λ′,R}. We call Y_{λ′,R} alternating source or Young module, following [12, 2.1].

Note that Y_{λ,K} = S_{K}^{λ}, but Y_{λ,R} is not a Specht module in general. In particular Y_{λ,K} ≇ K ⊗ Y_{λ,O}, where tensoring is as usual tensoring over O. However, since O is a complete discrete valuation ring we may lift idempotents from End_{H_{F,s}}(E_{F,s}) ≅ F ⊗ End_{H_{s,O}}(E_{s,O}) to End_{H_{s,O}}(E_{s,O}) and we have therefore:

Y_{λ,F} ≅ F ⊗_{O} Y_{λ,O}. (6.29)

Let μ ⊢ n_{s} and suppose that S_{K}^{μ} occurs as composition factor in K ⊗ Y_{λ,O}, then S_{K}^{μ} occurs as compostion factor of 𝔶_{λ}′H_{K,s} = K ⊗ 𝔶_{λ}′H_{s,O} as well. We apply [19, 4.12] observing that Specht modules here are irreducible and selfdual by [19, 4.15], since H_{K,s} is semisimple, and conclude from [20, 3.1 and 3.5] that μ′ ⊵ λ′ and hence μ ⊴ λ. We order the set 𝔓_{s} of multipartitions of n_{s} compatibly with the dominance ordering downward (e.g. lexicographically downward) and let d_{μλ} to be the multiplicity of S_{K}^{μ} as composition factor of K ⊗ Y_{λ,O}. We define the square matrix Δ̂_{s} to be:

Δ̂ = Δ̂_{s} = (d_{μλ}). (6.30)

We shall see below that our notation here is consistent with our defintion of decomposition numbers of Hecke algebras in the previous section.

Let X be an indecomposable direct summand of E_{R,s} and let π ∈ S_{R,s} be the projection of E_{R,s} onto X. Thus πS_{R,s} is a projective indecomposable S_{R,s}-module, which has a simple head for fields R. We denote the irreducible S_{K,s}-module corresponding to the indecomposable direct summand Y_{λ,K} ≅ S_{K}^{λ} of E_{K,s} by W_{K}^{λ}, and call this module Weyl module to highest weight λ. Indeed this is the Weyl module to weight λ as investigated in [23, 3.8] (using dual weights). It is defined universally, that is for any commutative ring R and any choice of parameters for H_{R,s}, as follows.

For ω = ((1^{n_{1}}), (1^{n_{2}}), …) ⊢ n_{s} we have 𝔶_{ω} = 1 ∈ H_{R,s} and so 𝔶_{ω}H_{R,s} is the regular H_{R,s}-module. Let π_{λ} be the idempotent of S_{R,s} projecting E_{R,s} onto 𝔶_{λ}H_{R,s}. Then H_{R,s} ≅ π_{ω}S_{R,s}π_{ω} and so multiplication by π_{ω} on the right is a functor Π_{R,s}, which takes S_{R,s}-modules to H_{R,s}-modules, (called the q-Schur functor). It is a special case of the constructions in [9] and this type of functor has been thoroughly investigated (see e.g [1] or for the special case of Schur algebras [30]). As endomorphism of E_{R,s}, an element h ∈ H_{r,s} acts by projecting first E_{R,s} onto H_{R,s} = 𝔶_{ω}H_{R,s}, and then by left multiplication by h denoted again by h abusing notation. Thus h is identified with hπ_{ω} ∈ S_{R,s}. Now the Weyl module W_{R}^{λ} is the right ideal of S_{R,s} generated by

ẑ_{λ} = z_{λ}π_{ω} = 𝔵_{λ}T_{w_{λ}}𝔶_{λ}′π_{ω}. (6.31)

For fields R we have an irreducible S_{R,s}-module L_{R}^{λ} taking the head of the principal projective S_{R,s}-module generated by the projection of E_{R,s} onto the Young module Y_{λ,R}.

We summarize some of the main results of [23]:

Theorem 6.32 Let s ∈ ℭ_{ss} be ℓ-stable. Then

(i) {Y_{λ,R} | λ ⊢ n_{s}} is a complete set of nonisomorphic indecomposable direct summands of E_{R,s}.

(ii) {W_{K}^{λ} | λ ⊢ n_{s}} is a complete set of nonisomorphic irreducibel S_{K,s}-modules. (iii) The Weyl module W_{R}^{λ} to highest weight λ ⊢ n_{s} is free as R-module with basis in-

dependent of the choice of R and the parameters occurring in H_{R,s}. It is a cyclic S_{R,s}-module generated by a highest weight vector. Moreover, if R is a field, then W_{R}^{λ} has a unique maximal ideal J(W_{R}^{λ}) and

W_{R}^{λ}/J(W_{R}^{λ}) ≅ L_{R}^{λ}.

Finally W_{R}^{λ} = L_{R}^{λ} for all λ ⊢ n_{s} if and only if H_{R,s} is semisimple. (iv) If R is a field, {L_{R}^{λ} | λ ⊢ n_{s}} is a complete set of nonisomorphic irreducible S_{R,s}-

modules.

(v) If R is a field, and L_{R}^{μ} is a composition factor of W_{R}^{λ}, then μ ⊵ λ for λ, μ ∈ 𝔓_{s}. Moreover the irreducible module L_{R}^{λ} has multiplicity one as composition factor of W_{R}^{λ}.

(vi) The decomposition matrix of S_{s,O}, describing the multiplicity of L^{μ} as composition factor of W_{R}^{λ}, is Δ̂_{s}. In particular ths matrix is a lower unitriangular square matrix.

Note that part vi) of the Theorem follows easily from Fittings lemma and Brauer reciprocity, observing the previous parts.

Remark 6.33 Our labelling of Weyl and irreducible modules of q-Schur algebras is the one of [11]. In [23] and related papers the labelling is dual to ours here, and consequently the decomposition matrix of S_{s,O} there is upper unitriangular.

Note that the functor Π_{R,s} : mod_{S_{R,s}} → mod_{H_{R,s}} has a right inverse, namely

Π̂_{R,s} = − ⊗_{H_{R,s}} π_{ω}S_{R,s}. (6.34)

Moreover, as remarked before, it is a special case of our constructions in [9] and the first section, considering the identity map as projective presentation of the S_{R,s}-module π_{ω}S_{R,s}. In particular for fields R, the functor Π_{R,s} maps an irreducible module for S_{R,s} either to the zero or to an irreducible H_{r,s}-module, and the decomposition matrix of H_{O,s} is part of the decomposition matrix of S_{O,s} by [9, 4.10]. We have:

Theorem 6.35 Let λ ∈ 𝔓_{s}. Then Π_{R,s}(W_{R}^{λ}) ≅ S_{R}^{λ}. Moreover Π_{R,s}(L_{R}^{λ}) is isomorphic to D_{R}^{λ} for λ ∈ 𝔓_{s}^{reg} and (0) otherwise. Finally Δ_{s} can be obtained from Δ̂_{s} by eliminating all columns from Δ̂_{s} labelled by λ ⊢ n_{s} with λ ∉ 𝔓^{reg}.

Proof By 6.31 the Weyl module W = W_{R}^{λ} is generated by 𝔷̂_{λ,R}, hence

Π_{R,s}(W_{R}^{λ}) = W_{R}^{λ} · π_{ω} = 𝔷̂_{ω}S_{R,s}π_{ω} = 𝔷_{λ,R}π_{ω}S_{R,s}π_{ω} = 𝔷_{λ,R}H_{R,s} = S_{R}^{λ}.

Now the theorem follows easily from the fact, that the unitriangular decomposition matrix Δ_{s} of H_{O,s} is by [9, 4.10] part of the decomposition matrix Δ̂_{s} of S_{O,s}, and from [9, 2.30]).

□

We apply now [9] using Theorems 6.19, 6.23, and the projective presentation β_{s,R} : P_{R,s} → N_{R,s} of 6.19 to get functors:

(6.36)  S_{R,s} : mod_{RG} → mod _{S_{R,s}}
Ŝ_{R,s} : mod_{S_{s,R}}→ mod _{RG},

where R ∈ {K, O, F }. Note that W_{λ,O}N_{O,s} is an OG-sublattice of N_{O,s} which might be not pure. By 1.2 and 6.18 however

W_{O}^{λ}N_{O,s} = Ŝ_{O,s}(W_{O}^{λ}), (6.37)

hence Ŝ_{O,s}(W_{O}^{λ}) ≤ N_{O,s} and the next definition makes sense.

Definition 6.38 Let λ ⊢ n_{s}. Then we define G-modules

(i) S_{R}(s, λ) = R ⊗_{O} \sqrt{Ŝ_{O,s}(W_{O}^{λ})} for R ∈ \{K, O, F \}. (ii) D(s, λ) = Ŝ_{F,s}(L^{λ}).

We have to explain why our notation here and in 5.12 and 5.11 is consistent. By construction the map H_{R,s} : End_{RG}(N_{R,s}) → End_{H_{R,s}}(E_{R,s}) respects projections onto direct summands. In particular, it takes the projection π̃_{ω} of N_{R,s} onto its direct summand M_{R,s} = N_{ω,R} to the projection π_{ω} of E_{R,s} onto its direct summand 𝔶_{ω}H_{R,s} = H_{R,s}. Omitting indices R, s we have:

Lemma 6.39 The following diagram is commutative:

mod_{H} \overline{\begin{matrix}− ⊗_{H} M \\  \end{matrix}}mod_{RG}
[figure: − ⊗_{S} N]
mod_{S}^{Π̂}

Proof We have by 6.34

(6.40)  (− ⊗_{S} N) ∘ Π̂ = (− ⊗_{S} N) ∘ (− ⊗_{H} π_{ω}S) = − ⊗_{H} π_{ω}S ⊗_{S} N
= − ⊗_{H} M,

since π_{ω}S ⊗_{S} N = π̃_{ω}N = M. □

Further relations between the functors Π, H, S and their various right inverses will be investigated elsewhere. We shall justify now our notation for the irreducible G-modules using the same labels for images under the functors Ŝ and Ĥ:

Theorem 6.41 Let λ ⊢ n_{s}. Then

Ĥ_{R,s}(S_{R}^{λ}) = Ŝ_{R,s}(W_{R}^{λ}) = S(s, λ),

and if in addition λ ∈ 𝔓_{s}^{reg}, then

Ĥ_{F,s}(D_{F}^{λ}) = Ŝ_{F,s}(L_{F}^{λ}) = D(s, λ).

If λ ∉ 𝔓_{s}^{reg}, then Π(L_{F}^{λ}) = (0).

Proof We apply Theorems 6.18, 5.3 and 1.2 to obtain

\begin{aligned}Ŝ_{R,s}(W_{R}^{λ}) & ≅ W_{R}^{λ}π_{ω} · N_{R,s} = W_{R}^{λ}π_{ω}π_{ω} · N_{R,s} ≅ W_{R}^{λ}π_{ω} · π̃_{ω}N_{R,s} \\ & ≅ W_{R}^{λ}π_{ω} · M_{R,s} ≅ S_{R}^{λ} · M_{R,s} ≅ Ĥ_{R,s}(S_{R}^{λ}) = S(s, λ), \end{aligned}

since W_{R}^{λ}π_{ω} = S_{R}^{λ} = S(s, λ) by Theorem 6.35. A similar argument applies for the F Gmodules D(s, λ). □

We choose now s ∈ ℭ_{ss}^{ℓ}, thus s is semisimple of order prime to ℓ, and let λ ⊢ n_{s}. Since s is ℓ-regular, the associated (n, ∞)-index 4.2

(s, λ) ↔ I_{ℓ}(s, λ) = ( \begin{matrix}d_{1} & d_{2} & … & n_{1} & n_{2} & … \\ s_{1} & s_{2} & … & λ^{(1)} & λ^{(2)} & … \end{matrix} ) (6.42)

is a head (n, ℓ)-index by Definition 5.18, where again s_{i} ∈ GF (q^{d_{i}}) is determined by 4.3, λ = (λ^{(1)}, λ^{(2)}, …), and n_{s} = (n_{1}, n_{2}, …) Moreover d_{i} is the degree of the minimum polynomial of s_{i} over GF (q). Thus we have a bijection between the set {(s, λ) | s ∈ ℭ_{ss}^{ℓ}, λ ⊢ n_{s}} and the set of head (n, ℓ)-indices. Recall that the latter parametrizes the ℓ-regular conjugacy classes by [21, 2.8]. Observe further that D(s, λ) ≅ D̸(t, μ) for s, t ∈ ℭ_{ss}^{ℓ}, λ ⊢ n_{s} and μ ⊢ n_{t} with (s, λ) ≠ (t, μ). This follows from 4.16 for s ≠ t and from [9, 2.28] if s = t and λ ≠ μ. Thus the number of elements in {D(s, λ) | s ∈ ℭ_{ss}^{ℓ}, λ ⊢ n_{s}} is precisely the number of ℓ-regular conjugacy classes of G, and we have shown (comp. [22, 4.9]):

Theorem 6.43 The set

{D(s, λ) | s ∈ ℭ_{ss}^{ℓ}, λ ⊢ n_{s}}

is a complete set of non-isomorphic irreducible F G-modules. The F G-module S_{F} (s, μ) for s ∈ ℭ_{ss}^{ℓ} and μ ⊢ n_{s} is a reduction modulo ℓ of the irreducible KG-module S_{K}(s, μ) and every composition factor of S_{F} (s, μ) is of the form D(s, λ) for some λ ⊢ n_{s}. The matrix Δ̂_{s} defined in 6.30 is the part of the ℓ-modular decomposition matrix of G which describes the composition factors in reductions modulo ℓ of irreducible KG-modules in the geometric conjugacy class E(s).

As an immediate consequence in view of Result 4.16 we have:

Corollary 6.44 The ordinary and ℓ-modular irreducible representations of G can be ordered such that the ℓ-modular decomposition matrix Δ̂ of G is lower unitriangular. Here the lower unitriangular matrices Δ̂_{s} with s ∈ ℭ_{ss}^{ℓ} are positioned down the main diagonal of Δ̂.

As remarked in 5.22 there is a bijection between the head and special foot (n, ℓ)-indices given by an e-ℓ-adic decomposition of multipartitions. It was shown in [21] that the corresponding irreducible F G-modules for corresponding indices coincide. More percisely, if s ∈ ℭ_{ss}^{ℓ} and λ ⊢ n_{s} and if I_{ℓ}(s, λ) is the corresponding head (n, ℓ)-index as defined in 6.42, we find y ∈ C_{G}(s)_{ℓ} (constructed by the procedure in Definition 4.36) such that the following holds: sy ∈ 𝔖𝔗_{ℓ} and there is λ̂ ∈ 𝔓_{sy}^{reg}, given by the e-ℓ-adic decomposition of λ, such that

I_{ℓ}(s, λ) ↦ I_{ℓ}(sy, λ̂) (6.45)

is a bijection between the set of head and the set of special foot (n, ℓ)-indices. Here the special foot (n, ℓ)-index I_{ℓ}(sy, λ̂) is defined as in 5.17. The following result was shown in [21, 5.1]:

Theorem 6.46 Let the (n, ℓ)-indices I_{ℓ}(s, λ) and I_{ℓ}(sy, λ̂) be defined as in 6.45. Then

D(s, λ) ≅ D(sy, λ̂).

In the representation theory of algebraic groups in positive characteristic Steinberg’s famous tensor product theorem [44] decomposes irreducible rational representations into twisted tensor products. On the level of highest weights, Theorem 6.46 looks precisely like an analogous result. In [14] it was shown, that behind this is indeed a tensor decomposition for the involved F G-modules.

We can now describe the Harish-Chandra series of the irreducible F G-modules in more detail. For s ∈ ℭ_{ss}^{ℓ} and λ ⊢ n_{s} we determine y ∈ C_{G}(s)_{ℓ} as above. Let

sy = (t_{1})^{×k_{1}} × (t_{2})^{×k_{2}} × ⋯, (6.47)

where t_{i} is of degree l_{i} over GF (q), then

L_{t} = \underbrace{GL_{l_{1}}(q) × ⋯ × GL_{l_{1}}(q)}_{k_{1}factors} ×GL_{l_{2}}(q) ⋯ (6.48)

by Formula 4.8 and L_{t} is the semisimple HC-vertex of D(s, λ) by Theorem 5.21. The outer tensor product

\underbrace{D(t_{1}, (1)) ⊗ ⋯ D(t_{1}, (1))}_{k_{1}factors} ⊗D(t_{2}, (1)) ⊗ ⋯ (6.49)

is a semisimple HC-source of D(s, λ) (replacing (t_{i}) in indices by t_{i}). Note that D(t_{i}, (1)) = C_{F,t_{i}}. Moreover, applying 6.46 once more we have D(t_{i}, (1)) = D(s_{i}, (1^{l_{i}})), where s_{i} is the ℓ-regular part of t_{i} (which coincides with one of the parts s_{j} in Formula 4.3 for s). Thus the semisimple HC-source of D(s, λ) can also be described as outer tensor product

\underbrace{D(s_{1}, (1^{l_{1}})) ⊗ ⋯ ⊗ D(s_{1}, (1^{l_{1}}))}_{k_{1}factors} ⊗D(s_{2}, (1^{l_{2}})) ⊗ · · · . (6.50)

In Theorems 5.14 and 6.43 we obtained also results on the ℓ-modular decomposition matrix of G. Indeed the information we have so far suffices to calculate all decomposition numbers of G from those of q-Schur algebras by an purley combinatorial algorithm. This was shown in [21] and [22]. For this, however, quotients of Hom-functors do not contribute anything new, hence we present here only a rough sketch of some ideas connected with Hom-functors and refer for details and proofs to the papers mentioned above. First one observes that the statement on decomposition numbers in Theorem 6.43 may be extended to all ℓ-stable elements s in ℭ_{ss} by [9, 4.10]:

Theorem 6.51 Let s ∈ 𝔖𝔗_{ℓ}. Then Δ̂_{s} describes the composition factors of the F Gmodules S_{F} (s, λ) of the form D(s, μ) for λ, μ ⊢ n_{s}.

Theorem 6.46 matches up (again by [21]) part the columns of Δ̂_{s′} and of Δ̂_{s}. However, in general, Δ̂_{s′} has more columns than Δ̂_{s}, and hence there may be composition factors in S_{F} (s, λ) which are not of the form D(s, μ), where λ, μ ⊢ n_{s}. However, it was shown in [40] that this is not the case under the additional hypothesis, that s has an irreducible minimum polynomial, or equivalently that s = (s_{1})^{×d_{1}} and d_{1}n_{1} = n. From this one derives the general case by an algorithm which involves only the Richardson Littlewood rule and inverting some decomposition matrices of q-Schur algebras. This takes care of all geometric conjugacy classes E(s) for s ∈ 𝔖𝔗_{ℓ}.

Thus let s ∈ ℭ_{ss}\𝔖𝔗_{ℓ}. Let s̃ be the unique element in 𝔖𝔗_{ℓ} which is ℓ-equivalent to s which exists by Lemma 4.37. Note that s′ = s̃′ by construction. Then we apply Lemma 4.29 to conclude that F ⊗_{O} End_{OG}(N_{s,O}) is a parabolic subalgebra of End_{F G}(N_{F,s}) ≅ End_{F G}(N_{F,s̃′}), extending the constructions of this section to elements s ∈ ℭ_{ss} which are not ℓ-stable. The Littlewood Richardson rule tells us, how Weyl modules decompose after inducing them to larger q-Schur algebras, and this corresponds to taking corresponding linear combinations in the decomposition matrix of E(s̃′) (of which Δ̂_{}_{s̃′} is a part) to get the decomposition matrix of E(s) yielding thus all decomposition numbers of OG in terms of the matrices Δ̂_{s}.

Recall Result 4.16 about the blocks of G. I particular for s ∈ ℭ_{ss}^{ℓ} we introduced the notation E_{ℓ}(s) to be the sum of blocks with semisimple part s. Since we vary G now we add a supperscript G, that is we write E_{ℓ}^{G}(s) = E_{ℓ}(s). Blocks whose semisimple part is the identity element of G are called unipotent. Here is another application of our pervious combinatorial results, which were first observed in [22, 6.2] and yields a Jordan decomposition of blocks and decomposition numbers of G (which is for arbitrary finite reductive groups still conjectural, see e.g. [17, 4.5]).

Corollary 6.52 Let s ∈ ℭ_{ss}^{ℓ}. Then the decomposition matrix of the sum E_{ℓ}^{G}(s) equals the decomposition matrix of the sum E_{ℓ}^{C_{G}(s)}(1) of unipotent blocks o the centralizer C_{G}(s) of s in G.

In fact it is conjectured that the block ideals of E_{ℓ}^{G}(s) and E_{ℓ}^{C_{G}(s)}(1) are Morita equivalent O-algebras.

Finally we want to apply the results of section 3 to generalize Takeuchi’s result [45]. Let s ∈ 𝔖𝔗_{ℓ}. In 3.4 applied to 4.27 we take as right ideals of H_{O} = H_{O,s} the ideals 𝔶_{λ}H_{O,s}, and these are pure right ideals by Lemma 6.20. The corresponding OG-lattices are the modules N_{λ,O}, so X_{R} of 3.4 becomes E_{R,s} and N_{R} there is replaced by N_{R,s} here. Furthermore the idempotent f of 3.4 is the idempotent f_{s} defined in Formula 6.15. Now Formulas 6.10, Quotients of Hom-functors II 57

6.16 and Theorem 6.11 imply that all assertions of 3.4 are satisfied. Moreover by Theorem 5.2 H_{F,s} is a symmetric hence self injective algebra. Thus we may apply Theorem 3.8 and Corollary 3.20 to get the following theorem, of which Takeuchi’s result [45] is a special case:

Theorem 6.53 Let s ∈ 𝔖𝔗_{ℓ}, and let f_{s} ∈ OG, E_{R,s}, M_{R,s}, and N_{R,s} for R ∈ {K, O, F } be defined as above. Then the following holds:

(i) M_{R,s}f_{s} ≅ ⨁_{λ⊢n_{s}} H_{R,s}𝔶_{λ} as left H_{R,s}-module. (ii) End_{H_{R,s}}(M_{R,s}f_{s}) ≅ S_{R,s} canonically. (iii) The ring f_{s}RGf_{s} acts on M_{R,s}f_{s} by right multiplication as H_{R,s}-endomorphisms

such that the canonical map f_{s}RGf_{s} → End_{H_{R,s}}(M_{R,s}f_{s}) is surjective. (iv) The f_{s}RGf_{s}-module M_{R,s}f_{s} has the double centralizer property. Thus in particular

End_{f_{s}RGf_{s}}(M_{R,s}f_{s}) ≅ H_{R,s}.

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