Quantum linear groups

and representations of GL_{n}(𝔽_{q})

Jonathan Brundan^{1}  Richard Dipper^{2}
Department of Mathematics  Mathematisches Institut B
University of Oregon  Universität Stuttgart
Eugene, OR 97403  Postfach 80 11 40
U.S.A.  70550 Stuttgart, Germany

brundan@darkwing.uoregon.edu    rdipper@mathematik.uni-stuttgart.de

Alexander Kleshchev^{1}

Department of Mathematics University of Oregon Eugene, OR 97403 U.S.A.

                    klesh@math.uoregon.edu

01991 subject classification: 20C20, 20C33, 20G05, 17B37.

1Partially supported by the NSF (grant nos DMS-9801442 and DMS-9600124).

2Partial support from the European TMR programme “Algebraic Lie Representations” (EC Network Contract No. ERB FMRX-CT97/0100).

Contents

Introduction 3

1 Quantum linear groups and polynomial induction 11 1.1 Symmetric groups and Hecke algebras . . . . . . . . . . . . . . . . . . . . . . 11 1.2 The q-Schur algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Tensor products and Levi subalgebras . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Polynomial induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 Schur algebra induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Classical results on GL_{n} 31 2.1 Conjugacy classes and Levi subgroups . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Harish-Chandra induction and restriction . . . . . . . . . . . . . . . . . . . . 33 2.3 Characters and Deligne-Lusztig operators . . . . . . . . . . . . . . . . . . . . 36 2.4 Cuspidal representations and blocks . . . . . . . . . . . . . . . . . . . . . . . 39 2.5 Howlett-Lehrer theory and the Gelfand-Graev representation . . . . . . . . . 43

3 Connecting GL_{n} with quantum linear groups 51 3.1 Schur functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 The cuspidal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 ‘Symmetric’ and ‘exterior’ powers . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 Endomorphism algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5 Standard modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Further connections and applications 71 4.1 Base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Connecting Harish-Chandra induction with tensor products . . . . . . . . . . 75 4.3 p-Singular classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4 Blocks and decomposition numbers . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5 The Ringel dual of the cuspidal algebra . . . . . . . . . . . . . . . . . . . . . 86

5 The affine general linear group 91 5.1 Levels and the branching rule from AGL_{n} to GL_{n} . . . . . . . . . . . . . . . 91 5.2 Affine induction operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3 The affine cuspidal algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.4 The branching rule from GL_{n} to AGL_{n−1} . . . . . . . . . . . . . . . . . . . . 105 5.5 A dimension formula for irreducibles . . . . . . . . . . . . . . . . . . . . . . . 109

Bibliography 113

Abstract

We give a self-contained account of the results originating in the work of James and the second author in the 1980s relating the representation theory of GL_{n}(𝔽_{q}) over fields of characteristic coprime to q to the representation theory of “quantum GL_{n}” at roots of unity.

The new treatment allows us to extend the theory in several directions. First, we prove a precise functorial connection between the operations of tensor product in quantum GL_{n} and Harish-Chandra induction in finite GL_{n}. This allows us to obtain a version of the recent Morita theorem of Cline, Parshall and Scott valid in addition for p-singular classes.

From that we obtain simplified treatments of various basic known facts, such as the computation of decomposition numbers and blocks of GL_{n}(𝔽_{q}) from knowledge of the same for the quantum group, and the non-defining analogue of Steinberg’s tensor product theorem. We also easily obtain a new double centralizer property between GL_{n}(𝔽_{q}) and quantum GL_{n}, generalizing a result of Takeuchi.

Finally, we apply the theory to study the affine general linear group, following ideas of Zelevinsky in characteristic zero. We prove results that can be regarded as the modular analogues of Zelevinsky’s and Thoma’s branching rules. Using these, we obtain a new dimension formula for the irreducible cross-characteristic representations of GL_{n}(𝔽_{q}), expressing their dimensions in terms of the characters of irreducible modules over the quantum group.

Introduction

This article is a contribution to the study of the modular representation theory of the finite general linear group GL_{n}(𝔽_{q}) over a field F of characteristic p coprime to q. We have attempted in the first place to give a self-contained account of the results originating in the work of James and the second author [D_{1}, D_{2}, DJ_{2}, DJ_{3}, J_{1}, J_{2}] in the 1980s relating representation theory of GL_{n}(𝔽_{q}) to representation theory of “quantum GL_{n}” at roots of unity. Since that time, there have been a number of conceptual simplifications to the theory, e.g. in [CPS_{3}, D_{3}, D_{4}, DDu_{1}, DDu_{2}, HL_{2}, T], which we have incorporated in the present approach from the outset. We mention above all the centrally important Morita theorem of Cline, Parshall and Scott from [CPS_{3}, §9]. We will reprove this Morita theorem in the present article in a selfcontained way, thus leading us to a new and independent approach to the results of [D_{1}, D_{2}, DJ_{2}, DJ_{3}, J_{1}, J_{2}] assumed in the Cline-Parshall-Scott argument. Along the way, we make many of these results more precise or more functorial, which is essential in order to prove the new results of the article described further below.

Our point of view has been to deduce as much as possible of the modular theory from standard, often purely character theoretic results in the characteristic zero theory of GL_{n}(𝔽_{q}), combined with knowledge of the highest weight representation theory of quantum linear groups. For the former, we have adopted the point of view of the Deligne-Lusztig theory, as described for GL_{n}(𝔽_{q}) by Fong and Srinivasan [FS], supplemented by various other basic results most of which can be found in Carter’s book [C]; we also appeal to the result of [DDu_{1}, §5],[HL_{2}] showing that Harish-Chandra induction is independent of the choice of parabolic subgroup, and the basic result of block theory proved in [BM] (also originally proved in [FS] though we do not use the full block classfication of Fong and Srinivasan). For quantum linear groups, we have followed the treatment by Parshall and Wang [PW] wherever possible, as well as [Cl, JM, DDo, Do_{4}] for various additional results.

We now summarize the main steps in the development, so that we can describe the new results of the article in more detail. We restrict our attention in this introduction to the case of unipotent representations, though there is no such restriction in the main body of the article. So let F be an algebraically closed field of characteristic p coprime to q, and let G_{n} = GL_{n}(𝔽_{q}). Let M denote the F G_{n}-module arising from the permutation representation of G_{n} on the cosets of a Borel subgroup. Following the idea of Cline, Parshall and Scott, we introduce the cuspidal algebra, which in the unipotent case is the quotient algebra C_{n} := F G_{n}/ ann_{F G_{n}}(M). The cuspidal algebra C_{n} is actually a quotient algebra just of the “unipotent block” B_{n} of F G_{n}, by which we mean the union of blocks of the algebra F G_{n} corresponding to the irreducible constituents of the module M.

Fixing some arbitrary m ≥ n, we let S_{m,n} denote the q-Schur algebra. Following [Du], this can be viewed as the quotient algebra S_{m,n} := U_{m}/ ann_{U_{m}}(V^{⊗n}) of Lusztig’s divided power version from [L_{2}] of the quantized enveloping algebra U_{m} := U_{q}(𝔤𝔩_{m}) (specialized over F at root of unity v = q^{1/2}), where V^{⊗n} is the nfold tensor power of the natural m-dimensional U_{m}-module. (Actually, in the main body of the article, we work with the quantized coordinate ring rather than with U_{m}.) At the heart of the theory is an explicit Morita equivalence (see (3.5a)):

β_{m,n} : mod(S_{m,n}) → mod(C_{n}).

The existence of this Morita equivalence was originally proved (in the unipotent case only) by Takeuchi [T], but we follow the quite different strategy of Cline, Parshall and Scott from [CPS_{3}, §9] for its construction. The idea is to exhibit an explicit projective generator for mod(C_{n}) with endomorphism algebra isomorphic to S_{m,n}. The projective generator used is roughly speaking the direct sum of the modules obtained by Harish-Chandra induction from all Steinberg representations of all standard Levi subgroups of G_{n} (the Steinberg representation does not in general remain irreducible on reduction modulo p so we refer the reader to §3.3 for the precise definition we use for this in characteristic p).

Write L_{m}(λ) (resp. Δ_{m}(λ)) for the irreducible (resp. standard or “Weyl”) U_{m}- module of highest weight λ, where λ is a partition of height at most m. If λ is a partition of n, these modules factor through the quotient S_{m,n}, so we obtain the C_{n}-modules:

L(1, λ) := β_{m,n}(L_{m}(λ′)),
Δ(1, λ) := β_{m,n}(Δ_{m}(λ′)),

where λ′ denotes the transpose partition. Since β_{m,n} is a Morita equivalence, the modules {L(1, λ) | λ ⊢ n} give a complete set of non-isomorphic irreducible C_{n}- modules. It turns out moreover that their inflations to the unipotent block B_{n} give a complete set of non-isomorphic irreducible B_{n}-modules, while the standard module Δ(1, λ) is a reduction modulo p of an irreducible ℂG_{n}-module affording the irreducible unipotent character χ_{λ} parametrized by the partition λ (cf. Theorem 4.1c, (4.4b)).

However, the unipotent block B_{n} has more irreducible characters in characteristic zero than just the characters {χ_{λ} | λ ⊢ n}, that is, one does not obtain complete information about the decomposition matrix of the unipotent block from the results described so far (only a square submatrix). To understand these additional irreducible characters in terms of the q-Schur algebra, our approach is to first prove an extension of the above Morita theorem to arbitrary elements σ ∈ 𝔽̄_{q}^{×} with p′-part equal to 1 (so the original unipotent case is then the special case that σ = 1).

The key new result in our proof of this extension relates the tensor product operator on U_{m}-modules to the Harish-Chandra induction operator ⋄ (more usually denoted by ∘) on GL-modules. To describe this, first recall that there is a natural notion of tensor product of two U_{m}-modules coming from the comultiplication of the Hopf algebra U_{m}. This means that for n_{1} + n_{2} = n, the tensor product M_{1} ⊗ M_{2} of an S_{m,n_{1}}-module with an S_{m,n_{2}}-module can be viewed as an S_{m,n}-module in a natural way. In other words, there is a bifunctor

? ⊗?′ : mod(S_{m,n_{1}}) × mod(S_{m,n_{2}}) → mod(S_{m,n}).

Also Harish-Chandra induction gives us a bifunctor

? ⋄?′ : mod(F G_{n_{1}}) × mod(F G_{n_{2}}) → mod(F G_{n}).

We show in Theorem 4.2a that these two bifunctors correspond under the Morita equivalence, i.e. that the bifunctors (β_{m,n_{1}}?) ⋄ (β_{m,n_{2}}?′) and β_{m,n}( ? ⊗ ?′ ) from mod(S_{m,n_{1}}) × mod(S_{m,n_{2}}) to mod(F G_{n}) are isomorphic. In order to prove this, we first prove a q-analogue of the main result of [BK, §2] concerning polynomial induction in quantum linear groups, see §§1.4–1.5.

Using the p-singular generalization of the Morita theorem, one obtains additional standard F G_{n}-modules of the form Δ(σ, λ), for σ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q} with p′-part equal to 1, and λ ⊢ k where n = kd. If the image of q is a primitive ℓ-th root of unity in F^{×}, there is an integer r ≥ 0 so that d = ℓp^{r}. We show that the module Δ(σ, λ) can be realized alternatively under the original (unipotent) Morita equivalence as

Δ(σ, λ) = β_{m,n}(Δ_{m}(λ′)^{[r]}),

where M^{[r]} denotes the S_{m,n}-module obtained by taking the rth Frobenius twist of M (cf. Theorem 4.3d). Finally, to construct a general standard module of B_{n}, i.e. a module equal to the reduction modulo p of an arbitrary irreducible complex character of B_{n}, one needs to consider modules of the form

Δ(σ_{1}, λ_{1}) ⋄ … ⋄ Δ(σ_{a}, λ_{a})

where the σ_{i} are non-conjugate elements of 𝔽̄_{q}^{×} with p′-part equal to 1 and the λ_{i} are partitions with ∑_{i} |λ_{i}| deg(σ_{i}) = n (cf. (4.4a)). In other words, the standard modules for the unipotent block B_{n} are lifts of C_{n}-modules which correspond under the Morita equivalence to Frobenius-twisted tensor products of standard modules for the quantum group of the form

Δ_{m}(λ′_{1})^{[r_{1}]} ⊗ ⋯ ⊗ Δ_{m}(λ′_{a})^{[r_{a}]}

(cf. Lemma 4.4c).

We then easily obtain alternative, functorial proofs of the main results of [FS, J_{2}, DJ_{3}], such as a description of the blocks of the algebra B_{n} (Theorem 4.4g) and an explicit formula for the decomposition numbers in terms of familiar (but unknown!) decomposition numbers and modular Littlewood-Richardson coefficients for the quantum algebra U_{m} (Theorem 4.4d). There is also a non-defining characteristic analogue of Steinberg’s tensor product theorem (Theorem 4.3e) for irreducible modules obtained originally by Du and the second author [DDu_{2}], which now follows immediately from the tensor product theorem for U_{m} and the other results described so far.

We mentioned earlier the work of Takeuchi. From our point of view, Takeuchi’s double centralizer property from [T] follows as an elementary consequence of a more general double centralizer property, which we regard as the non-defining characteristic analogue of Donkin-Howe duality [Do_{3}]. This has a particularly simple formulation in the unipotent case. Let T be the permutation representation of F G_{n} on the set of all m-step flags

0 = f_{0} ⊆ f_{1} ⊆ ⋯ ⊆ f_{m} = W_{n},

where W_{n} = 𝔽_{q}^{n} is the natural representation of G_{n}. It turns out that the action of F G_{n} on T factors through the quotient C_{n} of F G_{n}, so that T is a C_{n}-module in a natural way, and moreover, the endomorphism algebra End_{C_{n}}(T) is isomorphic to S_{m,n} (see Theorem 3.4a). This was originally proved in [DJ_{3}, Theorem 2.24] (see also the construction in [BLM]). In Theorem 4.5e, we prove the following double centralizer property:

End_{S_{m,n}}(T) = C_{n}.

This result is very natural from the point of view of quasi-hereditary algebras; since C_{n} is Morita equivalent to S_{m,n}, and the latter is a quasi-hereditary algebra according to [PW], C_{n} is itself quasi-hereditary. The permutation representation T is in fact a full tilting module for the quasi-hereditary algebra C_{n} in the sense of Ringel [R], so that S_{m,n} is a Ringel dual of C_{n} (cf. Theorem 4.5d).

The final results of the article are concerned with the affine general linear group H_{n} := AGL_{n}(𝔽_{q}), that is, the semi-direct product G_{n}W_{n} of the elementary Abelian group W_{n} by G_{n}. Our results here were motivated by the ideas of Zelevinsky [Z, §13] in characteristic 0. Corresponding to the natural embedding U_{m} ↪ U_{m+1} there is a Levi subalgebra of the q-Schur algebra S_{m+1,n} which we denote by

S_{m,≤n} ≅ ⨁_{j=0}^{n} S_{m,j}.

We define an affine analogue D_{n} of the cuspidal algebra C_{n}, namely, the algebra F H_{n}/ ann_{F H_{n}}(N) where N is the permutation representation of F H_{n} on the cosets of a Borel subgroup of G_{n} ⊂ H_{n} (“affine flags”). There is an explicit Morita equivalence (cf. (5.3d)):

β_{m,≤n} : mod(S_{m,≤n}) → mod(D_{n}).

Regarding β_{m+1,n} (resp. β_{m,≤n}) now as a functor from mod(S_{m+1,n}) to F G_{n} (resp. from mod(S_{m,≤n}) to F H_{n}), our main result on the affine linear group (Theorem 5.4c) shows that there is an isomorphism of functors:

ind_{G_{n}}^{H_{n}} ∘β_{m+1,n} ≅ β_{m,≤n} ∘ res_{S_{m,≤n}}^{S_{m+1,n}} .

This allows us to relate the problem of decomposing an induced module of the form ind_{G_{n}}^{H_{n}} L for an irreducible F G_{n}-module L to the problem of decomposing the restriction of an irreducible U_{m+1}-module to U_{m}. From this, we also explain how to calculate the composition multiplicities of the restriction res_{H_{n−1}}^{G_{n}} L from knowledge of the composition multiplicities of restrictions of irreducibles from U_{m} to U_{m−1}, a result which we regard as a modular analogue of Zelevinsky’s branching rule from [Z, Theorem 13.5] (in turn, this is really an extension of Thoma’s branching rule [Th]).

Finally, there is a well-known hook formula (2.3.3) for the degrees of the irreducible complex characters of G_{n}, but in positive characteristic remarkably little is known about the dimensions of the irreducible F G_{n}-modules (see e.g. [GT]). As an application of our branching rule for the affine linear group, we obtain a dimension formula (Theorem 5.5d) for the irreducible F G_{n}-modules in terms of (unknown!) characters of irreducible U_{m}-modules. To state this for unipotent representations, define the polynomial S_{λ}(t) ∈ ℤ[t] for a partition λ = (l_{1} ≥ l_{2} ≥ ⋯ ≥ l_{h} > 0) of height h by

S_{λ}(t) := ∑_{(m_{1},...,m_{h})} [∏_{i=1}^{n}(t^{i} − 1)/ ∏_{i=1}^{h}(t^{m_{1}+···+m_{i}} − 1)] ,

where the sum is over all h-tuples (m_{1}, …, m_{h}) that can be obtained by reordering the non-zero parts l_{1}, …, l_{h} of λ. Then, we show that

dim L(1, λ′) = ∑_{μ⊢n} m_{λ,μ}S_{μ}(q)

where m_{λ,μ} is the weight multiplicity of the weight μ in the irreducible U_{m}-module L_{m}(λ) of highest weight λ. Even though the latter weight multiplicities m_{λ,μ} are unknown in general, we expect that this result can be used to improve the known bounds for the low dimensional representations of GL_{n}(𝔽_{q}) in cross characteristics of [GT].

We now give a brief description of the layout of the article. First, we mention that there is a very brief guide to some of our non-standard notation for the expert reader, immediately following this introduction. Then, there are five further chapters. The first gives a rapid review of the basic results concerning quantum linear groups and the q-Schur algebra that we need, as well as proving the q-analogue of the results on polynomial induction from [BK]. Chapter 2 gives a similar review of the basic results from the characteristic zero theory of GL_{n}(𝔽_{q}).

The core of the theory is explained in chapter 3, where in particular we prove the most important special case of the Morita theorem and use it to introduce the various basic modules. Note that throughout chapter 3 (and the early part of chapter 4), we make certain standing assumptions (A1) and (A2) on the element σ ∈ 𝔽̄_{q}^{×}, which we only know a priori are satisfied if σ is p-regular. Only in chapter 4 are we able to show that these assumptions are satisfied in general, so that the earlier results are true without restriction, when we prove the extension of the Morita theorem to p-singular classes. Also in chapter 4 we discuss an integral version of the Morita theorem allowing us to understand base change and prove the result relating tensor products to Harish-Chandra induction mentioned earlier. We then give the applications to reprove the results on decomposition numbers of [DJ_{3}] and the non-defining tensor product theorem of [DDu_{2}], and end by proving the double centralizer property. Chapter 5 contains the results on the affine linear group.

Acknowledgements. We would like to thank the organizers of the “Algebraic Representation Theory” conference in Aarhus in 1998, where part of the research for this article was carried out.

Notation

Conventions:

Unless we specify otherwise, F is a commutative, unital ring (from chapter 2 onwards F will always be an algebraically closed field of characteristic p) and ⊗ denotes tensor product over F.

Given an algebra C over F, a C-module always means a left C-module that is finitely generated over F unless we explicitly say otherwise. We will write mod(C) for the category of all such finitely generated left C-modules. Given a C-module M, the endomorphism algebra End_{C}(M) is always assumed to act on M on the right.

For an F -coalgebra A, an A-comodule will always mean a right A-comodule, not necessarily finitely generated. So an A-comodule M is an F -module together with a structure map τ : M → M ⊗ A satisfying the usual axioms, see Sweedler [Sw]. We write comod(A) for the category of all such right A-comodules.

Notation overview:

G_{n} The general linear group GL_{n}(𝔽_{q}) over the finite field 𝔽_{q} of q elements, where q is a prime power not divisible by p.

h, k, d Positive integers such that h ≥ k and n = kd.

H_{k} The Hecke algebra H_{F,q^{d}}(Σ_{k}) associated to the symmetric group Σ_{k}, with standard basis {T_{w} | w ∈ Σ_{k}} (p.12).

S_{h,k} The q^{d}-Schur algebra S_{F,q^{d}}(h, k), which is the quotient of Lusztig’s divided power version of the quantum algebra U_{F,q^{d/2}}(𝔤𝔩_{h}) (over F at root of unity q^{d/2}) under its representation on the kth tensor power of its h-dimensional natural module (p.14).

σ An element of 𝔽̄_{q}^{×} of degree d over 𝔽_{q}, with associated companion matrix (σ) ∈ G_{d} (p.31).

M(σ) The irreducible, cuspidal F G_{d}-module associated to σ (p.39).

M^{k}(σ) The left F G_{n}-module obtained by Harish-Chandra induction from the outer tensor product M(σ) ⊠ ⋯ ⊠ M(σ) (k times); e.g. in the unipotent case, M^{n}(1) is the permutation representation of F G_{n} on cosets of a Borel subgroup. There is an explicit, fixed isomorphism End_{F G_{n}}(M^{k}(σ)) ≅ H_{k} (p.40).

C_{σ,k} The cuspidal algebra, namely, the image of F G_{n} under its representation on M^{k}(σ) (p.55).

10  Representations of general linear groups

Z^{k}(σ) The largest submodule of M^{k}(σ) on which each T_{w} ∈ H_{k} acts as (−1)^{ℓ(w)}; e.g. in the unipotent case, Z^{n}(1) is a modular reduction of the Steinberg module (p.58).

Λ^{k}(σ) The irreducible F G_{n}-module M^{k}(σ)(∑_{w∈Σ_{k}} T_{w}); e.g. in the unipotent case, Λ^{n}(1) is the trivial representation of F G_{n} (p.59).

Ż^{ν}(σ) For ν = (k_{1}, …, k_{a}) ⊨ k, Ż^{ν}(σ) is the F G_{n}-module obtained by Harish-Chandra induction from Z^{ν}(σ) = Z^{k_{1}}(σ) ⊠ ⋯ ⊠ Z^{k_{a}}(σ). Regarded as a C_{σ,k}-module, Ż^{ν}(σ) is projective and ⨁_{ν} Ż^{ν}(σ), summing over all compositions ν of k with at most h rows, is a projective generator for C_{σ,k}. The endomorphism algebra End_{C_{σ,k}}(⨁_{ν} Ż^{ν}(σ)) is isomorphic to S_{h,k}, so S_{h,k} and C_{σ,k} are Morita equivalent (p.61).

Λ̇^{ν}(σ) For ν = (k_{1}, …, k_{a}) ⊨ k, Λ̇^{ν}(σ) is the F G_{n}-module Harish-Chandra induced from Λ^{ν}(σ) = Λ^{k_{1}}(σ) ⊠ ⋯ ⊠ Λ^{k_{a}}(σ). The endomorphism algebra End_{C_{σ,k}}(⨁_{ν} Λ̇^{ν}(σ)), summing over all compositions ν of k with at most h rows, is again isomorphic to S_{h,k}. In addition, the double centralizer property holds, that is, End_{S_{h,k}}(⨁_{ν} Λ̇^{ν}(σ)) is isomorphic to C_{σ,k}. From this point of view, ⨁_{ν} Λ̇^{ν}(σ) is a full tilting module for the quasi-hereditary algebra C_{σ,k}, and S_{h,k} and C_{σ,k} are Ringel duals (p.61).

α_{σ,h,k} The functor Hom_{C_{σ,k}}(⨁_{ν} Ż^{ν}(σ),?) which yields the equivalence of categories between mod(C_{σ,k}) and mod(S_{h,k}) (p.67).

β_{σ,h,k} The functor (⨁_{ν} Ż^{ν}(σ))⊗_{S_{h,k}}? which is inverse to the Morita equivalence α_{σ,h,k} (p.67).

χ_{σ,λ} For F = ℂ, the ordinary irreducible characters arising as constituents of M^{k}(σ) are the characters {χ_{σ,λ} | λ ⊢ k} (p.38).

Δ(σ, λ) The modules {Δ(σ, λ) | λ ⊢ k} are the standard C_{σ,k}-modules (in the sense of quasihereditary algebras). The standard module Δ(σ, λ) is a modular reduction of a ℂG_{n}- module affording the character χ_{σ,λ}, and corresponds under the Morita equivalence to the standard S_{h,k}-module of highest weight λ′ (transpose partition). Explicitly, Δ(σ, λ) can be defined as the image of any non-zero element of the one dimensional space Hom_{F G_{n}}(Ż^{λ′}(σ), Λ̇^{λ}(σ)) (p.67).

L(σ, λ) The simple head of Δ(σ, λ). The {L(σ, λ)|λ ⊢ k} give a complete set of non-isomorphic irreducible C_{σ,k}-modules (p.67).

(s, λ) An arbitrary semisimple element s ∈ G_{n} can be written up to conjugacy in blockdiagonal form as s = diag((σ_{1})^{k_{1}} … (σ_{a})^{k_{a}}) for non-conjugate elements σ_{1}, …, σ_{a} of 𝔽̄_{q}^{×} and k_{1}, …, k_{a} ≥ 1. For such an s, fix also a multi-partition λ = (λ_{1}, …, λ_{a}) with each λ_{i} ⊢ k_{i} (p.32).

χ_{s,λ} The ordinary irreducible character obtained from χ_{σ_{1},λ_{1}} … χ_{σ_{a},λ_{a}} by Harish-Chandra induction (p.37).

Δ(s, λ) The module obtained by Harish-Chandra induction from Δ(σ_{1}, λ_{1}) ⊠ ⋯ ⊠ Δ(σ_{a}, λ_{a}), and a modular reduction of a ℂG_{n}-module affording the character χ_{s,λ} (p.83).

L(s, λ) Suppose in addition that s is p-regular. Then, L(s, λ) is the module obtained by Harish-Chandra induction from L(σ_{1}, λ_{1}) ⊠ ⋯ ⊠ L(σ_{a}, λ_{a}), and is isomorphic to the simple head of Δ(s, λ). All irreducible F G_{n}-modules have this form (p.83).

Chapter 1

Quantum linear groups and polynomial induction

In this first chapter, we collect together all the known results about quantum linear groups and the q-Schur algebra that we will need later. Then we prove some new results about polynomial induction in quantum linear groups, generalizing results of [BK, §2] in the classical setting.

1.1. Symmetric groups and Hecke algebras

Fix an integer k ≥ 1. We write ν ⊨ k if ν is a composition of k, so that ν = (k_{1}, k_{2}, …) for non-negative integers k_{1}, k_{2}, … such that k_{1} + k_{2} + ⋯ = k. If in addition k_{1} ≥ k_{2} ≥ …, then ν is a partition of k, written ν ⊢ k. The height h(ν) of a composition ν = (k_{1}, k_{2}, …) ⊨ k is the smallest integer a ≥ 1 such that k_{a+1} = k_{a+2} = ⋯ = 0. For λ ⊢ k, write λ′ for the transpose partition, i.e. the partition whose Young diagram is obtained by reflecting the Young diagram of λ in the main diagonal. Let ≤ denote the usual dominance order on the set of all compositions of k, namely, μ = (m_{1}, m_{2}, …) ≤ λ = (l_{1}, l_{2}, …) if and only if ∑_{i=1}^{j} m_{i} ≤ ∑_{i=1}^{j} l_{i} for all j ≥ 1. We also write μ ≥ λ if λ ≤ μ, and μ < λ if μ ≤ λ but μ ≠ λ.

We write Σ_{k} for the symmetric group on k letters. For w ∈ Σ_{k}, ℓ(w) is the length of w, that is, the minimal number ℓ of basic transpositions s_{1}, …, s_{ℓ} such that w = s_{1}s_{2} … s_{ℓ}. For ν = (k_{1}, …, k_{a}) ⊨ k, Σ_{ν} denotes the Young subgroup of Σ_{k} isomorphic to Σ_{k_{1}} × ⋯ × Σ_{k_{a}}.

For λ, μ ⊨ k, the set D_{λ} (resp. D_{μ}^{−1}) of elements of Σ_{k} which are of minimal length in their Σ_{k}/Σ_{λ}-coset (resp. their Σ_{μ}\Σ_{k}-coset) gives a set of distinguished Σ_{k}/Σ_{λ}- coset representatives (resp. Σ_{μ}\Σ_{k}-coset representatives). We set D_{μ,λ} = D_{μ}^{−1} ∩D_{λ}, to obtain a set of distinguished Σ_{μ}\Σ_{k}/Σ_{λ}-coset representatives. Moreover, if both Σ_{μ} and Σ_{λ} are subgroups of Σ_{ν} for some ν ⊨ k, the set D_{μ,λ}^{ν} = D_{μ,λ} ∩ Σ_{ν} is a set of Σ_{μ}\Σ_{ν}/Σ_{λ}-coset representatives.

We will freely use well-known properties of these distinguished double coset representatives, all of which can be found in [C, §2.7] or [DJ_{1}, §1]. We note in particular the following:

(1.1a) Given λ, μ ⊨ k and any w ∈ D_{μ,λ}, Σ_{μ} ∩^{w}Σ_{λ} is also a Young subgroup of Σ_{k}.

Now suppose that F is an arbitrary commutative ring and q ∈ F is arbitrary. The Iwahori-Hecke algebra H_{F,q}(Σ_{k}) associated to the symmetric group Σ_{k} over F at parameter q is a certain F -free F -algebra with basis {T_{w} | w ∈ Σ_{k}} and satisfying the relations

T_{w}T_{s} = \{ \begin{matrix}T_{ws} & if ℓ(ws) = ℓ(w) + 1, \\ qT_{ws} + (q − 1)T_{w} & if ℓ(ws) = ℓ(w) − 1 \end{matrix}

for all w ∈ Σ_{k} and all basic transpositions s ∈ Σ_{k}. For an indeterminate t, regard the ring F as a ℤ[t]-module by letting t act on F by multiplication by q. We have that:

H_{F,q}(Σ_{k}) ≅ F ⊗_{ℤ[t]} H_{ℤ[t],t}(Σ_{k}),  (1.1.1)

the isomorphism being the obvious one sending the basis element 1 ⊗ T_{w} of F ⊗_{ℤ[t]} H_{ℤ[t],t}(Σ_{k}) to the corresponding basis element T_{w} of H_{F,q}(Σ_{k}).

Write simply H_{k} for H_{F,q}(Σ_{k}). There are two F -free H_{k}-modules of rank one, and we next recall their definitions. We let

x_{k} = ∑_{w∈Σ_{k}} T_{w}  and  y_{k} = (−1)^{ℓ(w_{0})} ∑_{w∈Σ_{k}}(−q)^{ℓ(w_{0})−ℓ(w)}T_{w}

where ℓ(w_{0}) = \frac{1}{2}k(k − 1). Note that our definition of y_{k} is different from the original definition in [DJ_{1}], but only up to a scalar; the present definition allows the basic results to be stated also if q is not a unit, and is more convenient in view of (1.1c). According to [DJ_{1}, pp.28–29]:

(1.1b) For w ∈ Σ_{k}, T_{w}x_{k} = x_{k}T_{w} = q^{ℓ(w)}x_{k} and T_{w}y_{k} = y_{k}T_{w} = (−1)^{ℓ(w)}y_{k}.

The left H_{k}-modules I_{H_{k}} and E_{H_{k}}, called the trivial module and the sign module respectively, can now be defined as the left ideals I_{H_{k}} = H_{k}x_{k} and E_{H_{k}} = H_{k}y_{k}. We will also write I_{H_{k}} and E_{H_{k}} for the functions from H_{k} to F that arise from the action of H_{k} on these rank one modules. It will be important to know that if F is a field and q ≠ 0, then the modules I_{H_{k}} and E_{H_{k}} are the only one dimensional H_{k}-modules (cf. [DJ_{1}, Lemma 3.1]).

The algebra H_{k} possesses an involutive automorphism # defined on generators by T_{s}^{#} = −T_{s} + q − 1 for a basic transposition s. For any left H_{k}-module M, we let M^{#} denote the module which is equal to M as an F -module, but with new action defined by h · m = h^{#}m for m ∈ M, h ∈ H_{k}. This is the analogue for the Hecke algebra of the module M tensored with the sign representation in the symmetric group setting. We record the elementary calculation, showing that # swaps the trivial and sign representations (e.g. see [T, Proposition 2.2]):

§1.2 The q-Schur algebra  13

(1.1c) x_{k}^{#} = y_{k} and y_{k}^{#} = x_{k}.

Let ν = (k_{1}, …, k_{a}) ⊨ k. The above definitions generalize easily to the Young subalgebra H_{ν} = H_{F,q}(Σ_{ν}) of H_{k}, which is the subalgebra spanned by {T_{w} | w ∈ Σ_{ν}}. Identifying H_{ν} and H_{k_{1}} ⊗ ⋯ ⊗ H_{k_{a}} in the natural way, we let x_{ν} (resp. y_{ν}) denote the element x_{k_{1}} ⊗⋯⊗x_{k_{a}} (resp. y_{k_{1}} ⊗⋯⊗y_{k_{a}}) of H_{ν}. The H_{ν}-modules I_{H_{ν}} = H_{ν}x_{ν} and E_{H_{ν}} = H_{ν}y_{ν} are then the trivial and sign representations of H_{ν}.

As in [DJ_{1}, §3], we define the permutation module M^{ν} and signed permutation module N^{ν} of H_{k} to be left ideals H_{k}x_{ν} and H_{k}y_{ν} respectively. By (1.1c), we have that (M^{ν})^{#} ≅ N^{ν}. Moreover, H_{k} is a free right H_{ν}-module with basis {T_{w}|w ∈ D_{ν}}. Therefore, M^{ν} can also be defined as the induced module ind_{H_{ν}}^{H_{k}} I_{H_{ν}} = H_{k} ⊗_{H_{ν}} I_{H_{ν}}. Similarly, N^{ν} ≅ ind_{H_{ν}}^{H_{k}} E_{H_{ν}} = H_{k} ⊗_{H_{ν}} E_{H_{ν}}.

Define u_{λ} to be the unique element of D_{λ′,λ} such that Σ_{λ′} ∩^{u_{λ}} Σ_{λ} = {1}. Note that u_{λ} is precisely the element denoted w_{λ′} in [DJ_{4}, p.258]. We need the following fact proved in [DJ_{1}, Lemma 4.1]:

(1.1d) For λ ⊢ k, the space y_{λ′}H_{k}x_{λ} is an F -free F -module of rank one, generated by the element y_{λ′}T_{u_{λ}}x_{λ}.

The Specht module Sp^{λ} is the left ideal Sp^{λ} = H_{k}y_{λ′}T_{u_{λ}}x_{λ}. Observe that Sp^{λ} is both a submodule of M^{λ} and a quotient of N^{λ′}. As is well-known (see e.g. [DJ_{1}, Theorem 4.15]) if F is a field of characteristic 0 and q is a positive integer, H_{k} is a semisimple algebra and the Specht modules {Sp^{λ} | λ ⊢ k} give a complete set of non-isomorphic irreducible H_{k}-modules. In this case, we also have the following well-known characterization of Specht modules:

(1.1e) For F a field of characteristic 0 and q a positive integer, Sp^{λ} is the unique irreducible H_{k}-module that is a constituent of both N^{λ′} and M^{λ} (having multiplicity one in each).

Of course, we can take the very special case with F = ℚ and q = 1. Then, H_{ℚ,1}(Σ_{k}) is just the group algebra ℚΣ_{k} of the symmetric group over ℚ and we see that the Specht modules {Sp^{λ} | λ ⊢ k} give a complete set of non-isomorphic irreducible ℚΣ_{k}-modules. We will write X(Σ_{k}) for the character ring of Σ_{k} over ℚ. For λ ⊢ k, let

ϕ_{λ} ∈ X(Σ_{k})  (1.1.2)

denote the irreducible character of the symmetric group corresponding to the Specht module Sp^{λ} over ℚ. Given in addition μ ⊢ k, we write ϕ_{λ}(μ) for the value of the character ϕ_{λ} on any element of Σ_{k} of cycle-type μ.

1.2. The q-Schur algebra

Continue initially with F denoting an arbitrary ring and q ∈ F being arbitrary. Fix h ≥ 1. We write Λ(h, k) for the set of all compositions ν = (k_{1}, …, k_{h}) ⊨ k of height at most h and Λ^{+}(h, k) for the set of all partitions ν ∈ Λ(h, k). Let Λ(h) be the set of all h-tuples of non-negative integers. Given λ, μ ∈ Λ(h), we write λ + μ for their coordinate-wise sum, and cλ denotes λ + ⋯ + λ (c times). Identifying Λ(h) with the union ⋃_{k≥0} Λ(h, k), let Λ^{+}(h) = ⋃_{k≥0} Λ^{+}(h, k) ⊆ Λ(h). We refer to the elements of Λ^{+}(h) as dominant weights.

Following [DJ_{3}, DJ_{4}], we define the q-Schur algebra S_{h,k} = S_{F,q}(h, k) to be the endomorphism algebra

End_{H_{k}} ( ⨁_{ν∈Λ(h,k)} M^{ν}) ,

writing endomorphisms commuting with the left action of H_{k} on the right. By convention, the algebra S_{0,k} = S_{F,q}(0, k) is the trivial algebra F.

The q-Schur algebra is F -free and has a natural basis corresponding to certain double coset sums in H_{k}, which we now describe. Fix initially λ, μ ∈ Λ(h, k). For u ∈ Σ_{k}, we note that

∑_{w∈Σ_{μ}uΣ_{λ}} T_{w} = ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{μ}^{−1}}x_{μ}T_{w} = ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{λ}}T_{w}x_{λ}.  (1.2.1)

So, right multiplication in H_{k} by the element ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{μ}^{−1}} T_{w} induces a welldefined homomorphism of left H_{k}-modules

ϕ_{μ,λ}^{u} : H_{k}x_{μ} → H_{k}x_{λ}.

Extending ϕ_{μ,λ}^{u} to all of ⨁_{ν∈Λ(h,k)} M^{ν} by letting it act as zero on M^{ν} for ν ≠ μ, we obtain a well-defined element ϕ_{μ,λ}^{u} of S_{h,k}. Now we can state the well-known result, proved originally in [DJ_{4}, Theorem 1.4], under slightly more restrictive assumptions than here; see also [Ma, Theorem 4.8] for an argument valid in general.

(1.2a) S_{h,k} is F -free with basis {ϕ_{μ,λ}^{u} | μ, λ ∈ Λ(h, k), u ∈ D_{μ,λ}}.

We refer to the basis for S_{h,k} of (1.2a) as the natural basis. One shows easily using (1.2a) that S_{h,k} behaves well under base change. To be precise, one has the analogue of (1.1.1):

S_{F,q}(h, k) ≅ F ⊗_{ℤ[t]} S_{ℤ[t],t}(h, k)

where we are regarding F as a ℤ[t]-algebra by letting t act on F as multiplication by q. We also note at this point the following well-known property, which is an immediate consequence of the definition of the natural basis of S_{h,k} given above:

(1.2b) For h ≥ k, the F -linear map κ : H_{k} → S_{h,k}, defined on a basis element T_{w} for w ∈ Σ_{k} by κ(T_{w}) = ϕ_{(1^{k}),(1^{k})}^{w}, is a ring embedding.

We could have chosen to define the q-Schur algebra S_{h,k} equally well using the signed permutation module N^{ν} instead of M^{ν}, and we will later need this alternative point of view. So consider instead the algebra End_{H_{k}} (⨁_{λ∈Λ(h,k)} N^{λ}) , writing endomorphisms on the right again. Applying # to (1.2.1), we note that

∑_{w∈Σ_{μ}uΣ_{λ}} T_{w}^{#} = ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{μ}^{−1}}y_{μ}T_{w}^{#} = ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{λ}}T_{w}^{#}y_{λ}.  (1.2.2)

Now, as in [DJ_{3}, p.26], the following fact follows easily:

(1.2c) The algebras S_{h,k} and End_{H_{k}}(⨁_{λ∈Λ(h,k)} N^{λ}) are isomorphic, the natural basis element ϕ_{μ,λ}^{u} of S_{h,k} corresponding under the isomorphism to the endomorphism which is zero on N^{ν} for ν ≠ μ and sends N^{μ} into N^{λ} via the homomorphism induced by right multiplication in H_{k} by ∑_{w∈Σ_{λ}uΣ_{μ}∩D_{μ}^{−1}} T_{w}^{#} ∈ H_{k}.

Henceforth, we assume that F is a field and that q ∈ F is non-zero. Now we briefly recall some basic facts about the representation theory of the finite dimensional F -algebra S_{h,k}. The irreducible S_{h,k}-modules are parametrized by the set Λ^{+}(h, k) of all partitions of k of height at most h. We will write L_{h}(λ) for the irreducible S_{h,k}-module corresponding to λ ∈ Λ^{+}(h, k) in the standard way. So L_{h}(λ) is the unique irreducible S_{h,k}-module L with highest weight λ, that is, ϕ_{λ,λ}^{1}L ≠ 0 and ϕ_{μ,μ}^{1}L = 0 for all μ ∈ Λ(h, k) with μ ≰ λ.

Then, S_{h,k} is a quasi-hereditary algebra with weight poset (Λ^{+}(h, k), ≤), in the sense of Cline, Parshall and Scott [CPS_{2}]. This was first proved by Parshall and Wang [PW]; for other proofs see [Gr] or [Ma]. For recent accounts of the theory of quasi-hereditary algebras, see [Do_{4}, Appendix] or [KK].

In particular, we have associated to λ ∈ Λ^{+}(h, k) (in a canonical way) the modules Δ_{h}(λ) and ∇_{h}(λ), which are the standard and costandard modules corresponding to L_{h}(λ). We record the well-known properties:

(1.2d) Δ_{h}(λ) (resp. ∇_{h}(λ)) has simple head (resp. socle) isomorphic to L_{h}(λ), and all other composition factors are of the form L_{h}(μ) with μ < λ.

In addition to being a quasi-hereditary algebra, the algebra S_{h,k} possesses an anti-automorphism τ defined on the standard basis element ϕ_{μ,λ}^{u} by τ(ϕ_{μ,λ}^{u}) = ϕ_{λ,μ}^{u^{−1}} (see [DJ_{4}, Theorem 1.11]). Using this, we define the contravariant dual M^{τ} of an S_{h,k}-module M to be the dual vector space M^{∗} with action defined by (s.f)(m) = f(τ(s)m) for all s ∈ S_{h,k}, m ∈ M, f ∈ M^{∗}. This duality fixes the simple modules, that is, L_{h}(λ) ≅ L_{h}(λ)^{τ} for all λ ∈ Λ^{+}(h, k). We also note that Δ_{h}(λ) ≅ ∇_{h}(λ)^{τ}.

Given a left S_{h,k}-module M, we will write M̃ for the right S_{h,k}-module equal to M as a vector space with right action defined by ms = τ(s)m for m ∈ M, s ∈ S_{h,k}. In particular, this gives us modules L̃_{h}(λ), Δ̃_{h}(λ) and ˜∇_{h}(λ) for each λ ∈ Λ^{+}(h, k). We will use the following well-known result:

(1.2e) S_{h,k} has a filtration as an (S_{h,k}, S_{h,k})-bimodule with factors isomorphic to Δ_{h}(λ) ⊗ Δ̃_{h}(λ), each appearing precisely once for each λ ∈ Λ^{+}(h, k) and ordered in any way refining the dominance order on partitions so that factors corresponding to most dominant λ appear at the bottom of the filtration.

Actually, (1.2e) is a special case of a general property of quasi-hereditary algebras with an anti-automorphism τ fixing the simple modules as in the previous paragraph. It follows directly from the definition of quasi-hereditary algebra in terms of heredity ideals (see e.g. [CPS_{2}, p.92] or [KK, §1]) together with [DR, Statement 7] (one needs to observe that the module Ae of loc. cit. is a standard module and, by our assumption that τ fixes the simple modules, that eA ≅ Ãe). That the resulting filtration can be reordered in any way as described follows from another general property of an arbitrary quasi-hereditary algebra A, namely, that Ext_{A}^{1}(Δ(λ), Δ(μ)) = 0 unless λ < μ.

1.3. Tensor products and Levi subalgebras

To describe further results about the q-Schur algebra, we need to relate it to the quantum linear group. Actually, we only need to work with the associated “quantum monoid”, which is a certain deformation of the coordinate ring of the algebraic monoid of all n×n matrices over F. The bialgebra structure of the quantum monoid will allow us to take tensor products of modules for q-Schur algebras in a natural way. We have chosen here to use Manin’s quantization of the coordinate ring, see [PW], though one could equally well work with the coordinate ring of [DDo].

Continue with F being a field of characteristic p and assume in addition that q ∈ F^{×} has a square root in F. Let ℓ be the smallest positive integer such that q^{ℓ} = 1 (i.e. q is a primitive ℓth root of unity), taking ℓ = 0 if no such positive integer exists. We fix a square root v of q in F such that if v is a primitive fth root of unity then f ≢ 2(mod 4). Note this is always possible: if ℓ is odd one of ±\sqrt{q} is again a primitive ℓth root of unity, and if ℓ is even both of ±\sqrt{q} are primitive 2ℓth roots of unity.

The quantized coordinate ring A_{h} = A_{F,v}(h) is the associative, unital F -algebra generated by {c_{i,j} | 1 ≤ i, j ≤ h} subject to the relations

c_{i,s}c_{j,t} = c_{j,t}c_{i,s}  (i > j, s < t)
c_{i,s}c_{j,t} = c_{j,t}c_{i,s} + (v − v^{−1})c_{i,t}c_{j,s}  (i > j, s > t)
c_{i,s}c_{i,t} = vc_{i,t}c_{i,s}  (s > t)
c_{i,s}c_{j,s} = vc_{j,s}c_{i,s}  (i > j)

for all admissible 1 ≤ i, j, s, t ≤ h. Let I(h, k) denote the set of all k-tuples i = (i_{1}, …, i_{k}) of integers between 1 and h as in [G_{2}]. Then, A_{h} is graded by degree as A_{h} = ⨁_{k≥0} A_{h,k}, and each homogeneous component A_{h,k} is spanned by monomials c_{i,j} = c_{i_{1},j_{1}}c_{i_{2},j_{2}} … c_{i_{k},j_{k}} for all ‘multi-indices’ i, j ∈ I(h, k).

§1.3 Tensor products and Levi subalgebras  17

There is an F -bialgebra structure on A_{h} with counit ε : A_{h} → F and comultiplication Δ : A_{h} → A_{h} ⊗ A_{h} satisfying

ε(c_{i,j}) = δ_{i,j}, Δ(c_{i,j}) = ∑_{k∈I(h,k)} c_{i,k} ⊗ c_{k,j}

for all i, j ∈ I(h, k) and k ≥ 0. The subspace A_{h,k} is a subcoalgebra of A_{h}. We define the category of all polynomial representations (resp. polynomial representations of degree k) to be the category comod(A_{h}) (resp. the category comod(A_{h,k})). Note that comod(A_{h,k}) is a full subcategory of comod(A_{h}). Moreover, any A_{h}-comodule M can be decomposed uniquely as M = M_{0} ⊕ ⋯ ⊕ M_{k} ⊕ … where each M_{k} is polynomial of degree k and Hom_{A_{h}}(M_{k}, M_{l}) = 0 for k ≠ l.

Let

I^{2}(h, k) = \{(i, j) ∈ I(h, k) × I(h, k) | \begin{matrix}j_{1} ≤ ⋯ ≤ j_{k} and i_{l} ≤ i_{l+1} \\ whenever j_{l} = j_{l+1} \end{matrix} \} .

This is a set of representatives of the orbits of Σ_{k} acting diagonally by place permutation on I(h, k) × I(h, k). The monomials {c_{i,j} | (i, j) ∈ I^{2}(h, k)} give a basis for the coalgebra A_{h,k}. We let {ξ_{i,j} | (i, j) ∈ I^{2}(h, k)} denote the corresponding dual basis of the F -linear dual A_{h,k}^{∗}. It is known that A_{h,k}^{∗}, endowed with the naturally induced algebra structure, is isomorphic to the q-Schur algebra S_{h,k} from §1.2. Moreover, copying the argument of [DDo, 3.2.5] but for the Manin quantization, the isomorphism A_{h,k}^{∗} → S_{h,k} can be chosen so that ξ_{i,j} corresponds to the natural basis element ϕ_{μ,λ}^{u} from (1.2a) for suitable μ, λ ⊨ k and u ∈ D_{μ,λ}, up to multiplying by some power of the unit v. To be precise:

(1.3a) There is an algebra isomorphism A_{h,k}^{∗} → S_{h,k} under which ξ_{i,j}, for (i, j) ∈ I^{2}(h, k), maps to v^{−ℓ(u)}ϕ_{μ,λ}^{u} where:

(i) μ is the weight of i (so μ = (m_{1}, …, m_{h}) with m_{l} equal to the number of times the integer l appears in the tuple (i_{1}, …, i_{k}));

(ii) λ is the weight of j;

(iii) u is the unique element of D_{μ}^{−1} such that for each l = 1, …, k, i_{u^{−1}l} is equal

to the lth entry of the k-tuple (\underbrace{1, …, 1}_{m_{1} times}, \underbrace{2, …, 2}_{m_{2} times}, …, \underbrace{h, …, h}_{m_{h} times}) (u is then automatically

an element of D_{μ,λ});

Henceforth, we always identify algebras A_{h,k}^{∗} and S_{h,k} according to (1.3a). This allows us to regard a left S_{h,k}-module as a right A_{h,k}-comodule, and vice versa, in a natural way. In other words, we identify the category mod(S_{h,k}) with the full subcategory of comod(A_{h}) consisting of all A_{h}-comodules that are finite dimensional and of polynomial degree k. We often switch between these two points of view without comment.

In particular, for λ ∈ Λ^{+}(h, k), we have A_{h,k}-comodules, hence A_{h}-comodules, L_{h}(λ), Δ_{h}(λ) and ∇_{h}(λ) induced by the corresponding S_{h,k}-modules. The modules {L_{h}(λ)|λ ∈ Λ^{+}(h, k)} give a complete set of non-isomorphic irreducibles in the category comod(A_{h,k}). It follows directly from this that the modules {L_{h}(λ)|λ ∈ Λ^{+}(h)} give a complete set of non-isomorphic irreducibles in the category comod(A_{h}).

The bialgebra structure of A_{h} endows the category comod(A_{h}) with a natural notion of tensor product. As a special case, algebra multiplication A_{h,k} ⊗ A_{h,l} → A_{h,k+l} allows us to view the tensor product M ⊗ M′ of an A_{h,k}-comodule M and an A_{h,l}-comodule M′ as an A_{h,k+l}-comodule. Dually, we have an algebra map

S_{h,k+l} → S_{h,k} ⊗ S_{h,l},  (1.3.1)

which enables us to view the tensor product M ⊗ M′ of an S_{h,k}-module M and an S_{h,l}-module M′ as an S_{h,k+l}-module.

We next introduce notation for various familiar modules. First, let V_{h} = L_{h}((1)) = Δ_{h}((1)) = ∇_{h}((1)) be the natural module, a right A_{h}-comodule of dimension h over F. The kth tensor power V_{h} ⊗ ⋯ ⊗ V_{h} is naturally a right A_{h,k}-comodule, so can be regarded as a left S_{h,k}-module. This gives us tensor space, which we denote by V_{h}^{⊗k}. We also have the symmetric, divided and exterior powers:

S^{k}(V_{h}) = ∇_{h}((k)),
Z^{k}(V_{h}) = Δ_{h}((k)),
Λ^{k}(V_{h}) = L_{h}((1^{k})) = Δ_{h}((1^{k})) = ∇_{h}((1^{k})).

All these modules actually have more natural direct realizations (which we do not need here) as quotients or submodules of V_{h}^{⊗k}, see for instance [DDo, §2.1]. More generally, given any composition ν = (k_{1}, …, k_{a}) ⊨ k, define

S^{ν}(V_{h}) = S^{k_{1}}(V_{h}) ⊗ ⋯ ⊗ S^{k_{a}}(V_{h}),  (1.3.2)
Z^{ν}(V_{h}) = Z^{k_{1}}(V_{h}) ⊗ ⋯ ⊗ Z^{k_{a}}(V_{h}),  (1.3.3)
Λ^{ν}(V_{h}) = Λ^{k_{1}}(V_{h}) ⊗ ⋯ ⊗ Λ^{k_{a}}(V_{h}),  (1.3.4)

all of which can be regarded as right A_{h}-comodules or left S_{h,k}-modules. Observe that in the special case ν = (1^{k}) all of S^{ν}(V_{h}), Z^{ν}(V_{h}) and Λ^{ν}(V_{h}) are isomorphic simply to V_{h}^{⊗k}. We will need the following known descriptions of these modules as left ideals of S_{h,k}:

(1.3b) For ν ∈ Λ(h, k), (i) the left ideal S_{h,k}ϕ_{ν,ν}^{1} of S_{h,k} is isomorphic to Z^{ν}(V_{h}) as an S_{h,k}-module; (ii) providing h ≥ k, the left ideal S_{h,k}κ(y_{ν}) of S_{h,k} is isomorphic to Λ^{ν}(V_{h}) as

an S_{h,k}-module, where κ : H_{k} → S_{h,k} is the embedding of (1.2b).

We say an A_{h}-comodule M has a Δ-filtration (resp. a ∇-filtration) if M has an ascending filtration 0 = M_{0} < M_{1} < … with ∑_{i} M_{i} = M such that each factor M_{i}/M_{i−1} is isomorphic to a direct sum of copies of Δ_{h}(λ) (resp. ∇_{h}(λ)) for some fixed λ ∈ Λ^{+}(h) (depending on i). The following important fact is well-known, see for instance [PW, (10.4.1)]:

(1.3c) If M, M′ are right A_{h}-comodules having ∇-filtrations (resp. Δ-filtrations) then M ⊗ M′ also has a ∇-filtration (resp. a Δ-filtration).

In particular, (1.3c) implies that for any ν ∈ Λ(h, k), the modules S^{ν}(V_{h}) and Λ^{ν}(V_{h}) have ∇-filtrations, while Z^{ν}(V_{h}) and Λ^{ν}(V_{h}) have Δ-filtrations. The following fact is well-known; it can be deduced easily from basic properties of modules with Δ- and ∇-filtrations together with the Littlewood-Richardson rule:

(1.3d) For λ ∈ Λ^{+}(h, k), regard the transpose partition λ′ as a composition of k. Then, the space Hom_{S_{h,k}}(Z^{λ}(V_{h}), Λ^{λ′}(V_{h})) is one dimensional, and the image of any non-zero such homomorphism is isomorphic to Δ_{h}(λ).

Suppose now that q is a root of unity (i.e. ℓ > 0). Then, there is an analogue for A_{h} (hence for the q-Schur algebras) of Steinberg’s tensor product theorem. This was proved in [PW, chapter 9] (for ℓ odd) and [Cl] (in general). Other proofs (for the Dipper-Donkin quantization) have been given in [DDu_{2}, 5.6] or [Do_{4}, §3.2].

Let Ā_{h} = A_{F,1}(h) denote the free polynomial algebra over F on generators {c̄_{i,j} | 1 ≤ i, j ≤ h} (which is just the above bialgebra A_{h} in the special case v = 1). The comodule representation theory of Ā_{h} is precisely the classical polynomial representation theory of GL_{h} as discussed by Green [G_{2}]. For λ ∈ Λ^{+}(h), we will write L̄_{h}(λ), Δ̄_{h}(λ) and ¯∇_{h}(λ) for the irreducible, standard and costandard comodules of Ā_{h}, to distinguish them from the ones above for A_{h}. Of course, these are just the usual polynomial representations of GL_{h} over F.

Recall that we chose v earlier so that it is a primitive fth root of unity with f ≢ 2(mod 4). So we can apply [PW, chapter 9] for ℓ odd and [Cl] for ℓ even to deduce that there is for each r ≥ 0 a unique bialgebra homomorphism

F_{r} : Ā_{h} → A_{h}  (1.3.5)

such that c̄_{i,j} ↦ c_{i,j}^{ℓp^{r}} for all 1 ≤ i, j ≤ h. This map is called the rth Frobenius morphism. We stress that if q = 1 then Ā_{h} = A_{h}, ℓ = 1 and the zeroth Frobenius map F_{0} is just the identity map.

Using the Frobenius map F_{r}, we can regard an Ā_{h}-comodule M with structure map τ : M → M ⊗ Ā_{h} as an A_{h}-comodule with structure map (id_{M} ⊗F_{r}) ∘ τ : M → M ⊗ A_{h}. This gives the rth Frobenius twist of M denoted M^{[r]}. Note that if M is an Ā_{h,k}-comodule, then M^{[r]} is an A_{h,kℓp^{r}}-comodule.

For any s > 1, we say that λ = (l_{1}, …, l_{h}) ∈ Λ^{+}(h) is s-restricted if l_{i} − l_{i+1} < s for i = 1, …, h − 1. By convention, λ is 1-restricted if and only if λ = (0). By an (ℓ, p)-adic expansion of λ we mean some (non-unique) way of writing

λ = λ_{−1} + ℓλ_{0} + ℓpλ_{1} + ℓp^{2}λ_{2} + …

such that λ_{−1} ∈ Λ^{+}(h) is ℓ-restricted and each λ_{i} ∈ Λ^{+}(h) is p-restricted for i ≥ 0.

Now we can state the tensor product theorem (after the first twist this is the usual Steinberg tensor product theorem):

(1.3e) Suppose that λ ∈ Λ^{+}(h) has (ℓ, p)-adic expansion λ = λ_{−1} +ℓλ_{0} +⋯+ℓp^{r}λ_{r}. Then, L_{h}(λ) ≅ L_{h}(λ_{−1}) ⊗ L̄_{h}(λ_{0})^{[0]} ⊗ L̄_{h}(λ_{1})^{[1]} ⋯ ⊗ L̄_{h}(λ_{r})^{[r]}.

The next lemma gives a technical character theoretic fact of importance in §4.3. In the statement, for λ ⊢ n, c_{λ} is the number of elements of Σ_{n} of cycle-type λ. We also recall that ϕ_{μ} denotes the irreducible character of the symmetric group defined in (1.1.2).

1.3f. Lemma. Let ℓ and p be as above. Set m = ℓp^{r} for some r ≥ 0 and k = ml for some l ≥ 1. Then, for any h ≥ k,

L_{h}((m^{l})) = \frac{(−1)^{k+l}}{l!} ∑_{λ⊢l} ∑_{μ⊢k} c_{λ}ϕ_{μ}(mλ)Δ_{h}(μ′),

where the equality is written in the Grothendieck group of mod(S_{h,k}).

Proof. This is a calculation involving symmetric functions; we refer the reader to [M, §I.2–I.3] for the basic notions used in the proof. In particular, for μ ⊢ n, we will write ε_{μ} for (−1)^{n−h(μ)}, which is the sign of any element of Σ_{n} of cycle-type μ.

By (1.3e), the module L_{h}((m^{l})) is isomorphic to the Frobenius twist L̄_{h}((1^{l}))^{[r]}. So, the formal character of L_{h}((m^{l})) is the symmetric function

ψ = ∑_{1≤i_{1}<···<i_{l}≤h} x_{i_{1}}^{m}x_{i_{2}}^{m} … x_{i_{l}}^{m}.

Now using [M, §I.2, ex.8,10] one sees that ψ can be written as

ψ = \frac{1}{l!} ∑_{λ⊢l} ε_{λ}c_{λ}p_{mλ}

where p_{mλ} is the power sum symmetric function of [M, §I.2]. Now applying [M, I.7.8], we deduce that

ψ = \frac{1}{l!} ∑_{λ⊢l} ∑_{μ⊢k} ε_{λ}c_{λ}ϕ_{μ}(mλ)s_{μ}

where s_{μ} is the Schur function. Now according to [M, §I.7, ex.2], ϕ_{μ}(mλ) = ε_{mλ}ϕ_{μ′}(mλ). Noting that ε_{λ}ε_{mλ} = (−1)^{k+l}, we deduce that

ψ = \frac{(−1)^{k+l}}{l!} ∑_{λ⊢l} ∑_{μ⊢k} c_{λ}ϕ_{μ}(mλ)s_{μ′}.

Finally, recall by Weyl’s character formula that the formal character of Δ_{h}(μ′) is precisely the Schur function s_{μ′}.

§1.4 Polynomial induction  21

Finally in this section, we review the definition of the analogues of Levi subgroups, following [Do_{4}, §4.6]. Fix now a ≥ 1 and μ = (h_{1}, …, h_{a}) ⊨ h. Let L denote the standard Levi subgroup of G = GL_{h}(F) consisting of all invertible block diagonal matrices of block sizes h_{1}, …, h_{a}. Let

Ω_{μ} = {(i, j) ∈ [1, h] × [1, h] | there is some g ∈ L with g_{ij} ≠ 0}.

Define A_{μ} to be the quotient of A_{h} by the biideal generated by {c_{i,j} | 1 ≤ i, j ≤ h, (i, j) ∉ Ω_{μ}}. So A_{μ} is the quantum analogue of the coordinate ring of the monoid corresponding to the Levi subgroup L. We note that

A_{μ} ≅ A_{h_{1}} ⊗ ⋯ ⊗ A_{h_{a}}  (1.3.6)

and that A_{μ} is graded by degree. So for k ≥ 0 we can talk about the subcoalgebra A_{μ,k}, which is a coalgebra quotient of A_{h,k}; we have the coalgebra isomorphism:

A_{μ,k} ≅ ⨁_{k_{1}+···+k_{a}=k} A_{h_{1},k_{1}} ⊗ ⋯ ⊗ A_{h_{a},k_{a}}.  (1.3.7)

We have the categories of polynomial and polynomial degree k representations for the Levi subgroup, namely, the categories comod(A_{μ}) and comod(A_{μ,k}) respectively.

There are Levi analogues of the q-Schur algebra, also discussed by Donkin in [Do_{4}, §4.6]. Define S_{μ,k} to be the dual space A_{μ,k}^{∗}, with natural algebra structure inherited from the comultiplication and counit of A_{μ,k}. So dualizing (1.3.7), we have that

S_{μ,k} ≅ ⨁_{k_{1}+···+k_{a}=k} S_{h_{1},k_{1}} ⊗ ⋯ ⊗ S_{h_{a},k_{a}}.  (1.3.8)

We identify mod(S_{μ,k}) with the full subcategory of comod(A_{μ}) consisting of all finite dimensional A_{μ}-comodules of polynomial degree k.

Dual to the surjective coalgebra map A_{h,k} → A_{μ,k} we have a natural embedding S_{μ,k} ↪ S_{h,k}. We will always regard S_{μ,k} as a subalgebra of S_{h,k} embedded in this way. Explicitly, we have that:

(1.3g) The subalgebra S_{μ,k} of S_{h,k} is spanned by the standard basis elements ξ_{i,j} for i, j ∈ I(h, k) such that (i_{l}, j_{l}) ∈ Ω_{μ} for all l = 1, …, k.

1.4. Polynomial induction

We recall briefly the definitions of induction and restriction functors for coalgebras, following [Do_{1}]. Let (A, Δ, ε) and (A′, Δ′, ε′) be coalgebras over our fixed field F, and ϕ : A → A′ be a fixed coalgebra homomorphism. We have the exact restriction functor

res_{A′}^{A} : comod(A) → comod(A′)

defined on a right A-comodule M with structure map τ : M → M ⊗ A by letting res_{A′}^{A} M be the right A′-comodule equal to M as a vector space but with new structure map (id_{M} ⊗ϕ) ∘ τ : M → M ⊗ A′. On an A-comodule homomorphism α : M → M′, res_{A′}^{A} α is the same linear map α, but regarded now as an A′-comodule map. There is a comodule induction functor that is right adjoint to res_{A′}^{A}, namely

ind_{A′}^{A} : comod(A′) → comod(A).

To define this on objects, fix M ∈ comod(A′) with structure map τ : M → M ⊗ A′. Write |M| ⊗ A for the right A-comodule which is equal to M ⊗ A as a vector space, with structure map id_{M} ⊗Δ : M ⊗ A → M ⊗ A ⊗ A. Then, ind_{A′}^{A} M is the subcomodule of |M| ⊗ A consisting of all elements f such that

(τ ⊗ id_{A})(f) = (id_{M} ⊗[(ϕ ⊗ id_{A}) ∘ Δ])(f).  (1.4.1)

On a morphism α : M → M′, ind_{A′}^{A} α is the restriction to ind_{A′}^{A} M of the morphism α ⊗ id_{A} : |M| ⊗ A → |M′| ⊗ A.

We wish to study induction and restriction (in the sense of coalgebras as just defined) between A_{h} and its Levi quotient A_{μ} introduced just before (1.3.6). So choose μ ⊨ k and consider the polynomial restriction and induction functors:

res_{A_{μ}}^{A_{h}} :comod(A_{h}) → comod(A_{μ}),
ind_{A_{μ}}^{A_{h}} :comod(A_{μ}) → comod(A_{h}).

We record some basic properties:

1.4a. Lemma. (i) The functor res_{A_{μ}}^{A_{h}} sends finite dimensional modules to finite dimensional modules, and A_{h,k}-comodules to A_{μ,k}-comodules.

(ii) The functor ind_{A_{μ}}^{A_{h}} sends finite dimensional modules to finite dimensional modules, and A_{μ,k}-comodules to A_{h,k}-comodules.

Proof. (i) This is obvious.

(ii) Let M be an A_{μ,k}-comodule with structure map τ : M → M ⊗ A_{μ,k}. Recall that A_{h} = ⨁_{l≥0} A_{h,l} as an A_{h}-comodule. For l ≥ 0,

τ ⊗ id_{A_{h}}(M ⊗ A_{h,l}) ⊆ M ⊗ A_{μ,k} ⊗ A_{h,l},
id_{M} ⊗[(ϕ ⊗ id_{A_{h}}) ∘ Δ](M ⊗ A_{h,l}) ⊆ M ⊗ A_{μ,l} ⊗ A_{h,l}.

So, recalling the definition of the functor ind_{A_{μ}}^{A_{h}}, we deduce that the only non-zero f ∈ M ⊗ A_{h} satisfying (1.4.1) in fact lie in M ⊗ A_{h,k}. Hence, ind_{A_{μ}}^{A_{h}} sends degree k modules to degree k modules.

Moreover, if M is a finite dimensional A_{μ,k}-module, ind_{A_{μ}}^{A_{h}} M is finite dimensional because it is a subspace of the finite dimensional space M ⊗ A_{h,k}. Since any finite dimensional A_{μ}-comodule can be written as a direct sum of finite dimensional A_{μ,k}-comodules for finitely many different k, we deduce that ind_{A_{μ}}^{A_{h}} sends finite dimensional modules to finite dimensional modules.

Now we focus on a special case. So, until just before the end of the section, fix l ≤ h, set l′ = h − l and consider A_{μ} with μ = (l, l′) ⊨ h. In this case, A_{μ} is a bialgebra quotient of A_{h} isomorphic to A_{l} ⊗ A_{l′}; let

ϕ : A_{h} → A_{l} ⊗ A_{l′}

be the quotient map. Corresponding to ϕ, we have the coalgebra induction and restriction functors res_{A_{l}⊗A_{l′}}^{A_{h}} and ind_{A_{l}⊗A_{l′}}^{A_{h}}.

We will also need truncated versions of these functors. There is a natural bialgebra embedding i : A_{l} ↪ A_{l} ⊗ A_{l′}, a ↦ a ⊗ 1, with image A_{l} ⊗ A_{l′,0}, recalling that A_{l′,0} ≅ F denotes the degree zero part of A_{l′}. Define the exact truncation functor

trunc : comod(A_{l} ⊗ A_{l′}) → comod(A_{l})

as follows. On an object M with structure map τ : M → M ⊗ A_{l} ⊗ A_{l′}, trunc M is defined as the subspace {m ∈ M |τ(m) ∈ M ⊗A_{l}⊗A_{l′,0}}, regarded as an A_{l}-comodule via the restriction of τ and the isomorphism id_{M} ⊗i : M ⊗ A_{l} → M ⊗ A_{l} ⊗ A_{l′,0}. On a morphism, trunc is defined simply as restriction. The functor trunc has an adjoint, namely, the inflation functor

infl : comod(A_{l}) → comod(A_{l} ⊗ A_{l′})

defined on an object M with structure map τ : M → M ⊗ A_{l} by letting infl M be M as a vector space, but with new structure map τ̂ = (id_{M} ⊗i)∘τ : M → M ⊗A_{l} ⊗A_{l′}. On a morphism θ, infl θ is the same linear map but regarded instead as an A_{l} ⊗ A_{l′}- comodule map. Now we define the truncated polynomial restriction and induction functors:

trunc_{A_{l}}^{A_{h}} : comod(A_{h}) → comod(A_{l}) by trunc_{A_{l}}^{A_{h}} = trunc ∘ res_{A_{l}⊗A_{l′}}^{A_{h}} ,
infl_{A_{l}}^{A_{h}} : comod(A_{l}) → comod(A_{h}) by infl_{A_{l}}^{A_{h}} = ind_{A_{l}⊗A_{l′}}^{A_{h}} ∘ infl .

Lemma 1.4a easily implies the analogous result for the truncated versions of the functors:

(1.4b) (i) The functor trunc_{A_{l}}^{A_{h}} sends finite dimensional modules to finite dimensional modules, and A_{h,k}-comodules to A_{l,k}-comodules.

(ii) The functor infl_{A_{l}}^{A_{h}} sends finite dimensional modules to finite dimensional modules, and A_{l,k}-comodules to A_{h,k}-comodules.

The main result of the section is as follows:

1.4c. Theorem. For non-negative integers l, l′ with l+l′ = h as above, the following bifunctors are isomorphic:

ind_{A_{l}⊗A_{l′}}^{A_{h}} ( ? ⊠?′ ) : comod(A_{l}) × comod(A_{l′}) → comod(A_{h}),
(infl_{A_{l}}^{A_{h}} ? ) ⊗ (infl_{A_{l′}}^{A_{h}} ?′ ) : comod(A_{l}) × comod(A_{l′}) → comod(A_{h}).

Proof. The proof proceeds in a number of steps. We fix an A_{l}-comodule M and an A_{l′}-comodule M′. Let τ : M → M ⊗ A_{l} and τ′ : M′ → M′ ⊗ A_{l′} be their respective structure maps. We denote the structure map of infl M (resp. infl M′) as an A_{l} ⊗ A_{l′}-comodule by τ̂ : M → M ⊗ A_{l} ⊗ A_{l′} (resp. τ̂′ : M′ → M′ ⊗ A_{l} ⊗ A_{l′}). Define

J = span{c_{i,j} | k ≥ 0, i, j ∈ I(h, k), 1 ≤ i_{1}, …, i_{k} ≤ l},
J′ = span{c_{i,j} | k ≥ 0, i, j ∈ I(h, k), l + 1 ≤ i_{1}, …, i_{k} ≤ h}.

Note that both J and J′ are subalgebras and right A_{h}-subcomodules of A_{h}. It is routine to check the following, using the fact that algebra multiplication μ : A_{h} ⊗ A_{h} → A_{h} is a coalgebra homomorphism:

(1.4d) Multiplication μ : J ⊗ J′ → A_{h} is an isomorphism of right A_{h}-comodules.

Consider the A_{h}-comodule maps

θ = (τ̂ ⊗ id_{A_{h}} − id_{M} ⊗[(ϕ ⊗ id_{A_{h}}) ∘ Δ]) : |M| ⊗ A_{h} → |M ⊗ A_{l} ⊗ A_{l′}| ⊗ A_{h},
θ′ = (τ̂′ ⊗ id_{A_{h}} − id_{M′} ⊗[(ϕ ⊗ id_{A_{h}}) ∘ Δ]) : |M′| ⊗ A_{h} → |M′ ⊗ A_{l} ⊗ A_{l′}| ⊗ A_{h}.

Note we are using the symbol |.| to emphasize that the A_{h}-comodule structure is coming just from the final term in these tensor products. We claim that:

(1.4e) ker θ ⊆ |M| ⊗ J and ker θ′ ⊆ |M′| ⊗ J′.

We prove (1.4e) just for θ, the proof in the case of θ′ being entirely similar. Note directly from the definition of τ̂ that (τ̂ ⊗ id_{A_{h}})(ker θ) ⊆ M ⊗ A_{l} ⊗ A_{l′,0} ⊗ A_{h}. Now take v ∈ ker θ with v ∈ M ⊗ A_{h,k} for some k ≥ 0. Then, v̄ := (τ̂ ⊗ id_{A_{h}})(v) = (id_{M} ⊗[(ϕ⊗id_{A_{h}})∘Δ])(v) ∈ M ⊗A_{l,k} ⊗A_{l′,0} ⊗A_{h,k}. Write v = ∑_{i,j∈I(h,k)} m_{i,j} ⊗c_{i,j}, for some m_{i,j} ∈ M. We have that

v̄ = ∑_{i,j,s∈I(h,k)} m_{i,j} ⊗ ϕ(c_{i,s}) ⊗ c_{s,j} ∈ M ⊗ A_{l,k} ⊗ A_{l′,0} ⊗ A_{h,k}.  (1.4.2)

Consider the projection π_{m} of M ⊗ ϕ(A_{h,k}) ⊗ A_{h,k} onto the mth term of the direct sum decomposition M ⊗ ϕ(A_{h,k}) ⊗ A_{h,k} = ⨁_{m=0}^{k} M ⊗ A_{l,k−m} ⊗ A_{l′,m} ⊗ A_{h,k} coming from the grading. We know by (1.4.2) that π_{m}(v̄) = 0 for m > 0, hence that

v̄ = π_{0}(v̄) = ∑_{i,j,s∈I(h,k)} m_{i,j} ⊗ ϕ(c_{i,s}) ⊗ c_{s,j}

summing only over s with 1 ≤ s_{1}, …, s_{k} ≤ l. For such s, c_{s,j} ∈ J, hence v̄ ∈ M ⊗ A_{l} ⊗ A_{l′,0} ⊗ J.

The preceding paragraph shows that

(τ̂ ⊗ id_{A_{h}})(ker θ) ⊆ M ⊗ A_{l} ⊗ A_{l′,0} ⊗ J.  (1.4.3)

Consider the map id_{M} ¯⊗ε : M ⊗ A_{l} ⊗ A_{l′} → M such that m ⊗ a ↦ mε(a) for m ∈ M, a ∈ A_{l} ⊗ A_{l′}, where ε is the counit of A_{l} ⊗ A_{l′}. By the comodule axioms, (id_{M} ¯⊗ε) ∘ τ̂ = id_{M}. Now applying id_{M} ¯⊗ε ⊗ id_{A_{h}} to both sides of (1.4.3), we deduce that ker θ ⊆ M ⊗ J to complete the proof of (1.4e).

Consider now the A_{h}-comodule map

ω : |M| ⊗ J ⊗ |M′| ⊗ J′ → |M ⊗ M′| ⊗ A_{h}

defined by m⊗j⊗m′⊗j′ ↦ m⊗m′⊗jj′. According to (1.4d), this is an isomorphism of A_{h}-comodules. Observe moreover that infl_{A_{l}}^{A_{h}} M is precisely ker θ by the definition (1.4.1), and similarly, infl_{A_{l′}}^{A_{h}} M′ = ker θ′. So by (1.4e) and the definitions, we have that

(infl_{A_{l}}^{A_{h}} M) ⊗ (infl_{A_{l′}}^{A_{h}} M′) ⊆ |M| ⊗ J ⊗ |M′| ⊗ J′,
ind_{A_{l}⊗A_{l′}}^{A_{h}} (M ⊠ M′) ⊆ |M ⊗ M′| ⊗ A_{h}.

We claim that the restriction of ω induces an isomorphism between (infl_{A_{l}}^{A_{h}} M) ⊗ (infl_{A_{l′}}^{A_{h}} M′) and ind_{A_{l}⊗A_{l′}}^{A_{h}} (M ⊠ M′). This will complete the proof of the theorem, functoriality being immediate as ω is clearly functorial in arguments M and M′.

To prove the claim, ω is a bijection, so we just need to check that

ω^{−1}(ind_{A_{l}⊗A_{l′}}^{A_{h}} (M ⊠ M′)) = ker θ ⊗ ker θ′.  (1.4.4)

Recall that ind_{A_{l}⊗A_{l′}}^{A_{h}} (M ⊠ M′) is the set of vectors in M ⊗ M′ ⊗ A_{h} satisfying the appropriate version of (1.4.1). So, ω^{−1}(ind_{A_{l}⊗A_{l′}}^{A_{h}} (M ⊠ M′)) = ker(α − β), where α and β are the maps from M ⊗ J ⊗ M′ ⊗ J′ to M ⊗ A_{l} ⊗ M′ ⊗ A_{l′} ⊗ A_{h} defined by

α = (τ ⊗ τ′ ⊗ id_{A_{h}}) ∘ ω and β = (id_{M⊠M′} ⊗[(ϕ ⊗ id_{A_{h}}) ∘ Δ]) ∘ ω,

respectively. Consider instead the maps α′ and β′ from M ⊗ J ⊗ M′ ⊗ J′ to M ⊗ A_{l} ⊗ A_{l′} ⊗ A_{h} ⊗ M′ ⊗ A_{l} ⊗ A_{l′} ⊗ A_{h} defined by

α′ = τ̂ ⊗ id_{J} ⊗τ̂′ ⊗ id_{J′} and β′ = id_{M} ⊗([ϕ ⊗ id_{A_{h}}] ∘ Δ) ⊗ id_{M′} ⊗([ϕ ⊗ id_{A_{h}}] ∘ Δ),

respectively. Define π : M⊗A_{l}⊗A_{l′}⊗A_{h}⊗M′⊗A_{l}⊗A_{l′}⊗A_{h} → M⊗A_{l}⊗M′⊗A_{l′}⊗A_{h}, m ⊗ a_{1} ⊗ a′_{1} ⊗ b_{1} ⊗ m′ ⊗ a_{2} ⊗ a′_{2} ⊗ b_{2} ↦ m ⊗ a_{1}a_{2} ⊗ m′ ⊗ a′_{1}a′_{2} ⊗ b_{1}b_{2}. Observe that α = π ∘ α′ and β = π ∘ β′. Moreover, the images of both α′ and β′ lie in Y = M ⊗ A_{l} ⊗ A_{l′,0} ⊗ J ⊗ M′ ⊗ A_{l,0} ⊗ A_{l′} ⊗ J′, which is obvious for α′ and an easy exercise for β′. Since the restriction of π to Y is injective, it follows that ker(α − β) = ker(α′ − β′). Writing η = (α′ − β′) for short, we have shown that ω^{−1}(ind_{A_{l}⊗A_{l′}}^{A_{h}} (M ⊠ M′)) = ker η. So (1.4.4) is equivalent to showing:

(1.4f) ker η = ker θ ⊗ ker θ′.

To prove (1.4f), first take a pure tensor k ⊗k′ ∈ ker θ ⊗ker θ′. Then, τ̂⊗id_{J}(k) = id_{M} ⊗([ϕ ⊗ id_{A_{h}}] ∘ Δ)(k) and analogously for k′, which immediately shows that k ⊗ k′ ∈ ker η. Conversely, let id_{M} ¯⊗ε ¯⊗ id_{J} : M ⊗ A_{l} ⊗ A_{l′} ⊗ J → M ⊗ J be the map m⊗c⊗j ↦ mε(c)⊗j = m⊗ε(c)j, where ε is the counit of A_{l} ⊗A_{l′}. Recalling that ϕ is a bialgebra map, the coalgebra and comodule axioms immediately give that both of the composites

(id_{M} ¯⊗ε ¯⊗ id_{J}) ∘ (τ̂ ⊗ id_{J}) :M ⊗ J → M ⊗ J,
(id_{M} ¯⊗ε ¯⊗ id_{J}) ∘ (id_{M} ⊗[(ϕ ⊗ id_{A_{h}}) ∘ Δ]) :M ⊗ J → M ⊗ J

are equal to the identity. So,

(id_{M} ¯⊗ε ¯⊗ id_{J} ⊗ id_{M′⊗A_{l}⊗A_{l′}⊗J′}) ∘ η = id_{M⊗J} ⊗θ′.

It follows directly that ker η ⊆ ker(id_{M⊗J} ⊗θ′) = M ⊗J ⊗ker θ′. A similar argument shows that ker η ⊆ ker θ ⊗ M′ ⊗ J′. Hence,

ker η ⊆ (ker θ ⊗ M′ ⊗ J′) ∩ (M ⊗ J ⊗ ker θ′).  (1.4.5)

Finally, by linear algebra the right hand side of (1.4.5) is precisely ker θ ⊗ ker θ′, which completes the proof of (1.4f) hence the theorem.

Finally, we extend Theorem 1.4c to the general case, to obtain the quantum analogue of [BK, Theorem 2.7]:

1.4g. Corollary. Fix μ = (h_{1}, …, h_{a}) ⊨ h. Then, the following functors are isomorphic:

ind_{A_{μ}}^{A_{h}}( ? ⊠ ⋯ ⊠? ) : comod(A_{h_{1}}) × ⋯ × comod(A_{h_{a}}) → comod(A_{h}),
(infl_{A_{h_{1}}}^{A_{h}} ? ) ⊗ ⋯ ⊗ (infl_{A_{ha}}^{A_{h}} ? ) : comod(A_{h_{1}}) × ⋯ × comod(A_{h_{a}}) → comod(A_{h}).

Proof. This follows from Theorem 1.4c by induction on a, using the fact that coalgebra induction is transitive.

We refer the reader to [BK, §2] for further properties of the polynomial induction and restriction functors, as well as some consequences of Corollary 1.4g, in the classical case. The proofs in loc. cit. carry over to the quantum case, now that we have Corollary 1.4g.

1.5. Schur algebra induction

The goal in this section is to reformulate Corollary 1.4g in terms of the q-Schur algebra. Fix throughout the section h, k ≥ 1 and μ = (h_{1}, …, h_{a}) ⊨ h. Recall that S_{μ,k} denotes the Levi subalgebra of S_{h,k}, defined as in (1.3.8), over the field F. We have the restriction and induction functors

res_{S_{μ,k}}^{S_{h,k}} :mod(S_{h,k}) → mod(S_{μ,k}),
ind_{S_{μ,k}}^{S_{h,k}} :mod(S_{μ,k}) → mod(S_{h,k})

§1.5 Schur algebra induction  27

in the usual sense of finite dimensional algebras; so ind_{S_{μ,k}}^{S_{h,k}} = S_{h,k}⊗_{S_{μ,k}}?. Note that

ind_{S_{μ,k}}^{S_{h,k}} is left adjoint to res_{S_{μ,k}}^{S_{h,k}}.

We also need truncated versions, so fix l ≤ h and embed Λ(l, k) (resp. I(l, k)) in Λ(h, k) (resp. I(h, k)) in the natural way. Let e be the idempotent

e = e_{h,l} = ∑_{μ∈Λ(l,k)} ϕ_{μ,μ}^{1} ∈ S_{h,k}.  (1.5.1)

Note this is the analogue of the idempotent defined by Green in [G_{2}, (6.5b)]. The subring eS_{h,k}e is spanned by all standard basis elements ξ_{i,j} with i, j ∈ I(l, k). Moreover, just as in the classical case [G_{2}, p. 103], S_{l,k} ≅ eS_{h,k}e, the natural basis element ξ_{i,j} of S_{l,k} for i, j ∈ I(l, k) mapping under the isomorphism to the corresponding natural basis element ξ_{i,j} of S_{h,k}. In what follows, we identify S_{l,k} with eS_{h,k}e in this way. Then, we have the Schur functor

trunc_{S_{l,k}}^{S_{h,k}} : mod(S_{h,k}) → mod(S_{l,k}),  (1.5.2)

defined on an object M by trunc_{S_{l,k}}^{S_{h,k}} M = eM and by restriction on morphisms. The

functor trunc_{S_{l,k}}^{S_{h,k}} has a left adjoint, the inverse Schur functor

infl_{S_{l,k}}^{S_{h,k}} : mod(S_{l,k}) → mod(S_{h,k}),  (1.5.3)

which is the functor S_{h,k}e⊗_{eS_{h,k}e}?. The truncation functor trunc_{S_{l,k}}^{S_{h,k}} is the quantum analogue of the functor d_{h,l} considered by Green [G_{2}, §6.5], and the argument of [G_{2}, (6.5g)] carries over to show:

(1.5a) Assuming that k ≤ l, the functors

trunc_{S_{l,k}}^{S_{h,k}} : mod(S_{h,k}) → mod(S_{l,k})  and  infl_{S_{l,k}}^{S_{h,k}} : mod(S_{l,k}) → mod(S_{h,k})

are mutually inverse equivalences of categories.

The effect of the truncation functor trunc_{S_{l,k}}^{S_{h,k}} on the standard S_{h,k}-modules is well-known; we record the basic facts, see e.g. [G_{2}, (6.5f)], [BK, 2.3, 2.4], [Do_{4}, §4.2]:

(1.5b) Take μ = (m_{1}, …, m_{h}) ∈ Λ^{+}(h, k).

(i) If m_{l+1} ≠ 0 then trunc_{S_{l,k}}^{S_{h,k}} L_{h}(μ) = trunc_{S_{l,k}}^{S_{h,k}} Δ_{h}(μ) = trunc_{S_{l,k}}^{S_{h,k}} ∇_{h}(μ) = 0. (ii) If m_{l+1} = 0, we may regard μ as an element of Λ^{+}(l, k) ⊆ Λ^{+}(h, k), and then

trunc_{S_{l,k}}^{S_{h,k}} L_{h}(μ) ≅ L_{l}(μ), trunc_{S_{l,k}}^{S_{h,k}} Δ_{h}(μ) ≅ Δ_{l}(μ) and trunc_{S_{l,k}}^{S_{h,k}} ∇_{h}(μ) ≅ ∇_{l}(μ).

Now we wish to relate these Schur algebra functors to the polynomial induction and restriction functors of the previous section.

First, recall from §1.2 that the algebra S_{h,k} has an anti-automorphism τ, which is easily seen to stabilize the subalgebra S_{μ,k}, for instance using (1.3g). This allows us to define the contravariant dual M^{τ} of an S_{μ,k}-module M, in the same way as we did for S_{h,k} in §1.2. Contravariant duality gives us functors mod(S_{h,k}) → mod(S_{h,k}), mod(S_{μ,k}) → mod(S_{μ,k}) and mod(S_{l,k}) → mod(S_{l,k}), all of which we will denote simply by τ. It is obvious that τ commutes with res_{S_{μ,k}}^{S_{h,k}} and trunc_{S_{l,k}}^{S_{h,k}}. To be precise, there are isomorphisms of functors:

τ ∘ res_{S_{μ,k}}^{S_{h,k}} ≅ res_{S_{μ,k}}^{S_{h,k}} ∘τ,  (1.5.4)
τ ∘ trunc_{S_{l,k}}^{S_{h,k}} ≅ trunc_{S_{l,k}}^{S_{h,k}} ∘τ  (1.5.5)

We remark that contravariant duality does not in general commute with the functor ind_{S_{μ,k}}^{S_{h,k}} (resp. infl_{S_{l,k}}^{S_{h,k}} unless k ≤ l).

Second, notice that, in view of Lemma 1.4a and (1.4b), we can restrict the functors res_{A_{μ}}^{A_{h}}, ind_{A_{μ}}^{A_{h}}, res_{A_{l}}^{A_{h}} and ind_{A_{l}}^{A_{h}} to finite dimensional, polynomial degree k modules to obtain well-defined functors which we will denote by the same names:

res_{A_{μ}}^{A_{h}} : mod(S_{h,k}) → mod(S_{μ,k}),  ind_{A_{μ}}^{A_{h}} : mod(S_{μ,k}) → mod(S_{h,k}),
trunc_{A_{l}}^{A_{h}} : mod(S_{h,k}) → mod(S_{l,k}),  infl_{A_{l}}^{A_{h}} : mod(S_{l,k}) → mod(S_{h,k}).

To connect all these functors, we have:

1.5c. Lemma. The following pairs of functors are isomorphic:

(i) res_{S_{μ,k}}^{S_{h,k}} and res_{A_{μ}}^{A_{h}} : mod(S_{h,k}) → mod(S_{μ,k});

(ii) trunc_{S_{l,k}}^{S_{h,k}} and trunc_{A_{l}}^{A_{h}} : mod(S_{h,k}) → mod(S_{l,k});

(iii) τ ∘ ind_{S_{μ,k}}^{S_{h,k}} ∘τ and ind_{A_{μ}}^{A_{h}} : mod(S_{μ,k}) → mod(S_{h,k});

(iv) τ ∘ infl_{S_{l,k}}^{S_{h,k}} ∘τ and infl_{A_{l}}^{A_{h}} : mod(S_{l,k}) → mod(S_{h,k}).

Proof. (i) This is obvious.

(ii) To see this, note that both of the functors trunc_{S_{l,k}}^{S_{h,k}} and trunc_{A_{l}}^{A_{h}} are defined on objects by taking certain weight spaces, and by restriction on morphisms. To be more precise, on an object M, both functors send M to ⨁_{λ∈Λ(l,k)} ϕ_{λ,λ}^{1}M. Hence, the functors are isomorphic.

(iii) Using (i) and the fact that ind_{A_{μ}}^{A_{h}} is right adjoint to res_{A_{μ}}^{A_{h}}, we just need to

check by uniqueness of adjoint functors that τ ∘ ind_{S_{μ,k}}^{S_{h,k}} ∘τ is right adjoint to res_{S_{μ,k}}^{S_{h,k}}. Well, for M ∈ mod(S_{h,k}), N ∈ mod(S_{μ,k}), we have

Hom_{S_{h,k}}(M, (ind_{S_{μ,k}}^{S_{h,k}} N^{τ})^{τ}) ≅ Hom_{S_{h,k}}(ind_{S_{μ,k}}^{S_{h,k}} N^{τ} , M^{τ})
≅ Hom_{S_{μ,k}}(N^{τ} , res_{S_{μ,k}}^{S_{h,k}}(M^{τ}))
≅ Hom_{S_{μ,k}}(res_{S_{μ,k}}^{S_{h,k}} M, N)

using the fact that contravariant duality commutes with restriction (1.5.4). (iv) This follows by (ii) and (1.5.5) by the same general argument as (iii). Finally, we can restate Corollary 1.4g in terms of Schur algebras. To explain

the notation in the statement, note that given modules M_{1}, …, M_{a} with each M_{i} ∈ mod(S_{h_{i},k_{i}}), we can regard the outer tensor product M_{1}⊠⋯⊠M_{a} as an S_{μ,k}-module where k = k_{1} + ⋯ + k_{a}. We obtain in this way a functor

? ⊠ ⋯ ⊠? : mod(S_{h_{1},k_{1}}) × ⋯ × mod(S_{h_{a},k_{a}}) → mod(S_{μ,k}).

Similarly, the tensor product operation on modules over Schur algebras, induced via the map (1.3.1), gives us a functor

? ⊗ ⋯ ⊗? : mod(S_{h,k_{1}}) × ⋯ × mod(S_{h,k_{a}}) → mod(S_{h,k})

where k = k_{1} + ⋯ + k_{a}. With this notation:

1.5d. Theorem. For μ = (h_{1}, …, h_{a}) ⊨ h and ν = (k_{1}, …, k_{a}) ⊨ k, the following functors are isomorphic:

ind_{S_{μ,k}}^{S_{h,k}}(? ⊠ ⋯ ⊠ ?) : mod(S_{h_{1},k_{1}}) × ⋯ × mod(S_{h_{a},k_{a}}) → mod(S_{h,k}),
(infl_{S_{h_{1},k_{1}}}^{S_{h,k_{1}}} ? ) ⊗ ⋯ ⊗ (infl_{S_{ha,ka}}^{S_{h,ka}} ? ) : mod(S_{h_{1},k_{1}}) × ⋯ × mod(S_{h_{a},k_{a}}) → mod(S_{h,k}).

Proof. Combine Corollary 1.4g and Lemma 1.5c(iii),(iv).

Later, Theorem 1.5d will be important because it gives a way to realize the tensor product operation induced by the bialgebra structure of A_{h} directly within the finite dimensional algebras S_{h,k} (for all k).

Classical results on GL_{n}

We next collect all the results about GL_{n}(𝔽_{q}) from the literature that we will need later. Most of these results are of a purely character theoretic nature. At the same time, we will deduce their basic consequences for the modular theory using elementary base change arguments.

2.1. Conjugacy classes and Levi subgroups

We begin with some basic notation that will be in place for the remainder of the article. Choose some prime power q and denote the finite general linear group GL_{n}(𝔽_{q}) by G_{n}. Let p be a prime not dividing q and fix a p-modular system (F, O, K). So O is a complete discrete valuation ring with field of fractions K of characteristic 0 and residue field F of characteristic p. We will assume in addition that K, hence F, is algebraically closed (e.g. see [Ka, §3.1]).

For σ ∈ 𝔽̄_{q}^{×}, we let (σ) denote the associated companion matrix. So if σ is of degree d over 𝔽_{q}, (σ) is the d × d-matrix corresponding to the automorphism of 𝔽_{q}[σ] induced by left multiplication by σ, when written in terms of the 𝔽_{q}-basis 1, σ, …, σ^{d−1}. We say that σ is p-regular if its multiplicative order is coprime to p; otherwise, we say that σ is p-singular. Given in addition τ ∈ 𝔽̄_{q}^{×}, we say that σ and τ are conjugate if they have the same minimal polynomial over 𝔽_{q}, or equivalently, if the matrices (σ) and (τ) are conjugate matrices.

For d ≥ 1, let ℓ(d) denote the smallest positive integer such that q^{dℓ(d)} ≡ 1(mod p) (i.e. the image of q^{d} in F is a primitive ℓ(d)th root of unity). We record the basic number theoretic fact (cf. [DJ_{2}, Lemma 2.3]):

(2.1a) Let σ ∈ 𝔽̄_{q}^{×} be a p-regular element of degree d over 𝔽_{q}. There exists a psingular element τ ∈ 𝔽_{q}^{×} of degree e over 𝔽_{q} with p-regular part conjugate to σ if and only if e = dℓ(d)p^{r} for some r ≥ 0.

For σ ∈ 𝔽̄_{q}^{×} and k ≥ 1, let (σ)^{k} denote the block diagonal matrix consisting of k copies of (σ) along the diagonal. Every semisimple conjugacy class of G_{n} can be represented as a block diagonal matrix

s = diag((σ_{1})^{k_{1}}, …, (σ_{a})^{k_{a}})  (2.1.1)

for σ_{i} ∈ 𝔽̄_{q}^{×} with σ_{i} not conjugate to σ_{j} for i ≠ j, and integers k_{1}, …, k_{a} ≥ 0. We will say s ∈ G_{n} is block-diagonal if it is a semisimple element of the form (2.1.1). For such an s, we associate the following compositions:

κ(s) = (k_{1}, …, k_{a}),  (2.1.2)
δ(s) = ((d_{1})^{k_{1}}, …, (d_{a})^{k_{a}}) ⊨ n,  (2.1.3)
π(s) = (d_{1}k_{1}, …, d_{a}k_{a}) ⊨ n,  (2.1.4)

where d_{i} is the degree of σ_{i} over 𝔽_{q}. The centralizer C_{G_{n}}(s) is isomorphic to

GL_{k_{1}}(𝔽_{q^{d_{1}}} ) × ⋯ × GL_{k_{a}}(𝔽_{q^{da}}).  (2.1.5)

Recalling that the unipotent classes of G_{n} are parametrized in a standard way by partitions λ ⊢ n, we see from (2.1.5) that the unipotent classes of C_{G_{n}}(s) are parametrized by multi-partitions λ = (λ_{1}, …, λ_{a}) where λ_{i} ⊢ k_{i} for i = 1, …, a. We will write λ ⊢ κ(s) if λ is such a multi-partition.

Let C_{ss} be a set of representatives of the semisimple classes of G_{n}, such that each s ∈ C_{ss} is block-diagonal, and let C_{ss,p′} denote the p-regular elements s ∈ C_{ss}, that is, the elements of C_{ss} of order coprime to p. By the Jordan decomposition, the conjugacy classes of G_{n} are parametrized by pairs (s, λ) for all s ∈ C_{ss} and λ ⊢ κ(s). Moreover, as p is coprime to q, the p-regular conjugacy classes are parametrized by pairs (s, λ) for all s ∈ C_{ss,p′} and λ ⊢ κ(s).

We turn next to describing various Levi subgroups of G_{n}. Let G_{n} = GL_{n}(𝔽̄_{q}) denote the corresponding algebraic group and

f_{q} : G_{n} → G_{n}  (2.1.6)

be the Frobenius map defined by raising all the entries of a matrix g ∈ G_{n} to the qth power. Then, the finite general linear group G_{n} is precisely the set of f_{q}-fixed points in G_{n}. By a Levi subgroup of G_{n}, we mean the set of f_{q}-fixed points in an f_{q}-stable Levi subgroup of G_{n}. For example, the centralizer C_{G_{n}}(s) of (2.1.5) is a Levi subgroup of G since it is the set of f_{q}-fixed points in the Levi subgroup C_{G_{n}}(s) of G_{n}.

Another special case gives us maximal tori: a maximal torus of G_{n} is the set of f_{q}-fixed points in an f_{q}-stable maximal torus of G_{n}. By Lang’s theorem, the conjugacy classes of maximal tori of G_{n} are parametrized by conjugacy classes of the symmetric group Σ_{n}, hence by partitions of n. For λ ⊢ n, we let T_{λ} be a representative of the corresponding conjugacy class of maximal tori of G_{n}, so

T_{λ} ≅ GL_{1}(𝔽_{q^{l_{1}}} ) × ⋯ × GL_{1}(𝔽_{q^{la}})  (2.1.7)

if λ = (l_{1}, …, l_{a}) ⊢ n.

§2.2 Harish-Chandra induction and restriction  33

Let G_{ν} = G_{n_{1}} ×⋯×G_{n_{a}} (embedded into G_{n} as block diagonal matrices) denote the standard Levi subgroup of G_{n} parametrized by the composition ν = (n_{1}, …, n_{a}) ⊨ n. We write U_{n} (resp., U_{ν}) for the subgroup of all upper uni-triangular matrices in G_{n} (resp., G_{ν}) and Y_{ν} for the unipotent radical of the standard parabolic subgroup of G_{n} with Levi factor G_{ν}, so U_{n} = U_{ν}Y_{ν} (semi-direct product).

We always henceforth identify the symmetric group Σ_{n} with the subgroup of G_{n} consisting of all permutation matrices, so that for ν ⊨ n, Σ_{ν} = Σ_{n} ∩ G_{ν}. The following fact follows from (1.1a):

(2.1b) Given λ, μ ⊨ n, where again n = kd, and w ∈ D_{μ,λ}, G_{μ} ∩^{w}G_{λ} is a standard Levi subgroup of G_{n}.

Now suppose that n = kd for integers k, d ≥ 1. There is an embedding of Σ_{k} into G_{n} as the subgroup of all “d × d-block permutation matrices”, so for example, the basic transposition (1 2) ∈ Σ_{3} corresponds to the matrix

( \begin{matrix}0 & I_{d} & 0 \\ I_{d} & 0 & 0 \\ 0 & 0 & I_{d} \end{matrix} )

in G_{3d}. For x ∈ Σ_{k}, let π_{x} denote the corresponding block permutation matrix in G_{n}. If ν = (k_{1}, …, k_{a}) ⊨ k, we write dν for the composition (dk_{1}, …, dk_{a}) ⊨ n. We will often appeal to the following observation. Although the proof is not immediate, it is a purely combinatorial statement about coset representatives in the symmetric group which we leave as an exercise for the reader, referring to [DJ_{1}, p. 23] for guidance.

(2.1c) Given λ, μ ⊨ k and w ∈ D_{dμ,dλ}, G_{dμ} ∩^{w}G_{dλ} contains G_{(d^{k})} if and only if w = π_{x} for some x ∈ D_{μ,λ}. In that case, G_{dμ} ∩^{w}G_{dλ} = G_{dν} where ν ⊨ k is determined by G_{μ} ∩^{x}G_{λ} = G_{ν}.

2.2. Harish-Chandra induction and restriction

Let G = G_{ν} for some ν ⊨ n and L be any standard Levi subgroup of G. Let P be the standard parabolic subgroup of G with Levi factor L. Let Y denote the unipotent radical of P, so that there is a surjection P → L with kernel Y. We write infl_{L}^{P} for the usual inflation functor from mod(F L) to mod(F P) along this surjection, and inv_{Y} for the truncation functor from mod(F P) to mod(F L) induced by taking Y -fixed points. Define the Harish-Chandra induction and restriction functors

R_{L}^{G} : mod(F L) → mod(F G)  by  R_{L}^{G} = ind_{P}^{G} ∘ infl_{L}^{P},
∗R_{L}^{G} : mod(F G) → mod(F L)  by  ^{∗}R_{L}^{G} = inv_{Y} ∘ res_{P}^{G} .

It is known (cf. [DDu_{1}, §5], [HL_{2}]) that if some other parabolic subgroup is used in these definitions, one obtains isomorphic functors. However, it is then not immediately obvious that our later definitions (specifically, the modules defined in §3.3) are independent of this choice. This is indeed the case, as shown in [Ac], but for simplicity, we almost always work with the fixed choice of the standard parabolic in this article.

We record two well-known basic properties of the functors R_{L}^{G} and ^{∗}R_{L}^{G}; the first depends on the assumption that (p, q) = 1. See [DF, §1] for proofs.

(2.2a) The functors ^{∗}R_{L}^{G} and R_{L}^{G} are both exact, and are both left and right adjoint to one another.

(2.2b) Given standard Levi subgroups L′ ≤ L ≤ G, the functors R_{L}^{G} ∘ R_{L′}^{L} and R_{L′}^{G} (resp., ^{∗}R_{L′}^{L} ∘^{∗}R_{L}^{G} and ^{∗}R_{L′}^{G}) are isomorphic.

We will need the fundamental Mackey decomposition theorem (see e.g. [DF, Theorem 1.14]), which makes sense in view of (2.1b):

(2.2c) Given standard Levi subgroups G_{μ}, G_{λ} of G = G_{ν} for λ, μ ⊨ n, there is an isomorphism of functors

∗R_{G_{μ}}^{G} ∘ R_{G_{λ}}^{G} ≅ ⨁_{w∈D_{μ,λ}^{ν}} R_{G_{μ}∩^{w}G_{λ}}^{G_{μ}} ∘ conj_{w} ∘^{∗}R_{G_{λ}∩^{w^{−1}}G_{μ}}^{G_{λ}},

where conj_{w} : mod(F (G_{λ} ∩^{w^{−1}}G_{μ})) → mod(F (G_{μ} ∩^{w}G_{λ})) denotes the functor induced by conjugation by w ∈ G.

The functors R_{L}^{G} and ^{∗}R_{L}^{G} can also be defined in the same way as above over the ground ring O (similarly, over K). This gives us functors which we will denote with the same names, namely, R_{L}^{G} : mod(OL) → mod(OG) and ^{∗}R_{L}^{G} : mod(OG) → mod(OL). The functors over O commute with base change, so this notation should not cause confusion:

2.2d. Lemma. (i) For any OL-module M_{O}, the F G-modules F ⊗_{O} (R_{L}^{G}M_{O}) and R_{L}^{G}(F ⊗_{O} M_{O}) are naturally isomorphic.

(ii) For any OG-module N_{O}, the F L-modules F ⊗_{O} (^{∗}R_{L}^{G}N_{O}) and ^{∗}R_{L}^{G}(F ⊗_{O} N_{O}) are naturally isomorphic.

Proof. These results are well-known, but since we could not find a suitable reference we include a proof. First, let H be any group and A (resp. B) be a right (resp. left) OH-module. Write A_{F} = F ⊗_{O} A and B_{F} = F ⊗_{O} B. Then

A_{F} ≅ A ⊗_{O} F ≅ (A ⊗_{OH} OH) ⊗_{O} F ≅ A ⊗_{OH} (OH ⊗_{O} F) = A ⊗_{OH} F H.

Similarly, B_{F} ≅ F H ⊗_{OH} B. Now

A_{F} ⊗_{F H} B_{F} ≅ (A ⊗_{OH} F H) ⊗_{F H} (F H ⊗_{OH} B) ≅ (A ⊗_{OH} F H) ⊗_{OH} B
≅ (A ⊗_{O} F) ⊗_{OH} B ≅ (F ⊗_{O} A) ⊗_{OH} B
≅ F ⊗_{O} (A ⊗_{OH} B).

Now take H = L, B = M_{O} and A = OGe where e is the idempotent \frac{1}{|Y |} ∑_{y∈Y} y. Then, R_{L}^{G}M_{O} = A ⊗_{OH} B, while R_{L}^{G}M_{F} = A_{F} ⊗_{F H} B_{F}. So (i) follows directly from the general fact just proved. The proof of (ii) is entirely similar, since Harish-Chandra restriction can also be interpreted as tensoring with a certain bimodule.

We also need to know that the basic properties of Harish-Chandra induction and restriction, namely, (2.2a), (2.2b) and (2.2c), are true also over O; indeed, the references cited above prove the results over any ground ring in which q is invertible.

In the category mod(F G), there is a notion of contravariant duality. Let τ : G → G denote the anti-automorphism given by matrix transposition. Given a left F G-module V let V^{τ} denote the left F G-module which as an F -space is equal to the dual V^{∗} = Hom_{F} (V, F), but with action defined by (g.f)(v) = f(τ(g)v) for v ∈ V, g ∈ G, f ∈ V^{∗}. Since τ leaves conjugacy classes of G invariant, V and V^{τ} have the same Brauer character. In particular, if V is an irreducible F G-module, then V ≅ V^{τ} (cf. [J_{2}, (7.27)]). The same remarks apply to the standard Levi subgroup L of G, since τ stabilizes any such subgroup.

2.2e. Lemma. For a left F L-module V, R_{L}^{G}(V^{τ}) ≅ (R_{L}^{G}V)^{τ}. Similarly, for a left F G-module U, ^{∗}R_{L}^{G}(U^{τ}) ≅ (^{∗}R_{L}^{G}U)^{τ}.

Proof. We first prove that the functors (?)^{τ} ∘^{∗}R_{L}^{G}∘(?)^{τ} and^{∗}R_{L}^{G} are isomorphic. Let P be the standard parabolic subgroup of G with Levi factor L and unipotent radical Y, and let e = \frac{1}{|Y |} ∑_{u∈Y} u ∈ F G. Then, the functor ^{∗}R_{L}^{G} is given on objects by left multiplication by the idempotent e, and by restriction on morphisms. The corresponding idempotent for the opposite parabolic subgroup τ(P) of G containing L is τ(e), and by [DDu_{1}, §5] or [HL_{2}] the functor given on objects by left multiplication by τ(e) is isomorphic to ^{∗}R_{L}^{G}. So now it suffices to show that for V ∈ mod(F G), the F L-modules τ(e)V and (eV^{τ})^{τ} are naturally isomorphic. Identifying V and V^{ττ}, we have the natural isomorphism (eV^{τ})^{τ} ≅ V/(eV^{τ})^{∘} where

(eV^{τ})^{∘} = {v ∈ V | (ef)(v) = 0 for all f ∈ V^{τ} }
= {v ∈ V | f(τ(e)v) = 0 for all f ∈ V^{τ} } = (1 − τ(e))V.

We see that (eV^{τ})^{τ} ≅ V/((1 − τ(e))V) ≅ τ(e)V as required.

Now to show that the functors (?)^{τ} ∘ R_{L}^{G} ∘ (?)^{τ} and R_{L}^{G} are isomorphic, we note that by what we have proved so far and (2.2a), R_{L}^{G} is left adjoint to (?)^{τ} ∘^{∗}R_{L}^{G} ∘ (?)^{τ}. So by uniqueness of adjoint functors, it suffices to check that (?)^{τ} ∘ R_{L}^{G} ∘ (?)^{τ} is also left adjoint to (?)^{τ} ∘^{∗}R_{L}^{G} ∘ (?)^{τ}. For any U ∈ mod(F L), V ∈ mod(F G), we have

Hom_{F G}((R_{L}^{G}(U^{τ}))^{τ} , V) ≅ Hom_{F G}(V^{τ} , R_{L}^{G}(U^{τ}))
≅ Hom_{F L}(^{∗}R_{L}^{G}(V^{τ}), U^{τ}) ≅ Hom_{F L}(U, (^{∗}R_{L}^{G}(V^{τ}))^{τ})

as required for adjointness.

2.2f. Corollary. Given any irreducible left F L-module M, the F G-module R_{L}^{G}M is isomorphic to (R_{L}^{G}M)^{τ}.

Proof. We have observed that M ≅ M^{τ}. So, R_{L}^{G}M ≅ R_{L}^{G}(M^{τ}) ≅ (R_{L}^{G}M)^{τ} by the lemma.

2.3. Characters and Deligne-Lusztig operators

Let G = G_{n} and write ℚ_{p} for the p-adic field with algebraic closure ℚ̄_{p}. We write X(G) for the character ring of G over ℚ̄_{p}, and similarly for any Levi subgroup of G. The irreducible characters in X(G) were originally constructed by Green [G_{1}]. We will adopt for brevity the point of view of Fong and Srinivasan [FS], where Green’s results are reformulated in terms of the Deligne-Lusztig theory; see also [LS], [DM_{1}] and [DM_{2}, §15.4] for this point of view. The very elegant approach of [Z] (which contains many but not all of the results we need) might also serve as a useful introduction.

The parametrization of irreducibles described in [FS, §1] depends on first fixing an embedding 𝔽̄_{q}^{×} ↪ ℚ̄_{p}^{×}, as well as on the choice of parametrization of the irreducible characters of the symmetric groups by partitions reviewed in §1.1. Having made these choices, we obtain an irreducible character χ_{s,λ} ∈ X(G) for each blockdiagonal element s ∈ G and each λ ⊢ κ(s). The irreducible characters in X(G) are the characters {χ_{s,λ} | s ∈ C_{ss}, λ ⊢ κ(s)}.

We say a little more about the construction of χ_{s,λ}. For a Levi subgroup L of G, R_{L}^{G} : X(L) → X(G) now denotes the Deligne-Lusztig operator as introduced in [DL, L_{1}]. Note we are abusing notation somewhat here: strictly speaking we should denote the operator R_{L}^{G} by R\begin{matrix}G \\ L \end{matrix}, where G is the algebraic group GL_{n}(𝔽̄_{q}) and L is a Levi subgroup of G stable under the Frobenius map f_{q} of (2.1.6), with the Levi subgroup L of G equal to the set of f_{q}-fixed points in L.

If L happens to be a standard Levi subgroup of G, we also have operators R_{L}^{G} : X(L) → X(G) and ^{∗}R_{L}^{G} : X(G) → X(L) induced by the Harish-Chandra induction and restriction functors of §2.2 over ℚ̄_{p}. It is known that the Harish-Chandra operator R_{L}^{G} coincides with the Deligne-Lusztig operator in this special case that L is standard.

Recall that for μ ⊢ n, T_{μ} denotes a representative of the corresponding conjugacy class of maximal tori of G_{n}, as in (2.1.7). Also, as in Lemma 1.3f, we write c_{μ} for the number of elements of Σ_{n} of cycle-type μ. For λ ⊢ n, the class function

χ_{λ} = \frac{1}{n!} ∑_{μ⊢n} c_{μ}ϕ_{λ}(μ)R_{T_{μ}}^{G}(1)  (2.3.1)

is an irreducible unipotent character of G. The formula (2.3.1) (which is [FS, (1.13)]) can be inverted to give the formula [FS, (1.14)]:

R_{T_{μ}}^{G}(1) = ∑_{λ⊢n} ϕ_{λ}(μ)χ_{λ}.  (2.3.2)

§2.3 Characters and Deligne-Lusztig operators  37

For λ = (l_{1}, …, l_{a}) with transpose λ′ = (l′_{1}, …, l′_{b}), set

m(λ) = ∑_{j=1}^{b}(l′_{j})^{2} − ∑_{i=1}^{a} il_{i}.

The degree of the unipotent character χ_{λ} is given by [FS, (1.15)]:

χ_{λ}(1) = \frac{q^{m(λ)}(q^{n} − 1)(q^{n−1} − 1) … (q − 1)}{∏_{h}(q^{h} − 1)}  (2.3.3)

where h runs over the hook lengths of λ.

Suppose now in general that s ∈ G is a block-diagonal element of the form (2.1.1). The choice of embedding 𝔽̄_{q}^{×} ↪ ℚ̄_{p}^{×} allows us to associate to s a linear character ŝ ∈ X(C_{G}(s)), as in [FS, (1.16)]. Recalling the structure of C_{G}(s) from (2.1.5), we have for each λ = (λ_{1}, …, λ_{a}) ⊢ κ(s) the irreducible unipotent character χ_{λ_{1}} … χ_{λ_{a}} (outer product) of C_{G}(s). Note we are abusing notation here (and later) by allowing q to vary to give the definition of the unipotent character χ_{λ_{i}} ∈ X(GL_{k_{i}}(q^{d_{i}})), i = 1, …, a. Define the sign ε_{s} by

ε_{s} = (−1)^{n+k_{1}+···+k_{a}}.  (2.3.4)

Then,

χ_{s,λ} = ε_{s}R_{C_{G}(s)}^{G}(ŝ.χ_{λ_{1}} … χ_{λ_{a}})  (2.3.5)

is precisely the irreducible character of G parametrized by the pair (s, λ).

Using [FS, (1.3)] (cf. [DM_{1}, 3.10]), we see that the degree of χ_{s,λ} can be computed from (2.3.3) by the formula:

χ_{s,λ}(1) = |G : C_{G}(s)|_{p′}χ_{λ_{1}}(1) … χ_{λ_{a}}(1).  (2.3.6)

We repeat: in calculating the degree χ_{λ_{i}}(1) using (2.3.3) q needs to be replaced by q^{d_{i}}, since χ_{λ_{i}} ∈ X(GL_{k_{i}}(q^{d_{i}})) not X(GL_{n}(q)) as there.

The following property, proved in [FS, p. 140], is of great importance to the modular theory:

(2.3a) Let t = sy be a block-diagonal element of G with p-regular part s. For any λ = (λ_{1}, …, λ_{b}) ⊢ κ(t), the generalized characters R_{C_{G}(t)}^{G}(t̂.χ_{λ_{1}} … χ_{λ_{b}}) and

R_{C_{G}(s)}^{G}(ŝ.R_{C_{G}(t)}^{C_{G}(s)}(χ_{λ_{1}} … χ_{λ_{b}})) agree on all p-regular conjugacy classes of G.

As our first application of (2.3a), we have:

2.3b. Lemma. With t = sy and λ as in (2.3a), the Brauer character obtained by restricting the character χ_{t,λ} to the p-regular classes of G can be written as a ℤlinear combination of the restrictions of the characters {χ_{s,μ} | μ ⊢ κ(s)} to p-regular classes.

Proof. Using (2.3a), we just need to observe that all of the irreducible constituents of R_{C_{G}(t)}^{C_{G}(s)}(χ_{λ_{1}} … χ_{λ_{b}}) are unipotent characters of C_{G}(s). Let T be a maximal split

torus of C_{G}(t). Then, χ_{λ_{1}} … χ_{λ_{b}} is a constituent of R_{T}^{C_{G}(t)}(1). So it suffices by transitivity of Deligne-Lusztig operators to observe that all constituents of R_{T}^{C_{G}(s)}(1) are unipotent characters of C_{G}(s), which follows by (2.3.2) (noting that T is also a maximal torus of C_{G}(s)).

Now fix σ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q} and suppose that n = kd for some k ≥ 1. Abbreviate χ_{s,λ} in the special case s = (σ)^{k} by χ_{σ,λ}, where λ = (λ) for λ ⊢ k. Recall again that c_{λ} denotes the number of elements of Σ_{k} of cycle-type λ. The second fundamental consequence of (2.3a) (cf. Lemma 1.3f and [DJ_{2}, Lemma 3.2]) is as follows:

2.3c. Lemma. Let τ ∈ 𝔽̄_{q}^{×} be of degree e = md over 𝔽_{q}, with p-regular part conjugate to σ. Then, for any l ≥ 1, k = ml and n = kd = le, the character χ_{τ,(l)} agrees with the generalized character

\frac{(−1)^{k+l}}{l!} ∑_{λ⊢l} ∑_{μ⊢k} c_{λ}ϕ_{μ}(mλ)χ_{σ,μ}

on all p-regular classes of G_{n}.

Proof. Let t = (τ)^{l} and s = (σ)^{k}, so that C_{G}(t) ≅ GL_{l}(q^{e}) and C_{G}(s) ≅ GL_{k}(q^{d}). By definition, R_{C_{G}(t)}^{G}(t̂.χ_{(l)}) = (−1)^{n+l}χ_{τ,(l)} and R_{C_{G}(s)}^{G}(ŝ.χ_{μ}) = (−1)^{n+k}χ_{σ,μ} for any μ ⊢ k. So applying (2.3a), it suffices to show that

R_{C_{G}(t)}^{C_{G}(s)}χ_{(l)} = \frac{1}{l!} ∑_{λ⊢l} ∑_{μ⊢k} c_{λ}ϕ_{μ}(mλ)χ_{μ}.

Now, ϕ_{(l)} is the trivial character of the symmetric group, so by (2.3.1), we can write the unipotent character χ_{(l)} of C_{G}(t) ≅ GL_{l}(q^{e}) as

χ_{(l)} = \frac{1}{l!} ∑_{λ⊢l} c_{λ}R_{T_{λ}}^{C_{G}(t)}(1)

where T_{λ} ≤ C_{G}(t) is a maximal torus isomorphic to GL_{1}(q^{el_{1}}) × ⋯ × GL_{1}(q^{el_{a}}) for λ = (l_{1}, …, l_{a}) ⊢ l. Observe that T_{λ} is also a maximal torus of C_{G}(s); we should then denote it instead by T_{mλ} ≤ C_{G}(s). Now, using transitivity of the Deligne-Lusztig operators, we deduce from (2.3.2) that

R_{C_{G}(t)}^{C_{G}(s)}χ_{(l)} = \frac{1}{l!} ∑_{λ⊢l} c_{λ}R_{T_{mλ}}^{C_{G}(s)}(1) = \frac{1}{l!} ∑_{λ⊢l} ∑_{μ⊢k} c_{λ}ϕ_{μ}(mλ)χ_{μ}

as required to complete the proof.

By transitivity of Deligne-Lusztig induction, we see immediately from the definition (2.3.5) that:

§2.4 Cuspidal representations and blocks  39

(2.3d) Suppose that s and t are block-diagonal element of G_{n} with C_{G_{n}}(t) ⊆ C_{G_{n}}(s). For λ = (λ_{1}, …, λ_{a}) ⊢ κ(t), χ_{t,λ} = R_{G_{π(s)}}^{G}χt,λ^{Levi} where χt,λ^{Levi} denotes the irreducible

character ε_{t}R_{C_{G}(t)}^{G_{π(s)}}(t̂.χ_{λ_{1}} … χ_{λ_{a}}) of G_{π(s)}.

As a special case of this (taking s = t), we see that an arbitrary irreducible character χ_{s,λ} can easily be expressed in terms of the χ_{σ,λ}’s:

(2.3e) For a block-diagonal s ∈ G_{n} of the form (2.1.1) and λ = (λ_{1}, …, λ_{a}) ⊢ κ(s), χ_{s,λ} = R_{G_{π(s)}}^{G_{n}}(χ_{σ_{1},λ_{1}} … χ_{σ_{a},λ_{a}}).

In view of (2.3e), it will simplify notation in what follows to always restrict our attention initially to characters of the form χ_{σ,λ}. Now that we have this notation in place, we will not need to use the Deligne-Lusztig operators again.

We record the following formula for the degree of χ_{σ,λ}, which follows easily from (2.3.3) and (2.3.6):

χ_{σ,λ}(1) = \frac{q^{dm(λ)}(q^{n} − 1)(q^{n−1} − 1) … (q − 1)}{∏_{h}(q^{dh} − 1)}  (2.3.7)

where h runs over the hook lengths of λ. We will also use the following inner product formula which is a special case of [FS, (1B), (1C)]:

(2.3f) Fix ν = (k_{1}, …, k_{a}) ⊨ k and a multi-partition λ = (λ_{1}, …, λ_{a}) ⊢ ν. Then, every irreducible constituent of R_{G_{dν}}^{G_{n}}χ_{σ,λ_{1}} … χ_{σ,λ_{a}} is of the form χ_{σ,μ} for μ ⊢ k. Moreover, given μ ⊢ k,

(χ_{σ,μ}, R_{G_{dν}}^{G_{n}}χ_{σ,λ_{1}} … χ_{σ,λ_{a}}) = (ϕ_{μ}, ind_{Σ_{ν}}^{Σ_{k}} ϕ_{λ_{1}} … ϕ_{λ_{a}})

where the inner products are the usual ones on X(G_{n}) and X(Σ_{k}) respectively.

2.4. Cuspidal representations and blocks

If G is a standard Levi subgroup of G_{n}, a cuspidal F G-module is an F G-module M with the property that ^{∗}R_{L}^{G}M = 0 for all standard Levi subgroups L < G. Similarly, a cuspidal character χ ∈ X(G) is a character with the property that ∗R_{L}^{G}χ = 0 for all standard Levi subgroups L < G.

The irreducible cuspidal characters of G_{n} are precisely the χ_{σ,(1)} for σ ∈ 𝔽̄_{q}^{×} of degree n over 𝔽_{q}. We will write simply χ_{σ} for the cuspidal character χ_{σ,(1)} for such σ. We fix a KG_{n}-module M(σ)_{K} affording the character χ_{σ}, for each σ. Choose an O-free OG_{n}-module M(σ)_{O} such that M(σ)_{K} ≅ K ⊗_{O} M(σ)_{O}, and set M(σ) = M(σ)_{F} = F ⊗_{O} M(σ)_{O}. Since Harish-Chandra restriction commutes with base change, M(σ) is a cuspidal F G_{n}-module. We will later see (cf. Lemma 2.5f and Theorem 4.3b) that M(σ) is actually an irreducible F G_{n}-module, so that its definition does not depend on the particular choice of the O-lattice M(σ)_{O}, but this is not yet clear. We just note for now that by (2.3.7),

dim M(σ) = (q − 1)(q^{2} − 1) … (q^{d−1} − 1),  (2.4.1)

interpreted as 1 if d = 1.

Now let R be one of the rings F, K or O. For σ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q} and n = kd, the module M(σ)_{R} ⊠ ⋯ ⊠ M(σ)_{R}, that is, the outer tensor product of k copies of M(σ)_{R}, is a cuspidal RG_{(d^{k})}-module. We define

M^{k}(σ)_{R} = R_{G_{(d^{k})}}^{G_{n}}M(σ)_{R} ⊠ ⋯ ⊠ M(σ)_{R}.  (2.4.2)

Since Harish-Chandra induction commutes with base change, M^{k}(σ)_{O} is an O-lattice in M^{k}(σ)_{K}, and M^{k}(σ) := M^{k}(σ)_{F} ≅ F ⊗_{O} M^{k}(σ)_{O}.

Observe that M^{k}(σ)_{K} is a module affording the character R_{G_{(d^{k})}}^{G_{n}}χ_{σ} … χ_{σ} (k

times). So according to (2.3f), the irreducible constituents of M^{k}(σ)_{K} have characters among {χ_{σ,λ} |λ ⊢ k}. Moreover, by (2.3f) again, (χ_{σ,λ}, R_{G_{(d^{k})}}^{G_{n}}χ_{σ} … χ_{σ}) = ϕ_{λ}(1). Since the only one dimensional characters of the symmetric group are the trivial and sign characters ϕ_{(k)} and ϕ_{(1^{k})} respectively, we deduce:

(2.4a) The irreducible constituents of M^{k}(σ)_{K} have characters {χ_{σ,λ} | λ ⊢ k}, and the only constituents appearing with multiplicity one are those with characters χ_{σ,(k)} and χ_{σ,(1^{k})}.

More generally, for a block-diagonal element s as in (2.1.1), set

M(s)_{R} = R_{G_{π(s)}}^{G_{n}}M^{k_{1}}(σ_{1})_{R} ⊠ ⋯ ⊠ M^{k_{a}}(σ_{a})_{R}  (2.4.3)

and write simply M(s) for M(s)_{F}. By transitivity of Harish-Chandra induction, M(s)_{R} is precisely the RG_{n}-module Harish-Chandra induced from the cuspidal RG_{δ(s)}-module

\overbrace{M(σ_{1})_{R} ⊠ ⋯ ⊠ M(σ_{1})_{R}}^{k_{1} times} ⊠ ⋯ ⊠ \overbrace{M(σ_{a})_{R} ⊠ ⋯ ⊠ M(σ_{a})_{R}}^{k_{a} times} .  (2.4.4)

Applying (2.4a) and (2.3e), we see immediately that the irreducible constituents of M(s)_{K} in characteristic 0 correspond to the irreducible characters {χ_{s,λ} | λ ⊢ κ(s)}. These irreducible constituents for fixed s constitute the geometric conjugacy class of irreducible KG_{n}-modules corresponding to s.

Now we can state a fundamental result from block theory, which follows as a consequence of the classification of blocks obtained by Fong and Srinavasan [FS]. It also has a short direct proof, valid for arbitrary type and independent of the full block classification of loc. cit., due to Broué and Michel [BM].

(2.4b) Suppose s and t are block-diagonal elements of G_{n} such that the p-regular parts of s and t are not conjugate in G_{n}. Then, the irreducible characters χ_{s,λ} and χ_{t,μ} belong to different p-blocks, for all λ ⊢ κ(s), μ ⊢ κ(t).

Let s be a p-regular block-diagonal element of G_{n}. By (2.4b), we can find a central idempotent e_{s} ∈ OG_{n} such that e_{s} acts as the identity on M(t)_{K} for all block-diagonal t ∈ G_{n} with p-regular part conjugate to s and as zero on M(t)_{K} for all other block-diagonal t ∈ G_{n}. The set {e_{s} | s ∈ C_{ss,p′}} is a set of mutually orthogonal central idempotents in OG_{n} summing to the identity. For R = K, F or O, let B_{s,R} be the union of blocks of RG_{n} corresponding to the central idempotent e_{s}, and write simply B_{s} for the algebra B_{s,F}. So, the set

⋃_{t}{χ_{t,λ} | λ ⊢ κ(t)},  (2.4.5)

as t runs over all elements of C_{ss} with p-regular part conjugate to s, gives a complete set of non-isomorphic irreducible characters belonging to the block B_{s,K}. Then,

RG_{n} = ⨁_{s∈C_{ss,p′}} B_{s,R}  (2.4.6)

is a decomposition of the group algebra as a direct sum of two-sided ideals. Let Par_{k} denote the number of partitions of k, and more generally for a composition κ = (k_{1}, …, k_{a}) write Par_{κ} for the number of multi-partitions of κ, so Par_{κ} = Par_{k_{1}} … Par_{k_{a}}. We have (see also [GH, §3]):

2.4c. Lemma. For a p-regular block-diagonal s ∈ G_{n}, the F -algebra B_{s} has precisely Par_{κ(s)} non-isomorphic irreducible modules, all of which appear as constituents of M(s). In particular, for a p-regular σ ∈ 𝔽̄_{q}^{×}, M^{k}(σ) has precisely Par_{k} nonisomorphic composition factors.

Proof. We may assume that s ∈ C_{ss,p′}. Using (2.4.5), the ring of Brauer characters of B_{s} is spanned by the restrictions to p-regular classes of the characters ⋃_{t}{χ_{t,λ}|λ ⊢ κ(t)} as t runs over all elements of C_{ss} with p-regular part conjugate to s.

Given this, Lemma 2.3b implies in fact that the ring of Brauer characters of B_{s} is spanned just by the restrictions of the {χ_{s,μ} |μ ⊢ κ(s)}. Hence, the algebra B_{s} has at most Par_{κ(s)} distinct irreducibles, for all s ∈ C_{ss,p′}. The total number of irreducible F G_{n}-modules is equal to the number of p-regular conjugacy classes, so now (2.4.6) and an elementary counting argument shows that in fact B_{s} has precisely Par_{κ(s)} distinct irreducibles.

We have now shown that the Brauer characters obtained by restricting {χ_{s,λ} |λ ⊢ κ(s)} to p-regular classes give a basis for the ring of Brauer characters of B_{s}. Since each χ_{s,λ} appears as a constituent of M(s)_{K}, it follows that all of the Par_{κ(s)} distinct irreducible B_{s}-modules definitely appear as composition factors of M(s).

We will also need to consider blocks of Levi subgroups of G_{n} in one special case. So fix now a p-regular block-diagonal element s ∈ G_{n}, suppose that π(s) = (n_{1}, …, n_{a}) ⊨ n is defined as in (2.1.4), and write s = s_{1} … s_{a} with s_{i} lying in the factor G_{n_{i}} of G_{π(s)}. For R = F, K or O, define Bs,R^{Levi} to be the block B_{s_{1},R}⊗⋯⊗B_{s_{a},R} of RG_{π(s)} ≅ RG_{n_{1}} ⊗ ⋯ ⊗ RG_{n_{a}}.

2.4d. Lemma. With notation as above, the Harish-Chandra operator R_{G_{π(s)}}^{G_{n}} gives a bijection between the set of irreducible characters of Bs,K^{Levi} and the set of irreducible characters of B_{s,K}.

Proof. If t is a block-diagonal element of G_{n} with p-regular part s, we observe

that C_{G_{n}}(t) ⊆ C_{G_{n}}(s) ⊆ G_{π(s)}. For λ = (λ_{1}, …, λ_{a}) ⊢ κ(t), let χt,λ^{Levi} denote

the irreducible character ε_{t}R_{C_{G}(t)}^{G_{π(s)}}(t̂.χ_{λ_{1}} … χ_{λ_{a}}) of G_{π(s)} as in (2.3d). Then, the irreducible characters of B_{s,K} (resp. Bs,K^{Levi}) are precisely the characters χ_{t,λ} (resp. χt,λ^{Levi}) for all λ ⊢ κ(t), as t runs over a set of representatives of the block-diagonal elements t ∈ G_{n} with p-regular part equal to s. So the lemma follows directly from (2.3d).

2.4e. Theorem. With notation as above and R = F, K or O, the Harish-Chandra induction functor R_{G_{π(s)}}^{G_{n}} induces a Morita equivalence between Bs,R^{Levi} and B_{s,R}

Proof. We just need to prove this in the case R = O, the other cases following immediately from this since Harish-Chandra induction commutes with base change. Let us first observe that R_{G_{π(s)}}^{G_{n}} does indeed restrict to a well-defined functor from Bs,O^{Levi} to B_{s,O}. Thanks to Lemma 2.4d, it certainly sends torsion free Bs,O^{Levi}-modules to B_{s,O}-modules. But an arbitrary Bs,O^{Levi}-module M is a quotient of a projective, which is torsion free, so by exactness, R_{G_{π(s)}}^{G_{n}}M is a quotient of a B_{s,O}-module, hence itself a B_{s,O}-module.

Now let G = G_{n} and L = G_{π(s)}. Denote the standard parabolic subgroup of G with Levi factor L by P. Let X be a projective generator for mod(Bs,O^{Levi}) and set Y = R_{L}^{G}(X) = OG ⊗_{OP} X. We will also write KX = K ⊗_{O} X, KY = K ⊗_{O} Y and identify KY with R_{L}^{G}(KX) = KG ⊗_{KP} KX. In view of the previous paragraph and (2.2a), Y is a projective B_{s,O}-module.

Writing endomorphisms on the right, consider the endomorphism algebras

E^{Levi} = End_{B_{s,O}^{Levi}} (X), E = End_{B_{s,O}}(Y).

§2.5 Howlett-Lehrer and Gelfand-Graev  43

The functor R_{L}^{G} determines an algebra homomorphism θ : E^{Levi} → E, which is clearly injective. We claim that θ is surjective, hence an isomorphism. Applying Lemma 2.4d, every irreducible constituent of KX is mapped to an irreducible constituent of KY and multiplicities are preserved. Consequently, comparing dimensions, R_{L}^{G} certainly induces an algebra isomorphism from End_{KL}(KX) to End_{KG}(KY). In particular, every KG-endomorphism of KY preserves the (KP - direct) summand 1 ⊗_{KP} KX of KY = KG ⊗_{KP} KX. We conclude that every OG-endomorphism of Y = R_{L}^{G}(X) also preserves the direct summand 1 ⊗_{OP} X of Y, since it preserves Y and the space 1 ⊗_{KP} KX when extended to an endomorphism of KY. In other words, every OG-endomorphism of Y is induced by an OL-endomorphism of X, that is, θ is surjective as claimed.

As a consequence, there is a one-to-one correspondence between the primitive idempotents in the endomorphism algebras E^{Levi} and E. We deduce that R_{L}^{G} takes projective indecomposable summands of X to projective indecomposable summands of Y, and preserves distinct isomorphism types. Applying Lemma 2.4c, the algebras Bs,F^{Levi} and B_{s,F} have the same number of non-isomorphic irreducibles, so Bs,O^{Levi} and B_{s,O} have the same number of non-isomorphic projective indecomposables. So since X was a projective generator, Y must also be a projective generator for mod(B_{s,O}). So now the fact that the endomorphism algebras E^{Levi} and E are isomorphic proved in the previous paragraph means that the algebras Bs,O^{Levi} and B_{s,O} are Morita equivalent.

It remains to check that the functor R_{L}^{G} itself gives the Morita equivalence. For this, we will identifying E^{Levi} and E via the isomorphism θ. Then, we can regard X (resp. Y) as a (Bs,O^{Levi}, E)-bimodule (resp. a (B_{s,O}, E)-bimodule). We have a diagram of functors:

mod(Bs,O^{Levi}) ⟶^{R_{L}^{G}} mod(B_{s,O})
X⊗_{E}?↑  ↑^{Y ⊗_{E}?}
mod(E)  mod(E)

where the vertical functors are Morita equivalences since X and Y are projective generators. Moreover, the diagram commutes, that is, the functors R_{L}^{G} ∘ (X⊗_{E}?) and (R_{L}^{G}X)⊗_{E}? are isomorphic, which follows directly by associativity of tensor product. So the top functor, R_{L}^{G}, is also a Morita equivalence.

Theorem 2.4e plays the role of (2.3e) in the modular theory: it allows us almost all of the time to restrict our attention to studying the blocks of B_{s,R} with s of the form (σ)^{k} for p-regular σ ∈ 𝔽̄_{q}^{×}, rather than the more general B_{s,R} for arbitrary s ∈ C_{ss,p′}. The following notation will be convenient: given p-regular σ ∈ 𝔽̄_{q}^{×} and k ≥ 1, let B_{σ,k,R} denote the algebra B_{s,R} in the special case that s = (σ)^{k}. Write simply B_{σ,k} for B_{σ,k,F}. We remark at this point that as a special case of a conjecture of Broué [B, p. 61], it is expected that the algebra B_{σ,k,O} is Morita equivalent to the ‘unipotent block’ B_{1,O} of GL_{k}(𝔽_{q^{d}}), where d is the degree of σ over 𝔽_{q}.

2.5. Howlett-Lehrer theory and the Gelfand-Graev representation

Suppose now that σ ∈ 𝔽̄_{q}^{×} is of degree d over 𝔽_{q} and that n = kd for some k ≥ 1. Let R denote one of the rings F, K or O throughout the section. Recalling the definition of the Hecke algebra from §1.1, we let H_{k,R} = H_{R,q^{d}}(Σ_{k}). Applying (1.1.1), we may identify H_{k,F} (resp. H_{k,K}) with the algebra F ⊗_{O} H_{k,O} (resp.

K ⊗_{O} H_{k,O}). We will usually write simply H_{k} in place of H_{k,F} over the modular field. We wish to construct a right action of H_{k,R} on the module M^{k}(σ)_{R}, as a very special case of the theory of Howlett and Lehrer [HL_{1}].

Write N_{R} = M(σ)_{R} ⊠ ⋯ ⊠ M(σ)_{R} (k times) and M_{R} = M^{k}(σ)_{R} = R_{G_{(d^{k})}}^{G_{n}}N_{R} for short. We identify M_{R} with RG_{n} ⊗_{RP} N_{R}, where P is the standard parabolic subgroup of G_{n} with Levi factor G_{(d^{k})} and unipotent radical Y_{(d^{k})}. Let f denote the idempotent

f = \frac{1}{|Y_{(d^{k})}|} ∑_{u∈Y_{(d^{k})}} u ∈ RP

so that^{∗}R_{G_{(d^{k})}}^{G_{n}}M_{R} = fM_{R}. Finally, recall the embedding π : Σ_{k} ↪ G_{n} as d×d-block permutation matrices, from §2.1.

2.5a. Lemma. fM_{R} = ⨁_{x∈Σ_{k}} fπ_{x} ⊗ N_{R}

Proof. One argues using the Bruhat decomposition as in [D_{1}, Lemma 3.4] to show that the sum

∑_{x∈Σ_{k}} fπ_{x} ⊗ N_{R} ⊆ fM_{R}

is direct and has a complement in fM_{R} as an R-module. It then just remains to check that the R-rank of fM_{R} is k!(rank N_{R}). But by the Mackey decomposition, writing L = G_{(d^{k})},

fM_{R} = ^{∗}R_{L}^{G_{n}}M_{R} ≅ ⨁_{w∈D_{(d^{k}),(d^{k})}} R_{L∩^{w}L}^{L} ∘ conj_{w} ∘^{∗}R_{L∩^{w^{−1}}L}^{L}(N_{R}).

Since N_{R} is a cuspidal RL-module, the summand corresponding to w is zero unless L ∩^{w^{−1}}L = L, in which case according to (2.1c) w = π_{x} for x ∈ Σ_{k}. So, fM_{R} ≅ ⨁_{x∈Σ_{k}} conj_{π_{x}}(N_{R}) which has the required R-rank.

Notice that each summand fπ_{w} ⊗ N_{R} appearing in the lemma is an RG_{(d^{k})}- submodule of fM_{R}. Moreover, given w ∈ Σ_{k}, there is an isomorphism of RG_{(d^{k})}- modules Ā_{w} : N_{R} → fπ_{w} ⊗ N_{R}, such that

v_{1} ⊗ ⋯ ⊗ v_{k} ↦ fπ_{w} ⊗ v_{w1} ⊗ ⋯ ⊗ v_{wk}

for all v_{1}, …, v_{k} ∈ M(σ)_{R}. Identifying ^{∗}R_{G_{(d^{k})}}^{G_{n}}M_{R} with ⨁_{x∈Σ_{k}} fπ_{x} ⊗ N_{R} according

to Lemma 2.5a, we extend the range of Ā_{w} to obtain an RG_{(d^{k})}-module monomorphism

A_{w} : N_{R} → ^{∗}R_{G_{(d^{k})}}^{G_{n}}M_{R},  (2.5.1)

for each w ∈ Σ_{k}. Clearly, the maps {A_{w} | w ∈ Σ_{k}} are linearly independent. Now for each w ∈ Σ_{k}, let

B_{w} : M_{R} → M_{R}

be the RG_{n}-module homomorphism induced by A_{w} under the isomorphism

Hom_{RG_{(d^{k})}}(N_{R},^{∗}R_{G_{(d^{k})}}^{G_{n}}M_{R}) ≅ Hom_{RG_{n}}(M_{R}, M_{R})

arising from adjointness (2.2a). So, we have constructed k! linearly independent RG_{n}-endomorphisms of M_{R}, namely, {B_{w} | w ∈ Σ_{k}}. The Howlett-Lehrer theory [HL_{1}] in our special case (see [D_{1}, Lemma 3.5] and [J_{2}, Theorem 4.12]) computes relations between the B_{w} (observe that knowing a relation over K implies the relation over O, hence over F) to show:

(2.5b) There is an algebra embedding θ : H_{k,R} ↪ End_{RG_{n}}(M_{R}) (writing endomorphisms on the right) with θ(T_{w}) = ((−1)^{o(σ)+1}q^{\frac{1}{2} d(d+1)})^{ℓ(w)}B_{w} for all w ∈ Σ_{k}, where o(σ) is the order of σ ∈ 𝔽̄_{q}^{×}. In the case R = K, θ is an isomorphism.

Later on we will show that θ is an isomorphism for R = F, O too, but we cannot prove this yet. In view of (2.5b), we will henceforth always regard M^{k}(σ)_{R} as an (RG_{n}, H_{k,R})-bimodule; it is clear from (2.5b) that the bimodule structure is compatible with base change.

Next, we focus on the case R = K. Then, M^{k}(σ)_{K} is a completely reducible KG_{n}-module and its endomorphism algebra

H_{k,K} ≅ End_{KG_{n}}(M^{k}(σ)_{K})

is semisimple. Fitting’s lemma gives a bijection between the irreducible KG_{n}- modules appearing as constituents of M^{k}(σ)_{K} and the irreducible H_{k,K}-modules. We are interested here in the constituents of M^{k}(σ)_{K} corresponding to the trivial representation I_{H_{k,K}} and the sign representation E_{H_{k,K}} of H_{k,K}, as defined in §1.1. These appear in M^{k}(σ)_{K} with multiplicity one since I_{H_{k,K}} and E_{H_{k,K}} are one dimensional. Over K, (1.1b) easily implies that x_{k} and y_{k} are idempotents up to multiplication by non-zero scalars, so these constituents of M^{k}(σ)_{K} are according to Fitting’s lemma precisely the irreducible submodules M^{k}(σ)_{K}x_{k} and M^{k}(σ)_{K}y_{k}, respectively.

2.5c. Lemma. The KG_{n}-module M^{k}(σ)_{K}x_{k} (resp. M^{k}(σ)_{K}y_{k}) corresponds to the irreducible character χ_{σ,(k)} (resp. χ_{σ,(1^{k})}).

Proof. We may assume k > 1, the case k = 1 being trivial. Note that according to (2.3.7), we know the degrees:

χ_{σ,(k)}(1) = ∏_{i=1}^{kd}(q^{i} − 1)/ ∏_{i=1}^{k}(q^{id} − 1),  (2.5.2)

χ_{σ,(1^{k})}(1) = q^{\frac{d}{2} k(k−1)} ∏_{i=1}^{kd}(q^{i} − 1)/ ∏_{i=1}^{k}(q^{id} − 1).  (2.5.3)

For k > 1 these degrees are different. Moreover, by Fitting’s lemma, the only irreducible constituents of M^{k}(σ)_{K} appearing with multiplicity one are the modules M^{k}(σ)_{K}x_{k} and M^{k}(σ)_{K}y_{k}. So applying (2.4a), we see that it suffices to check that dim M^{k}(σ)_{K}x_{k} = χ_{σ,(k)}(1) not χ_{σ,(1^{k})}(1).

The dimension of M(σ)_{K} is given by (2.4.1). Using this, an easy calculation gives the dimension of the induced module:

dim M^{k}(σ)_{K} = ∏_{i=1}^{kd}(q^{i} − 1)/(q^{d} − 1)^{k}.

Now [C, Proposition 10.9.6] tells us that

dim M^{k}(σ)_{K}x_{k} = dim M^{k}(σ)_{K}/ ∑_{w∈Σ_{k}}(q^{d})^{ℓ(w)}.

It is well-known that ∑_{w∈Σ_{k}} t^{ℓ(w)} = ∏_{i=1}^{k}(t^{i} − 1)/(t − 1)^{k}, and one easily verifies now that dim M^{k}(σ)_{K}x_{k} = χ_{σ,(k)}(1).

We conclude the chapter by reviewing the fundamental properties of the Gelfand-Graev representation introduced in [GG]. Fix a non-trivial homomorphism

χ_{K} : (𝔽_{q}, +) → K^{×}.  (2.5.4)

Note that the values of χ_{K} lie in O, so we can restrict χ_{K} to χ_{O} : (𝔽_{q}, +) → O^{×}, then reduce module p to obtain the non-trivial character χ = χ_{F} : (𝔽_{q}, +) → F^{×}.

For u ∈ U_{n} and R = F, K or O, let

θ_{n,R}(u) = χ_{R}(∑_{i=1}^{n−1} u_{i,i+1})

where u_{i,i+1} denotes the (i, i + 1)-entry of the matrix u. We associate to this linear character of RU_{n} the idempotent

γ_{n,R} = \frac{1}{|U_{n}|} ∑_{u∈U_{n}} θ_{n,R}(u^{−1})u ∈ RG_{n}.  (2.5.5)

In particular, γ_{n} := γ_{n,F}. The left ideal F G_{n}γ_{n} is the Gelfand-Graev representation Γ_{n} of F G_{n}. Observe that Γ_{n} is a projective F G_{n}-module. Moreover, it is the reduction modulo p of Γ_{n,K} = KG_{n}γ_{n,K} via the O-lattice Γ_{n,O} = OG_{n}γ_{n,O}.

More generally, given ν = (n_{1}, …, n_{a}) ⊨ n, we have the analogous left F G_{ν}- module denoted Γ_{ν}, which is just the outer tensor product Γ_{n_{1}} ⊠ ⋯ ⊠ Γ_{n_{a}}. This is the left ideal F G_{ν}γ_{ν} where

γ_{ν} = γ_{n_{1}} ⊗ ⋯ ⊗ γ_{n_{a}} ∈ F G_{ν} = F G_{n_{1}} ⊗ ⋯ ⊗ F G_{n_{a}}.

We also write Γ_{ν,O} and Γ_{ν,K} for the Gelfand-Graev representations over O and K, defined in the same way using the idempotents γ_{ν,O} and γ_{ν,K}.

We will need the following key facts, due originally to I. Gelfand and M. Graev [GG] and S.Gelfand [Ge] (see e.g. [Z, Proposition 9.4]), about the Gelfand-Graev representation in characteristic 0, all of which we deduce here from standard results in [C]:

2.5d. Theorem. For any μ ⊨ n, (i) ^{∗}R_{G_{μ}}^{G_{n}}Γ_{n,K} ≅ Γ_{μ,K}; (ii) Γ_{μ,K} is a multiplicity-free KG_{μ}-module; (iii) the number of irreducible constituents of Γ_{n,K} is equal to the number of

semisimple conjugacy classes in G_{n}, namely, (q − 1)q^{n−1}; (iv) for any block-diagonal element s ∈ G_{n}, dim Hom_{KG_{n}}(Γ_{n,K}, M(s)_{K}) = 1;

(v) given σ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q} and n = kd, the image of any non-zero homomorphism from Γ_{n,K} to M^{k}(σ)_{K} is precisely the irreducible module M^{k}(σ)_{K}y_{k}.

Proof. (i) By [C, Theorem 8.1.5], ^{∗}R_{G_{μ}}^{G_{n}}Γ_{n,K} and Γ_{μ,K} have the same character. (ii) This is [C, Theorem 8.1.3]. (iii) That the number of semisimple classes in G_{n} is equal to (q−1)q^{n−1} is proved

for example in [C, Theorem 3.7.6(ii)]. So now the result follows using (ii) and [C, Proposition 8.3.1].

(iv) Note that by (i) and adjointness,

Hom_{KG_{n}}(Γ_{n,K}, M(s)_{K}) ≅ Hom_{KG_{δ(s)}}(Γ_{δ(s),K}, N_{K})

where N_{K} is the cuspidal module of (2.4.4). The right hand side is either 0 or 1-dimensional by (ii). To see that it is actually always 1-dimensional, use (iii) and the observation that the sum as s runs over all semisimple classes of G_{n} of dim Hom_{KG_{n}}(Γ_{n,K}, M(s)_{K}) must count the total number of irreducible constituents of Γ_{n,K}.

(v) By (iv) and Frobenius reciprocity, the space γ_{n,K}M^{k}(σ)_{K} is one dimensional. So γ_{n,K}M^{k}(σ)_{K} is a one dimensional right module for the Hecke algebra H_{k,K}. Since there are just two one dimensional right H_{k,K}-modules, namely I_{H_{k,K}} and E_{H_{k,K}} (or rather, the analogous right modules), we deduce that H_{k,K} acts on KG_{n}γ_{n,K}M^{k}(σ)_{K} either by I_{H_{k,K}} or by E_{H_{k,K}}. So, as x_{k} and y_{k} are idempotents up to a scalar over K, the image Y of Γ_{n,K} in M^{k}(σ)_{K} under any non-zero homomorphism satisfies either Y = M^{k}(σ)_{K}x_{k} or Y = M^{k}(σ)_{K}y_{k}. But the dimension of Y is calculated in [C, Theorem 8.4.9] (thanks to [C, Propositions 8.4.4-8.4.5] and (iii)):

dim Y = |G_{n} : C_{G_{n}}(s)|_{p′}|C_{G_{n}}(s)|_{p}

where s = (σ)^{k}. Recalling that C_{G_{n}}(s) ≅ GL_{k}(𝔽_{q^{d}}), this is easily checked to be the same as dim M^{k}(σ)_{K}y_{k} = χ_{σ,(1^{k})}(1) as in (2.5.3), not dim M^{k}(σ)_{K}x_{k} = χ_{σ,(k)}(1) which is (2.5.2).

Since Γ_{n} is a projective F G_{n}-module, many of the properties in Theorem 2.5d generalize to the modular case:

2.5e. Corollary. For any μ ⊨ n and R equal to F or O, (i) ^{∗}R_{G_{μ}}^{G_{n}}Γ_{n,R} ≅ Γ_{μ,R}; (ii) for any block-diagonal s ∈ G_{n}, Hom_{RG_{n}}(Γ_{n,R}, M(s)_{R}) is R-free of rank 1; (iii) given σ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q} and n = kd, we have mT_{w} = (−1)^{ℓ(w)}m

for all w ∈ Σ_{k} and for all m lying in the image of any non-zero homomorphism from Γ_{n,R} to M^{k}(σ)_{R}.

Proof. (i) Since Harish-Chandra restriction commutes with base change, we see easily from Theorem 2.5d(i) that ^{∗}R_{G_{μ}}^{G_{n}}Γ_{n} and Γ_{μ} have the same Brauer characters. By (2.2a), ^{∗}R_{G_{μ}}^{G_{n}} sends projectives to projectives. Therefore ^{∗}R_{G_{μ}}^{G_{n}}Γ_{n} and Γ_{μ} are projective F G_{μ}-modules having the same Brauer character, so they are isomorphic. The result over O follows immediately since any projective F G_{μ}-module has a unique lift to O.

(ii) Taking R = O, Hom_{OG_{n}}(Γ_{n,O}, M(s)_{O}) is an O-lattice in the corresponding space over K, hence is O-free of rank 1 by Theorem 2.5d(iv). Now, for example by the universal coefficient theorem,

F ⊗_{O} Hom_{OG_{n}}(Γ_{n,O}, M(s)_{O}) ≅ Hom_{F G_{n}}(Γ_{n}, M(s))

as Γ_{n,O} is projective. This implies the result over F.

(iii) Let θ_{O} be a generator of Hom_{OG_{n}}(Γ_{n,O}, M^{k}(σ)_{O}), applying (ii). Extending scalars, we obtain non-zero maps

θ_{K} ∈ Hom_{KG_{n}}(Γ_{n,K}, M^{k}(σ)_{K}),
θ_{F} ∈ Hom_{F G_{n}}(Γ_{n}, M^{k}(σ)).

By Theorem 2.5d(v), the image of θ_{K} is M^{k}(σ)_{K}y_{k}, so by (1.1b), for any v ∈ Γ_{n,K},

θ_{K}(v)T_{w} = (−1)^{ℓ(w)}θ_{K}(v)

for all w ∈ Σ_{k}. In particular, we see that for v ∈ Γ_{n,O} ⊂ Γ_{n,K},

θ_{O}(v)T_{w} = (−1)^{ℓ(w)}θ_{O}(v)

for all w ∈ Σ_{k}. This proves the result in case R = O.

Over F, θ_{F} (Γ_{n}) is certainly a quotient of F ⊗_{O} θ_{O}(Γ_{n,O}) (though we cannot yet assert that the two are isomorphic). Since the action of H_{k,R} on M^{k}(σ)_{R} is compatible with base change, the conclusion follows directly.

Finally, we can now prove that M(σ) is an irreducible F G_{n}-module, as we mentioned earlier, at least if σ is a p-regular element. Later, we will also prove this if σ is not p-regular. We remark that the first proof of this fact (for arbitrary σ) was given in [D_{1}, D_{2}] using the classification of irreducible F G_{n}-modules. There is also a simple direct proof due to James [J_{2}, §3], which depends on the earlier work of Gelfand [Ge].

2.5f. Lemma. If σ ∈ 𝔽̄_{q}^{×} is p-regular of degree n over 𝔽_{q}, the F G_{n}-module M(σ) is irreducible.

Proof. In view of Lemma 2.4c, we know that there is just one irreducible F G_{n}- module, N say, in the same block as M(σ). In other words, all composition factors of M(σ) are isomorphic to N. But by Corollary 2.5e(ii), Γ_{n} is a projective F G_{n}- module such that Hom_{F G_{n}}(Γ_{n}, M(σ)) is one dimensional. So, Γ_{n} must contain the projective cover of N as a summand and now the one dimensionality implies that N appears with multiplicity one as a constituent of M(σ), hence M(σ) = N is irreducible.

Connecting GL_{n} with quantum linear groups

In this chapter, we prove the Morita theorem at the heart of the modular theory. This was first proved by Cline, Parshall and Scott [CPS_{3}, §9]. The proof in loc. cit. depends fundamentally on the work of James [J_{2}] and Dipper-James [DJ_{3}], whereas the approach here is self-contained, independent of this earlier work.

3.1. Schur functors

We begin with a short review of the “Schur functors” used heavily throughout the remainder of the chapter. The results described here are based on the results in [G_{2}, chapter 6] and [JS], though we work in terms of projective modules instead of idempotents. We also mention the work of Auslander [A], where a thorough study of the functors in this section was made in a more general setting.

Fix a finite dimensional F -algebra C (for this section only, F can be taken to be an arbitrary field). Assume that we are given a fixed projective module P ∈ mod(C). We set H = End_{C}(P), writing endomorphisms commuting with the left C-action on the right. Define the functors

α : mod(C) → mod(H),  α = Hom_{C}(P,?),  (3.1.1)
β : mod(H) → mod(C),  β = P ⊗_{H}?.  (3.1.2)

Since P is projective, α is exact. Moreover, by adjointness of ‘tensor’ and ‘hom’, β is left adjoint to α. The basic example to keep in mind (e.g. as in (1.5.2)) is the case that P = Ce for some idempotent e ∈ C, when H ≅ eCe. Then the functor α is the familiar Schur functor, since Hom_{C}(Ce, V) ≅ eV for any left C-module V, and β is the inverse Schur functor.

Given a left C-module V, let O_{P} (V) denote the largest submodule V′ of V with the property that Hom_{C}(P, V′) = 0. Let O^{P} (V) denote submodule of V generated by the images of all C-homomorphisms from P to V. Since P is projective, these have alternative descriptions: O_{P} (V) (resp. O^{P} (V)) is the largest (resp. smallest) submodule V′ of V such that no composition factor of V′ (resp. V/V′) appears in the head of P. We remark that in [A], Auslander refers to O^{P} (V) as the P -trace and O_{P} (V) as the P -torsion part of V.

Clearly any C-module homomorphism V → W sends O_{P} (V) into O_{P} (W) and O^{P} (V) into O^{P} (W), so we can view O_{P} and O^{P} as functors mod(C) → mod(C), by defining their action on morphisms to be restriction. Finally, any homomorphism V → W induces a well-defined C-module homomorphism V/O_{P} (V) → W/O_{P} (W). We thus obtain an exact functor A_{P} : mod(C) → mod(C) defined on objects by V ↦ V/O_{P} (V).

3.1a. Lemma. The composite functors α ∘ β and α ∘ A_{P} ∘ β are both isomorphic to the identity.

Proof. For any left H-module U, the fact that P is projective and [AF, 20.10] implies that

Hom_{C}(P, P ⊗_{H} U) ≅ Hom_{C}(P, P) ⊗_{H} U = H ⊗_{H} U ≅ U.

All isomorphisms are natural, so this proves that α ∘ β is isomorphic to the identity. Now apply the exact functor α to the exact sequence 0 → O_{P} ∘ β(U) → β(U) → A_{P} ∘β(U) → 0, using the fact that Hom_{C}(P, O_{P} ∘β(U)) = 0, to deduce that α∘β(U) and α ∘ A_{P} ∘ β(U) are naturally isomorphic, completing the proof.

3.1b. Lemma. For V ∈ mod(C), let V̂ = β ∘ α(V) = P ⊗_{H} Hom_{C}(P, V) and let ω : V̂ → V be the natural C-homomorphism defined by p ⊗ φ ↦ φ(p) for φ ∈ Hom_{C}(P, V) and p ∈ P. Then, im ω = O^{P} (V) and ker ω ⊆ O_{P} (V̂).

Proof. It follows directly from the definitions that im ω = O^{P} (V). Let Z = ker ω; we need to show that α(Z) = Hom_{C}(P, Z) = 0. Using the short exact sequence

0 ⟶ α(Z) ⟶ α(V̂) ⟶ α(O^{P} (V)) ⟶ 0,

we just need to check that α(V̂) ≅ α(O^{P} (V)). By Lemma 3.1a, α(V̂) ≅ Hom_{C}(P, V) which is obviously isomorphic to α(O^{P} (V)) = Hom_{C}(P, O^{P} (V)) by definition of O^{P}.

3.1c. Corollary. If V, W ∈ mod(C) satisfy O^{P} V = V and O_{P} W = 0, then

Hom_{C}(V, W) ≅ Hom_{H}(α(V), α(W)).

Proof. By adjointness, Hom_{H}(α(V), α(W)) ≅ Hom_{C}(β ∘ α(V), W). By the lemma, there is a natural homomorphism β ∘ α(V) → V which is surjective as O^{P} V = V, and has kernel Z contained in O_{P} (β ∘ α(V)). Moreover, any homomorphism from β ∘ α(V) to W must act as zero on Z since O_{P} (W) = 0, hence factors through the quotient V of β ∘ α(V). Thus, Hom_{C}(β ∘ α(V), W) ≅ Hom_{C}(V, W).

Now we have the main result about Schur functors, see [JS, §2]:

§3.2 Schur functors  53

3.1d. Theorem. The restrictions of the functors α and A_{P} ∘ β induce mutually inverse equivalences of categories between mod(H) and the full subcategory ℭ of mod(C) consisting of all V ∈ mod(C) such that O_{P} (V) = 0, O^{P} (V) = V.

Proof. We first note that A_{P} ∘ β is a well-defined functor from mod(H) to ℭ. Take U ∈ mod(H). Then, A_{P} ∘ β(U) is a quotient of β(U) which is a quotient of the left C-module P ⊗_{F} U ≅ P^{⊕ dim U}. The latter is certainly generated by the images of all C-homomorphisms from P, so O^{P} (A_{P} ∘ β(U)) = A_{P} ∘ β(U). Moreover, O_{P} (A_{P} ∘ β(U)) = 0, so we do indeed have that A_{P} ∘ β(U) ∈ ℭ.

Now for the theorem, take V ∈ ℭ and consider A_{P} ∘ β ∘ α(V). By Lemma 3.1b and the assumption that O^{P} (V) = V, we know that β ∘ α(V) has a submodule Z ⊆ O_{P} (β ∘ α(V)) such that (β ∘ α(V))/Z ≅ V. Since O_{P} (V) = 0, we see that in fact Z = O_{P} (β ∘ α(V)), so A_{P} ∘ β ∘ α(V) ≅ V, and this isomorphism is certainly functorial. Finally, by Lemma 3.1a, we know already that α ∘ A_{P} ∘ β is isomorphic to the identity, completing the proof.

3.1e. Corollary. Let {E_{i} | i ∈ I} be a complete set of non-isomorphic irreducible C-modules appearing in the head of P. For i ∈ I, set D_{i} = α(E_{i}). Then, the set {D_{i} | i ∈ I} is a complete set of non-isomorphic irreducible H-modules, and A_{P} ∘ β(D_{i}) ≅ E_{i}.

Proof. Note that each E_{i} (i ∈ I) is an irreducible module belonging to the Abelian category ℭ. Consequently, by Theorem 3.1d, each D_{i} is an irreducible H-module with A_{P} ∘ β(D_{i}) ≅ E_{i}, and the D_{i}’s are pairwise non-isomorphic. To see that all irreducible H-modules arise in this way, use Fitting’s lemma.

Finally, we include a useful lemma which gives a more explicit description of the effect of the composite functor A_{P} ∘ β on left ideals of H:

3.1f. Lemma. Suppose that O_{P} (P) = 0, that is, that every composition factor of the socle of P also appears in its head. Then for any left ideal J of H, A_{P} ∘ β(J) ≅ P J.

Proof. Our assumption on P implies that O_{P} (P J) = 0 hence A_{P} (P J) ≅ P J. Now we prove the more general result that A_{P} ∘β(J) ≅ A_{P} (P J), without any assumption on P. There is a short exact sequence 0 ⟶ Z ⟶ P ⊗_{H} J ⟶^{μ} P J ⟶ 0 where μ is the natural multiplication map. Applying the exact functor A_{P} to this, we see that it suffices to show that A_{P} (Z) = 0, or equivalently, that Hom_{C}(P, Z) = 0.

Now apply α to this short exact sequence, using Lemma 3.1a, to obtain the exact sequence: 0 ⟶ Hom_{C}(P, Z) ⟶ J ⟶^{μ̄} Hom_{C}(P, P J) ⟶ 0. The second map μ̄ maps j ∈ J to the homomorphism P → P J given by right multiplication by j. Since P is a faithful H-module, μ̄ is injective. This implies that Hom_{C}(P, Z) = 0 as required.

3.2. The cuspidal algebra

Now we come to a key technical lemma underlying the modular theory; it will allow us to apply the results of §3.1 to study the module M^{k}(σ) from (2.4.2). This lemma was noticed originally (in a slightly different form) by Cline, Parshall and Scott [CPS_{3}, Lemma 9.1] and simplifies the original theory of [DJ_{3}] considerably. The version we present here is due to V. Schubert [S, 12.4/1]; we are grateful to Schubert for allowing us to include the proof of this lemma.

3.2a. Lemma. Let A be a ring and 0 ⟶ Z ⟶ P ⟶^{π} M ⟶ 0 be a short exact sequence of A-modules with P projective. If every A-module homomorphism from P to M annihilates Z, then M is a projective A/ ann_{A}(M)-module, where ann_{A}(M) denotes the annihilator of M in A.

Proof. We need to show that every A/ ann_{A}(M)-module homomorphism from M to a quotient of an A/ ann_{A}(M)-module V can be lifted to a homomorphism to V. Equivalently, we show that every A-module homomorphism α : M → U, where U is a quotient of an A-module V with ann_{A}(M) ⊆ ann_{A}(V), can be lifted to a homomorphism β : M → V. Well, α ∘ π gives an A-module homomorphism from P to U so can be lifted (as P is projective) to a map γ : P → V. So the result will follow if we can show that γ annihilates Z, so that γ factors through P ⟶^{π} M to induce the required map β. In other words, we need to check that every A-module homomorphism γ : P → V annihilates Z, for every A-module V with ann_{A}(M) ⊆ ann_{A}(V).

Given such a module V, the map Hom_{A}(P, A) ⊗_{A} V → Hom_{A}(P, V) sending a generator f ⊗ v ∈ Hom_{A}(P, A) ⊗_{A} V to the map p ↦ f(p)v for p ∈ P, is an isomorphism by [AF, 20.10] (this requires the projectivity of P). Therefore, we just need to check that f(p)v = 0 for all f ∈ Hom_{A}(P, A), p ∈ Z and v ∈ V, or in other words, that f(Z) ⊆ ann_{A}(V) for all f ∈ Hom_{A}(P, A). Now take such an f ∈ Hom_{A}(P, A). For m ∈ M, the map P → M defined by p ↦ f(p)m for p ∈ P is an A-module homomorphism, so by hypothesis annihilates Z. That is, f(Z) ⊆ ann_{A}(M). By our initial assumption on V, this finally implies that f(Z) ⊆ ann_{A}(V) as required.

3.2b. Remark. In [S], Schubert also proves the converse to this lemma, namely, if M is a projective A/ ann_{A}(M)-module, then every A-module homomorphism from P to M annihilates Z. The latter condition is easily checked to be equivalent to the hypothesis adopted in the earlier work of Dipper [D_{3}, D_{4}].

Now for the remainder of the chapter, we fix σ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q}. We assume moreover that the following two properties are satisfied:

(A1) M(σ) is an irreducible F G_{d}-module;

(A2) the F G_{kd}-module M^{k}(σ) has exactly Par_{k} non-isomorphic composition factors for all k ≥ 1.

§3.2 The cuspidal algebra  55

We know for instance by Lemma 2.4c and Lemma 2.5f that the assumptions (A1) and (A2) are satisfied if σ is p-regular. In fact, we will prove later in Theorem 4.3b that (A1) and (A2) are satisfied for arbitrary σ, so these assumptions will turn out to be redundant.

Fix also now some k ≥ 1 and set n = kd. As our first consequence of the assumptions on σ, we have:

3.2c. Lemma. The module M^{k}(σ) is self-dual, that is, M^{k}(σ) ≅ M^{k}(σ)^{τ}. In particular, the socle and the head of M^{k}(σ) are isomorphic.

Proof. Since M(σ) is irreducible by assumption, this follows immediately as a special case of Corollary 2.2f.

Introduce the cuspidal algebra

C_{σ,k} = C_{F,(σ)^{k}}(GL_{n}(𝔽_{q})) = F G_{n}/ ann_{F G_{n}}(M^{k}(σ)).  (3.2.1)

So, C_{σ,k} is the image of F G_{n} under the representation afforded by the F G_{n}-module M^{k}(σ). Note in view of (2.4b) that for p-regular σ, C_{σ,k} is a quotient algebra of the corresponding block algebra B_{σ,k} of F G_{n} (as defined at the end of §2.4).

3.2d. Theorem. M^{k}(σ) is a projective C_{σ,k}-module with endomorphism algebra isomorphic to H_{k} = H_{F,q^{d}}(Σ_{k}), acting as in (2.5b).

Proof. Write G = G_{n}, L = G_{(d^{k})}, N = M(σ)⊠⋯⊠M(σ) and M = M^{k}(σ) ≅ R_{L}^{G}N for short. By our assumption (A1), N is an irreducible cuspidal F L-module. Let Q be the projective cover of N in the category mod(F L). By the Mackey formula,

α := dim Hom_{F G}(R_{L}^{G}(Q), M) = dim Hom_{F L}(Q,^{∗}R_{L}^{G} ∘ R_{L}^{G}(N))

= ∑_{w∈D_{(d^{k}),(d^{k})}}dim Hom_{F L}(Q, R_{L∩^{w}L}^{L} ∘ conj_{w} ∘^{∗}R_{L∩^{w^{−1}}L}^{L}(N)).

Now since N is cuspidal, (2.1c) gives that the summand corresponding to w is zero unless w = π_{x} for some x ∈ Σ_{k}. So, α = ∑_{x∈Σ_{k}} dim Hom_{F L}(Q, conj_{π_{x}}(N)). Finally, observe that conj_{π_{x}}(N) ≅ N as an F L-module (the isomorphism sends v_{1} ⊗ ⋯ ⊗ v_{k} ∈ N to v_{x1} ⊗ ⋯ ⊗ v_{xk} ∈ conj_{π_{x}}(N)). So since N is irreducible and Q is its projective cover, we see that α = ∑_{x∈Σ_{k}} dim Hom_{F L}(Q, N) = k!.

As Q is projective and maps surjectively onto N, R_{L}^{G}Q is projective and maps surjectively onto M. So, R_{L}^{G}Q contains the projective cover P of M as a summand. Also recall from (2.5b) that H_{k}, which has dimension k!, embeds into End_{F G}(M). Combining these remarks with our calculation that α = k!, we have shown that

dim Hom_{F G}(P, M) ≤ dim Hom_{F G}(R_{L}^{G}Q, M) = k! ≤ dim Hom_{F G}(M, M),

whence that equality holds everywhere. Now the criterion of Lemma 3.2a implies that M is a projective C_{σ,k}-module, while by dimension, the embedding of H_{k} into End_{F G}(M^{k}(σ)) from (2.5b) is an isomorphism.

We mention the following immediate corollary, which is relevant to the point of view (not pursued further here) of modular Harish-Chandra theory:

3.2e. Corollary. There is a bijection between the isomorphism classes of irreducible H_{k}-modules and the isomorphism classes of irreducible F G_{n}-modules appearing as constituents of the head (resp. socle) of M^{k}(σ).

Proof. By Lemma 3.2c, a copy of every composition factor of the socle of M^{k}(σ) appears in its head. So the corollary follows from the theorem by Corollary 3.1e, taking C = C_{σ,k}, H = H_{k}, P = M^{k}(σ).

Suppose now that ν = (k_{1}, …, k_{a}) ⊨ k. Regard M^{k_{i}}(σ) as an (F G_{dk_{i}}, H_{k_{i}})- bimodule for each i = 1, …, a in the same way as explained after (2.5b). Then, identifying F G_{dν} with F G_{dk_{1}} ⊗ ⋯ ⊗ F G_{dk_{a}} and H_{ν} with H_{k_{1}} ⊗ ⋯ ⊗ H_{k_{a}}, we obtain an (F G_{dν}, H_{ν})-bimodule:

M^{ν}(σ) = M^{k_{1}}(σ) ⊠ ⋯ ⊠ M^{k_{a}}(σ) ≅ R_{G_{(d^{k})}}^{G_{dν}}( \underbrace{M(σ) ⊠ ⋯ ⊠ M(σ)}_{k times} ).  (3.2.2)

We have the Levi analogue of the cuspidal algebra, namely,

C_{σ,ν} = F G_{dν}/ ann_{F G_{dν}}(M^{ν}(σ)) ≅ C_{σ,k_{1}} ⊗ ⋯ ⊗ C_{σ,k_{a}}.  (3.2.3)

Theorem 3.2d immediately gives that M^{ν}(σ) is projective as a C_{σ,ν}-module, and that H_{ν} is precisely the endomorphism algebra End_{C_{σ,ν}}(M^{ν}(σ)).

Now, M^{ν}(σ) is an (F G_{dν}, H_{ν})-bimodule, so we can regard R_{G_{dν}}^{G_{n}}M^{ν}(σ) as an (F G_{n}, H_{ν})-bimodule. Similarly, M^{k}(σ) is an (F G_{n}, H_{k})-bimodule, so ^{∗}R_{G_{dν}}^{G_{n}}M^{k}(σ) is an (F G_{dν}, H_{k})-bimodule in a precise way. On the other hand, M^{k}(σ) is an (F G_{n}, H_{k})-bimodule, hence also an (F G_{n}, H_{ν})-bimodule, restricting the H_{k}-action to the subalgebra H_{ν} of H_{k}, while M^{ν}(σ) ⊗_{H_{ν}} H_{k} is an (F G_{dν}, H_{k})-bimodule. The following basic lemma identifies these various bimodule structures:

3.2f. Lemma. For ν ⊨ k, (i) R_{G_{dν}}^{G_{n}}M^{ν}(σ) is isomorphic to M^{k}(σ) as an (F G_{n}, H_{ν})-bimodule; (ii) ^{∗}R_{G_{dν}}^{G_{n}}M^{k}(σ) is isomorphic to M^{ν}(σ) ⊗_{H_{ν}} H_{k} as an (F G_{dν}, H_{k})-bimodule.

In particular, the F G_{dν}-action on ^{∗}R_{G_{dν}}^{G_{n}}M^{k}(σ) factors through the quotient C_{σ,ν}, so ∗R_{G_{dν}}^{G_{n}}M^{k}(σ) is a C_{σ,ν}-module in the natural way.

Proof. (i) Let N = M(σ) ⊠ ⋯ ⊠ M(σ) (k times). As in (3.2.2), M^{ν}(σ) ≅ R_{G_{(d^{k})}}^{G_{dν}}N.

So by transitivity of Harish-Chandra induction, we see that M^{k}(σ) ≅ R_{G_{dν}}^{G_{n}}M^{ν}(σ) as F G_{n}-modules. We need to check that the isomorphism is compatible with the right H_{ν}-module structures. We have a canonical embedding of N into M^{ν}(σ) (resp. into M^{k}(σ)) as an F G_{(d^{k})}-module. Take w ∈ Σ_{ν}. Then the action of B_{w} (hence T_{w} ∈ H_{k}) on M^{k}(σ) is by definition the unique F G_{n}-endomorphism of M^{k}(σ) whose restriction to N is the map A_{w} of (2.5.1). Similarly, the action of B_{w} (hence T_{w} ∈ H_{ν}) on M^{ν}(σ) is the unique F G_{dν}-endomorphism of M^{ν}(σ) whose restriction to N is the map A_{w}. So the induced action of T_{w} ∈ H_{ν} on R_{G_{dν}}^{G_{n}}M^{ν}(σ) has the same restriction to N as the action of T_{w} ∈ H_{k} on M^{k}(σ), identifying R_{G_{dν}}^{G_{n}}M^{ν}(σ) with M^{k}(σ). This shows that the two actions of T_{w} coincide.

(ii) Writing N = M(σ) ⊠ ⋯ ⊠ M(σ) and L = G_{(d^{k})}, we have

∗R_{G_{dν}}^{G_{n}} ∘ R_{L}^{G_{n}}(N) ≅ ⨁_{w∈D_{dν,(d^{k})}} R_{G_{dν}∩^{w}L}^{G_{dν}} ∘ conj_{w} ∘^{∗}R_{L∩^{w^{−1}}G_{dν}}^{L}(N).

As N is a cuspidal F L-module the summand corresponding to w is zero unless L ∩^{w^{−1}}G_{dν} = L. In that case, by (2.1c), w = π_{x} for x ∈ D_{ν,(1^{k})} = D_{ν}^{−1}. Now recalling that conj_{π_{x}}(N) ≅ N as an F L-module, we have shown that

∗R_{G_{dν}}^{G_{n}}M^{k}(σ) ≅ ⨁_{x∈D_{ν}^{−1}} R_{L}^{G_{dν}}(N) ≅ M^{ν}(σ)^{⊕|D_{ν}|}

as an F G_{dν}-module. In particular, this shows that the F G_{dν}-action on ^{∗}R_{G_{dν}}^{G_{n}}M^{k}(σ) factors through C_{σ,ν}.

By (i), we can identify M^{k}(σ) and F G_{n} ⊗_{F P} M^{ν}(σ) as (F G_{n}, H_{ν})-bimodules, where P is the standard parabolic subgroup with Levi factor G_{dν}. This allows us to identify M^{ν}(σ) with the (F G_{dν}, H_{ν})-subbimodule 1 ⊗_{F P} M^{ν}(σ) of M^{k}(σ). Multiplication then gives us an (F G_{dν}, H_{k})-bimodule map μ : M^{ν}(σ) ⊗_{H_{ν}} H_{k} → M^{k}(σ) whose image M^{ν}(σ)H_{k} is clearly contained in the fixed point set^{∗}R_{G_{dν}}^{G_{n}}M^{k}(σ). Moreover, the calculation in the preceding paragraph shows that

dim M^{ν}(σ) ⊗_{H_{ν}} H_{k} = dim^{∗}R_{G_{dν}}^{G_{n}}M^{k}(σ).

So it just remains to check that μ is injective.

We have an exact sequence

0 ⟶ Z ⟶ M^{ν}(σ) ⊗_{H_{ν}} H_{k} ⟶^{μ} M^{ν}(σ)H_{k} ⟶ 0

of C_{σ,ν}-modules. Since M^{ν}(σ) ⊗_{H_{ν}} H_{k} ≅ M^{ν}(σ)^{⊕|D_{ν}|} as an F G_{dν}-module, and M^{ν}(σ) is self-dual by Corollary 2.2f, every constituent of the socle of M^{ν}(σ) ⊗_{H_{ν}} H_{k} appears in the head of M^{ν}(σ). Therefore to show that Z = 0, it suffices to show that Hom_{C_{σ,ν}}(M^{ν}(σ), Z) = 0. Applying the exact functor Hom_{C_{σ,ν}}(M^{ν}(σ),?) to the above exact sequence using Lemma 3.1a, we obtain the exact sequence

0 ⟶ Hom_{C_{σ,ν}}(M^{ν}(σ), Z) ⟶ H_{k} ⟶^{μ̄} Hom_{C_{σ,ν}}(M^{ν}(σ), M^{ν}(σ)H_{k}) ⟶ 0.

We just need to check that the map μ̄, which sends h ∈ H_{k} to the map M^{ν}(σ) → M^{ν}(σ)H_{k} given by right multiplication by h, is injective. Suppose for some h ∈ H_{k} that μ̄(h) = 0 so M^{ν}(σ)h = 0. Then, as M^{ν}(σ) generates M^{k}(σ) = R_{G_{dν}}^{G_{n}}M^{ν}(σ) as an F G_{n}-module, we deduce that h annihilates all of M^{k}(σ), whence h = 0.

3.2g. Corollary. The following pairs of functors are isomorphic:

(i) R_{G_{dν}}^{G_{n}} ∘ (M^{ν}(σ)⊗_{H_{ν}}?) and (M^{k}(σ)⊗_{H_{k}}?) ∘ ind_{H_{ν}}^{H_{k}} : mod(H_{ν}) → mod(F G_{n}); (ii) ^{∗}R_{G_{dν}}^{G_{n}} ∘ (M^{k}(σ)⊗_{H_{k}}?) and (M^{ν}(σ)⊗_{H_{ν}}?) ∘ res_{H_{ν}}^{H_{k}} : mod(H_{k}) → mod(F G_{dν}).

Proof. (i) Take N ∈ mod(H_{ν}). Let P denote the standard parabolic subgroup of G_{n} with Levi factor G_{dν}. Using Lemma 3.2f(i) and associativity of tensor product, we have natural isomorphisms

M^{k}(σ) ⊗_{H_{k}} ind_{H_{ν}}^{H_{k}} N = M^{k}(σ) ⊗_{H_{k}} (H_{k} ⊗_{H_{ν}} N) ≅ (M^{k}(σ) ⊗_{H_{k}} H_{k}) ⊗_{H_{ν}} N
≅ M^{k}(σ) ⊗_{H_{ν}} N ≅ (F G_{n} ⊗_{F P} M^{ν}(σ)) ⊗_{H_{ν}} N
≅ F G_{n} ⊗_{F P} (M^{ν}(σ) ⊗_{H_{ν}} N) = R_{G_{dν}}^{G_{n}}(M^{ν}(σ) ⊗_{H_{ν}} N),

as required.

(ii) We first claim that the functors^{∗}R_{G_{dν}}^{G_{n}} ∘(M^{k}(σ)⊗_{H_{k}}?) and (^{∗}R_{G_{dν}}^{G_{n}}M^{k}(σ))⊗_{H_{k}}? are isomorphic. Recall that ^{∗}R_{G_{dν}}^{G_{n}} is defined by taking Y_{dν}-fixed points, so can be viewed as the functor Hom_{F Y_{dν}}(I,?) where I is the trivial representation of Y_{dν}. Now, F Y_{dν} is a semisimple algebra so I is a projective F Y_{dν}-module. So e.g. [AF, 20.10] immediately gives a natural isomorphism

Hom_{F Y_{dν}}(I, M^{k}(σ) ⊗_{H_{k}} N) ≅ Hom_{F Y_{dν}}(I, M^{k}(σ)) ⊗_{H_{k}} N

for any H_{k}-module N, to prove the claim. Hence, using Lemma 3.2f(ii) as well, there are natural isomorphisms

∗R_{G_{dν}}^{G_{n}}(M^{k}(σ) ⊗_{H_{k}} N) ≅ (^{∗}R_{G_{dν}}^{G_{n}}M^{k}(σ)) ⊗_{H_{k}} N ≅ (M^{ν}(σ) ⊗_{H_{ν}} H_{k}) ⊗_{H_{k}} N
≅ M^{ν}(σ) ⊗_{H_{ν}} (H_{k} ⊗_{H_{k}} N) ≅ M^{ν}(σ) ⊗_{H_{ν}} res_{H_{ν}}^{H_{k}} N

for any N ∈ mod(H_{k}). This completes the proof.

3.3. ‘Symmetric’ and ‘exterior’ powers

Now we are ready to introduce modules which play the role of symmetric and exterior powers in the non-defining characteristic theory. As motivation, recall the defining characteristic theory, where V is a finite dimensional vector space over F and Σ_{k} acts on the right on V^{⊗k} by permuting tensors. The symmetric power S^{k}(V) can be defined as the largest quotient of V^{⊗k} on which Σ_{k} acts trivially. The dual notion, the divided power Z^{k}(V), is the largest submodule of V^{⊗k} on which Σ_{k} acts trivially; in positive characteristic, S^{k}(V) and Z^{k}(V) need not be isomorphic as left GL(V)-modules. We define the analogues of symmetric and divided powers in our theory by

S^{k}(σ) = M^{k}(σ)/{mh − E_{H_{k}}(h)m | h ∈ H_{k}, m ∈ M^{k}(σ)},
Z^{k}(σ) = {m ∈ M^{k}(σ) | mh = E_{H_{k}}(h)m for all h ∈ H_{k}}.

§3.3 ‘Symmetric’ and ‘exterior’ powers  59

Observe these are left F G_{n}-modules which factor through the quotient C_{σ,k} to induce well-defined C_{σ,k}-modules.

To define the exterior power in defining characteristic, the definition as the largest quotient (resp. submodule) of V^{⊗r} on which Σ_{k} acts as sign is of course wrong in characteristic 2, so the best definition from the point of view of the symmetric group is Λ^{k}(V) = V^{⊗k}(∑_{w∈Σ_{k}}(−1)^{ℓ(w)}w). We define our analogue, a C_{σ,k}-module again, by

Λ^{k}(σ) = M^{k}(σ)x_{k}.

It is perhaps unfortunate that our “symmetric powers” correspond now to the sign representation and our “exterior power” corresponds to the trivial representation, unlike in the classical case.

As an example, in the special case σ = 1 when k = n, the module Λ^{n}(1) is precisely the trivial F G_{n}-module, whereas Z^{n}(1) is a modular reduction of the Steinberg module (these statements follow as special cases of Theorem 3.5e and Theorem 4.1c).

3.3a. Lemma. Λ^{k}(σ) is an irreducible F G_{n}-module.

Proof. We know that M^{k}(σ) is a projective C_{σ,k}-module and that every composition factor of the socle of M^{k}(σ) appears in its head, by Theorem 3.2d and Lemma 3.2c respectively. Also, the left ideal H_{k}x_{k} is an irreducible H_{k}-module. Using these remarks, the lemma follows at once from the general theory of Schur functors; see Lemma 3.1f and Corollary 3.1e.

The structure of S^{k}(σ) and Z^{k}(σ) is more subtle. Observe though that by definition and (1.1b), M^{k}(σ)y_{k} is a quotient of S^{k}(σ) and Z^{k}(σ) contains M^{k}(σ)y_{k} as a submodule.

3.3b. Lemma. (i) S^{k}(σ) ≅ M^{k}(σ) ⊗_{H_{k}} E_{H_{k}};

(ii) S^{k}(σ) has simple head equal to the irreducible quotient M^{k}(σ)y_{k} of M^{k}(σ), and no other composition factors of S^{k}(σ) are isomorphic to quotients of M^{k}(σ);

(iii) Z^{k}(σ) ≅ S^{k}(σ)^{τ};

(iv) Z^{k}(σ) has simple socle equal to the irreducible submodule M^{k}(σ)y_{k} of M^{k}(σ), and no other composition factors of Z^{k}(σ) are isomorphic to submodules of M^{k}(σ).

Proof. (i) By definition, M^{k}(σ) ⊗_{H_{k}} E_{H_{k}} is the quotient of M^{k}(σ) ⊗_{F} E_{H_{k}} by

{mh ⊗ 1 − E_{H_{k}}(h)m ⊗ 1 | h ∈ H_{k}, m ∈ M^{k}(σ)}.

If we identify M^{k}(σ) ⊗_{F} E_{H_{k}} and M^{k}(σ) as vector spaces in the natural way, this immediately gives (i).

(ii) Let α, β and A_{P} be the functors defined in §3.1, taking the projective module P to be the C_{σ,k}-module M^{k}(σ) of Theorem 3.2d. Then, (i) shows that S^{k}(σ) ≅ β(E_{H_{k}}). Just as in Lemma 3.3a, Lemma 3.1f shows that

A_{P} ∘ β(E_{H_{k}}) ≅ A_{P} (S^{k}(σ)) ≅ M^{k}(σ)y_{k}

is an irreducible C_{σ,k}-module. Hence, M^{k}(σ)y_{k} appears in the head of S^{k}(σ) and no other composition factors of S^{k}(σ) appear in the head of M^{k}(σ). Since S^{k}(σ) is a quotient of M^{k}(σ), this means that S^{k}(σ) actually has simple head.

(iii) Write M = M^{k}(σ) for short. We can by Corollary 2.2f choose some isomorphism i : M → M^{τ} as F G_{n}-modules. This choice induces an isomorphism j : End_{F G_{n}}(M) → End_{F G_{n}}(M^{τ}) with fj(θ) = i((i^{−1}f)θ) for all f ∈ M^{τ} , θ ∈ End_{F G_{n}}(M) (recall we are writing endomorphisms on the right). On the other hand, there is a natural anti-isomorphism # : End_{F G_{n}}(M) → End_{F G_{n}}(M^{τ}) defined simply by letting θ^{#} be the dual map to θ ∈ End_{F G_{n}}(M), that is, (fθ^{#})(m) = f(mθ) for all m ∈ M, f ∈ M^{τ} = M^{∗}. Now if we set τ = j^{−1} ∘ #, we have defined an antiautomorphism of H_{k} = End_{F G_{n}}(M). Define a non-degenerate bilinear form on M by (m, n) = i(m)(n) for m, n ∈ M. For any h ∈ H_{k} we have

(mτ(h), n) = (i^{−1}(i(m)h^{#}), n) = (i(m)h^{#})(n) = i(m)(nh) = (m, nh).

In other words, the bilinear form (., .) is ‘contravariant’ for the action of H_{k} with respect to the anti-automorphism τ.

By definition, S^{k}(σ) = M/J where J = {mh − E_{H_{k}}(h)m | h ∈ H_{k}, m ∈ M}. So:

S^{k}(σ)^{τ} ≅ J^{∘} = {n ∈ M | (n, mh − E_{H_{k}}(h)m) = 0 for all m ∈ M, h ∈ H_{k}}
= {n ∈ M | (nτ(h) − E_{H_{k}}(h)n, m) = 0 for all m ∈ M, h ∈ H_{k}}
= {n ∈ M | nh = E_{H_{k}}(τ(h))n for all h ∈ H_{k}}.

It now just remains to show that E_{H_{k}}(τ(h)) = E_{H_{k}}(h) for all h ∈ H_{k}.

Certainly E_{H_{k}}∘τ is a linear representation of H_{k}, so since the only one dimensional H_{k}-modules are E_{H_{k}} and I_{H_{k}}, we either have that E_{H_{k}} ∘ τ = E_{H_{k}} as required, or that E_{H_{k}} ∘ τ = I_{H_{k}}. In the latter case, we see from (1.1b) that S^{k}(σ)^{τ} contains Mx_{k} as an irreducible submodule, whence that S^{k}(σ) contains Mx_{k} in its head. But this is not so according to (ii) unless in fact Mx_{k} ≅ My_{k}, in which case applying Corollary 3.1e, I_{H_{k}} ≅ E_{H_{k}} and we are done.

(iv) This follows from (ii) on dualizing, using (iii).

Slightly more generally, for ν = (k_{1}, …, k_{a}) ⊨ k, we have analogous left C_{σ,ν}- modules S^{ν}(σ), Z^{ν}(σ) and Λ^{ν}(σ):

S^{ν}(σ) = M^{ν}(σ)/{mh − E_{H_{ν}}(h)m | h ∈ H_{ν}, m ∈ M^{ν}(σ)},  (3.3.1)
Z^{ν}(σ) = {m ∈ M^{ν}(σ) | mh = E_{H_{ν}}(h)m for all h ∈ H_{ν}},  (3.3.2)
Λ^{ν}(σ) = M^{ν}(σ)x_{ν}.  (3.3.3)

Recalling (3.2.2) and (3.2.3), if ν = (k_{1}, …, k_{a}) then S^{ν}(σ) ≅ S^{k_{1}}(σ) ⊠ ⋯ ⊠ S^{k_{a}}(σ), and similarly for Z, Λ. In view of this observation, the basic properties of S^{ν}(σ), Z^{ν}(σ) and Λ^{ν}(σ) follow directly from Lemma 3.3a and Lemma 3.3b.

3.3c. Lemma. For any ν ⊨ k, ^{∗}R_{G_{dν}}^{G_{n}}S^{k}(σ) ≅ S^{ν}(σ) and ^{∗}R_{G_{dν}}^{G_{n}}Z^{k}(σ) ≅ Z^{ν}(σ).

§3.4 Endomorphism algebras  61

Proof. We just need to prove the first statement, the second following directly from Lemma 3.3b(iii) since Harish-Chandra restriction commutes with contravariant duality. For the first, by Lemma 3.3b(i), S^{k}(σ) ≅ M^{k}(σ) ⊗_{H_{k}} E_{H_{k}}. So using Corollary 3.2g(ii), we have immediately that

∗R_{G_{dν}}^{G_{n}}S^{k}(σ) ≅ M^{ν}(σ) ⊗_{H_{ν}} res_{H_{ν}}^{H_{k}} E_{H_{k}} ≅ M^{ν}(σ) ⊗_{H_{ν}} E_{H_{ν}}.

So by Lemma 3.3b(i) (or rather its Levi analogue) we see that ^{∗}R_{G_{dν}}^{G_{n}}S^{k}(σ) ≅ S^{ν}(σ).

We are mainly interested in what follows in the C_{σ,k}-modules obtained from S^{ν}(σ), Z^{ν}(σ) and Λ^{ν}(σ) by Harish-Chandra induction. Set

Ṡ^{ν}(σ) = M^{k}(σ)/{mh − E_{H_{ν}}(h)m | h ∈ H_{ν}, m ∈ M^{k}(σ)},  (3.3.4)
Ż^{ν}(σ) = {m ∈ M^{k}(σ) | mh = E_{H_{ν}}(h)m for all h ∈ H_{ν}},  (3.3.5)
Λ̇^{ν}(σ) = M^{k}(σ)x_{ν}.  (3.3.6)

If we identify M^{k}(σ) with R_{G_{dν}}^{G_{n}}M^{ν}(σ) as (F G_{n}, H_{ν})-bimodules as in Lemma 3.2f(i), it is easy to check that the quotient Ṡ^{ν}(σ) of M^{k}(σ) is identified with the quotient R_{G_{dν}}^{G_{n}}S^{ν}(σ) of R_{G_{dν}}^{G_{n}}M^{ν}(σ). Similarly, we can identify R_{G_{dν}}^{G_{n}}Z^{ν}(σ) with Ż^{ν}(σ) and R_{G_{dν}}^{G_{n}}Λ^{ν}(σ) with Λ̇^{ν}(σ).

Note by (1.1b) again that Ż^{ν}(σ) contains M^{k}(σ)y_{ν} as a submodule. Recall the definitions of the H_{k}-modules M^{ν} = H_{k}x_{ν} and N^{ν} = H_{k}y_{ν} from §1.1. The next result generalizes Lemma 3.3b:

3.3d. Lemma. For any ν ⊨ k, we have:

(i) Ṡ^{ν}(σ) ≅ M^{k}(σ) ⊗_{H_{k}} N^{ν}.

(ii) Ż^{ν}(σ) ≅ Ṡ^{ν}(σ)^{τ}.

(iii) No composition factors of Ż^{ν}(σ)/M^{k}(σ)y_{ν} are isomorphic to submodules of M^{k}(σ).

Proof. (i) Since Ṡ^{ν}(σ) ≅ R_{G_{dν}}^{G_{n}}S^{ν}(σ), this is immediate from Corollary 3.2g(i) and the Levi analogue of Lemma 3.3b(i).

(ii) This follows from the Levi analogue of Lemma 3.3b(iii), since Harish-Chandra induction commutes with contravariant duality.

(iii) Let α, β and A_{P} be the functors defined in §3.1 taking the projective module P to be the C_{σ,k}-module M^{k}(σ) of Theorem 3.2d. Now, N^{ν} is the left ideal H_{k}y_{ν} of H_{k}. So, using Lemma 3.2c and Lemma 3.1f, we know that

A_{P} ∘ β(N^{ν}) ≅ M^{k}(σ)y_{ν}.

So using (i) and the definition of the functor A_{P} ∘ β, we see that Ṡ^{ν}(σ) ≅ β(N^{ν}) is an extension of M^{k}(σ)y_{ν} and a module having no composition factors in common with the head (or equivalently by Lemma 3.2c the socle) of M^{k}(σ). Now (iii) follows on dualizing using (ii).

3.4. Endomorphism algebras

Now fix in addition an integer h ≥ 1 and let S_{h,k} denote the q^{d}-Schur algebra S_{F,q^{d}}(h, k) of §1.2. Our first connection between G_{n} and the q^{d}-Schur algebra arises as follows (cf. [DJ_{3}, Theorem 2.24(iii)]):

3.4a. Theorem. There is an algebra isomorphism

S_{h,k} ⟶^{∼} End_{C_{σ,k}} ( ⨁_{ν∈Λ(h,k)} Λ̇^{ν}(σ))

under which the natural basis element ϕ_{μ,λ}^{u} of S_{h,k} maps to the endomorphism which is zero on the summand Λ̇^{ν}(σ) for ν ≠ μ and sends Λ̇^{μ}(σ) into Λ̇^{λ}(σ) via the homomorphism induced by right multiplication in M^{k}(σ) by ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{μ}^{−1}} T_{w}.

Proof. Let A_{P} ∘ β denote the equivalence of categories from Theorem 3.1d, for the projective C_{σ,k}-module P = M^{k}(σ). According to Lemma 3.1f and Lemma 3.2c,

⨁_{ν∈Λ(h,k)} Λ̇^{ν}(σ) ≅ A_{P} ∘ β ( ⨁_{ν∈Λ(h,k)} M^{ν}) .

We deduce that the endomorphism algebras of ⨁_{ν∈Λ(h,k)} Λ̇^{ν}(σ) and ⨁_{ν∈Λ(h,k)} M^{ν} are isomorphic; the latter is S_{h,k} by definition. It remains to check that the image of ϕ_{μ,λ}^{u} under the functor A_{P} ∘ β is precisely the endomorphism described, which follows using Lemma 3.1f.

Recalling from (1.2c) that S_{h,k} can also be described as the endomorphism algebra End_{H_{k}} (⨁_{λ∈Λ(h,k)} N^{λ}) , the same argument as the proof of Theorem 3.4a shows (cf. [DJ_{3}, Theorem 2.24(iv)]):

(3.4b) There is an algebra isomorphism S_{h,k} ⟶^{∼} End_{C_{σ,k}} (⨁_{ν∈Λ(h,k)} M^{k}(σ)y_{ν}) under which the natural basis element ϕ_{μ,λ}^{u} of S_{h,k} maps to the endomorphism which is zero on the summand M^{k}(σ)y_{ν} for ν ≠ μ and sends M^{k}(σ)y_{μ} into M^{k}(σ)y_{λ} via the homomorphism induced by right multiplication in M^{k}(σ) by ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{μ}^{−1}} T_{w}^{#}.

The next theorem gives the second important connection between G_{n} and the q^{d}-Schur algebra (cf. [DJ_{3}, Theorem 2.24(vi)]):

3.4c. Theorem. There is an algebra isomorphism

S_{h,k} ⟶^{∼} End_{C_{σ,k}} ( ⨁_{ν∈Λ(h,k)} Ż^{ν}(σ))

under which the natural basis element ϕ_{μ,λ}^{u} of S_{h,k} maps to the endomorphism which is zero on the summand Ż^{ν}(σ) for ν ≠ μ and sends Ż^{μ}(σ) into Ż^{λ}(σ) via the homomorphism induced by right multiplication in M^{k}(σ) by ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{μ}^{−1}} T_{w}^{#}.

Proof. Let us first check that the endomorphisms in the statement are well-defined. We need to observe that as submodules of M^{k}(σ), Ż^{μ}(σ)h ⊆ Ż^{λ}(σ) where

h = ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{μ}^{−1}} T_{w}^{#}.

To prove this, it suffices by (3.3.5) to show that T_{s} − 1 annihilates Ż^{μ}(σ)h for all basic transpositions s ∈ Σ_{λ}. Right multiplication by h(T_{s} − 1) gives an F G_{n}- module homomorphism from Ż^{μ}(σ) to M^{k}(σ). Obviously, h(T_{s} − 1) annihilates the submodule M^{k}(σ)y_{μ} of Ż^{μ}(σ) by (1.1b) and (1.2.2). So in fact, h(T_{s} − 1) must annihilate all of Ż^{μ}(σ) by Lemma 3.3d(iii).

Now let S be the subalgebra of End_{F G_{n}} (⨁_{ν∈Λ(h,k)} Ż^{ν}(σ)) consisting of all endomorphisms which stabilize the subspace ⨁_{ν∈Λ(h,k)} M_{k}(σ)y_{ν}. Restriction gives an algebra homomorphism

S → End_{F G_{n}} ( ⨁_{ν∈Λ(h,k)} M^{k}(σ)y_{ν})

which is injective by Lemma 3.3d(iii) and surjective by the previous paragraph and (3.4b). This shows in particular that the endomorphisms of ⨁_{ν∈Λ(h,k)} Ż^{ν}(σ) defined in the statement of the theorem are linearly independent and span S. It remains to check using dimension that S equals all of End_{F G_{n}} (⨁_{ν∈Λ(h,k)} Ż^{ν}(σ)). On expanding the direct sums, this will follow if we can show that

dim Hom_{F G_{n}}(Ż^{μ}(σ), Ż^{λ}(σ)) = dim Hom_{H_{k}}(N^{λ}, N^{μ})

for all λ, μ ∈ Λ(h, k). Let α and β be the functors defined in (3.1.1) and (3.1.2) for P = M^{k}(σ). We calculate using Lemma 3.3d and Lemma 3.1a:

Hom_{F G_{n}}(Ż^{μ}(σ), Ż^{λ}(σ)) ≅ Hom_{F G_{n}}(Ṡ^{λ}(σ), Ṡ^{μ}(σ))
≅ Hom_{F G_{n}}(β(N^{λ}), β(N^{μ}))
≅ Hom_{H_{k}}(N^{λ}, α ∘ β(N^{μ})) ≅ Hom_{H_{k}}(N^{λ}, N^{μ}).

The result follows.

To proceed further, we need to utilize the properties of the Gelfand-Graev representation from §2.5. Recall the idempotent γ_{n} ∈ F G_{n} from (2.5.5) and its Levi analogue γ_{dν} ∈ F G_{dν} for ν = (k_{1}, …, k_{a}) ⊨ k. Define

Y^{k}(σ) = F G_{n}γ_{n}M^{k}(σ),
Y^{ν}(σ) = F G_{dν}γ_{dν}M^{ν}(σ).

Note these definitions are only temporary: at the end of the section, we will see that Y^{k}(σ) = Z^{k}(σ). It is obvious that Y^{ν}(σ) ≅ Y^{k_{1}}(σ) ⊠ ⋯ ⊠ Y^{k_{a}}(σ), so this will also give that Y^{ν}(σ) = Z^{ν}(σ).

Recall for the next lemma that by definition, Z^{k}(σ) (resp. Z^{ν}(σ)) is a submodule of M^{k}(σ) (resp. M^{ν}(σ)).

3.4d. Lemma. For ν ⊨ k,

(i) Y^{ν}(σ) is the image of any non-zero element of the one dimensional space Hom_{F G_{dν}}(Γ_{dν}, Z^{ν}(σ));

(ii) ^{∗}R_{G_{dν}}^{G_{n}}Y^{k}(σ) ≅ Y^{ν}(σ).

Proof. (i) According to Corollary 2.5e, the space Hom_{F G_{n}}(Γ_{n}, M^{k}(σ)) is one dimensional and the image of a non-zero such homomorphism lies in Z^{k}(σ). So, by Frobenius reciprocity and the fact that γ_{n} is idempotent, Y^{k}(σ) is precisely the image of any non-zero element of the one dimensional space Hom_{F G_{n}}(Γ_{n}, Z^{k}(σ)). Generalizing to the Levi analogue in the obvious way gives (i).

(ii) We first observe that ^{∗}R_{G_{dν}}^{G_{n}}Y^{k}(σ) is non-zero. For this, note by (i) that Y^{k}(σ) is non-zero and a submodule of M^{k}(σ), so Hom_{F G_{n}}(Y^{k}(σ), M^{k}(σ)) ≠ 0. Since Hom_{F G_{dν}}(^{∗}R_{G_{dν}}^{G_{n}}Y^{k}(σ), M^{ν}(σ)) ≅ Hom_{F G_{n}}(Y^{k}(σ), M^{k}(σ)), ^{∗}R_{G_{dν}}^{G_{n}}Y^{k}(σ) must therefore also be non-zero.

Now let θ : Γ_{n} → Z^{k}(σ) be a non-zero homomorphism, with image Y^{k}(σ) by (i). Applying the exact functor ^{∗}R_{G_{dν}}^{G_{n}}, recalling that ^{∗}R_{G_{dν}}^{G_{n}}Γ_{n} ≅ Γ_{dν} and ^{∗}R_{G_{dν}}^{G_{n}}Z^{k}(σ) ≅ Z^{ν}(σ) by Corollary 2.5e(i) and Lemma 3.3c, we obtain a homomorphism θ̄: Γ_{dν} → Z^{ν}(σ) with image ^{∗}R_{G_{dν}}^{G_{n}}Y^{k}(σ). But θ̄ is non-zero, and by (i) again, the image of such a non-zero homomorphism is precisely Y^{ν}(σ).

3.4e. Lemma. Suppose that we are given λ, μ ⊨ k and an F G_{dμ}-submodule M of M^{μ}(σ). If w ∈ D_{dλ,dμ} is such that ^{∗}R_{G_{dμ}∩^{w^{−1}}G_{dλ}}^{G_{dμ}}(M) ≠ 0, then w is of the form π_{x} for some x ∈ D_{λ,μ}.

Proof. By exactness of Harish-Chandra restriction, it suffices to prove this in the special case M = M^{μ}(σ). As usual, write L = G_{(d^{k})} and N = M(σ) ⊠ ⋯ ⊠ M(σ). Define ν ⊨ n by G_{ν} = G_{dμ} ∩^{w^{−1}}G_{dλ}. Then a Mackey calculation gives that

∗R_{G_{ν}}^{G_{dμ}}M ≅ ⨁_{y∈D_{ν,(d^{k})}^{dμ}} R_{G_{ν}∩^{y}L}^{G_{ν}} ∘ conj_{y} ∘^{∗}R_{L∩^{y^{−1}}G_{ν}}^{L}(N).

Since this is non-zero and N is cuspidal, there must be some y ∈ D_{ν,(d^{k})}^{dμ} such that

L = L ∩^{y^{−1}}G_{ν}, in which case ^{y}L = G_{ν} ∩^{y}L is a standard Levi subgroup of G_{n} conjugate to L. But the only such Levi subgroup is L itself, so in fact ^{y}L = L and we have that L ⊆ G_{ν} = G_{dμ} ∩^{w^{−1}}G_{dλ}. We deduce from (2.1c) that w^{−1} = π_{x} for some x ∈ D_{μ,λ}.

The next Mackey calculation is of central importance.

3.4f. Lemma. For any λ, μ ⊨ k, all of the spaces

Hom_{F G_{n}}(R_{G_{dλ}}^{G_{n}}Y^{λ}(σ), R_{G_{dμ}}^{G_{n}}Y^{μ}(σ)),  (3.4.1)
Hom_{F G_{n}}(R_{G_{dλ}}^{G_{n}}Γ_{dλ}, R_{G_{dμ}}^{G_{n}}Y^{μ}(σ)),  (3.4.2)
Hom_{F G_{n}}(R_{G_{dλ}}^{G_{n}}Γ_{dλ}, R_{G_{dμ}}^{G_{n}}Z^{μ}(σ))  (3.4.3)

have the same dimension, namely, |D_{λ,μ}|.

Proof. Consider (3.4.1). By the Mackey theorem and Lemma 3.4e, we have that

Hom_{F G_{n}}(R_{G_{dλ}}^{G_{n}}Y^{λ}(σ), R_{G_{dμ}}^{G_{n}}Y^{μ}(σ)) ≅ ⨁_{w∈D_{dλ,dμ}} H_{w} ≅ ⨁_{x∈D_{λ,μ}} H_{π_{x}},

where for w ∈ Σ_{n},

H_{w} = Hom_{F G_{dλ}}(Y^{λ}(σ), R_{G_{dλ}∩^{w}G_{dμ}}^{G_{dλ}} ∘ conj_{w} ∘^{∗}R_{G_{dμ}∩^{w^{−1}}G_{dλ}}^{G_{dμ}}(Y^{μ}(σ))).

So to prove the result for (3.4.1), we need to show that each H_{π_{x}} is one dimensional. Fix x ∈ D_{λ,μ}. Applying (2.1c), G_{dλ} ∩^{π_{x}} G_{dμ} = G_{dν} and G_{dμ} ∩^{π_{x}^{−1}} G_{dλ} = G_{dν′} for some ν, ν′ ⊨ k. Using Lemma 3.4d(ii) and adjointness,

dim H_{π_{x}} = dim Hom_{F G_{dν}}(Y^{ν}(σ), conj_{π_{x}} Y^{ν′}(σ)).

Conjugating the standard Levi subgroup G_{dν′} by π_{x} simply rearranges the diagonal blocks to obtain G_{dν}. Since Y^{ν′}(σ) is an outer tensor product over these diagonal blocks, we see that conj_{π_{x}} Y^{ν′}(σ) ≅ Y^{ν}(σ). Finally, Y^{ν}(σ) is a quotient of Γ_{dν} and a submodule of Z^{ν}(σ) so Lemma 3.4d(i) implies that dim H_{π_{x}} = dim Hom_{F G_{dν}}(Y^{ν}(σ), Y^{ν}(σ)) is one dimensional.

This proves the lemma for (3.4.1), and the proofs for (3.4.2) and (3.4.3) are entirely similar, using instead that ^{∗}R_{G_{dλ}}^{G_{n}}Z^{k}(σ) ≅ Z^{λ}(σ) and ^{∗}R_{G_{dλ}}^{G_{n}}Γ_{n} ≅ Γ_{dλ} according to Lemma 3.3c and Corollary 2.5e(i) respectively.

Now we can prove what we regard as the fundamental theorem:

3.4g. Theorem. Maintaining assumptions (A1) and (A2) from §3.2, Ż^{ν}(σ) is a projective C_{σ,k}-module, for all ν ⊨ k. Moreover for any h ≥ k, ⨁_{ν∈Λ(h,k)} Ż^{ν}(σ) is a projective generator for mod(C_{σ,k}).

Proof. Fix some h ≥ k and set

Z = ⨁_{ν∈Λ(h,k)} R_{G_{dν}}^{G_{n}}Z^{ν}(σ) ≅ ⨁_{ν∈Λ(h,k)} Ż^{ν}(σ),
Y = ⨁_{ν∈Λ(h,k)} R_{G_{dν}}^{G_{n}}Y^{ν}(σ),
Q = ⨁_{ν∈Λ(h,k)} R_{G_{dν}}^{G_{n}}Γ_{dν}.

As Y^{ν}(σ) is a non-zero submodule of Z^{ν}(σ), it contains the simple socle M^{ν}(σ)y_{ν} of Z^{ν}(σ) as a submodule (see Lemma 3.3b). Applying R_{G_{dν}}^{G_{n}} to the inclusions M^{ν}(σ)y_{ν} ⊆ Y^{ν}(σ) ⊆ Z^{ν}(σ), we deduce that

M^{k}(σ)y_{ν} ⊆ R_{G_{dν}}^{G_{n}}Y^{ν}(σ) ⊆ R_{G_{dν}}^{G_{n}}Z^{ν}(σ)

as naturally embedded submodules of M^{k}(σ). So, ⨁_{ν∈Λ(h,k)} M^{k}(σ)y_{ν} ⊆ Y ⊆ Z. Also observe that Y is a quotient of Q, since each Y^{ν}(σ) is a quotient of Γ_{dν}.

It follows easily from Lemma 3.4f (on expanding the direct sums) that all of

Hom_{F G_{n}}(Y, Y), Hom_{F G_{n}}(Q, Y), Hom_{F G_{n}}(Q, Z)

have the same dimension, namely, ∑_{λ,μ∈Λ(h,k)} |D_{μ,λ}| which is precisely the dimension of S_{h,k} by (1.2a). Since Q is projective it contains the projective cover P of Y as a summand. Now the equality dim Hom_{F G_{n}}(Y, Y) = dim Hom_{F G_{n}}(Q, Y) implies that dim Hom_{F G_{n}}(Y, Y) = dim Hom_{F G_{n}}(P, Y). This verifies the condition in Lemma 3.2a, showing that Y is a projective F G_{n}/ ann_{F G_{n}}(Y)-module.

Now we compute End_{F G_{n}}(Y). The fact that Hom_{F G_{n}}(Q, Y) and Hom_{F G_{n}}(Q, Z) have the same dimension implies that every F G_{n}-homomorphism from Q to Z has image lying in Y. So since Y is certainly a quotient of Q, we can describe Y alternatively as the subspace of Z spanned by the images of all F G_{n}-homomorphisms from Q to Z. This alternative description makes it clear that Y is stable under all F G_{n}-endomorphisms of Z. So, restriction gives a well-defined map

End_{F G_{n}}(Z) → End_{F G_{n}}(Y).

It is injective since we know from Theorem 3.4c and (3.4b) that the homomorphism End_{F G_{n}}(Z) → End_{F G_{n}}(⨁_{ν∈Λ(h,k)} M^{k}(σ)y_{ν}) induced by restriction is injective. Since End_{F G_{n}}(Z) ≅ S_{h,k} and End_{F G_{n}}(Y) has the same dimension as S_{h,k}, we deduce that End_{F G_{n}}(Y) ≅ S_{h,k}.

For h ≥ k, the algebra S_{h,k} has Par_{k} non-isomorphic irreducible modules. Combining the previous two paragraphs and Fitting’s lemma, we deduce that Y has precisely Par_{k} non-isomorphic irreducible modules appearing in its head. Since Y is a direct sum of submodules of M^{k}(σ), assumption (A2) now gives that every irreducible constituent of M^{k}(σ) appears in the head of Y. Hence every irreducible constituent of M^{k}(σ) appears in the head of the projective F G_{n}-module Q. Now we know that every homomorphism from Q to Z has image lying in Y, while every composition factor of Z/Y appears in the head of the projective module Q. This shows Z = Y.

Then, observe that M^{k}(σ) is a summand of Y = Z, so

ann_{F G_{n}}(Y) = ann_{F G_{n}}(M^{k}(σ)).

In other words, F G_{n}/ ann_{F G_{n}}(Y) = C_{σ,k}. We have already shown that Y is a projective F G_{n}/ ann_{F G_{n}}(Y)-module, which means that Z and all its summands are projective C_{σ,k}-modules. Taking h large enough, this shows in particular that R_{G_{dν}}^{G_{n}}Z^{ν}(σ) ≅ Ż^{ν}(σ) is projective for each ν ⊨ k.

It remains to show that every irreducible C_{σ,k}-module appears in the head of Z. As M^{k}(σ) is a faithful C_{σ,k}-module, every irreducible C_{σ,k}-module appears as some composition factor of M^{k}(σ). Now we know that a copy of every composition factor of M^{k}(σ) does appear in the head of Z.

§3.5 Standard modules  67

We also record at this point the following fact obtained in the proof of the theorem:

(3.4h) The submodules Z^{k}(σ) and Y^{k}(σ) of M^{k}(σ) coincide. So Z^{k}(σ) can be characterized as the image of any non-zero homomorphism from Γ_{n} to M^{k}(σ).

In view of (3.4h), we will not need the notation Y^{k}(σ) again.

3.5. Standard modules

Choose an integer h ≥ k and let S_{h,k} = S_{F,q^{d}}(h, k). Let Z = ⨁_{ν∈Λ(h,k)} Ż^{ν}(σ). We always regard Z as a (C_{σ,k}, S_{h,k})-bimodule, with S_{h,k} acting as in Theorem 3.4c. Define the functors

α_{σ,h,k} : mod(C_{σ,k}) → mod(S_{h,k}),  α_{σ,h,k} = Hom_{C_{σ,k}}(Z,?),  (3.5.1)
β_{σ,h,k} : mod(S_{h,k}) → mod(C_{σ,k}),  β_{σ,h,k} = Z⊗_{S_{h,k}}?.  (3.5.2)

Because of Theorem 3.4c, Theorem 3.4g and our standing assumptions (A1) and (A2) on σ, Z is a projective generator for mod(C_{σ,k}) with endomorphism algebra S_{h,k}, so:

(3.5a) The functors α_{σ,h,k} and β_{σ,h,k} are mutually inverse equivalences of categories.

Recall now the basic facts about the representation theory of S_{h,k} described in §1.2. In particular, S_{h,k} is a quasi-hereditary algebra with weight poset Λ^{+}(h, k) partially ordered by ≤. Also, for λ ∈ Λ^{+}(h, k), we have the S_{h,k}-modules L_{h}(λ), Δ_{h}(λ) and ∇_{h}(λ). For λ ⊢ k, we can regard its transpose λ′ as an element of Λ^{+}(h, k), since h ≥ k. Define the C_{σ,k}-modules (hence F G_{n}-modules, inflating in the usual way):

L(σ, λ) = β_{σ,h,k}(L_{h}(λ′)),  (3.5.3)
Δ(σ, λ) = β_{σ,h,k}(Δ_{h}(λ′)),  (3.5.4)
∇(σ, λ) = β_{σ,h,k}(∇_{h}(λ′)),  (3.5.5)

for any partition λ ⊢ k. For example, as we shall see shortly, if σ = 1 then L(1, (n)) = Δ(1, (n)) = ∇(1, (n)) is the trivial F G_{n}-module, while the (not necessarily irreducible) modules Δ(1, (1^{n})) and ∇(1, (1^{n})) are modular reductions of the Steinberg module.

Since β_{σ,h,k} is a Morita equivalence, we see at once that the algebra C_{σ,k} is a quasi-hereditary algebra with weight poset {λ ⊢ k} partially ordered by ≥ (the opposite order to S_{h,k} since we have transposed partitions). Moreover, {L(σ, λ)}, {Δ(σ, λ)} and {∇(σ, λ)} for all λ ⊢ k give the irreducible, standard and costandard C_{σ,k}-modules. Recall also that F G_{n} possesses the anti-automorphism τ, induced by taking transpose matrices. Since M^{k}(σ)^{τ} ≅ M^{k}(σ) by Corollary 2.2f, τ factors to induce an anti-automorphism of the quotient C_{σ,k} = F G_{n}/ ann_{F G_{n}}(M^{k}(σ)). So we also have a notion of contravariant duality on mod(C_{σ,k}), and moreover, as this is true even as F G_{n}-modules, L(σ, λ)^{τ} ≅ L(σ, λ) for each λ ⊢ k. The following basic facts now follow immediately from the Morita equivalence or from standard properties of quasi-hereditary algebras (cf. (1.2d)):

(3.5b) (i) Δ(σ, λ) has simple head isomorphic to L(σ, λ), and all other composition factors are of the form L(σ, μ) for μ > λ.

(ii) For λ, μ ⊢ k, [Δ(σ, λ) : L(σ, μ)] = [Δ_{h}(λ′) : L_{h}(μ′)].

(iii) For λ ⊢ k, L(σ, λ)^{τ} ≅ L(σ, λ) and Δ(σ, λ)^{τ} ≅ ∇(σ, λ).

We pause to explain why the definitions (3.5.3)–(3.5.5) are independent of the particular choice of h ≥ k. Take h ≥ l ≥ k and, as explained in §1.5, identify S_{l,k} with the subring eS_{h,k}e of S_{h,k}, where e = e_{h,l} is the idempotent of (1.5.1). Recall the equivalence of categories infl_{S_{l,k}}^{S_{h,k}} : mod(S_{l,k}) → mod(S_{h,k}) from (1.5.3) and (1.5a).

3.5c. Lemma. The functors

β_{σ,h,k} ∘ infl_{S_{l,k}}^{S_{h,k}} : mod(S_{l,k}) → mod(C_{σ,k}) and β_{σ,l,k} : mod(S_{l,k}) → mod(C_{σ,k})

are isomorphic.

Proof. The module ⨁_{λ∈Λ(l,k)} Ż^{λ}(σ) is precisely the (C_{σ,k}, S_{l,k})-subbimodule Ze of Z. So, the functor β_{σ,l,k} is by definition the functor Ze⊗_{eS_{h,k}e}?. Now associativity of tensor product gives the isomorphism

Z ⊗_{S_{h,k}} (S_{h,k}e ⊗_{eS_{h,k}e} M) ≅ (Z ⊗_{S_{h,k}} S_{h,k}e) ⊗_{eS_{h,k}e} M ≅ Ze ⊗_{eS_{h,k}e} M

for any M ∈ mod(S_{l,k}). The isomorphism is clearly functorial.

Now by (1.5a) and (1.5b), L_{h}(λ′) ≅ infl_{S_{l,k}}^{S_{h,k}} L_{l}(λ′). So Lemma 3.5c shows immediately that β_{σ,l,k}(L_{l}(λ′)) ≅ β_{σ,h,k}(L_{h}(λ′)) as C_{σ,k}-modules. Hence, the definition (3.5.3) is independent of the choice of h, and the same argument gives independence of h for (3.5.4) and (3.5.5).

The next goal is to give two alternative definitions of the standard module Δ(σ, λ) without reference to the Schur algebra. Since L(σ, λ) is the simple head of Δ(σ, λ), this also gives a more implicit realization of irreducibles. First, recalling the definitions (1.3.3) and (1.3.4), we have the following lemma which motivates our choice of notation:

3.5d. Lemma. For ν ⊨ k, Ż^{ν}(σ) ≅ β_{σ,h,k}(Z^{ν}(V_{h})) and Λ̇^{ν}(σ) ≅ β_{σ,h,k}(Λ^{ν}(V_{h})).

Proof. By (1.3b)(i), Z^{ν}(V_{h}) is isomorphic to the left ideal S_{h,k}ϕ_{ν,ν}^{1} of S_{h,k}. By Lemma 3.1f, β_{σ,h,k}(S_{h,k}ϕ_{ν,ν}^{1}) ≅ Zϕ_{ν,ν}^{1}, which is precisely the summand Ż^{ν}(σ) of Z by the definition of the action of ϕ_{ν,ν}^{1} from Theorem 3.4c.

Similarly, by (1.3b)(ii), Λ^{ν}(V_{h}) ≅ S_{h,k}κ(y_{ν}), so β_{σ,h,k}(S_{h,k}κ(y_{ν})) ≅ Zκ(y_{ν}). By definition of the embedding κ from (1.2b), together with (1.1c) and Theorem 3.4c, Zκ(y_{ν}) = M^{k}(σ)y_{ν}^{#} = M^{k}(σ)x_{ν} = Λ̇^{ν}(σ) as required.

Now we obtain the desired characterizations of Δ(σ, λ). Recall the definition of u_{λ} ∈ Σ_{k} from (1.1d).

3.5e. Theorem. For λ ⊢ k,

(i) the space Hom_{C_{σ,k}}(Ż^{λ′}(σ), Λ̇^{λ}(σ)) is one dimensional, and the image of any non-zero such homomorphism is isomorphic to Δ(σ, λ);

(ii) Δ(σ, λ) is isomorphic to the submodule Ż^{λ′}(σ)T_{u_{λ}}x_{λ} of M^{k}(σ).

Proof. (i) This is immediate from Lemma 3.5d, the definition (3.5.4) and (1.3d), since β_{σ,h,k} is an equivalence of categories.

(ii) First observe that Ż^{λ′}(σ)T_{u_{λ}}x_{λ} is both a homomorphic image of Ż^{λ′}(σ) and a submodule of Λ^{λ}(σ). So in view of (i), the result will follow once we show that Ż^{λ′}(σ)T_{u_{λ}}x_{λ} is non-zero. Well, Ż^{λ′}(σ) contains M^{k}(σ)y_{λ′} as a submodule. Moreover, M^{k}(σ)y_{λ′}T_{u_{λ}}x_{λ} is non-zero, as M^{k}(σ) is a faithful H_{k}-module and y_{λ′}T_{u_{λ}}x_{λ} ≠ 0 by (1.1d).

3.5f. Remark. In [J_{2}, Definition 7.7], James defines right F G_{n}-modules S(σ, λ) for each λ ⊢ k. In view of (3.4h), Theorem 3.5e(ii) is a left module analogue of James’ definition.

As we did in §1.2, we will write M̃ for the right C_{σ,k}-module obtained from M ∈ mod(C_{σ,k}) by twisting the left action into a right action using τ. In this way, we obtain right C_{σ,k}-modules Δ̃(σ, λ), L̃(σ, λ) and ˜∇(σ, λ). Now we conclude the section with some extensions of (1.2e):

3.5g. Theorem. (i) C_{σ,k} has a filtration as a (C_{σ,k}, C_{σ,k})-bimodule with factors isomorphic to Δ(σ, λ) ⊗ Δ̃(σ, λ), each appearing precisely once for each λ ⊢ k and ordered in any way refining the dominance order on partitions so that factors corresponding to most dominant λ appear at the top of the filtration.

(ii) Z = ⨁_{ν∈Λ(h,k)} Ż^{ν}(σ) has a filtration as a (C_{σ,k}, S_{h,k})-bimodule with factors Δ(σ, λ) ⊗ Δ̃_{h}(λ′) appearing precisely once for each λ ⊢ k and ordered in any way refining the dominance order so that factors corresponding to most dominant λ appear at the top of the filtration.

Proof. (i) This follows immediately from the general theory of quasi-hereditary algebras, in the same way as explained after (1.2e).

(ii) The functor Z⊗_{S_{h,k}}? can also be viewed as an exact functor from the category of (S_{h,k}, S_{h,k})-bimodules to the category of (C_{σ,k}, S_{h,k})-bimodules. Clearly,

Z ⊗_{S_{h,k}} (Δ_{h}(λ) ⊗ Δ̃_{h}(λ)) ≅ (Z ⊗_{S_{h,k}} Δ_{h}(λ)) ⊗ Δ̃_{h}(λ) ≅ Δ(σ, λ′) ⊗ Δ̃_{h}(λ).

So now applying Z⊗_{S_{h,k}}? to the filtration of (1.2e) gives the result.

Further connections and applications

In this chapter, we prove a number of results that supplement the main Morita theorem of the previous chapter. In particular, we make precise the idea that tensor products in the quantum linear group correspond to Harish-Chandra induction under the Morita equivalence, and extend the Morita theorem both to the ground ring O and to p-singular elements σ.

4.1. Base change

Let σ ∈ 𝔽̄_{q}^{×} be of degree d over 𝔽_{q} satisfying the conditions (A1) and (A2) from §3.2. Write n = kd for some k ≥ 1 and let R denote one of the rings F, K or O. Recall that we have defined the (RG_{n}, H_{k,R})-bimodule M^{k}(σ)_{R} in (2.4.2), with M^{k}(σ) = M^{k}(σ)_{F}. Define

Z^{k}(σ)_{R} = {m ∈ M^{k}(σ)_{R} | mh = E_{H_{k,R}}(h)m for all h ∈ H_{k,R}},

so our original module Z^{k}(σ) from (3.3.2) is precisely Z^{k}(σ)_{F}.

4.1a. Lemma. Z^{k}(σ)_{O} is an O-free O-module of finite rank, with

K ⊗_{O} Z^{k}(σ)_{O} ≅ Z^{k}(σ)_{K},
F ⊗_{O} Z^{k}(σ)_{O} ≅ Z^{k}(σ)_{F} .

Proof. Recalling that M^{k}(σ)_{O} is an O-lattice in M^{k}(σ)_{K}, we have by definition that

Z^{k}(σ)_{O} = Z^{k}(σ)_{K} ∩ M^{k}(σ)_{O}.

It is immediate from this that Z^{k}(σ)_{O} is an O-lattice in Z^{k}(σ)_{K}, and also that it is a pure submodule of M^{k}(σ)_{O} (see [La, 17.1(i)]). Consequently, the natural map i : F ⊗_{O} Z^{k}(σ)_{O} → M^{k}(σ)_{F} induced by the embedding Z^{k}(σ)_{O} ↪ M^{k}(σ)_{O} is also an embedding. Clearly, since the action of H_{k} on M^{k}(σ) is compatible with base change, the image of i is contained in Z^{k}(σ)_{F}.

Now for R = F, K or O, let Y^{k}(σ)_{R} denote the submodule of M^{k}(σ)_{R} spanned by the images of all RG_{n}-homomorphisms from Γ_{n,R} to Z^{k}(σ)_{R}. Since Γ_{n} is projective, any F G_{n}-homomorphism from Γ_{n} to M^{k}(σ) is induced by base change from some OG_{n}-homomorphism from Γ_{n,O} to M^{k}(σ)_{O}. It follows directly that the natural (not necessarily injective) map from F ⊗_{O} Y^{k}(σ)_{O} to M^{k}(σ)_{F} has image Y^{k}(σ)_{F}. Now by Corollary 2.5e(iii), Y^{k}(σ)_{O} ⊆ Z^{k}(σ)_{O}, so we deduce that the image of i contains Y^{k}(σ)_{F}.

Finally, we appeal to (3.4h), where we observed that Y^{k}(σ)_{F} = Z^{k}(σ)_{F}. So the two previous paragraphs show in fact that i : F ⊗_{O} Z^{k}(σ)_{O} → Z^{k}(σ)_{F} is an isomorphism.

For ν ⊨ k, let H_{ν,R} denote the parabolic subalgebra of H_{k,R} with sign representation E_{H_{ν,R}}. Define

Ż^{ν}(σ)_{R} = {m ∈ M^{k}(σ)_{R} | mh = E_{ℌ_{ν,R}}(h)m for all h ∈ H_{ν,R}},  (4.1.1)

so our original module Ż^{ν}(σ) from (3.3.5) is precisely Ż^{ν}(σ)_{F}. For ν = (k_{1}, …, k_{a}), Ż^{ν}(σ) ≅ R_{G_{dν}}^{G_{n}}Z_{k_{1}}(σ)⊠⋯⊠Z_{k_{a}}(σ) (cf. the comments after (3.3.6)). So since Harish-Chandra induction commutes with base change, we deduce from Lemma 4.1a that:

(4.1b) Ż^{ν}(σ)_{O} is an O-free O-module of finite rank, with

K ⊗_{O} Ż^{ν}(σ)_{O} ≅ Ż^{ν}(σ)_{K},
F ⊗_{O} Ż^{ν}(σ)_{O} ≅ Ż^{ν}(σ)_{F} .

Now we can define standard modules over O. For R = F, K or O and λ ⊨ k, set

Δ(σ, λ)_{R} = Ż^{λ′}(σ)_{R}T_{u_{λ}}x_{λ}.

By Theorem 3.5e, Δ(σ, λ)_{F} ≅ Δ(σ, λ).

4.1c. Theorem. Δ(σ, λ)_{O} is O-free of finite rank with

K ⊗_{O} Δ(σ, λ)_{O} ≅ Δ(σ, λ)_{K},
F ⊗_{O} Δ(σ, λ)_{O} ≅ Δ(σ, λ)_{F} .

Moreover, the character of the KG_{n}-module Δ(σ, λ)_{K} is precisely χ_{σ,λ}, so Δ(σ, λ) is the reduction modulo p of a KG_{n}-module affording the character χ_{σ,λ}.

Proof. First observe that Δ(σ, λ)_{O} is torsion free as it is a submodule of the torsion free O-module M^{k}(σ)_{O}. There is a natural map K ⊗_{O} Δ(σ, λ)_{O} → M^{k}(σ)_{K}, which is injective as K is flat over O. One easily checks that its image is precisely

§4.1 Base change  73

Δ(σ, λ)_{K}, hence proving that Δ(σ, λ)_{O} is an O-lattice in Δ(σ, λ)_{K}. Hence, in particular, Δ(σ, λ)_{O} has rank equal to dim Δ_{k}(σ, λ)_{K}. Now applying Theorem 3.5g(ii) (or rather, its easier analogue over K), taking h = k for definiteness, we see that

∑_{ν∈Λ(k,k)} dim Ż^{ν}(σ)_{K} = ∑_{λ⊢k}(dim Δ(σ, λ)_{K})(dim Δ_{k}(λ′)_{K})

where Δ_{k}(λ′)_{K} denotes the (irreducible) standard module for the Schur algebra algebra S_{K,q^{d}}(k, k) over K. Using (4.1b), dim Ż^{ν}(σ)_{K} = dim Ż^{ν}(σ)_{F}, while it is well-known that the dimension of standard modules for the Schur algebra do not depend on the ground field. So, we see that

∑_{ν∈Λ(k,k)} dim Ż^{ν}(σ)_{F} = ∑_{λ⊢k}(dim Δ(σ, λ)_{K})(dim Δ_{k}(λ′)_{F}).

Now again, there is a natural map i : F ⊗_{O} Δ(σ, λ)_{O} → M^{k}(σ)_{O} induced by the embedding Δ(σ, λ)_{O} ↪ M^{k}(σ)_{O}, with image Δ(σ, λ)_{F}. This shows that dim Δ(σ, λ)_{K} ≥ dim Δ(σ, λ)_{F}. On the other hand, applying Theorem 3.5g(ii) over F, we have that

∑_{ν∈Λ(k,k)} dim Ż^{ν}(σ)_{F} = ∑_{λ⊢k}(dim Δ(σ, λ)_{F})(dim Δ_{k}(λ′)_{F}).

Comparing with our previous expression, we see that dim Δ(σ, λ)_{K} must actually equal dim Δ(σ, λ)_{F} for all λ ⊢ k, hence that i is injective.

It remains to explain why Δ(σ, λ)_{K} has character χ_{σ,λ}. Write λ = (l_{1}, …, l_{a}) and λ′ = (l′_{1}, …, l′_{b}). First observe by (1.1e) and (2.3f) that χ_{σ,λ} is the unique irreducible character that is a constituent of both of the characters R_{G_{dλ}}^{G_{n}}(χ_{σ,(l_{1})} … χ_{σ,(l_{a})}) and R_{G_{dλ′}}^{G_{n}} (χ_{σ,(1^{l′_{1}})} … χ_{σ,(1^{l′_{a}})}). So recalling Lemma 2.5c, the modules

M^{k}(σ)_{K}x_{λ} ≅ R_{G_{dλ}}^{G_{n}}(M^{λ}(σ)_{K}x_{λ}),
M^{k}(σ)_{K}y_{λ′} ≅ R_{G_{dλ′}}^{G_{n}} (M^{λ′}(σ)_{K}y_{λ′})

have a unique irreducible composition factor in common, with character χ_{σ,λ}. But in characteristic zero, Ż^{λ′}(σ)_{K} = M^{k}(σ)_{K}y_{λ′} as then y_{λ′} is an idempotent (up to a non-zero scalar). So by definition, Δ(σ, λ)_{K} = M^{k}(σ)_{K}y_{λ′}T_{u_{λ}}x_{λ} so is an irreducible quotient of M^{k}(σ)_{K}y_{λ′} and an irreducible submodule of M^{k}(σ)_{K}x_{λ}. So Δ(σ, λ)_{K} does indeed have character χ_{σ,λ}.

Using Theorem 4.1c, we can extend all our earlier results to the ground ring O. For R = F, K or O, set

C_{σ,k,R} = C_{R,(σ)^{k}}(GL_{n}(𝔽_{q})) = RG_{n}/ ann_{RG_{n}}(M^{k}(σ)_{R}).

So, C_{σ,k,F} is precisely the algebra C_{σ,k} as defined in (3.2.1). We call C_{σ,k,R} the cuspidal algebra over R. Note that if σ is p-regular, C_{σ,k,R} is in fact a quotient of the block algebra B_{σ,k,R} introduced at the end of §2.4.

4.1d. Theorem. Maintaining the assumptions (A1) and (A2) on σ from §3.2,

(i) C_{σ,k,O} is O-free of finite rank with K ⊗_{O} C_{σ,k,O} ≅ C_{σ,k,K} and F ⊗_{O} C_{σ,k,O} ≅ C_{σ,k,F};

(ii) for each ν ⊨ k, Ż^{ν}(σ)_{O} is a projective C_{σ,k,O}-module;

(iii) the endomorphism algebra End_{C_{σ,k,O}}(⨁_{ν∈Λ(h,k)} Ż^{ν}(σ)_{O}) is isomorphic to the q^{d}-Schur algebra S_{O,q^{d}}(h, k);

(iv) for h ≥ k, ⨁_{ν∈Λ(h,k)} Ż^{ν}(σ)_{O} is a projective generator for mod(C_{σ,k,O}), so C_{σ,k,O} is Morita equivalent to S_{O,q^{d}}(h, k).

Proof. (i) By definition, C_{σ,k,R} is the R-submodule of End_{R}(M^{k}(σ)_{R}) spanned by the images of the elements of G_{n}. So it is finitely generated, C_{σ,k,O} is contained in C_{σ,k,K} and spans C_{σ,k,K} over K. This shows that C_{σ,k,O} is an O-lattice in C_{σ,k,K}. Tensoring the inclusion C_{σ,k,O} ↪ End_{O}(M^{k}(σ)_{O}) with F, we obtain a natural map F ⊗_{O} C_{σ,k,O} → F ⊗_{O} End_{O}(M^{k}(σ)_{O}) ≅ End_{F} (M^{k}(σ)_{F}) whose image is clearly C_{σ,k,F}. To show that this surjection is injective, we need to check that dim C_{σ,k,K} = dim C_{σ,k,F}. But by Theorem 3.5g(i), its analogue over K and Theorem 4.1c:

dim C_{σ,k,F} = ∑_{λ⊢k}(dim Δ(σ, λ)_{F})^{2} = ∑_{λ⊢k}(dim Δ(σ, λ)_{K})^{2} = dim C_{σ,k,K}.

(ii) As Ż^{ν}(σ)_{F} is a projective C_{σ,k,F} -module from Theorem 3.4g, we see from (i) and lifting idempotents (e.g. see [La, 14.4]) that it has a unique lift to a projective C_{σ,k,O}-module. This must be Ż^{ν}(σ)_{O} thanks to (4.1b).

(iii) Write E_{R} for End_{C_{σ,k,R}} (⨁_{ν∈Λ(h,k)} Ż^{ν}(σ)_{R}). We know by Theorem 3.4c (or its analogue over K) that E_{F} ≅ S_{F,q^{d}}(h, k) and E_{K} ≅ S_{K,q^{d}}(h, k). Moreover, E_{O} is an O-lattice in E_{K} and there is a natural embedding F ⊗_{O} E_{O} ↪ E_{F} which is an isomorphism by dimension. So we can identify K ⊗_{O} E_{O} with E_{K} and F ⊗_{O} E_{O} with E_{F}

Now, the basis element ϕ_{μ,λ}^{u} of E_{K} ≅ S_{K,q^{d}}(h, k) acts as zero on all summands except Ż^{μ}(σ)_{K} where it is induced by right multiplication by h = ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{μ}^{−1}} T_{w}^{#}. By (4.1.1), Ż^{ν}(σ)_{O} = Ż^{ν}(σ)_{K} ∩ M^{k}(σ)_{O}. Also, h lies in H_{O,q^{d}}(Σ_{k}) so stabilizes M^{k}(σ)_{O}. Hence, Ż^{μ}(σ)_{O}h ⊆ Ż^{λ}(σ)_{O}, so each ϕ_{μ,λ}^{u} ∈ E_{K} restricts to give a welldefined element of E_{O}. We have constructed an isomorphic copy S_{O} of S_{O,q^{d}}(h, k) in E_{O}, namely, the O-span of the standard basis elements ϕ_{μ,λ}^{u} ∈ S_{K,q^{d}}(h, k).

It remains to show that S_{O} = E_{O}. We have a short exact sequence 0 → S_{O} → E_{O} → Q_{O} → 0 of O-modules. To prove that Q_{O} = 0, it suffices to show that F ⊗_{O} Q_{O} = 0. Tensoring with F, we have an exact sequence

F ⊗_{O} S_{O} ⟶^{i} E_{F} ⟶ F ⊗_{O} Q_{O} ⟶ 0.

Now, the map i sends 1 ⊗ ϕ_{μ,λ}^{u} to the corresponding endomorphism ϕ_{μ,λ}^{u} defined as in Theorem 3.4c. Hence, i is injective so an isomorphism by dimension. We deduce that F ⊗_{O} Q_{O} = 0 to complete the proof.

§4.2 Connecting Harish-Chandra induction with tensor products 75

(iv) We have seen in (ii) that ⨁_{ν∈Λ(h,k)} Ż^{ν}(σ)_{O} is a projective C_{σ,k,O}-module. It is a generator because this is so on tensoring with F, using (i) and Theorem 3.4g. The statement about Morita equivalence follows directly from this and (iii).

4.1e. Remark. We remark that the fundamental theorem of Cline, Parshall and Scott [CPS_{3}, §9] follows easily at this point from Theorem 4.1d and the results of §2.4. We refer the reader to loc. cit. for the precise statement, as well as for applications to cohomology of F G_{n}.

4.2. Connecting Harish-Chandra induction with tensor products

Let σ ∈ 𝔽̄_{q}^{×} be of degree d over 𝔽_{q} satisfying the conditions (A1) and (A2) from §3.2. For any h ≥ k ≥ 1, S_{h,k} denotes the algebra S_{F,q^{d}}(h, k) and β_{σ,h,k} is the functor of (3.5.2), but regarded now as a functor from mod(S_{h,k}) to mod(F G_{kd}) via the evident full embedding mod(C_{σ,h,k}) ↪ mod(F G_{kd}). Now fix integers h ≥ k ≥ 1, a composition ν = (k_{1}, …, k_{a}) ⊨ k and set n = kd.

The main result of the section relates tensor products in the quantum linear group to Harish-Chandra induction in the finite linear group, as follows:

4.2a. Theorem. The following functors are isomorphic:

R_{G_{dν}}^{G_{n}}(β_{σ,h,k_{1}} ? ⊠ ⋯ ⊠ β_{σ,h,k_{a}} ? ) : mod(S_{h,k_{1}}) × ⋯ × mod(S_{h,k_{a}}) → mod(F G_{n}),
β_{σ,h,k}( ? ⊗ ⋯ ⊗? ) : mod(S_{h,k_{1}}) × ⋯ × mod(S_{h,k_{a}}) → mod(F G_{n}).

Proof. Choose μ = (h_{1}, …, h_{a}) ⊨ h with each h_{i} ≥ k_{i}. Let S_{μ,ν} denote the algebra S_{h_{1},k_{1}} ⊗ ⋯ ⊗ S_{h_{a},k_{a}} for short. Write Λ(μ, ν) for the set of all compositions γ = (g_{1}, …, g_{h}) ∈ Λ(h, k) such that, defining γ_{1} = (g_{1}, …, g_{h_{1}}), γ_{2} = (g_{h_{1}+1}, …, g_{h_{1}+h_{2}}), …, γ_{a} = (g_{h_{1}+···+h_{a−1}+1}, …, g_{h}), we have that γ_{i} ∈ Λ(h_{i}, k_{i}) for each i = 1, …, a. Consider the set of triples:

Ω = {(γ, δ, u) | γ, δ ∈ Λ(μ, ν), u ∈ D_{γ,δ}^{ν}}.

For a triple (γ, δ, u) ∈ Ω, so that γ_{i}, δ_{i} ∈ Λ(h_{i}, k_{i}) for each i = 1, …, a, we have that u = (u_{1}, …, u_{a}) ∈ Σ_{ν} = Σ_{k_{1}} × ⋯ × Σ_{k_{a}} with each u_{i} ∈ D_{γ_{i},δ_{i}}; so we can associate the element

ϕ̄_{γ,δ}^{u} = ϕ_{γ_{1},δ_{1}}^{u_{1}} ⊗ ⋯ ⊗ ϕ_{γ_{a},δ_{a}}^{u_{a}}

of S_{μ,ν}. The set {ϕ̄_{γ,δ}^{u} | (γ, δ, u) ∈ Ω} gives a natural basis for the algebra S_{μ,ν}. Now define

Z = ⨁_{λ∈Λ(h,k)} R_{G_{dλ}}^{G_{n}}Z^{λ}(σ) ≅ ⨁_{λ∈Λ(h,k)} Ż^{λ}(σ),

Z_{ν} = ⨁_{λ∈Λ(μ,ν)} R_{G_{dλ}}^{G_{dν}}Z^{λ}(σ) ≅ [ ⨁_{λ_{1}∈Λ(h_{1},k_{1})}Ż^{λ_{1}}(σ)] ⊠ ⋯ ⊠ [ ⨁_{λ_{a}∈Λ(h_{a},k_{a})}Ż^{λ_{a}}(σ)] .

We regard Z as an (F G_{n}, S_{h,k})-bimodule in the usual way, and also view Z_{ν} as an (F G_{dν}, S_{μ,ν})-bimodule, where the action of S_{μ,ν} = S_{h_{1},k_{1}} ⊗ ⋯ ⊗ S_{h_{a},k_{a}} on Z_{ν} is as described in Theorem 3.4c for each term in the above outer tensor product.

Then, R_{G_{dν}}^{G_{n}}Z_{ν} is an (F G_{n}, S_{μ,ν})-bimodule in a natural way. Moreover, by transitivity of Harish-Chandra induction,

R_{G_{dν}}^{G_{n}}Z_{ν} ≅ ⨁_{λ∈Λ(μ,ν)} Ż^{λ}(σ),

so R_{G_{dν}}^{G_{n}}Z_{ν} can be identified with the summand Ze_{ν} of the (F G_{n}, S_{h,k})-bimodule Z, where e_{ν} is the idempotent

e_{ν} = ∑_{λ∈Λ(μ,ν)} ϕ_{λ,λ}^{1} ∈ S_{h,k}.

Identifying e_{ν}S_{h,k}e_{ν} with End_{F G_{n}}(Ze_{ν}), we obtain an algebra embedding of S_{μ,ν} into e_{ν}S_{h,k}e_{ν}. By definition of the actions of S_{μ,ν} and e_{ν}S_{h,k}e_{ν} on Ze_{ν} and Lemma 3.2f(i), this embedding maps the basis element ϕ̄_{γ,δ}^{u} of S_{μ,ν} to ϕ_{γ,δ}^{u} ∈ e_{ν}S_{h,k}e_{ν}, for all (γ, δ, u) ∈ Ω. In other words:

(4.2b) Identifying S_{μ,ν} with a subalgebra of e_{ν}S_{h,k}e_{ν} via the map ϕ̄_{γ,δ}^{u} ↦ ϕ_{γ,δ}^{u}, the (F G_{n}, S_{μ,ν})-bimodule R_{G_{dν}}^{G_{n}}Z_{ν} is isomorphic to Ze_{ν}, regarding the latter as an (F G_{n}, S_{μ,ν})-bimodule by restricting the natural action of e_{ν}S_{h,k}e_{ν} to S_{μ,ν}.

Now let S_{μ,k} denote the Levi subalgebra of S_{h,k} as in (1.3g). Recalling the decomposition (1.3.8), the idempotent e_{ν} ∈ S_{h,k} is precisely the central idempotent of S_{μ,k} such that e_{ν}S_{μ,k}e_{ν} is isomorphic to S_{μ,ν}. So in fact, the embedding of S_{μ,ν} into e_{ν}S_{h,k}e_{ν} from (4.2b) identifies S_{μ,ν} with the summand e_{ν}S_{μ,k}e_{ν} of S_{μ,k}. Making this identification, define the functor

I : mod(S_{μ,ν}) → mod(S_{h,k}), I = S_{h,k}e_{ν}⊗_{e_{ν}S_{μ,k}e_{ν}}?.

Using associativity of tensor product, the functor I can be thought of as the composite of the natural inflation functor S_{μ,k}e_{ν}⊗_{e_{ν}S_{μ,k}e_{ν}}? : mod(S_{μ,ν}) → mod(S_{μ,k}) followed by ordinary induction ind_{S_{μ,k}}^{S_{h,k}} : mod(S_{μ,k}) → mod(S_{h,k}) as defined in §1.5. In view of this, the following fundamental fact follows immediately from Theorem 1.5d:

(4.2c) The following functors are isomorphic:

I( ? ⊠ ⋯ ⊠? ) : mod(S_{h_{1},k_{1}}) × ⋯ × mod(S_{h_{a},k_{a}}) → mod(S_{h,k}),
infl_{S_{h_{1},k_{1}}}^{S_{h,k_{1}}} ? ⊗ ⋯ ⊗ infl_{S_{ha,ka}}^{S_{h,ka}} ? : mod(S_{h_{1},k_{1}}) × ⋯ × mod(S_{h_{a},k_{a}}) → mod(S_{h,k}).

We next claim that the diagram

mod(S_{μ,ν}) ⟶^{Z_{ν}⊗_{Sμ,ν}?} mod(F G_{dν})
I↓  ↓^{R_{G_{dν}}^{Gn}}
mod(S_{h,k}) ⟶_{Z⊗_{S_{h,k}}?} mod(F G_{n})

(4.2.1)

commutes, i.e. that the following Schur algebra analogue of Corollary 3.2g(i) holds:

(4.2d) The functors R_{G_{dν}}^{G_{n}} ∘ Z_{ν}⊗_{S_{μ,ν}}? and β_{σ,h,k} ∘ I : mod(S_{μ,ν}) → mod(F G_{n}) are isomorphic.

To prove (4.2d), (4.2b) and associativity of tensor product gives the natural isomorphisms

R_{G_{dν}}^{G_{n}}(Z_{ν} ⊗_{S_{μ,ν}} N) ≅ (R_{G_{dν}}^{G_{n}}Z_{ν}) ⊗_{S_{μ,ν}} N ≅ Ze_{ν} ⊗_{S_{μ,ν}} N = Ze_{ν} ⊗_{e_{ν}S_{μ,k}e_{ν}} N
≅ Z ⊗_{S_{h,k}} S_{h,k}e_{ν} ⊗_{e_{ν}S_{μ,k}e_{ν}} N = β_{σ,h,k} ∘ I(N)

for any N ∈ mod(S_{μ,ν}).

Now, we obviously have the isomorphism of functors:

β_{σ,h_{1},k_{1}} ? ⊠ ⋯ ⊠ β_{σ,h_{a},k_{a}} ? ≅ Z_{ν} ⊗_{S_{μ,ν}} ( ? ⊠ ⋯ ⊠? )

as functors from mod(S_{h_{1},k_{1}}) × ⋯ × mod(S_{h_{a},k_{a}}) to mod(F G_{dν}). In view of this, (4.2c) and (4.2d), we deduce:

(4.2e) There is an isomorphism

R_{G_{dν}}^{G_{n}}(β_{σ,h_{1},k_{1}} ? ⊠ ⋯ ⊠ β_{σ,h_{a},k_{a}} ? ) ≅ β_{σ,h,k}(infl_{S_{h_{1},k_{1}}}^{S_{h,k_{1}}} ? ⊗ ⋯ ⊗ infl_{S_{ha,ka}}^{S_{h,ka}} ? )

as functors from mod(S_{h_{1},k_{1}}) × ⋯ × mod(S_{h_{a},k_{a}}) to mod(F G_{n}).

Finally, by (1.5a), the functor infl_{S_{h_{i},k_{i}}}^{S_{h,k_{i}}} : mod(S_{h_{i},k_{i}}) → mod(S_{h,k_{i}}) is an equivalence of categories, for each i = 1, …, a. By Lemma 3.5c, the functors β_{σ,h,k_{i}} ∘ infl_{S_{h_{i},k_{i}}}^{S_{h,k_{i}}} and β_{σ,h_{i},k_{i}} are isomorphic. The theorem follows on combining these statements and (4.2e).

We now give an immediate application of Theorem 4.2a. Say that a C_{σ,k}-module M has a Δ-filtration (resp. a ∇-filtration) if it has a filtration

0 = M_{0} < M_{1} < ⋯ < M_{b} = M

such that for each i = 1, …, b, the factor M_{i}/M_{i−1} is isomorphic to Δ(σ, λ) (resp. ∇(σ, λ)) for some partition λ ⊢ k (depending on i). Analogously, we say that a C_{σ,ν}-module M has a Δ-filtration (resp. a ∇-filtration) if it has a filtration

0 = M_{0} < M_{1} < ⋯ < M_{b} = M

such that for each i = 1, …, b, the factor M_{i}/M_{i−1} is isomorphic to Δ(σ, λ_{1}) ⊠ ⋯ ⊠ Δ(σ, λ_{a}) (resp. ∇(σ, λ_{1}) ⊠ ⋯ ⊠ ∇(σ, λ_{a})) for some partitions λ_{1} ⊢ k_{1}, …, λ_{a} ⊢ k_{a}.

4.2f. Theorem. (i) The functor R_{G_{dν}}^{G_{n}} sends C_{σ,ν}-modules with Δ-filtrations (resp. ∇-filtrations) to C_{σ,k}-modules with Δ-filtrations (resp. ∇-filtrations).

(ii) The functor ^{∗}R_{G_{dν}}^{G_{n}} sends C_{σ,k}-modules with Δ-filtrations (resp. ∇-filtrations) to C_{σ,ν}-modules with Δ-filtrations (resp. ∇-filtrations).

Proof. (i) We first need to observe that the functor R_{G_{dν}}^{G_{n}} sends C_{σ,ν}-modules to C_{σ,k}-modules. It suffices to check this on projective C_{σ,ν}-modules. In turn, since according to the Levi analogue of Theorem 3.4g every projective C_{σ,ν}-module is a submodule of M^{ν}(σ), we just need to check that R_{G_{dν}}^{G_{n}}M^{ν}(σ) is a C_{σ,k}-module. But this is clear since by transitivity of Harish-Chandra induction, R_{G_{dν}}^{G_{n}}M^{ν}(σ) ≅ M^{k}(σ). So it makes sense to regard the functor R_{G_{dν}}^{G_{n}} as a functor from mod(C_{σ,ν}) to mod(C_{σ,k}).

Now to show that R_{G_{dν}}^{G_{n}} sends modules with Δ-filtrations to modules with Δfiltrations, we just need to check by exactness that

R_{G_{dν}}^{G_{n}}(Δ(σ, λ_{1}) ⊠ ⋯ ⊠ Δ(σ, λ_{a}))

has a Δ-filtration, for any λ_{1} ⊢ k_{1}, …, λ_{a} ⊢ k_{a}. According to Theorem 4.2a,

R_{G_{dν}}^{G_{n}}(Δ(σ, λ_{1}) ⊠ ⋯ ⊠ Δ(σ, λ_{a})) ≅ β_{σ,h,k}(Δ_{h}(λ′_{1}) ⊗ ⋯ ⊗ Δ_{h}(λ′_{a})).

So the result follows since Δ_{h}(λ′_{1})⊗⋯⊗Δ_{h}(λ′_{a}) has a Δ-filtration as an S_{h,k}-module thanks to (1.3c).

This proves (i) in the case of Δ-filtrations, and the result for ∇-filtrations follows immediately on taking duals, since contravariant duality commutes with Harish-Chandra induction.

(ii) Again we first check that ^{∗}R_{G_{dν}}^{G_{n}} sends C_{σ,k}-modules to C_{σ,ν}-modules. Making the same reductions as before, we need to observe that^{∗}R_{G_{dν}}^{G_{n}}M^{k}(σ) is a C_{σ,ν}-module, which we checked in Lemma 3.2f. So it makes sense to regard the functor ^{∗}R_{G_{dν}}^{G_{n}} as a functor from mod(C_{σ,k}) to mod(C_{σ,ν}).

Now we prove (ii) in the case of ∇-filtrations; the analogous result for Δfiltrations follows on dualizing as before. So take N ∈ mod(C_{σ,k}) with a ∇-filtration. Using the cohomological criterion for ∇-filtrations [Do_{4}, A2.2(iii)], we need to show that

Ext_{C_{σ,ν}}^{1}(M,^{∗}R_{G_{dν}}^{G_{n}}N) = 0

for all M ∈ mod(C_{σ,ν}) with a Δ-filtration. So take such a module M. By (i) and the cohomological criterion for ∇-filtrations, we know that

Ext_{C_{σ,k}}^{1}(R_{G_{dν}}^{G_{n}}M, N) = 0.

So the result will follow if we can prove:

(4.2g) For all M ∈ mod(C_{σ,ν}), N ∈ mod(C_{σ,k}) and i ≥ 0,

Ext_{C_{σ,ν}}^{i}(M,^{∗}R_{G_{dν}}^{G_{n}}N) ≅ Ext_{C_{σ,k}}^{i}(R_{G_{dν}}^{G_{n}}M, N).

§4.3 p-Singular classes  79

Fix M ∈ mod(C_{σ,ν}). The adjoint functor property gives us an isomorphism of functors Hom_{C_{σ,ν}}(M,?) ∘^{∗}R_{G_{dν}}^{G_{n}} ≅ Hom_{C_{σ,k}}(R_{G_{dν}}^{G_{n}}M,?). Since ^{∗}R_{G_{dν}}^{G_{n}} is exact and sends injectives to injectives (being adjoint to the exact functor R_{G_{dν}}^{G_{n}}), a standard degree shifting argument (cf. [Ja, I.4.1(3)]) now gives (4.2g).

4.2h. Corollary. Take λ ⊢ k and partitions λ_{1} ⊢ k_{1}, …, λ_{a} ⊢ k_{a}. Then, both of (i) the multiplicity of Δ(σ, λ) in a Δ-filtration of R_{G_{dν}}^{G_{n}}Δ(σ, λ_{1}) ⊠ ⋯ ⊠ Δ(σ, λ_{a}), (ii) the multiplicity of Δ(σ, λ_{1}) ⊠ ⋯ ⊠ Δ(σ, λ_{a}) in a Δ-filtration of ^{∗}R_{G_{dν}}^{G_{n}}Δ(σ, λ),

are the same as in characteristic zero, i.e. are given by the Littlewood-Richardson rule.

Proof. The fact that the modules in (i) and (ii) have Δ-filtrations follows from Theorem 4.2f. That the multiplicities are the same as in characteristic zero follows in case (i) from the analogous well-known fact about tensor product multiplicities over the q^{d}-Schur algebra, using Theorem 4.2a. The conclusion in case (ii) follows from (i) and adjointness, together with the usual properties of Δ- and ∇-filtrations.

4.3. p-Singular classes

Throughout the section fix a p-singular element τ ∈ 𝔽̄_{q}^{×} of degree e over 𝔽_{q}. Let σ be the p-regular part of τ, of degree d over 𝔽_{q}. By (2.1a), e = dm where m = ℓ(d)p^{r} for some r ≥ 0. Choose l ≥ 1, set k = lm and let n = kd = le.

Fixing an integer h ≥ k, let S_{h,k} (resp. S̄_{h,l}) denote the q^{d}-Schur algebra S_{F,q^{d}}(h, k) (resp. the q^{e}-Schur algebra S_{F,q^{e}}(h, l)). We observe that the image of q^{e} in F is 1, so in fact S̄_{h,l} is just the classical Schur algebra S_{F,1}(h, l). Also let A_{h} (resp. Ā_{h}) denote the quantized coordinate ring A_{F,v^{d}}(h) (resp. A_{F,1}(h)), choosing a square root v^{d} of q^{d} in F in the same way as in §1.3. Let V_{h} (resp. V̄_{h}) denote the natural h-dimensional A_{h}-comodule (resp. the natural Ā_{h}-comodule).

Recall that for a right Ā_{h}-comodule M, we have defined its rth Frobenius twist M^{[r]}, which is a right A_{h}-comodule, by inflation along the bialgebra map F_{r} : Ā_{h} → A_{h} of (1.3.5). If M is an S̄_{h,l}-module, then the Frobenius twist M^{[r]} is an S_{h,k}- module. Since F_{r} is a bialgebra map, the operation of taking Frobenius twists commutes with tensor products.

Now, σ is p-regular, so the theory of the previous sections holds for σ, thanks to Lemma 2.4c and Lemma 2.5f. Let β_{σ,h,k} : mod(S_{h,k}) → mod(C_{σ,k}) denote the equivalence of categories of (3.5.2). Our first goal is to extend our theory to the p-singular element τ.

4.3a. Lemma. The Brauer character of L(σ, (l^{m})) agrees with the restriction of χ_{τ,(l)} to p-regular classes of G_{n}.

Proof. Recall that β_{σ,h,k}(L_{h}(m^{l})) = L(σ, (l^{m})) by (3.5.3). Applying the Morita equivalence β_{σ,h,k} to Lemma 1.3f, using Theorem 4.1c, we see that the Brauer character of L(σ, (l^{m})) is equal to the restriction of the generalized character

\frac{(−1)^{k+l}}{l!} ∑_{λ⊢l} ∑_{μ⊢k} c_{λ}ϕ_{μ}(mλ)χ_{σ,μ}

to p-regular classes. So the lemma follows from Lemma 2.3c.

4.3b. Theorem. For any l ≥ 1, there is an isomorphism of F G_{n}-modules:

M^{l}(τ) ≅ β_{σ,h,k}(\underbrace{V̄_{h}^{[r]} ⊗ ⋯ ⊗ V̄_{h}^{[r]}}_{l times}).

Hence, M(τ) is an irreducible F G_{e}-module and, for any l ≥ 1, M^{l}(τ) has precisely Par_{l} non-isomorphic composition factors.

Proof. Consider first the case that l = 1. By Lemma 4.3a, M(τ) has the same Brauer character as L(σ, (1^{m})). Since the latter is an irreducible module, this implies immediately that M(τ) ≅ L(σ, (1^{m})), so that M(τ) is also irreducible. Since L(σ, (1^{m})) = β_{σ,h,k}(L_{h}((m))) = β_{σ,h,k}(V̄_{h}^{[r]}), this proves the result in the case l = 1.

Now for l > 1, the isomorphism M^{l}(τ) ≅ β_{σ,h,k}(V̄_{h}^{[r]} ⊗ ⋯ ⊗ V̄_{h}^{[r]}) is immediate using Theorem 4.2a and the definition of M^{l}(τ).

Finally, to see that M^{l}(τ) has precisely Par_{l} non-isomorphic composition factors, observe that

\underbrace{V̄_{h}^{[r]} ⊗ ⋯ ⊗ V̄_{h}^{[r]}}_{l times} ≅ (\underbrace{V̄_{h} ⊗ ⋯ ⊗ V̄_{h}}_{l times})^{[r]}.

This has precisely Par_{l} non-isomorphic composition factors as an S_{h,k}-module, as the untwisted tensor space V̄_{h}^{⊗l} has precisely Par_{l} non-isomorphic composition factors as an S̄_{h,l}-module.

Theorem 4.3b (combined with Lemma 2.4c and Lemma 2.5f) verifies the assumptions (A1) and (A2) from §3.2 for every σ ∈ 𝔽̄_{q}. So the main Morita theorem of (3.5a), and the subsequent results obtained so far, are true for general σ, hence for our fixed p-singular element τ. In particular, we associate to τ the (p-singular) cuspidal algebra C_{τ,l}, defined as in (3.2.1). By the analogue of (3.5a), there is an equivalence of categories

β_{τ,h,l} : mod(S̄_{h,l}) → mod(C_{τ,l}),

defined as in (3.5.2). We have now recovered the main results from the paper of James [J_{2}] with our approach.

4.3c. Remarks. (i) One can also consider an analogue of the cuspidal algebra for an arbitrary block-diagonal element s ∈ G_{n}, namely, the quotient algebra

C_{s} = C_{F,s}(GL_{n}(𝔽_{q})) = F G_{n}/ ann_{F G_{n}}(M(s))

where M(s) is as in (2.4.3). If s is p-regular, the Cline-Parshall-Scott theorem mentioned in Remark 4.1e gives an analogue of the Morita theorem (even over O) for the algebra C_{s}: the algebra C_{s} is Morita equivalent to ⨂_{i=1}^{a} S_{F,q^{d_{i}}} (h_{i}, k_{i}) for integers h_{i} ≥ k_{i}, where s is of the form (2.1.1). The same is true for somewhat more general s, namely, if s is “reduction stable” (see [DF] or [D_{4}, 4.26]), but certainly false in general.

(ii) Theorem 4.3b also yields an alternative proof of [D_{2}, 3.5]: for an arbitrary block-diagonal element s ∈ G_{n}, the associated cuspidal G_{δ(s)}-module of (2.4.4) remains irreducible modulo p, since it is an outer tensor product of factors all of which remain irreducible.

We now associate to τ the C_{τ,l}- (hence F G_{n}-) modules L(τ, λ), Δ(τ, λ) and ∇(τ, λ) for each λ ⊢ l, defined as in (3.5.3)–(3.5.5) but using the functor β_{τ,h,l} instead. The next result realizes these modules alternatively as modules in the category mod(C_{σ,k}) (for (iv), see also [DJ_{2}, [Lemma 2.3]).

4.3d. Theorem. For any ν ⊨ l and λ ⊢ l, we have the following F G_{n}-module isomorphisms:

(i) Λ̇^{ν}(τ) ≅ β_{σ,h,k}(Λ^{ν}(V̄_{h})^{[r]});

(ii) Ż^{ν}(τ) ≅ β_{σ,h,k}(Z^{ν}(V̄_{h})^{[r]});

(iii) Δ(τ, λ) ≅ β_{σ,h,k}(Δ̄_{l}(λ′)^{[r]});

(iv) L(τ, λ) ≅ β_{σ,h,k}(L̄_{l}(λ′)^{[r]}) ≅ L(σ, (mλ′)′).

Proof. (i) Since Λ̇^{ν}(τ) ≅ R_{G_{eν}}^{G_{n}}Λ^{ν}(τ), it suffices applying Theorem 4.2a to prove this in the special case ν = (l). Recall (e.g. by Theorem 3.5e) that Λ^{l}(τ) is isomorphic to the module Δ(τ, (l)). So by Theorem 4.1c, the Brauer character of Λ^{l}(τ) is equal to the restriction of χ_{τ,(l)} to p-regular classes, which by Lemma 4.3a is the same as the Brauer character of L(σ, (l^{m})). Since the latter is an irreducible module, we deduce that

Λ^{l}(τ) ≅ L(σ, (l^{m})) = β_{σ,h,k}(L_{h}((m^{l}))) ≅ β_{σ,h,k}(L̄_{h}((1^{l}))^{[r]}) ≅ β_{σ,h,k}(Λ^{l}(V̄_{h})^{[r]}).

(ii) Again, we just need to prove this in the special case ν = (l). So, we need to prove that

Z^{l}(τ) ≅ β_{σ,h,k}(Z^{l}(V̄_{h})^{[r]}).

We first observe that Z^{l}(V̄_{h})^{[r]} is a quotient of Z^{k}(V_{h}) as S_{h,k}-modules. This follows from the universal property of standard modules [PW, (8.10.2)]: Z^{l}(V̄_{h})^{[r]} ≅ Δ̄_{h}((l))^{[r]} is generated by a highest weight vector of weight (k) so is a quotient of the universal highest weight module Z^{k}(V_{h}) ≅ Δ_{h}((k)). Now, Z^{l}(V̄_{h})^{[r]} is a submodule of (V̄_{h}^{⊗l})^{[r]}. So we see that there is a non-zero homomorphism

Z^{k}(V_{h}) → (V̄_{h}^{⊗l})^{[r]}

with image Z^{l}(V̄_{h})^{[r]}. Now apply the Morita equivalence β_{σ,h,k} and Lemma 4.3a to deduce that there is a non-zero homomorphism

Z^{k}(σ) → M^{l}(τ)

with image β_{σ,h,k}(Z^{l}(V̄_{h})^{[r]}). Now, Z^{k}(σ) is a quotient of Γ_{n}, so we have shown that the image of some non-zero homomorphism

Γ_{n} → M^{l}(τ)

is isomorphic to β_{σ,h,k}(Z^{l}(V̄_{h})^{[r]}). But by (3.4h), the image of any such homomorphism is precisely the submodule Z^{l}(τ) of M^{l}(τ).

(iii) By Theorem 3.5e(i), Δ(τ, λ) is isomorphic to the image of any non-zero homomorphism from Ż^{λ′}(τ) to Λ̇^{λ}(τ). So using (i) and (ii), Δ(τ, λ) is isomorphic to the image of any non-zero homomorphism

β_{σ,h,k}(Z^{λ′}(V̄_{h})^{[r]}) → β_{σ,h,k}(Λ^{λ}(V̄_{h})^{[r]}).

So using the Morita equivalence, we just need to check that the image of any non-zero homomorphism

Z^{λ′}(V̄_{h})^{[r]} → Λ^{λ}(V̄_{h})^{[r]}

is isomorphic to Δ̄_{h}(λ′)^{[r]}. But by (1.3d), the image of any non-zero homomorphism

Z^{λ′}(V̄_{h}) → Λ^{λ}(V̄_{h})

is isomorphic to Δ̄_{h}(λ′), so the conclusion follows on taking Frobenius twists.

(iv) The simple head of Δ̄_{h}(λ′) is isomorphic to L̄_{h}(λ′). So the simple head of Δ̄_{h}(λ′)^{[r]} is isomorphic to L̄_{h}(λ′)^{[r]}. Now applying (iii), we see that β_{σ,h,k}(L̄_{h}(λ′)^{[r]}) is isomorphic to the simple head of Δ(τ, λ), namely, L(τ, λ). This proves the first isomorphism. For the second isomorphism, note that

β_{σ,h,k}(L̄_{h}(λ′)^{[r]}) ≅ β_{σ,h,k}(L_{h}(mλ′)) = L(σ, (mλ′)′)

using a special case of (1.3e).

Now we can deduce the non-defining characteristic analogue of Steinberg’s tensor product theorem (cf. [DDu_{2}]).

4.3e. Theorem. Suppose that σ ∈ 𝔽̄_{q}^{×} is a p-regular element of degree d over 𝔽_{q}. Let λ be a partition of k and let λ_{−1}, λ_{0}, …, λ_{a} be the partitions such that

λ′ = λ′_{−1} + ℓ(d)λ′_{0} + ℓ(d)pλ′_{1} + ⋯ + ℓ(d)p^{a}λ′_{a}

is the (ℓ(d), p)-adic expansion of λ′ (the λ_{i} are uniquely determined as h ≥ k). For each i = 0, …, a, choose a p-singular element σ_{i} ∈ 𝔽̄_{q}^{×} of degree dℓ(d)p^{i} over 𝔽_{q} with p-regular part conjugate to σ (such elements exist by (2.1a)). Then, L(σ, λ) is isomorphic to the module obtained by Harish-Chandra induction from

L(σ, λ_{−1}) ⊠ L(σ_{0}, λ_{0}) ⊠ L(σ_{1}, λ_{1}) ⊠ ⋯ ⊠ L(σ_{a}, λ_{a}).

§4.4 Blocks and decomposition numbers  83

Proof. Because of Theorem 4.2a and Theorem 4.3d(iv), the theorem is a direct restatement of (1.3e) under the Morita equivalence.

4.3f. Remark. There are two different parametrizations of the irreducible F G_{n}- modules that appear in the literature. The first arises from Harish-Chandra theory and was introduced in [D_{1}, D_{2}]. It involves pairs (s, λ), where s runs over all pregular semisimple classes of G_{n} together with certain p-singular classes, and λ runs over p-regular multipartitions of κ(s). The second parametrization is described in the next section and was introduced by James in [J_{2}]. It involves pairs (s, λ) where s runs over representatives for the p-regular semisimple classes only but now λ runs over all multipartitions of κ(s). The combinatorics to translate between the two parametrizations was explained originally in [DJ_{2}]. Theorem 4.3e can be used to give an alternative, more representation theoretic proof of main result of loc. cit.. We refer the reader to [DDu_{2}, D_{4}] for further details of these matters.

4.4. Blocks and decomposition numbers

Let us now introduce notation for irreducible and standard modules of F G_{n} associated to an arbitrary semisimple element. So suppose that s is a block-diagonal element of G_{n} written in the form (2.1.1). For λ = (λ_{1}, …, λ_{a}) ⊢ κ(s) and R equal to one of K, F or O, define the standard F G_{n}-module:

Δ(s, λ)_{R} = R_{G_{π(s)}}^{G_{n}}(Δ(σ_{1}, λ_{1})_{R} ⊠ ⋯ ⊠ Δ(σ_{a}, λ_{a})_{R})

We will write simply Δ(s, λ) for Δ(s, λ)_{F} over F. Since Harish-Chandra induction commutes with base change, we have immediately by Theorem 4.1c and (2.3e):

(4.4a) Δ(s, λ)_{O} is O-free of finite rank with

Δ(s, λ)_{K} ≅ K ⊗_{O} Δ(s, λ)_{O},
Δ(s, λ)_{F} ≅ F ⊗_{O} Δ(s, λ)_{O}.

Moreover, the character of Δ(s, λ)_{K} is χ_{s,λ}, so Δ(s, λ) is a p-modular reduction of a KG_{n}-module affording the character χ_{s,λ}.

To describe all irreducible F G_{n}-modules, suppose now in addition that s is pregular. Define

L(s, λ) = R_{G_{π(s)}}^{G_{n}}(L(σ_{1}, λ_{1}) ⊠ ⋯ ⊠ L(σ_{a}, λ_{a})).  (4.4.1)

The F G_{n_{i}}-module L(σ_{i}, λ_{i}) is an irreducible module for the block B_{σ_{i},k_{1}}, and is isomorphic to the simple head of Δ(σ_{i}, λ_{i}). So, in the notation of Theorem 2.4e, L(σ_{1}, λ_{1}) ⊠ ⋯ ⊠ L(σ_{a}, λ_{a}) is an irreducible module for Bs^{Levi}, and is isomorphic the simple head of Δ(σ_{1}, λ_{1}) ⊠ ⋯ ⊠ Δ(σ_{a}, λ_{a}). Now Theorem 2.4e immediately gives that L(s, λ) is irreducible and is isomorphic to the simple head of Δ(s, λ). Moreover, recalling (2.4.6) we have:

(4.4b) For p-regular s, the modules {L(s, λ) | λ ⊢ κ(s)} give a complete set of nonisomorphic irreducible B_{s}-modules. So, the modules {L(s, λ) | s ∈ C_{ss,p′}, λ ⊢ κ(s)} give a complete set of non-isomorphic irreducible F G_{n}-modules.

Now we discuss two closely related problems. First, we would like to describe the p-blocks of the group G_{n}. So we would like to know precisely when the ordinary irreducible characters χ_{s,λ} and χ_{t,μ} belong to the same p-block, for any s, t ∈ C_{ss}, λ ⊢ κ(s), μ ⊢ κ(t). In view of (4.4a), we can equivalently consider when the modules Δ(s, λ) and Δ(t, μ) belong to the same block of the algebra F G_{n}. Second, we would like to say something about the p-modular decomposition numbers of G_{n}, that is, composition multiplicities of the form [Δ(s, λ) : L(t, μ)] for any s ∈ C_{ss}, t ∈ C_{ss,p′}, λ ⊢ κ(s), μ ⊢ κ(t).

We can make some basic reductions to both of these problems which will simplify notation considerably. First, we apply (2.4.6) to see that we just need to describe the p-blocks and the decomposition numbers of the algebra B_{t}, for some fixed t ∈ C_{ss.p′}. Then we can apply the Morita equivalence of Theorem 2.4e to reduce further to the special case that t = (σ)^{k} for some p-regular σ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q}, where n = kd. In other words, we just need to describe the blocks and the decomposition numbers of the algebra B_{σ,k}.

So fix now a p-regular σ ∈ 𝔽̄_{q}^{×} and h ≥ k ≥ 1. We recall from (2.4b) that the standard B_{σ,k}-modules are precisely the modules Δ(s, λ) (corresponding to the characters χ_{s,λ}) for λ ⊢ κ(s) and s ∈ C_{ss} of the form (2.1.1) with the p-regular part of each σ_{i} conjugate to σ. Our first lemma relates this module Δ(s, λ) to the q^{d}-Schur algebra S_{h,k} = S_{F,q^{d}}(h, k).

4.4c. Lemma. For any block-diagonal element s = (σ_{1})^{k_{1}} … (σ_{a})^{k_{a}} of the form (2.1.1), with the p-regular part of each σ_{i} conjugate to σ, and any λ = (λ_{1}, …, λ_{a}) ⊢ κ(s),

Δ(s, λ) ≅ β_{σ,h,k} (Δ_{h}(λ′_{1})^{[r_{1}]} ⊗ ⋯ ⊗ Δ_{h}(λ′_{a})^{[r_{a}]})

where each r_{i} is determined by the equation deg(σ_{i}) = dℓ(d)p^{r_{i}}.

Proof. By definition, Δ(σ, λ) is the F G_{n}-module obtained by Harish-Chandra induction from the F G_{π(s)}-module Δ(σ_{1}, λ_{1}) ⊠ ⋯ ⊠ Δ(σ_{a}, λ_{a}). By Theorem 4.3d(iii), each Δ(σ_{i}, λ_{i}) is isomorphic to β_{σ,h,k_{i}}(Δ_{h}(λ′_{i})^{[r_{i}]}). The lemma now follows on applying Theorem 4.2a.

Now we can explain the algorithm in [DJ_{3}, §7] showing how to calculate the p-modular decomposition numbers of G_{n} from knowledge of the decomposition numbers of quantum linear groups. Actually, we will give a somewhat more precise formula relating the two. We need a little notation for the statement: for λ, μ, ν_{1}, …, ν_{a} ∈ Λ^{+}(h) and e ≥ 1, D_{λ,μ}^{e} denotes the decomposition number [Δ_{h}(λ) : L_{h}(μ)] of the quantized coordinate ring A_{F,q^{e}}(h) and LR_{ν_{1},...,ν_{a};μ}^{e} denotes the tensor product composition multiplicity [L_{h}(ν_{1}) ⊗ ⋯ ⊗ L_{h}(ν_{a}) : L_{h}(μ)] for A_{F,q^{e}}(h).

4.4d. Theorem. Suppose that s = (σ_{1})^{k_{1}} … (σ_{a})^{k_{a}} is a block-diagonal element of G_{n} of the form (2.1.1), with each σ_{i} of degree d_{i} = m_{i}d over 𝔽_{q} having p-regular part conjugate to σ. Then, for any λ = (λ_{1}, …, λ_{a}) ⊢ κ(s) and any μ ⊢ k, we have the equality:

[Δ(s, λ) : L(σ, μ)] = ∑_{ν_{1}⊢k_{1}} ∑_{ν_{2}⊢k_{2}} ⋯ ∑_{ν_{a}⊢k_{a}} [∏_{i=1}^{a} D_{λ′_{i},ν′_{i}}^{d_{i}}] LR_{m_{1}ν′_{1},...,m_{a}ν′_{a};μ′}^{d} .

Proof. Using Lemma 4.4c and the Morita equivalence, we need to prove that for any λ_{i} ∈ Λ^{+}(h, k_{i}) and μ ∈ Λ^{+}(h, k),

[Δ̄_{h}(λ_{1})^{[r_{1}]} ⊗ ⋯ ⊗ Δ̄_{h}(λ_{a})^{[r_{a}]} : L_{h}(μ)] = ∑_{ν_{1}⊢k_{1}} ⋯ ∑_{ν_{a}⊢k_{a}} [∏_{i=1}^{a} D_{λ_{i},ν_{i}}^{d_{i}}] LR_{m_{1}ν_{1},...,m_{a}ν_{a};μ}^{d}

where the r_{i} are defined by m_{i} = ℓ(d)p^{r_{i}}. For ν ∈ Λ^{+}(h, m_{i}k_{i}), the composition multiplicity [Δ̄_{h}(λ_{1})^{[r_{1}]} : L_{h}(ν)] is zero unless ν = m_{i}ν_{i} for some ν_{i} ∈ Λ^{+}(h, k_{i}), when it equals D_{λ_{i},ν_{i}}^{d_{i}}. It follows that Δ̄_{h}(λ_{1})^{[r_{1}]} ⊗ ⋯ ⊗ Δ̄_{h}(λ_{a})^{[r_{a}]} has the same composition factors as

⨁_{ν_{1}⊢k_{1}} ⋯ ⨁_{ν_{a}⊢k_{a}} [∏_{i=1}^{a} D_{λ_{i},ν_{i}}^{d_{i}}] L_{h}(m_{1}ν_{1}) ⊗ ⋯ ⊗ L_{h}(m_{a}ν_{a}).

Now the theorem follows.

Now consider the block theory. We first recall the description of the blocks of the q^{d}-Schur algebra S_{h,k} from [JM, Theorem 4.24]. For s ≥ 1, recall that the s-core of a partition λ is the partition obtained by successively removing as many rim hooks of length s as possible. Then, for λ, μ ∈ Λ^{+}(h, k) (and h ≥ k as always), the irreducible S_{h,k}-modules L_{h}(λ) and L_{h}(μ) belong to the same block if and only if either:

(i) ℓ(d) = 1 and λ and μ have the same p-core;

(ii) ℓ(d) > 1 and λ and μ have the same ℓ(d)-core. Now, since C_{σ,k} is Morita equivalent to S_{h,k} (and the cores of λ and λ′ are always transpose to one another), this combinatorics also determines when the modules L(σ, λ) and L(σ, μ) belong to the same block of the cuspidal algebra C_{σ,k}. Recalling that C_{σ,k} is a quotient of the algebra B_{σ,k} and that the two algebras have the same irreducible modules, B_{σ,k} can in general have fewer blocks than C_{σ,k}.

4.4e. Lemma. If ℓ(d) = 1 (i.e. q^{d} ≡ 1 (mod p)), then the algebra B_{σ,k} has just one block.

Proof. Let λ = (l_{1}, …, l_{a}) and μ = (l_{1}+l_{a}, l_{2}, …, l_{a−1}) be partitions of k. Following the strategy of [DJ_{3}, Lemma 7.10], we show that L(σ, λ) and L(σ, μ) are linked. This is enough to complete the proof, for then by induction on a, we see that L(σ, λ) belongs to the same block as L(σ, (k)), whence all L(σ, λ) belong to the same block.

Since ℓ(d) = 1, we can by (2.1a) choose some p-singular τ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q} with p-regular part conjugate to σ. Writing ν = (l_{1}, …, l_{a−1}), the F G_{n}-module M obtained from Δ(σ, ν) ⊠ Δ(τ, (l_{a})) by Harish-Chandra induction is a standard module for B_{σ,k}. So to prove that L(σ, λ) and L(σ, μ) are linked, it suffices to show that they are both composition factors of M.

Equivalently, thanks to Lemma 4.4c and the Morita equivalence, we need to show that the S_{h,k}-module Δ_{h}(ν′) ⊗ Δ_{h}((1^{l_{a}})) contains both L_{h}(λ′) and L_{h}(μ′) as composition factors. But now Δ_{h}(ν′) ⊗ Δ_{h}((1^{l_{a}})) has a Δ-filtration, and a calculation involving the Littlewood-Richardson rule shows that this filtration has factors isomorphic to Δ_{h}(λ′) and to Δ_{h}(μ′). Hence, it certainly has composition factors L_{h}(λ′) and L_{h}(μ′) as desired.

4.4f. Lemma. Suppose that s = (σ_{1})^{k_{1}} … (σ_{a})^{k_{a}} is of the form (2.1.1), with σ_{1} conjugate to σ, and all other σ_{i} having p-regular part conjugate to σ. If L(σ, μ) is a composition factor of Δ(s, λ), for some λ = (λ_{1}, …, λ_{a}) ⊢ κ(s) and μ ⊢ k, then μ and λ_{1} have the same ℓ(d)-cores.

Proof. If ℓ(d) = 1, the lemma is trivially true. So assume that ℓ(d) > 1. Then by (2.1a), all of σ_{1}, …, σ_{a} are of degree strictly greater than d. So applying Lemma 4.4c, we see that Δ(s, λ) corresponds under the Morita equivalence to an S_{h,k}-module of the form Δ_{h}(λ′_{1}) ⊗ M where M is a pure Frobenius twist. Now it suffices to prove that L_{h}(μ′) is a composition factor of Δ_{h}(λ′_{1}) ⊗ M only if μ′ and λ′_{1} have the same ℓ(d)-cores.

Using Steinberg’s tensor product theorem (1.3e) and the known block theory of S_{h,k}, all composition factors of Δ_{h}(λ′_{1}) are of the form L_{h}(ν) ⊗ N where ν is ℓ(d)- restricted and has the same ℓ(d)-core as λ′_{1}, and N is a pure Frobenius twist. Hence, using the tensor product theorem once more, all composition factors of Δ_{h}(λ′_{1}) ⊗ M are of the form L_{h}(ν) ⊗ L̄_{h}(γ)^{[0]} for such ν. So, if L_{h}(μ′) is a composition factor of Δ_{h}(λ′_{1})⊗M then μ′ = ν +ℓ(d)γ, which has the same ℓ(d)-core as ν hence as λ′_{1}.

Finally, we can determine the blocks of B_{σ,k}. The following theorem is equivalent, after making the elementary reductions described above, to Fong and Srinivasan’s theorem [FS, (7A)].

4.4g. Theorem. The blocks of the algebra B_{σ,k} are parametrized by the set of ℓ(d)- cores of partitions of k. Moreover, given s = (σ_{1})^{k_{1}} … (σ_{a})^{k_{a}} of the form (2.1.1), with σ_{1} conjugate to σ and all other σ_{i} having p-regular part conjugate to σ, and λ = (λ_{1}, …, λ_{a}) ⊢ κ(s), the standard module Δ(s, λ) belongs to the block parametrized by the ℓ(d)-core of λ_{1}.

Proof. If ℓ(d) = 1, this is immediate from Lemma 4.4e. For ℓ(d) > 1, use Lemma 4.4f and the block theory of S_{h,k} as described above.

§4.5 The Ringel dual of the cuspidal algebra  87

4.5. The Ringel dual of the cuspidal algebra

If S is any quasi-hereditary algebra over F with weight poset (Λ^{+}, ≤), an Smodule M has a Δ-filtration (resp. a ∇-filtration) if it has a filtration with sections isomorphic to modules of the form Δ(λ) (resp. ∇(λ)), λ ∈ Λ^{+}. Recall that in any such Δ-filtration of M, the multiplicity of a particular Δ(λ) for fixed λ ∈ Λ^{+} us uniquely determined; we write [M : Δ(λ)]_{Δ} for this multiplicity. Say that an Smodule is tilting if it has both a Δ-filtration and a ∇-filtration. Ringel [R] has shown that for each λ ∈ Λ^{+}, there is a unique indecomposable tilting module T (λ) such that [T (λ) : Δ(λ)]_{Δ} = 1 and, for μ ∈ Λ^{+}, [T (λ) : Δ(μ)]_{Δ} = 0 unless μ ≤ λ. Furthermore, every tilting module is isomorphic to a direct sum of these indecomposable tilting modules modules T (λ), λ ∈ Λ^{+}.

Following the language of [Do_{4}, Appendix], a full tilting module is a tilting module that contains every T (λ), λ ∈ Λ^{+} as a summand with non-zero multiplicity. Given such a full tilting module T, the Ringel dual of S relative to T is the algebra S^{⋆} = End_{S}(T)^{op}. Here, we are writing endomorphisms on the right, so T is naturally a right End_{S}(T)-module, hence a left S^{⋆}-module. Ringel [R] showed that S^{⋆} is also a quasi-hereditary algebra with weight poset Λ^{+}, but ordered with the opposite order to the original partial ordering on Λ^{+}. We briefly indicate one approach to the proof of this, since we will need the notation shortly. We define the contravariant functor

γ : mod(S) → mod(S^{⋆}),  γ = Hom_{S}(?, T).  (4.5.1)

By definition, γ(T) = S^{⋆} so, using Fitting’s lemma, γ takes indecomposable tilting modules over S to indecomposable projectives over S^{⋆}. Moreover, the functor γ is exact on short exact sequences of modules with Δ-filtrations, so if we define Δ′(λ) = γ(Δ(λ)), we see that P′(λ) := γ(T (λ)) has a filtration with sections Δ′(λ). Then, as in [Do_{4}, A.4.7], we deduce that S^{⋆} is a quasi-hereditary algebra with indecomposable projectives {P′(λ)} and standard modules {Δ′(λ)}.

We will need the following known result. Although a proof of this can be deduced from the literature (using [HR, Theorem 2.1] or [CPS_{1}, Theorem 2.1(d)] together with [Do_{2}] to verify that our tilting modules coincide with the original notion of tilting module in [HR, CPS_{1}]), we include a short direct proof working purely in the framework of quasi-hereditary algebras. The argument was explained to us by S. Donkin, and we are grateful for his permission to include it here.

4.5a. Lemma. Regarded as a left S^{⋆}-module, T is a full tilting module for S^{⋆}. Moreover, the Ringel dual End_{S^{⋆}}(T)^{op} of S^{⋆} relative to T is isomorphic to S.

Proof. Let γ be the contravariant functor defined in (4.5.1) and Δ′(λ) = γ(Δ(λ)) for each λ ∈ Λ^{+}. For M, N ∈ mod(S) with Δ-filtrations, we have that Ext_{S}^{i}(M, N) ≅ Ext_{S^{⋆}}^{i}(γ(N), γ(M)) by [Do_{4}, A.4.8] (or rather its analogue for the functor γ). In particular, Ext_{S^{⋆}}^{1}(Δ′(λ), γ(S)) = Ext_{S}^{1}(S, Δ(λ)) = 0, so that γ(S) has a ∇′-filtration by the cohomological criterion [Do_{4}, A2.2(iii)]. Also S has a Δ-filtration so, using the fact that γ is exact on short exact sequences of modules with Δ-filtrations, we see that γ(S) also has a Δ′-filtration. So γ(S) is a tilting module, and more generally, passing to indecomposable summands of S, γ takes each indecomposable projective P (λ) to a tilting module. Since P (λ) has a Δ-filtration with Δ(λ) appearing with multiplicity one and all other factors being of the form Δ(μ) for μ > λ, γ(P (λ)) also has a Δ′-filtration with Δ′(λ) being the most dominant appearing (for the opposite ordering). So, the tilting module γ(P (λ)) definitely contains T′(λ) as a summand. We deduce finally that γ(S) is a full tilting module for S^{⋆}, and from the definition of γ, γ(S) = T. This shows that T is a full tilting module for S^{⋆}. Finally,

End_{S^{⋆}}(T)^{op} = End_{S^{⋆}}(γ(S))^{op} ≅ End_{S}(S) = S

using [Do_{4}, A.4.8] once more (recall that γ is a contravariant functor).

Now for the remainder of the section, we choose σ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q} and n = kd for some k ≥ 1. Let C_{σ,k} denote the cuspidal algebra of (3.2.1). Thanks to Theorem 4.3b, the hypotheses (A1) and (A2) of §3.2 are satisfied, so C_{σ,k} is Morita equivalent to the q^{d}-Schur algebra S_{h,k} = S_{F,q^{d}}(h, k), for some fixed h ≥ k.

Applying Ringel’s theorem first to the q^{d}-Schur algebra S_{h,k} = S_{F,q^{d}}(h, k), for any h, k ≥ 1, we obtain the indecomposable tilting modules of S_{h,k}. We denote these modules by {T_{h}(λ)|λ ∈ Λ^{+}(h, k)}. Applying Ringel’s theorem to C_{σ,k} instead, we obtain indecomposable tilting modules {T (σ, λ) | λ ⊢ k} for C_{σ,k}. It follows immediately from (3.5a) and the definitions that

T (σ, λ) ≅ β_{σ,h,k}(T_{h}(λ′)).

Now, in view of Lemma 3.5d, the following statement follows from the elementary weight argument of [Do_{4}, §3.3(1)], on applying the functor β_{σ,h,k}:

(4.5b) The indecomposable tilting modules for C_{σ,k} are precisely the indecomposable summands of Λ̇^{ν}(σ) for all ν ⊨ k. Furthermore, for λ ⊢ k, the module T (σ, λ) occurs exactly once as a summand of Λ̇^{λ}(σ), and if T (σ, μ) is a summand of Λ̇^{λ}(σ) for some μ ⊢ k then μ ≥ λ.

As a corollary of (4.5b), we obtain the following alternative description of T (σ, λ):

4.5c. Lemma. For λ ⊢ k, the module T (σ, λ) can be characterized as the unique indecomposable summand of Λ̇^{λ}(σ) containing a submodule isomorphic to Δ(σ, λ).

Proof. By Theorem 3.5e(i), Λ̇^{λ}(σ) has a unique submodule isomorphic to Δ(σ, λ). By (4.5b), Λ̇^{λ}(σ) has a unique summand isomorphic to T (σ, λ) and for any other summand M of Λ̇^{λ}(σ), Hom_{C_{σ,k}}(Δ(σ, λ), M) = 0. These two statements imply the lemma.

Next, we explain the role of Theorem 3.4a in the theory. Recall that h ≥ k.

4.5d. Theorem. The C_{σ,k}-module T = ⨁_{ν∈Λ(h,k)} Λ̇^{ν}(σ) is a full tilting module. Moreover, the Ringel dual C_{σ,k}^{⋆} of C_{σ,k} relative to T is precisely the algebra S_{h,k}^{op}, where S_{h,k} acts on T as in Theorem 3.4a.

Proof. (4.5b) shows immediately that T is a full tilting module. The second statement is a restatement of Theorem 3.4a.

As consequences of Theorem 4.5d, we have the following double centralizer properties; of these, (ii) is a generalization of Takeuchi’s theorem [T].

4.5e. Theorem. The following double centralizer properties hold, for h ≥ k:

(i) End_{C_{σ,k}} (⨁_{ν∈Λ(h,k)} Λ̇^{ν}(σ)) ≅ S_{h,k} and End_{S_{h,k}} (⨁_{ν∈Λ(h,k)} Λ̇^{ν}(σ)) ≅ C_{σ,k}, where the right S_{h,k}-action is as in Theorem 3.4a;

(ii) End_{C_{σ,k}}(M^{k}(σ)) ≅ H_{k} and End_{H_{k}}(M^{k}(σ)) ≅ C_{σ,k}, where H_{k} = H_{F,q^{d}}(Σ_{k}) and the right H_{k} action is as in (2.5b).

Proof. (i) Combine Theorem 4.5d with Lemma 4.5a.

(ii) Let e ∈ S_{h,k} be the idempotent ϕ_{(1^{k}),(1^{k})}^{1}. We note that as in [Do_{4}, §3.3(1)], the indecomposable tilting module T_{h}(λ) is a summand of Λ^{λ′}(V_{h}), for any λ ∈ Λ^{+}(h, k). Consequently, by (1.3b)(ii), T_{h}(λ) is both a submodule and a quotient of the S_{h,k}-module S_{h,k}e. Moreover, S_{h,k}e is isomorphic to V_{h}^{⊗k}, so is (contravariantly) self-dual. These observations imply that every composition factor of both the socle and the head of T_{h}(λ) belong to the head of the projective S_{h,k}-module S_{h,k}e.

Now let T = ⨁_{ν∈Λ(h,k)} Λ̇^{ν}(σ). Writing T̃ for the left S_{h,k}-module obtained from the right module T by twisting with τ, T̃ is a full tilting module for S_{h,k} by Lemma 4.5a and Theorem 4.5d. The previous paragraph therefore shows that every composition factor of the socle and the head of T̃ belong to the head of S_{h,k}e. In other words, V = W = T̃ satisfy the conditions of Corollary 3.1c, taking C = S_{h,k}, P = S_{h,k}e and H = eS_{h,k}e. We deduce at once from Corollary 3.1c that End_{S_{h,k}}(T̃) ≅ End_{eS_{h,k}e}(eT̃). Switching to right actions, and using (i), we have now shown that

C_{σ,k} ≅ End_{eS_{h,k}e}(T e).  (4.5.2)

Now we can prove the theorem. We know already that H_{k} = End_{C_{σ,k}}(M^{k}(σ)). As a left C_{σ,k}-module, M^{k}(σ) ≅ T e. Moreover, by the way the action of S_{h,k} on T was defined in (3.4b), the (C_{σ,k}, eS_{h,k}e)-bimodule T e is isomorphic to the (C_{σ,k}, H_{k})- bimodule M^{k}(σ), if we identify H_{k} with eS_{h,k}e so that T_{w} ↦ κ(T_{w}^{#}) for each w ∈ Σ_{k}, where κ is as in (1.2b). In view of this,

End_{H_{k}}(M^{k}(σ)) ≅ End_{eS_{h,k}e}(T e)

which is isomorphic to C_{σ,k} thanks to (4.5.2).

We end with the non-defining characteristic analogues of [Do_{3}, Lemma 3.4(i)] and [MP, Corollary 2.3]. For λ ⊢ k, let P (σ, λ) denote the projective cover of L(σ, λ) in the category mod(C_{σ,k}). Similarly, for μ ∈ Λ^{+}(h, k), let P_{h}(μ) denote the projective cover of L_{h}(μ) in the category mod(S_{h,k}). Obviously we have that:

P (σ, λ) ≅ β_{σ,h,k}(P_{h}(λ′)).  (4.5.3)

4.5f. Theorem. For ν ∈ Λ(h, k),

Ż^{ν}(σ) ≅ ⨁_{λ⊢k} P (σ, λ′)^{⊕m_{λ,ν}},  (4.5.4)
Λ̇^{ν}(σ) ≅ ⨁_{λ⊢k} T (σ, λ)^{⊕m_{λ,ν}},  (4.5.5)

where m_{λ,ν} is equal to the dimension of the ν-weight space of L_{h}(λ).

Proof. Using Fitting’s lemma, there is a unique decomposition 1 = ∑_{λ∈Λ^{+}(h,k)} e_{λ} of the identity of S_{h,k} into orthogonal idempotents such that for each λ,

S_{h,k}e_{λ} ≅ P_{h}(λ)^{⊕ dim L_{h}(λ)}.

We observe that dim L_{h}(λ) = dim Hom_{S_{h,k}}(S_{h,k}e_{λ}, L_{h}(λ)) = dim e_{λ}L_{h}(λ). This shows that e_{λ}L_{h}(λ) = L_{h}(λ). Now, for any ν ∈ Λ(h, k), ϕ_{ν,ν}^{1} ∈ S_{h,k} is an idempotent which obviously commutes with each e_{λ}. So, ϕ_{ν,ν}^{1}e_{λ} is an idempotent. Observe that

Hom_{S_{h,k}}(S_{h,k}ϕ_{ν,ν}^{1}e_{λ}, L_{h}(λ)) ≅ ϕ_{ν,ν}^{1}e_{λ}L_{h}(λ) = ϕ_{ν,ν}^{1}L_{h}(λ),

which is the ν-weight space of L_{h}(λ). So, S_{h,k}ϕ_{ν,ν}^{1}e_{λ} is a direct sum of precisely m_{λ,ν} copies of P_{h}(λ).

To prove (4.5.4), we have so far shown that S_{h,k}ϕ_{ν,ν}^{1} ≅ ⨁_{λ⊢k} P_{h}(λ)^{m_{λ,ν}}. Now apply the equivalence of categories β_{σ,h,k}, using (4.5.3) and the fact observed in the proof of Lemma 3.5d that β_{σ,h,k}(S_{h,k}ϕ_{ν,ν}^{1}) ≅ Ż^{ν}(σ).

Now consider (4.5.5). Certainly by (4.5b), all indecomposable summands of T = ⨁_{ν∈Λ(h,k)} Λ̇^{ν}(σ) are of the form T (σ, λ), λ ⊢ k. Combining this with Theorem 3.4a and Fitting’s lemma, we see that there is some permutation π of the set of partitions of k such that, for each λ ⊢ k, T e_{λ} is the largest summand of T isomorphic to a direct sum of copies of T (σ, π(λ)). Recall from the definition of the action of ϕ_{ν,ν}^{1} from Theorem 3.4a that T ϕ_{ν,ν}^{1} = Λ̇^{ν}(σ). So, T ϕ_{ν,ν}^{1}e_{λ} is the largest summand of Λ̇^{ν}(σ) isomorphic to a direct sum of copies of T (σ, π(λ)). Moreover, using the first paragraph, we know that in fact T ϕ_{ν,ν}^{1}e_{λ} is a direct sum of precisely m_{λ,ν} copies of T (σ, π(λ)). In other words, we know that for each ν ⊨ k:

Λ̇^{ν}(σ) ≅ ⨁_{λ⊢k} T (σ, π(λ))^{⊕m_{λ,ν}}.  (4.5.6)

It therefore remains to prove that π(λ) = λ for each λ ⊢ k. We prove this by downward induction on the dominance order on λ. The induction starts with λ = (k); here, π((k)) = (k) immediately from (4.5.6), since Λ^{k}(σ) = T (σ, (k)). Now take any μ < (k) and suppose we have proved inductively that π fixes all more dominant partitions. Using the inductive hypothesis, (4.5.6) tells us that

Λ̇^{μ}(σ) ≅ T (σ, π(μ)) ⊕ ⨁_{λ>μ} T (σ, λ)^{⊕m_{λ,μ}}.

Finally, we know by (4.5b) that T (σ, μ) appears as a summand of Λ̇^{μ}(σ), so we must have that T (σ, μ) ≅ T (σ, π(μ)), whence π(μ) = μ as required.

The affine general linear group

In this chapter we prove results that can be regarded as the modular analogues of the branching rules of Zelevinsky [Z, Theorem 13.5] and Thoma [Th]. Following the idea of Zelevinsky, we study the affine general linear group AGL_{n}(𝔽_{q}) with the same methods as we have developed so far for GL_{n}(𝔽_{q}). Roughly speaking, our main result relates restriction from GL_{n}(𝔽_{q}) to AGL_{n−1}(𝔽_{q}) to restriction from quantum GL_{n} to quantum GL_{n−1}. As an application, we obtain a new dimension formula for irreducible modular representations of GL_{n}(𝔽_{q}), in terms of weight space dimensions of irreducible modules over quantum GL_{n}.

5.1. Levels and the branching rule from AGL_{n} to GL_{n}

For a group G, I_{G} will denote the trivial F G-module, where F is as usual our fixed algebraically closed field of characteristic p coprime to q. Also, for a subgroup H ⊂ G and an F H-module M (resp. an F G-module N), we will often write M ↑^{G} (resp. N ↓_{H}) for ind_{H}^{G} M (resp. res_{H}^{G} N).

Let W_{n} denote an n-dimensional vector space over 𝔽_{q} with basis w_{1}, …, w_{n}. Let G_{n} denote GL_{n}(𝔽_{q}) acting naturally on W_{n}. Let H_{n} denote the affine general linear group AGL_{n}(𝔽_{q}), which is the semi-direct product G_{n}W_{n} of W_{n} (regarded now just as an Abelian group) by G_{n}. By convention, we allow the notations G_{0}, H_{0} and W_{0}, all of which denote groups with one element.

We make some remarks about the representations of the Abelian group W_{n} over F. Let X_{n} denote the set of irreducible characters W_{n} → F^{×}, regarded as an ndimensional 𝔽_{q}-vector space via (ε + χ)(w) = ε(w)χ(w) and (cε)(w) = ε(cw) for ε, χ ∈ X_{m}, c ∈ 𝔽_{q}, w ∈ W_{n}. To describe a parameterization of these characters, let χ : 𝔽_{q} → F^{×} be the non-trivial character fixed after (2.5.4). For any element θ of the 𝔽_{q}-linear dual W_{n}^{∗}, let ε_{θ} denote the character sending w ∈ W_{n} to χ(θ(w)) ∈ F^{×}. Then, X_{n} is precisely the set {ε_{θ} |θ ∈ W_{n}^{∗}}. We let θ_{1}, …, θ_{n} denote the basis of W_{n}^{∗} dual to w_{1}, …, w_{n}, and set ε_{i} = χ_{θ_{i}} to obtain a basis ε_{1}, …, ε_{n} for the 𝔽_{q}-vector space X_{n}.

The group G_{n} acts on the characters X_{n} by (gε)(w) = ε(g^{−1}w), for g ∈ G_{n}, w ∈ W_{n}, ε ∈ X_{n}. Under this action, the trivial character 0 ∈ X_{n} is fixed by G_{n} and all other characters are permuted transitively. An elementary calculation shows that the group C_{G_{n}}(ε_{n}) is isomorphic to H_{n−1}, embedded into G_{n} as the subgroup of matrices of the form

[ \begin{matrix}& ∗ & & ∗ \\ 0 & … & 0 & 1 \end{matrix} ] .

We always identify H_{n−1} with this subgroup of G_{n}. Thus, we have a chain of subgroups

1 = H_{0} ⊂ G_{1} ⊂ H_{1} ⊂ G_{2} ⊂ H_{2} ⊂ ….

Given an F H_{n}-module M and ε ∈ X_{n}, let

M_{ε} = {m ∈ M | wm = ε(w)m for all w ∈ W_{n}}

be the corresponding weight space. Since F W_{n} is a semisimple algebra, M decomposes as a direct sum of such weight spaces. Observe that the action of G_{n} on M induces an action on the weight spaces, so that gM_{ε} = M_{gε} for all g ∈ G_{n}, ε ∈ X_{n}. In particular, the 0-weight space M_{0} is stable under the action of G_{n}, while the ε_{n}-weight space M_{ε_{n}} of M is stable under the action of the subgroup H_{n−1} < G_{n}, since H_{n−1} centralizes ε_{n}.

Now we introduce some functors. First, for n ≥ 0, we have functors

f_{0}^{n} :mod(F H_{n}) → mod(F G_{n}),
e_{0}^{n} :mod(F G_{n}) → mod(F H_{n}).

For these, f_{0}^{n} is defined on an object M by M ↦ M_{0}, the zero weight space of M, which we observed in the previous paragraph is G_{n}-stable. On a morphism, f_{0}^{n} is defined simply to be its restriction to zero weight spaces. The functor e_{0}^{n} is defined on an object M to be the same vector space, but regarded as an F H_{n}-module by extending the action of G_{n} on M to H_{n} by letting W_{n} act trivially, and on a morphism by simply regarding the morphism as a homomorphism over H_{n} instead of G_{n}. Next, for n ≥ 1, we have functors

f_{+}^{n} :mod(F H_{n}) → mod(F H_{n−1}),
e_{+}^{n} :mod(F H_{n−1}) → mod(F H_{n}).

On an object M, f_{+}^{n} sends M to its ε_{n}-weight space M_{ε_{n}}, which is stable under the action of H_{n−1}; on a morphism, f_{+}^{n} is defined by restriction. The functor e_{+}^{n} is defined to be the composite of the inflation functor from H_{n−1} to H_{n−1}W_{n}, with the action of W_{n} being via the character ε_{n}, followed by ordinary induction from H_{n−1}W_{n} to H_{n}. Finally, for n ≥ 1 and 1 ≤ i ≤ n, we have functors

f_{i}^{n} :mod(F H_{n}) → mod(F G_{n−i}),
e_{i}^{n} :mod(F G_{n−i}) → mod(F H_{n}).

§5.1 Levels and the branching rule from AGL_{n} to GL_{n}  93

These are defined inductively by f_{i}^{n} = f_{i−1}^{n−1} ∘ f_{+}^{n} and e_{i}^{n} = e_{+}^{n} ∘ e_{i−1}^{n−1}. By convention, if i > n, the functor f_{i}^{n} denotes the zero functor. It will usually be obvious from context which group H_{n} we have in mind, so we will from now on drop the index n from our notation for the functors e_{+}^{n}, e_{i}^{n}, f_{i}^{n} and f_{+}^{n}.

5.1a. Lemma. (i) For any n ≥ 1, the functors

f_{0} ⊕ f_{+} : mod(F H_{n}) → mod(F G_{n}) × mod(F H_{n−1}),
e_{0} ⊕ e_{+} : mod(F G_{n}) × mod(F H_{n−1}) → mod(F H_{n})

are mutually inverse equivalences of categories. Here, f_{0} ⊕ f_{+} denotes the functor ? ↦ (f_{0} ?, f_{+} ?) and e_{0} ⊕ e_{+} is the functor (?,?′) ↦ (e_{0} ?) ⊕ (e_{+} ?′).

(ii) For any n ≥ 0, the functors

f_{0} ⊕ f_{1} ⊕ ⋯ ⊕ f_{n} : mod(F H_{n}) → mod(F G_{n}) × mod(F G_{n−1}) × ⋯ × mod(F G_{0}),
e_{0} ⊕ e_{1} ⊕ ⋯ ⊕ e_{n} : mod(F G_{n}) × mod(F G_{n−1}) × ⋯ × mod(F G_{0}) → mod(F H_{n})

are mutually inverse equivalences of categories.

Proof. (i) As in [Z, §13.1], this is a special case of a general result about representations of a semi-direct product GW where W is an Abelian normal subgroup of GW, see [Se, 8.2]. Although the argument in loc. cit. is in characteristic 0, it applies equally well to our case since F W is a semisimple algebra.

(ii) Apply (i) and induction on n.

With Lemma 5.1a as our motivation, we now make the basic definition for understanding the representation theory of H_{n}. Say that an F H_{n}-module M belongs to the ith level if f_{j}M = 0 for all j ≠ i. We will also refer to the F H_{n}-module e_{i} ∘ f_{i}(M) as the ith level of M. By the lemma, any F G_{n}-module M splits uniquely as the direct sum of its levels.

We will now write C_{ss,p′}^{n} for the set C_{ss,p′} of representatives of the p-regular classes of semisimple elements G_{n} chosen in §2.1. We allow the notation C_{ss,p′}^{0}, which is the set containing just one element, namely, the identity element 1 ∈ G_{0}; the composition κ(1) is then just the zero composition (0).

Applying Lemma 5.1a(ii), the irreducible F H_{n}-modules belonging to the ith level are precisely the modules e_{i}L as L runs over the irreducible F G_{n−i}-modules. We obtain immediately from (4.4b) the following description of the irreducible F H_{n}- modules:

(5.1b) The set {e_{i}L(s, λ) | 0 ≤ i ≤ n, s ∈ C_{ss,p′}^{n−i}, λ ⊢ κ(s)} is a complete set of nonisomorphic irreducible F H_{n}-modules.

Our aim in the remainder of the chapter is to understand induction and restriction of irreducibles between GL and AGL. To start off with, we have the basic:

5.1c. Lemma. The following pairs of functors are isomorphic:

(i) res_{H_{n−1}}^{G_{n}} and f_{+} ∘ ind_{G_{n}}^{H_{n}} : mod(F G_{n}) → mod(F H_{n−1});

(ii) ind_{G_{n}}^{H_{n}} and e_{0} ⊕ (e_{+} ∘ res_{H_{n−1}}^{G_{n}}) : mod(F G_{n}) → mod(F H_{n});

(iii) ind_{H_{n−1}}^{G_{n}} and res_{G_{n}}^{H_{n}} ∘e_{+} : mod(F H_{n−1}) → mod(F G_{n});

(iv) res_{G_{n}}^{H_{n}} and f_{0} ⊕ (ind_{H_{n−1}}^{G_{n}} ∘f_{+}) : mod(F H_{n}) → mod(F G_{n}).

Proof. We only need to prove (i) and (ii); then, (iii) and (iv) follow immediately on taking adjoints. The subgroup W_{n} ⊂ H_{n} is a set of H_{n}/G_{n}-coset representatives. Since F W_{n} is a semisimple algebra, we can pick a basis {z_{ε} | ε ∈ X_{n}} for F W_{n} such that wz_{ε} = ε(w)z_{ε} for each w ∈ W_{n} and ε ∈ X_{n}. Then, for any F G_{n}-module M, we see that ind_{G_{n}}^{H_{n}} M = F H_{n} ⊗_{F G_{n}} M can be written as a direct sum:

ind_{G_{n}}^{H_{n}} M = ⨁_{ε∈X_{n}} z_{ε} ⊗ M.

Then, f_{0}(ind_{G_{n}}^{H_{n}} M) = z_{0} ⊗ M ≅ M and f_{+}(ind_{G_{n}}^{H_{n}} M) = z_{ε_{n}} ⊗ M ≅ M ↓_{H_{n−1}}. We deduce that there are isomorphisms of functors:

f_{0} ∘ ind_{G_{n}}^{H_{n}} ≅ id_{mod(F G_{n})}, f_{+} ∘ ind_{G_{n}}^{H_{n}} ≅ res_{H_{n−1}}^{G_{n}} .

In particular, this proves (i), while (ii) follows on applying Lemma 5.1a(i) and the properties above.

Before the next theorem, we introduce some further notation. Given m, n ≥ 0, let S_{m,n} and Z_{m,n} be the subgroups of G_{m+n} consisting of all matrices of the form:

S_{m,n} : [ \begin{matrix}∗_{m} & & ∗ \\ & 1 & & ∗ \\ 0 & & ... \\ & 0 & & 1 \end{matrix} ] , Z_{m,n} : [ \begin{matrix}I_{m} & ∗ \\ 0 & I_{n} \end{matrix} ] .

We view the group U_{n} of all upper uni-triangular matrices in G_{n} as a subgroup of S_{m,n}, embedded in the obvious way into the bottom right hand corner of the matrices. Doing this, we can regard the Gelfand-Graev idempotent γ_{n} ∈ F U_{n} of (2.5.5) as an element of F S_{m,n}. We also recall for use shortly that F γ_{n} is a one dimensional F U_{n}-module, and the induced module (F γ_{n}) ↑^{G_{n}} is the Gelfand-Graev representation Γ_{n} of F G_{n}.

Now, there is an obvious surjective homomorphism

S_{m,n} → G_{m} × U_{n}  (5.1.1)

with kernel Z_{m,n}. Define the functor

infl_{G_{m}}^{S_{m,n}} : mod(F G_{m}) → mod(F S_{m,n})

to be the composite of the functor ? ⊠ F γ_{n} : mod(F G_{m}) → mod(F (G_{m} × U_{n})) followed by natural inflation functor along the surjection (5.1.1). Note that S_{m,n+1} is a subgroup of H_{m+n}.

5.1d. Lemma. For any m, n ≥ 0, the functor e_{n} : mod(F G_{m}) → mod(F H_{m+n}) is isomorphic to the composite functor ind_{S_{m,n+1}}^{H_{m+n}} ∘ infl_{G_{m}}^{S_{m,n+1}}.

Proof. We proceed by induction on n, the case n = 0 being trivial. Recalling that e_{n} = e_{+} ∘ e_{n−1}, the inductive step reduces at once to proving that there is an isomorphism of functors:

e_{+} ∘ ind_{S_{m,n}}^{H_{m+n−1}} ∘ infl_{G_{m}}^{S_{m,n}} ≅ ind_{S_{m,n+1}}^{H_{m+n}} ∘ infl_{G_{m}}^{S_{m,n+1}} .

To see this, it is easier to show instead that the adjoints of each of these functors are isomorphic.

The adjoint functor to e_{+} is f_{+} : mod(F H_{m+n}) → mod(F H_{m+n−1}). This can be viewed simply as left multiplication by the idempotent

z_{ε_{m+n}} = ∑_{w∈W_{m+n}} ε_{m+n}(−w)w ∈ F W_{n+m} ⊂ F H_{m+n}

corresponding to the character ε_{m+n} ∈ X_{m+n}. The adjoint of ind_{S_{m,n}}^{H_{m+n−1}} ∘ infl_{G_{m}}^{S_{m,n}} is given by first restriction from H_{m+n−1} to S_{m,n}, followed by multiplication by the idempotent ζ_{m,n}γ_{n}, where

ζ_{m,n} = \frac{1}{|Z_{m,n}|} ∑_{z∈Z_{m,n}} z ∈ F S_{m,n}

and we are viewing γ_{n} as an element of F S_{m,n} as explained above. Hence, the adjoint of the composite functor e_{+} ∘ ind_{S_{m,n}}^{H_{m+n−1}} ∘ infl_{G_{m}}^{S_{m,n}} is given on objects just by multiplication by the idempotent ζ_{m,n}γ_{n}z_{ε_{m+n}} ∈ F H_{m+n}, and by restriction on morphisms.

On the other hand, the adjoint of the functor ind_{S_{m,n+1}}^{H_{m+n}} ∘ infl_{G_{m}}^{S_{m,n+1}} is given on objects by multiplication by the idempotent ζ_{m,n+1}γ_{n+1} ∈ F H_{m+n}. Now an easy matrix calculation in the group algebra F H_{m+n} reveals that ζ_{m,n+1}γ_{n+1} = ζ_{m,n}γ_{n}z_{ε_{m+n}}. Hence our functors are isomorphic.

Now we obtain the following fundamental result:

5.1e. Theorem. For any m, n ≥ 0, the following functors are isomorphic:

R_{G_{m}×G_{n}}^{G_{m+n}}(? ⊠ Γ_{n}) : mod(F G_{m}) → mod(F G_{m+n})
res_{G_{m+n}}^{H_{m+n}} ∘e_{n} : mod(F G_{m}) → mod(F G_{m+n}).

Proof. We will view all of the groups G_{m+n}, S_{m,n}, S_{m,n+1} as naturally embedded subgroups of H_{m+n} ⊂ G_{m+n+1}. Also, write P_{m,n} for the standard parabolic subgroup of G_{m+n} with Levi factor G_{m} × G_{n} and unipotent radical Z_{m,n}.

Take any F G_{m}-module M. We first observe that as (F γ_{n}) ↑^{U_{n}} ≅ Γ_{n}, there is an isomorphism of functors:

R_{G_{m}×G_{n}}^{G_{m+n}}( ? ⊠ Γ_{n}) ≅ ind_{S_{m,n}}^{G_{m+n}} ∘ infl_{G_{m}}^{S_{m,n}} .  (5.1.2)

Now, there is a factorization H_{m+n} = G_{m+n}S_{m,n+1}, and G_{m+n} ∩ S_{m,n+1} = S_{m,n}. So the Mackey theorem gives an isomorphism of functors

res_{G_{m+n}}^{H_{m+n}} ∘ ind_{S_{m,n+1}}^{H_{m+n}} ≅ ind_{S_{m,n}}^{G_{m+n}} ∘ res_{S_{m,n}}^{S_{m,n+1}} .  (5.1.3)

Since (F γ_{n+1}) ↓_{U_{n}} ≅ F γ_{n}, there is an isomorphism of functors

res_{S_{m,n}}^{S_{m,n+1}} ∘ infl_{G_{m}}^{S_{m,n+1}} ≅ infl_{G_{m}}^{S_{m,n}} .  (5.1.4)

Now combine (5.1.2), (5.1.3), (5.1.4) and Lemma 5.1d to complete the proof.

Theorem 5.1e has a number of important consequences. First, we obtain a proof of Gelfand’s theorem [Ge]:

5.1f. Corollary. Take σ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q}. Then, (i) M(σ) ↑^{H_{d}} ≅ e_{0}M(σ) ⊕ e_{d}I_{G_{0}}; (ii) M(σ) ↓_{H_{d−1}} ≅ e_{d−1}I_{G_{0}}.

Proof. (i) For any 0 ≤ i ≤ d and any irreducible F G_{d−i}-module L, compute using Frobenius reciprocity and Theorem 5.1e:

Hom_{H_{d}}(e_{i}L, M(σ) ↑^{H_{d}}) ≅ Hom_{G_{d}}((e_{i}L) ↓_{G_{d}}, M(σ))
≅ Hom_{G_{d}}(R_{G_{d−i}×G_{i}}^{G_{d}}(L ⊠ Γ_{i}), M(σ))
≅ Hom_{G_{d−i}×G_{i}}(L ⊠ Γ_{i},^{∗}R_{G_{d−i}×G_{i}}^{G_{d}}M(σ)).

Since M(σ) is cuspidal and irreducible, this is zero unless either i = 0, L = M(σ) or i = d, L = I_{G_{0}}. In the former case, the resulting hom space is obviously one dimensional, while in the latter case, it is one dimensional thanks to Corollary 2.5e(ii). This shows that the socle of M(σ) ↑^{H_{d}} has just two irreducible constituents, namely, e_{0}M(σ) and e_{d}I_{G_{0}}. Finally, by a dimension calculation using (2.4.1), we must have in fact that M(σ) ↑^{H_{d}} is equal to its socle.

(ii) Apply (i) and Lemma 5.1c.

The next corollary describes the restriction of an arbitrary irreducible F H_{n}- module to F G_{n}.

5.1g. Corollary. If M = e_{i}L is an irreducible F H_{n}-module belonging to the ith level,

M ↓_{G_{n}} ≅ R_{G_{n−i}×G_{i}}^{G_{n}}(L ⊠ Γ_{i}).

§5.2 Affine induction operators  97

Proof. This is immediate from Theorem 5.1e.

5.1h. Remark. Using Corollary 5.1g in characteristic 0, one can obtain alternative proofs of the branching rules of Zelevinsky [Z, Theorem 13.5] and Thoma [Th]. We sketch the argument. First, using (2.3f) one can decompose R_{G_{n−i}×G_{i}}^{G_{n}}(L ⊠ L′) for any irreducible modules L, L′: in characteristic zero this reduces to the Littlewood-Richardson rule. Second, one knows how to decompose Γ_{i} into a direct sum of irreducibles in characteristic 0: as in Theorem 2.5d, one gets all irreducible modules L(s, λ_{s}) for all s ∈ C_{ss}, where for s of the form (2.1.1) λ_{s} = ((1^{k_{1}}), …, (1^{k_{a}})), each appearing with multiplicity one. Hence, using Corollary 5.1g, one can decompose the restriction of any irreducible KH_{n}-module to KG_{n}. Finally, one uses Frobenius reciprocity and Lemma 5.1c(i) to deduce the formula for the restriction of any irreducible KG_{n}-module to KH_{n−1}. This gives the branching rules of Zelevinsky, and some further combinatorial argument deduces Thoma’s branching rule from this.

5.2. Affine induction operators

As motivation, we recall the definition of Green’s induction operator ⋄, originally introduced in [G_{1}]. Take integers m, n ≥ 0. Given an F G_{m}-module M and an F G_{n}- module N, let M ⋄ N denote the F G_{m+n}-module

M ⋄ N = R_{G_{n}×G_{m}}^{G_{n+m}}(M ⊠ N).

The resulting operator ⋄ allows one to ‘multiply’ two GL-modules to obtain a new GL-module. We will define two new operators ⋄_{ℓ} and ⋄_{r}, which will allow one to ‘multiply’ an AGL-module by a GL-module on the left and on the right, respectively, to obtain a new AGL-module.

To define ⋄_{ℓ}, let M be an F G_{m}-module and N be an F H_{n}-module. Consider the subgroups Q_{m,n} and X_{m,n} of H_{m+n} ⊂ G_{m+n+1} consisting of all matrices of the form

Q_{m,n} : [ \begin{matrix}& ∗_{m} & & & ∗ & & ∗ \\ & 0 & & & ∗_{n} & & ∗ \\ 0 & … & 0 & 0 & … & 0 & 1 \end{matrix} ] X_{m,n} : [ \begin{matrix}& I_{m} & & & ∗ & & ∗ \\ & 0 & & & I_{n} & & 0 \\ 0 & … & 0 & 0 & … & 0 & 1 \end{matrix} ]

There is an obvious surjective homomorphism

Q_{m,n} → G_{m} × H_{n}

with kernel X_{m,n}. Now define M ⋄_{ℓ} N to be the F H_{m+n}-module obtained by first inflating the F (G_{m}×H_{n})-module M ⊠N to Q_{m,n} along this surjection, then inducing as usual from Q_{m,n} to H_{m+n}.

Next, to define ⋄_{r}, let M be an F H_{m}-module and N be an F G_{n}-module. Consider the subgroups R_{m,n} and Y_{m,n} of H_{m+n} ⊂ G_{m+n+1} consisting of all matrices of the form

R_{m,n} : [ \begin{matrix}& ∗_{m} & & & ∗ & & ∗ \\ & 0 & & & ∗_{n} & & 0 \\ 0 & … & 0 & 0 & … & 0 & 1 \end{matrix} ] Y_{m,n} : [ \begin{matrix}& I_{m} & & & ∗ & & 0 \\ & 0 & & & I_{n} & & 0 \\ 0 & … & 0 & 0 & … & 0 & 1 \end{matrix} ]

This time, there is a surjective homomorphism

R_{m,n} → H_{m} × G_{n}

with kernel Y_{m,n}. Define M ⋄_{r} N to be the F H_{m+n}-module obtained by first inflating the F (H_{m}×G_{n})-module M ⊠N to R_{m,n} along this surjection, then inducing as usual from R_{m,n} to H_{m+n}.

The next lemma explains our interest in the operators ⋄_{ℓ} and ⋄_{r}.

5.2a. Lemma. Given an F G_{m}-module M and an F G_{n}-module N,

(M ⋄ N) ↓_{H_{m+n−1}} ≅ M ⋄_{ℓ} (N ↓_{H_{n−1}}) ⊕ (M ↓_{H_{m−1}}) ⋄_{r} N.

Proof. Let P_{m,n} denote the standard parabolic subgroup of G_{m+n} with Levi factor G_{m} ×G_{n}, which is the stabilizer of the subspace of W_{m+n} spanned by {w_{1}, …, w_{m}}. The subgroup H_{m+n−1} has precisely two orbits on the set of m-dimensional subspaces of W_{m+n}, with representatives the subspaces spanned by {w_{1}, …, w_{m−1}, w_{m}} and {w_{1}, …, w_{m−1}, w_{m+n}}. So there are two H_{m+n−1}\G_{m+n}/P_{m,n}-double cosets in G_{m+n}, with representatives 1 and π, where π is the permutation matrix corresponding to the cycle (m + n m + n − 1 … m) (so πw_{m} = w_{m+n}). Now the Mackey formula gives us that (M ⋄ N) ↓_{H_{m+n−1}} is isomorphic to

(M#N) ↓_{P_{m,n}∩H_{m+n−1}}↑^{H_{m+n−1}} ⊕(conj_{π}(M#N)) ↓_{^{π}P_{m,n}∩H_{m+n−1}}↑^{H_{m+n−1}},

where M#N denotes the F P_{m,n}-module obtained by inflation from the F (G_{m}×G_{n})- module M ⊠ N. Observe that P_{m,n} ∩ H_{m+n−1} = Q_{m,n−1}, so that the first term gives us precisely M ⋄_{ℓ} (N ↓_{H_{n−1}}), and ^{π}P_{m,n} ∩ H_{m+n−1} = R_{m−1,n} so the second term is isomorphic to (M ↓_{H_{m−1}}) ⋄_{r} N.

5.2b. Theorem. (i) Given an F G_{m}-module M and an F H_{n}-module N,

f_{0}(M ⋄_{ℓ} N) ≅ M ⋄ (f_{0}N),
f_{+}(M ⋄_{ℓ} N) ≅ M ⋄_{ℓ} (f_{+}N).

(ii) Given an F H_{m}-module M and an F G_{n}-module N,

f_{0}(M ⋄_{r} N) ≅ (f_{0}M) ⋄ N,
f_{+}(M ⋄_{r} N) ≅ (f_{0}M) ⋄_{ℓ} (N ↓_{H_{n−1}}) ⊕ (f_{+}M) ⋄_{r} N.

Proof. We just explain how to prove the final formula

f_{+}(M ⋄_{r} N) ≅ (f_{0}M) ⋄_{ℓ} (N ↓_{H_{n−1}}) ⊕ (f_{+}M) ⋄_{r} N  (5.2.1)

appearing in (ii). The other three formulae are proved in similar (actually much easier) ways and we omit the details.

We shall identify W_{m} (resp. W_{n}) with the subspace of W_{m+n} spanned by w_{1}, …, w_{m} (resp. w_{m+1}, …, w_{m+n}), so W_{m+n} = W_{m} ⊕ W_{n}. Identify X_{m+n} with X_{m} ⊕ X_{n} in a similar way. We also need some notation for coset representatives, so write P_{m,n} for the standard parabolic subgroup of G_{m+n} with Levi factor G_{m} × G_{n}. Let Ω_{m,n} denote the set of (m + n)-tuples:

Ω_{m,n} = \{i = (i_{1}, i_{2}, …, i_{m+n}) | \begin{matrix}\{i_{1}, i_{2}, …, i_{m+n}\} = \{1, 2, …, m + n\}, \\ i_{k} < i_{k+1} unless k = m \end{matrix} \} .

For i = (i_{1}, …, i_{m+n}) ∈ Ω_{m,n}, let w_{i} ∈ Σ_{m+n} ⊂ G_{m+n} be the permutation matrix corresponding to the permutation j ↦ i_{j} for 1 ≤ j ≤ m + n. We note that {w_{i} | i ∈ Ω_{m,n}} is precisely the set of distinguished Σ_{m+n}/Σ_{m}×Σ_{n}-coset representatives. Also define

U_{i} = {u ∈ U_{m+n} | u_{i_{j}i_{k}} = 0 for every 1 ≤ j < k ≤ m + n with i_{j} < i_{k}}.

Finally, let π be the permutation matrix for the cycle (m+n m+n−1 … m+1 m) of Σ_{m+n}. Now, regarding all of H_{m+n−1}, Q_{m,n−1} and R_{m−1,n} as naturally embedded subgroups of G_{m+n} ⊂ H_{m+n}, we claim:

(5.2c) (i) {uw_{i}y | i ∈ Ω_{m,n}, u ∈ U_{i}, y ∈ W_{n}} is a set of H_{m+n}/R_{m,n}-coset representatives.

(ii) {uw_{i} | i ∈ Ω_{m,n}, i_{m+n} = m + n, u ∈ U_{i}} is a set of H_{m+n−1}/Q_{m,n−1}-coset representatives.

(iii) {uw_{i}π^{−1} | i ∈ Ω_{m,n}, i_{m} = m + n, u ∈ U_{i}} is a set of H_{m+n−1}/R_{m−1,n}-coset representatives.

To prove (5.2c), we first observe by the Bruhat decomposition that {uw_{i} | i ∈ Ω_{m,n}, u ∈ U_{i}} is a set of G_{m+n}/P_{m,n}-coset representatives. Parts (i) and (ii) follow easily from this. To see (iii), let W′_{n} denote the set

{w ∈ U_{m+n} | for 1 ≤ j < k ≤ m + n, w_{jk} = 0 unless j ≥ m, k = m + n}.

As a variant of (i), the set {uw_{i}y | i ∈ Ω_{m−1,n+1}, i_{m+n} = m + n, u ∈ U_{i}, y ∈ W′_{n}} is a set of H_{m+n−1}/R_{m−1,n}-coset representatives, working inside the group G_{m+n}. An explicit matrix calculation reveals that this is precisely the same set as in (iii).

Now we can prove (5.2.1). Let M and N be as in (ii). Choose a basis n_{1}, …, n_{b} for N and, for each ε ∈ X_{m}, a basis m_{ε,1}, …, m_{ε,a_{ε}} for M_{ε}. Also let {z_{ζ} | ζ ∈ X_{n}} be a basis of the semisimple group algebra F W_{n} such that wz_{ζ} = ζ(w)z_{ζ} for each w ∈ W_{n}. Using (5.2c)(i) and the definition of induced module, we can write down a basis of M ⋄_{r} N = F H_{m+n} ⊗_{F R_{m,n}} (M ⊗ N) as follows:

{uw_{i}z_{ζ} ⊗ m_{ε,j} ⊗ n_{k} | i ∈ Ω_{m,n}, u ∈ U_{i}, ε ∈ X_{m}, ζ ∈ X_{n}, 1 ≤ j ≤ a_{ε}, 1 ≤ k ≤ b}.

The basis vector uw_{i}z_{ζ} ⊗ m_{ε,j} ⊗ n_{k} has weight uw_{i}(ζ + ε). Now, for i ∈ Ω_{m,n}, u ∈ U_{i}, we have that w_{i}^{−1}u^{−1}ε_{m+n} = w_{i}^{−1}ε_{m+n} which either equals ε_{m+n} if i_{m+n} = m + n, or ε_{m} if i_{m} = m + n. Hence:

(5.2d) For i ∈ Ω_{m,n}, u ∈ U_{i}, ζ ∈ X_{n}, ε ∈ X_{m}, we have that uw_{i}(ζ + ε) = ε_{m+n} if and only if either ε = 0, ζ = ε_{m+n}, i_{m+n} = m + n or ε = ε_{m}, ζ = 0, i_{m} = m + n.

We deduce that the ε_{m+n}-weight space of M ⋄_{r} N splits as a direct sum of H_{m+n−1}-modules, where the first has basis

\{(uw_{i})z_{ε_{m+n}} ⊗ m_{0,j} ⊗ n_{k} | \begin{matrix}i ∈ Ω_{m,n}, i_{m+n} = m + n, u ∈ U_{i}, \\ 1 ≤ j ≤ a_{0}, 1 ≤ k ≤ b \end{matrix} \}  (5.2.2)

and the second has basis

\{(uw_{i}π^{−1})πz_{0} ⊗ m_{ε_{m},j} ⊗ n_{k} | \begin{matrix}i ∈ Ω_{m,n}, i_{m} = m + n, u ∈ U_{i}, \\ 1 ≤ j ≤ a_{ε_{m}}, 1 ≤ k ≤ b \end{matrix} \}.  (5.2.3)

Now, the z_{ε_{m+n}} ⊗m_{0,j} ⊗n_{k} span an F Q_{m,n−1}-submodule of M ⋄_{r} N isomorphic to the module obtained by inflating (f_{0}M) ⊠ (N ↓ H_{n−1}) along the natural quotient map Q_{m,n−1} → G_{m} ×H_{n−1}. Combining this with (5.2c)(ii), the definition of the operator ⋄_{ℓ} and the characterization of induced modules, we see that the vectors (5.2.2) span a module isomorphic to (f_{0}M) ⋄_{ℓ} (N ↓_{H_{n−1}}). Similarly, the πz_{0} ⊗ m_{ε_{m},j} ⊗ n_{k} span an F R_{m−1,n}-submodule of M ⋄_{r} N isomorphic to the module obtained by inflating (f_{+}M)⊠N along the natural quotient map R_{m−1,n} → H_{m−1}×G_{n}. So using (5.2c)(iii) and the characterization of induced modules, we see that the vectors in (5.2.3) span a module isomorphic to (f_{+}M) ⋄_{r} N.

5.2e. Corollary. (i) Given an F G_{m}-module M and an F H_{n}-module N,

f_{i}(M ⋄_{ℓ} N) ≅ M ⋄ (f_{i}N).

(ii) Given an F H_{m}-module M and an F G_{n}-module N,

f_{i}(M ⋄_{r} N) ≅ (f_{i}M) ⋄ N ⊕ ⨁_{j=0}^{i−1}(f_{j}M) ⋄ (f_{i−j−1}(N ↓_{H_{n−1}})).

Proof. Use the corresponding parts of Theorem 5.2b and induction on i.

5.2f. Corollary. Suppose that for i = 1, 2, 3, we are given an F G_{n_{i}}-module M_{i} and an F H_{n_{i}}-module N_{i}. Then, there are isomorphisms:

M_{1} ⋄ (M_{2} ⋄ M_{3}) ≅ (M_{1} ⋄ M_{2}) ⋄ M_{3},  (5.2.4)
N_{1} ⋄_{r} (M_{2} ⋄ M_{3}) ≅ (N_{1} ⋄_{r} M_{2}) ⋄_{r} M_{3},  (5.2.5)
M_{1} ⋄_{ℓ} (N_{2} ⋄_{r} M_{3}) ≅ (M_{1} ⋄_{ℓ} N_{2}) ⋄_{r} M_{3},  (5.2.6)
(M_{1} ⋄ M_{2}) ⋄_{ℓ} N_{3} ≅ M_{1} ⋄_{ℓ} (M_{2} ⋄_{ℓ} N_{3}).  (5.2.7)

Proof. Of these, (5.2.4) is a consequence of associativity of Harish-Chandra induction (2.2b). The remaining formulae follow in very similar ways from (5.2.4) and Corollary 5.2e, so we just prove (5.2.5) (the hardest) to illustrate the argument. In view of Lemma 5.1a, it suffices to check that for each i ≥ 0,

f_{i}(N_{1} ⋄_{r} (M_{2} ⋄ M_{3})) ≅ f_{i}((N_{1} ⋄_{r} M_{2}) ⋄_{r} M_{3}).

One then expands both sides using (5.2.4), Corollary 5.2e and Lemma 5.2a; in both cases, one obtains the formula

(f_{i}N_{1}) ⋄ M_{2} ⋄ M_{3} ⊕ ⨁_{j=0}^{i−1} ⨁_{k=0}^{i−j−2}(f_{j}N_{1}) ⋄ f_{k}(M_{2} ↓_{H_{n_{2}−1}}) ⋄ f_{i−j−k−2}(M_{3} ↓_{H_{n_{3}−1}})

⊕ ⨁_{j=0}^{i−1}(f_{j}N_{1}) ⋄ M_{2} ⋄ f_{i−j−1}(M_{3} ↓_{H_{n_{3}−1}})

⊕ ⨁_{j=0}^{i−1}(f_{j}N_{1}) ⋄ f_{i−j−1}(M_{2} ↓_{H_{n_{2}}−1}) ⋄ M_{3}.

Hence, the two are isomorphic.

In view of Corollary 5.2f, any multiple product involving the operators ⋄, ⋄_{r} and ⋄_{ℓ} can always be rewritten as an expression containing the operators ⋄_{r} and ⋄_{ℓ} at most once, and there is never any ambiguity over brackets.

5.2g. Corollary. Let M be an F G_{m}-module and N be an F G_{n}-module. Write M ↑^{H_{m}}= ⨁_{i≥0} e_{i}M_{i} and N ↑^{H_{n}}= ⨁_{j≥0} e_{j}N_{j}. Then:

(M ⋄ N) ↑^{H_{m+n}} ≅ ⨁_{i,j≥0} e_{i+j}(M_{i} ⋄ N_{j}),  (5.2.8)
(M ⋄ N) ↓_{H_{m+n−1}} ≅ ⨁_{{i,j≥0 \atop (i,j)≠(0,0)}} e_{i+j−1}(M_{i} ⋄ N_{j}).  (5.2.9)

Proof. We prove (5.2.9); (5.2.8) then follows immediately by Lemma 5.1c(ii). It suffices by Lemma 5.1a(ii) and Lemma 5.2a to show that for all k ≥ 0,

f_{k}((M ↓_{H_{m−1}}) ⋄_{r} N) ⊕ M ⋄_{ℓ} (N ↓_{H_{n−1}})) ≅ ⨁_{i,j≥0,i+j−1=k} M_{i} ⋄ N_{j}.

Note that M = M_{0} and M ↓_{H_{m−1}} ≅ ⨁_{i≥0} e_{i}M_{i+1}, and similarly for N, thanks to Lemma 5.1c. So, using in addition Corollary 5.2e, we see

f_{k}((M ↓_{H_{m−1}}) ⋄_{r} N) ⊕ M ⋄_{ℓ} (N ↓_{H_{n−1}})) ≅ M_{0} ⋄ N_{k+1} ⊕ M_{k+1} ⋄ N_{0} ⊕ ⨁_{j=0}^{k−1} M_{j+1} ⋄ N_{k−j}.

The proof follows.

5.3. The affine cuspidal algebra

Now we introduce an affine analogue of the cuspidal algebra. Fix σ ∈ 𝔽̄_{q}^{×} of degree d over 𝔽_{q} and k ≥ 0. Set n = kd throughout the section. Recall the definition of the F G_{n}-module M^{k}(σ) from (2.4.2); in the case k = 0, this is the trivial module over the trivial group G_{0}. Note that for each 0 ≤ j ≤ k, the module e_{dj}M^{k−j}(σ) is an F H_{n}-module. Define the affine cuspidal algebra

D_{σ,k} = D_{F,(σ)^{k}}(GL_{n}(𝔽_{q})) = F H_{n}/ ⋂_{j=0}^{k} ann_{F H_{n}}(e_{dj}M^{k−j}(σ)).

So D_{σ,k} can be thought of as the image of F H_{n} under the representation afforded by the module ⨁_{j=0}^{k} e_{dj}M^{k−j}(σ). The following lemma should serve as motivation for this definition:

5.3a. Lemma. M^{k}(σ) ↑^{H_{n}} ≅ ⨁_{j=0}^{k} (\begin{matrix}k \\ j \end{matrix})e_{dj}M^{k−j}(σ).

Proof. Use induction on k. The case k = 1 follows from Corollary 5.1f(i). For the induction step, write M^{k}(σ) = M^{k−1}(σ) ⋄ M(σ) and apply Corollary 5.2g.

5.3b. Corollary. If M is a C_{σ,k}-module, then M ↑^{H_{n}} is a D_{σ,k}-module.

Proof. It suffices to check this on projective indecomposables. In turn, since every projective indecomposable C_{σ,k}-module is a submodule of M^{k}(σ) according to Theorem 3.4g, we just need to check this in the special case M = M^{k}(σ). But in that case, the corollary follows from Lemma 5.3a and the definition of D_{σ,k}.

Now fix h ≥ k for the remainder of the chapter. Write S_{h,k} for the q^{d}-Schur algebra S_{F,q^{d}}(h, k) as in §1.2. We need to work now with the algebra

S_{h,≤k} = ⨁_{i=0}^{k} S_{h,i}.  (5.3.1)

This is a quasi-hereditary algebra with weight poset Λ^{+}(h, ≤ k) = ⋃_{i=0}^{k} Λ^{+}(h, i), partially ordered by the union of the dominance orders on each Λ^{+}(h, i). For

§5.3 The affine cuspidal algebra  103

λ ∈ Λ^{+}(h, ≤ k), we write L_{h}(λ), Δ_{h}(λ), etc... for the usual modules over S_{h,≤k} corresponding to the partition λ. So if λ ∈ Λ^{+}(h, i) then these are the usual modules over the summand S_{h,i} of S_{h,≤k}, with all other summands acting as zero.

Also, for any 0 ≤ j ≤ k, define

Z_{j} = ⨁_{ν∈Λ(h,k−j)} Ż^{ν}(σ),  (5.3.2)

which is an (F G_{n−dj}, S_{h,k−j})-bimodule. Applying the functor e_{dj}, we obtain an (F H_{n}, S_{h,k−j})-bimodule e_{dj}Z_{j}. Hence, the module

Ẑ = ⨁_{j=0}^{k} e_{dj}Z_{j}  (5.3.3)

is naturally an (F H_{n}, S_{h,≤k})-bimodule.

5.3c. Theorem. With notation as above, Ẑ is a D_{σ,k}-module which is a projective generator for mod(D_{σ,k}), and the endomorphism algebra End_{D_{σ,k}}(Ẑ) is precisely the algebra S_{h,≤k}.

Proof. Let us first recall some known facts from Theorem 3.4g. First, for each 0 ≤ j ≤ k, S_{h,k−j} is the endomorphism algebra End_{F G_{n−dj}}(Z_{j}). Second, Z_{j} contains a copy of each of its composition factors in its head. Finally, as explained in the proof of Theorem 3.4g, Lemma 3.4f gives that

dim Hom_{F G_{n−dj}}(P_{j}, Z_{j}) = dim Hom_{F G_{n−dj}}(Z_{j}, Z_{j})

where P_{j} is the projective cover of Z_{j} in the category mod(F G_{n−dj}).

Now, for 0 ≤ j ≤ k, we regard Z_{j} (resp. P_{j}) as an F G_{n} ⊕ F G_{n−1} ⊕ ⋯ ⊕ F G_{0}- module so that the summand F G_{n−dj} acts on Z_{j} (resp. P_{j}) as given and all other summands act as zero. Then, applying the Morita equivalence e_{0} ⊕ ⋯ ⊕ e_{n} of Lemma 5.1a(ii) to the statements in the previous paragraph, we deduce immediately that:

(i) End_{F H_{n}}(Ẑ) = S_{h,≤k};

(ii) Ẑ contains a copy of every composition factor in its head;

(iii) dim Hom_{F H_{n}}(P̂, Ẑ) = dim Hom_{F H_{n}}(Ẑ, Ẑ), where P̂ is the projective cover of Ẑ in the category mod(F H_{n}).

Now we prove the theorem. Since each Z_{j} is a direct sum of submodules of M^{k−j}(σ) by definition, e_{dj}Z_{j} is a direct sum of submodules of e_{dj}M^{k−j}(σ). Hence, ann_{F H_{n}}(e_{dj}M^{k−j}(σ)) annihilates e_{dj}Z_{j}. This shows that the action of F H_{n} on Ẑ factors through the quotient D_{σ,k} to induce a well-defined D_{σ,k}-module.

Moreover, Ẑ contains a summand isomorphic to ⨁_{j=0}^{k} e_{dj}M^{k−j}(σ), hence Ẑ is a faithful D_{σ,k}-module. The endomorphism algebra End_{D_{σ,k}}(Ẑ) is just End_{F H_{n}}(Ẑ), hence isomorphic to S_{h,≤k} by (i). Using (iii), Lemma 3.2a and faithfulness, Ẑ is a projective D_{σ,k}-module. Finally, it is a projective generator for mod(D_{σ,k}) by (ii) and faithfulness, using the same argument as in the last paragraph of the proof of Theorem 3.4g.

Now we proceed as we did in §3.5. Introduce functors

α_{σ,h,≤k} : mod(D_{σ,k}) → mod(S_{h,≤k}),  α_{σ,h,≤k} = Hom_{D_{σ,k}}(Ẑ,?),  (5.3.4)
β_{σ,h,≤k} : mod(S_{h,≤k}) → mod(D_{σ,k}),  β_{σ,h,≤k} = Ẑ⊗_{S_{h,≤k}}?.  (5.3.5)

Theorem 5.3c immediately implies that:

(5.3d) The functors α_{σ,h,≤k} and β_{σ,h,≤k} are mutually inverse equivalences of categories between mod(D_{σ,k}) and mod(S_{h,≤k}).

The next lemma identifies the various standard modules for D_{σ,k} coming from the above Morita equivalence:

5.3e. Lemma. For any 0 ≤ j ≤ k and λ ⊢ (k − j),

e_{dj}L(σ, λ) ≅ β_{σ,h,≤k}(L_{h}(λ′)),
e_{dj}Δ(σ, λ) ≅ β_{σ,h,≤k}(Δ_{h}(λ′)),
e_{dj}∇(σ, λ) ≅ β_{σ,h,≤k}(∇_{h}(λ′)),
e_{dj}T (σ, λ) ≅ β_{σ,h,≤k}(T_{h}(λ′)).

Proof. We prove something more general. For each 0 ≤ j ≤ k, the algebra S_{h,k−j} is a summand of S_{h,≤k}. So there is a natural inflation functor

infl_{j} : mod(S_{h,k−j}) → mod(S_{h,≤k}).

The lemma follows immediately from the claim:

(5.3f) The following functors are isomorphic:

e_{dj} ∘ β_{σ,h,k−j} : mod(S_{h,k−j}) → mod(F H_{n}),
β_{σ,h,≤k} ∘ infl_{j} : mod(S_{h,k−j}) → mod(F H_{n}).

To prove (5.3f), let i_{j} ∈ S_{h,≤k} be the central idempotent corresponding to the identity of the summand S_{h,k−j} of S_{h,≤k}. So, S_{h,k−j} = i_{j}S_{h,≤k}i_{j}, and the functor infl_{j} is the functor S_{h,≤k}i_{j}⊗_{S_{h,k−j}}?. Also observe that Ẑi_{j} = e_{dj}Z_{j}, so there is an isomorphism of functors

β_{σ,h,≤k} ∘ infl_{j} = Ẑ ⊗_{S_{h,≤k}} (S_{h,≤k}i_{j}⊗_{S_{h,k−j}}?) ≅ Ẑi_{j}⊗_{S_{h,k−j}}? = (e_{dj}Z_{j})⊗_{S_{h,k−j}}?.

Now use Lemma 5.1d and associativity of tensor product to deduce that

(e_{dj}Z_{j})⊗_{S_{h,k−j}}? ≅ e_{dj} ∘ (Z_{j}⊗_{S_{h,k−j}}?) = e_{dj} ∘ β_{σ,h,k−j},

§5.4 The branching rule from GL_{n} to AGL_{n−1}  105

completing the proof of (5.3f).

To summarize: the algebra D_{σ,k} is a quasi-hereditary algebra with weight poset {λ ⊢ (k −j)|0 ≤ j ≤ k} partially ordered by ≥ (the opposite order to S_{h,≤k} since we have transposed partitions). Moreover, the {e_{dj}L(σ, λ)}, {e_{dj}Δ(σ, λ)}, {e_{dj}∇(σ, λ)} and {e_{dj}T (σ, λ)} for all 0 ≤ j ≤ k, λ ⊢ (k − j) give the irreducible, standard, costandard and indecomposable tilting D_{σ,k}-modules.

5.4. The branching rule from GL_{n} to AGL_{n−1}

According to Lemma 5.1c(ii), understanding induction from G_{n} to H_{n} or restriction from G_{n} to H_{n−1} are essentially equivalent problems. It turns out that it is more convenient to study induction first and then deduce the consequences for restriction. Continue with the notation set up in the previous section. Also, let S_{h+1,k} denote the q^{d}-Schur algebra S_{F,q^{d}}(h + 1, k) and Z denote the (F G_{n}, S_{h+1,k})-bimodule

Z = ⨁_{ν∈Λ(h+1,k)} Ż^{ν}(σ).

So Z is a projective generator for mod(C_{σ,k}). Now, the Levi subalgebra S_{ν,k} of (1.3g), in the special case ν = (h, 1), is a subalgebra of S_{h+1,k}, isomorphic by (1.3.8) to

⨁_{j=0}^{k} S_{h,k−j} ⊗ S_{1,j}.

For any j, the algebra S_{1,j} is one dimensional, so we can identify this subalgebra with the algebra S_{h,≤k} of (5.3.1). To be explicit, for μ ∈ Λ(h, k−j), write μ[j] for the (h + 1)-tuple in Λ(h + 1, k) with (h + 1)-entry equal to j, and all other entries being the same as in the h-tuple μ. Also view Σ_{k−j} as the naturally embedded subgroup of Σ_{k}. Then:

(5.4a) The embedding S_{h,≤k} ↪ S_{h+1,k} maps the standard basis element ϕ_{μ,λ}^{u} of S_{h,k−j} ⊂ S_{h,≤k}, for μ, λ ∈ Λ(h, k−j), u ∈ D_{μ,λ} to the standard basis element ϕ_{μ[j],λ[j]}^{u′} of S_{h+1,k}, where u′ is the image of u under the natural embedding Σ_{k−j} ↪ Σ_{k}.

For the next lemma, we regard the restriction Ẑ ↓_{G_{n}} of the (F H_{n}, S_{h,≤k})- bimodule Ẑ of (5.3.3) as an (F G_{n}, S_{h,≤k})-bimodule in the natural way.

5.4b. Lemma. There is a surjective (F G_{n}, S_{h,≤k})-bimodule homomorphism

θ : Ẑ ↓_{G_{n}}→ Z,

such that all other F G_{n}-homomorphisms Ẑ ↓_{G_{n}}→ Z factor through θ.

Proof. Consider

Z′ = ⨁_{j=0}^{k} Z_{j} ⋄ Z^{j}(σ),  (5.4.1)

where Z_{j} is as in (5.3.2). Each Z_{j} is an (F G_{n−dj}, S_{h,k−j})-bimodule, so the summand Z_{j} ⋄Z^{j}(σ) of Z′ is an (F G_{n}, S_{h,k−j})-bimodule (we have applied the functor ?⋄Z^{j}(σ)). This makes Z′ into an (F G_{n}, S_{h,≤k})-bimodule. Now let ω : Z′ → Z be the evident F G_{n}-module isomorphism that identifies the summand Ż^{ν}(σ) ⋄ Z^{j}(σ) of Z′, for any 0 ≤ j ≤ k and ν ∈ Λ(h, k − j), with the summand Ż^{ν[j]}(σ) of Z. We claim that ω is a right S_{h,≤k}-module homomorphism, hence an isomorphism of (F G_{n}, S_{h,≤k})- bimodules.

For the claim, we consider a standard basis element ϕ_{μ,λ}^{u} of the summand S_{h,k−j} of S_{h,≤k}, for μ, λ ∈ Λ(h, k − j) and u ∈ D_{μ,λ}. By (5.4a), this coincides with the element ϕ_{μ[j],λ[j]}^{u′} of S_{h+1,k}, so by Theorem 3.4c it acts as zero on all summands of Z except for the ones of the form Ż^{ν[j]}(σ) for ν ∈ Λ(h, k − j). Moreover, the action of ϕ_{μ[j],λ[j]}^{u} on such a summand Ż^{ν[j]}(σ) is zero unless ν = μ, in which case the action is

induced by right multiplication in M^{k}(σ) by the element ∑_{w∈Σ_{μ[j]}u′Σ_{λ[j]}∩D_{μ[j]}^{−1}} T_{w}^{#} ∈

H_{k}. On the other hand, the action of the element ϕ_{μ,λ}^{u} on Z′ is zero on all summands of Z′ except Z_{j} ⋄Z^{j}(σ), while on any summand Ż^{ν}(σ)⋄Z^{j}(σ) of Z_{j} ⋄Z^{j}(σ) the action of ϕ_{μ,λ}^{u} is induced by applying the functor ? ⋄ Z^{j}(σ) to the action coming from right multiplication in M^{k−j}(σ) by the element ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{μ}^{−1}} T_{w}^{#} ∈ H_{k−j}. To prove the

claim, it just remains to observe that the element ∑_{w∈Σ_{μ[j]}u′Σ_{λ[j]}∩D_{μ[j]}^{−1}} T_{w}^{#} of H_{k} is

the same as the element ∑_{w∈Σ_{μ}uΣ_{λ}∩D_{μ}^{−1}} T_{w}^{#} of the naturally embedded subalgebra H_{k−j} ⊂ H_{k−j} ⊗ H_{j} ⊂ H_{k}. So the two actions do indeed agree under the bijection ω because of Lemma 3.2f(i).

Now we can prove the lemma. For each 0 ≤ j ≤ k, fix a non-zero surjection θ_{j} : Γ_{dj} → Z^{j}(σ), which exists and is unique up to scalars by Lemma 3.4d(i) and (3.4h). According to Theorem 5.1e, the (F G_{n}, S_{h,≤k})-bimodule Ẑ ↓_{G_{n}} can be identified with the bimodule ⨁_{j=0}^{k} Z_{j} ⋄ Γ_{dj}. Also, by the claim, we can use the map ω to identify the (F G_{n}, S_{h,≤k})-bimodule Z with ⨁_{j=0}^{k} Z_{j} ⋄Z^{j}(σ). Now define θ = ⨁_{j=0}^{k} id_{Z_{j}} ⊗ϕ_{j}, to give the desired a surjective (F G_{n}, S_{h,≤k})-bimodule homomorphism

θ : Ẑ ↓_{G_{n}}= ⨁_{j=0}^{k} Z_{j} ⋄ Γ_{dj} → ⨁_{j=0}^{k} Z_{j} ⋄ Z^{j}(σ) = Z.

It just remains to check that any other F G_{n}-homomorphism ⨁_{j=0}^{k} Z_{j} ⋄ Γ_{dj} → ⨁_{j=0}^{k} Z_{j} ⋄ Z^{j}(σ) factors through θ. Expanding the direct sums in the definition of Z_{j}, this follows if we can show that

dim Hom_{F G_{n}}(Ż^{ν}(σ) ⋄ Γ_{dj}, Ż^{μ}(σ)) = dim Hom_{F G_{n}}(Ż^{ν}(σ) ⋄ Z^{j}(σ), Ż^{μ}(σ))

for every ν ∈ Λ(h, k − j), μ ∈ Λ(h + 1, k). But given (3.4h), this follows at once from the stronger statement of Lemma 3.4f.

We remark that C_{σ,k} is not a subalgebra of D_{σ,k} in any natural way. However, by Corollary 5.3b, we can still define a functor

ind_{C_{σ,k}}^{D_{σ,k}} : mod(C_{σ,k}) → mod(D_{σ,k}),

namely, the restriction of the functor ind_{G_{n}}^{H_{n}} : mod(F G_{n}) → mod(F H_{n}) to the full subcategories mod(C_{σ,k}) ⊂ mod(F G_{n}) and mod(D_{σ,k}) ⊂ mod(F H_{n}). The following theorem relates this induction functor to the restriction functor from S_{h+1,k} to the Levi subalgebra S_{h,≤k}:

5.4c. Theorem. The following functors are isomorphic:

ind_{C_{σ,k}}^{D_{σ,k}} ∘β_{σ,h+1,k} : mod(S_{h+1,k}) → mod(D_{σ,k}),
β_{σ,h,≤k} ∘ res_{S_{h,≤k}}^{S_{h+1,k}} : mod(S_{h+1,k}) → mod(D_{σ,k}).

Proof. Since Z is an (F G_{n}, S_{h+1,k})-bimodule, we can view Hom_{F G_{n}}(Z, Z) as an (S_{h+1,k}, S_{h+1,k})-bimodule, where the actions are defined by (sψ)(z) = ψ(zs) and (ψs)(z) = ψ(z)s for s ∈ S_{h+1,k}, z ∈ Z and ψ : Z → Z. Then we can restrict the left action to the subalgebra S_{h,≤k} ⊂ S_{h+1,k} to view Hom_{F G_{n}}(Z, Z) an (S_{h,≤k}, S_{h+1,k})- bimodule. We first show that

ϕ : Hom_{F G_{n}}(Z, Z) ⟶^{∼} Hom_{F G_{n}}(Ẑ ↓_{G_{n}}, Z), f ↦ f ∘ θ,

is an (S_{h,≤k}, S_{h+1,k})-bimodule isomorphism, where θ is the surjection of Lemma 5.4b. Well, ϕ is bijective since θ is a surjection and every F G_{n}-homomorphism Ẑ ↓_{G_{n}}→ Z factors through θ. It is obviously a right S_{h+1,k}-module map, and a routine check using the fact that θ is S_{h,≤k}-equivariant shows that it is a left S_{h,≤k}-module map.

Now note that the (S_{h+1,k}, S_{h+1,k})-bimodule Hom_{F G_{n}}(Z, Z) is just the regular bimodule S_{h+1,k} itself, in view of Theorem 3.4c. So, viewing S_{h+1,k} as an (S_{h,≤k}, S_{h+1,k})-bimodule by restricting the left action, we have shown that the bimodules Hom_{F G_{n}}(Ẑ ↓_{G_{n}}, Z) and S_{h+1,k} are isomorphic. Hence, by Frobenius reciprocity, there is an isomorphism S_{h+1,k} ≅ Hom_{F G_{n}}(Ẑ, Z ↑^{H_{n}}) as (S_{h,≤k}, S_{h+1,k})- bimodules. Since Z is a C_{σ,k}-module, we even have by Corollary 5.3b that

S_{h+1,k} ≅ Hom_{D_{σ,k}}(Ẑ, ind_{C_{σ,k}}^{D_{σ,k}} Z)  (5.4.2)

as (S_{h,≤k}, S_{h+1,k})-bimodules.

Now regard the functors α_{σ,h,≤k} and β_{σ,j,≤k} of (5.3.4)–(5.3.5) as bimodule functors in the obvious way to obtain functors:

α_{σ,h,≤k} : bimod(D_{σ,k}, S_{h+1,k}) → bimod(S_{h,≤k}, S_{h+1,k}),
β_{σ,h,≤k} : bimod(S_{h,≤k}, S_{h+1,k}) → bimod(D_{σ,k}, S_{h+1,k}).

By (5.3d) and naturality, β_{σ,h,≤k} ∘ α_{σ,h,≤k} is isomorphic to the identity functor on bimod(D_{σ,k}, S_{h+1,k}). So if we apply β_{σ,h,≤k} to (5.4.2) we deduce at once that there is a (D_{σ,k}, S_{h+1,k})-bimodule isomorphism

Ẑ ⊗_{S_{h,≤k}} S_{h+1,k} ≅ ind_{C_{σ,k}}^{D_{σ,k}} Z.  (5.4.3)

Now we can prove the theorem. For any S_{h+1,k}-module M, (5.4.3) and associativity of tensor products gives us natural isomorphisms:

β_{σ,h,≤k} ∘ res_{S_{h,≤k}}^{S_{h+1,k}}(M) = Ẑ ⊗_{S_{h,≤k}} (res_{S_{h,≤k}}^{S_{h+1,k}} M)
≅ Ẑ ⊗_{S_{h,≤k}} (S_{h+1,k} ⊗_{S_{h+1,k}} M)
≅ (Ẑ ⊗_{S_{h,≤k}} S_{h+1,k}) ⊗_{S_{h+1,k}} M
≅ (ind_{C_{σ,k}}^{D_{σ,k}} Z) ⊗_{S_{h+1,k}} M
≅ ind_{C_{σ,k}}^{D_{σ,k}}(Z ⊗_{S_{h+1,k}} M) = ind_{C_{σ,k}}^{D_{σ,k}} ∘β_{σ,h,k}(M).

Hence, the functors β_{σ,h,≤k} ∘ res_{S_{h,≤k}}^{S_{h+1,k}} and ind_{C_{σ,k}}^{D_{σ,k}} ∘β_{σ,h,k} are isomorphic.

Using Theorem 5.4c, we obtain the following description of composition multiplicities:

5.4d. Corollary. Suppose that λ ⊢ k.

(i) All composition factors of L(σ, λ) ↑^{H_{n}} are of the form e_{dj}L(σ, μ) for 0 ≤ j ≤ k and μ ⊢ (k − j). Moreover, for such μ, j,

[L(σ, λ) ↑^{H_{n}}: e_{dj}L(σ, μ)] = [res_{S_{h,≤k}}^{S_{h+1,k}} L_{h+1}(λ′) : L_{h}(μ′)].

(ii) All composition factors of L(σ, λ) ↓_{H_{n−1}} are of the form e_{dj−1}L(σ, μ) for 1 ≤ j ≤ k and μ ⊢ (k − j). Moreover, for such μ, j,

[L(σ, λ) ↓_{H_{n−1}}: e_{dj−1}L(σ, μ)] = [res_{S_{h,≤k}}^{S_{h+1,k}} L_{h+1}(λ′) : L_{h}(μ′)].

Proof. (i) By the theorem, (3.5.3) and the definition of ind_{C_{σ,k}}^{D_{σ,k}},

L(σ, λ) ↑^{H_{n}} ≅ ind_{C_{σ,k}}^{D_{σ,k}} ∘β_{σ,h+1,k}(L_{h+1}(λ′)) ≅ β_{σ,h,≤k} ∘ res_{S_{h,≤k}}^{S_{h+1,k}}(L_{h+1}(λ′)).

Now the result follows immediately from Lemma 5.3e.

(ii) Apply Lemma 5.1c(i) to (i).

5.4e. Remarks. (i) Corollary 5.4d can be generalized easily to give a formula for the composition multiplicities in L ↑^{H_{n}} (resp. L ↓_{H_{n−1}}) for an arbitrary irreducible F G_{n}-module of the form (4.4.1), using Corollary 5.2g and induction.

(ii) Theorem 5.4c can also be used to reduce other natural questions about the structure of L(σ, λ) ↓_{H_{n−1}} to analogous problems about L_{h+1}(λ′) ↓_{S_{h,≤k}}. The latter can in turn be answered by studying the branching rule from quantum GL_{h+1} to quantum GL_{h}. For instance, the problem of determining the socle of L(σ, λ) ↓_{H_{n−1}}, or a criterion for complete reducibility of L(σ, λ) ↓_{H_{n−1}}, can be tackled in this way.

§5.5 A dimension formula for irreducibles  109

5.5. A dimension formula for irreducibles

In this section, we prove a dimension formula for irreducible F G_{n}-modules in terms of the characters of the irreducible S_{h,k}-modules (the latter being conjecturally determined by the Lusztig conjecture in many cases). Continue with the notation of the previous section. In addition, for λ ∈ Λ^{+}(h + 1, k), μ ∈ Λ^{+}(h, ≤ k) and ν ∈ Λ(h + 1, k), we will write

b_{λ,μ} = [res_{S_{h,≤k}}^{S_{h+1,k}} L_{h+1}(λ) : L_{h}(μ)],  (5.5.1)
m_{λ,ν} = dim L_{h+1}(λ)_{ν},  (5.5.2)

i.e. b_{λ,μ} is the branching coefficient for restricting from the quantized enveloping algebra A_{h+1} to A_{h}, and m_{λ,ν} is the dimension of the ν-weight space of L_{h+1}(λ).

We will need the following inductive description of the weight multiplicities:

5.5a. Lemma. For 0 ≤ j ≤ k, λ ∈ Λ^{+}(h+1, k) and μ = (m_{1}, …, m_{h}) ∈ Λ(h, k−j),

m_{λ,μ[j]} = ∑_{ν∈Λ^{+}(h,k−j)} b_{λ,ν}m_{ν,μ}

where μ[j] = (m_{1}, …, m_{h}, j) ∈ Λ(h + 1, k).

Proof. The restriction res_{S_{h,≤k}}^{S_{h+1,k}} L_{h+1}(λ) splits as a direct sum of ‘levels’

res_{S_{h,≤k}}^{S_{h+1,k}} L_{h+1}(λ) = ⨁_{j=0}^{k} L_{h+1}(λ)^{(j)},

where L_{h+1}(λ)^{(j)} is a module for the summand S_{h,k−j} of S_{h,≤k}. Explicitly, L_{h+1}(λ)^{(j)} is the sum of all weight spaces of L_{h+1}(λ) corresponding to weights of the form ν = (n_{1}, …, n_{h+1}) ∈ Λ(h + 1, k) with n_{h+1} = j. Equivalently, it is the largest submodule of res_{S_{h,≤k}}^{S_{h+1,k}} L_{h+1}(λ) all of whose composition factors are of the form L_{h}(ν) for ν ∈ Λ^{+}(h, k − j). This means that L_{h+1}(λ)^{(j)} has the same composition factors over S_{h,k−j} as

⨁_{ν∈Λ^{+}(h,k−j)} b_{λ,ν}L_{h}(ν).

We see that the μ[j]-weight space of L_{h+1}(λ) has the same dimension as the μ-weight space of ⨁_{ν∈Λ^{+}(h,k−j)} b_{λ,ν}L_{h}(ν). The lemma follows.

The next theorem describes of the restriction of the F G_{n}-module L(σ, λ′) to F G_{n−1}:

5.5b. Theorem. For λ ⊢ k,

L(σ, λ′) ↓_{G_{n−1}} ≅ ⨁_{j=1}^{k} M_{j} ⋄ Γ_{dj−1}

where M_{j} is an F G_{n−dj}-module having the same composition factors as the semisimple module ⨁_{μ⊢(k−j)} b_{λ,μ}L(σ, μ′).

Proof. By Corollary 5.4d(ii),

L(σ, λ′) ↓_{H_{n−1}} ≅ ⨁_{j=1}^{k} e_{dj−1}M_{j}.

Now apply Theorem 5.1e.

5.5c. Corollary. For λ ⊢ k,

dim L(σ, λ′) = ∑_{j=1}^{k} [ ∏_{i=1}^{dj−1}(q^{n−i} − 1)] [ ∑_{μ⊢(k−j)} b_{λ,μ} dim L(σ, μ′)] .

Proof. If M is an F G_{m}-module, then an elementary calculation shows that

dim M ⋄ Γ_{n} = dim M [∏_{i=1}^{n}(q^{m+i} − 1)] .

Now the corollary follows easily from Theorem 5.5b.

Now we introduce some polynomials. For a composition μ = (m_{1}, m_{2}, …) ⊨ n, define the polynomial R_{μ}(t) ∈ ℤ[t] by

R_{μ}(t) = ∏_{i=1}^{n}(t^{i} − 1)/ ∏_{i>0 with m_{i}>0}(t^{m_{1}+···+m_{i}} − 1).

Let λ = (l_{1}, …, l_{a}) be a partition of n of height a. We will write μ ∼ λ if μ is any composition of height a obtained from λ by reordering the non-zero parts. For example, if λ = (3, 2, 2), there are precisely three compositions μ with μ ∼ λ, namely, μ = (3, 2, 2), (2, 3, 2) and (2, 2, 3). Define the polynomial S_{λ}(t) ∈ ℤ[t] by the formula

S_{λ}(t) = ∑_{μ∼λ} R_{μ}(t).  (5.5.3)

For some very simple examples:

S_{(1^{n})}(t) = 1,
S_{(n)}(t) = (t^{n−1} − 1)(t^{n−2} − 1) … (t − 1),
S_{(d^{k})}(t) = ∏_{i=1}^{kd}(t^{i} − 1)/ ∏_{i=1}^{k}(t^{di} − 1),
S_{(2,1^{n−2})}(t) = (t^{n} − 1)/(t − 1) − n.

We note in particular that S_{(d^{k})}(q) = |GL_{kd}(q) : GL_{k}(q^{d})|_{p′}. Our main result is as follows:

5.5d. Theorem. Suppose that σ ∈ 𝔽̄_{q}^{×} is of degree d over 𝔽_{q}, and n = kd for some k ≥ 1. Then, for any λ ⊢ k,

dim L(σ, λ′) = S_{(d^{k})}(q) ∑_{μ⊢k} m_{λ,μ}S_{μ}(q^{d}),

where m_{λ,μ} is as in (5.5.2).

Proof. For the proof, we will use the notation ν ⊩ k to mean that ν ∼ μ for some μ ⊢ k. So if ν ⊩ k, ν is a composition of k with as many non-zero parts as its height. Now, if ν ∼ μ, then m_{λ,μ} = m_{λ,ν}. So we can rewrite the statement we are trying to prove equivalently as

dim L(σ, λ′) = S_{(d^{k})}(q) ∑_{ν ⊩k} m_{λ,ν}R_{ν}(q^{d}).  (5.5.4)

We will prove (5.5.4) by induction on k, the case k = 1 following from (2.4.1). For k > 1, we use Corollary 5.5c and the inductive hypothesis to obtain:

dim L(σ, λ′) = ∑_{j=1}^{k} [ ∏_{i=1}^{dj−1}(q^{n−i} − 1)] ∑_{μ⊢k−j} b_{λ,μ}S_{(d^{k−j})}(q) ∑_{ν ⊩k−j} m_{μ,ν}R_{ν}(q^{d})

= ∑_{j=1}^{k} ∑_{ν ⊩k−j} [S_{(d^{k−j})}(q)R_{ν}(q^{d}) ∏_{i=1}^{dj−1}(q^{n−i} − 1)] [ ∑_{μ⊢k−j} b_{λ,μ}m_{μ,ν}] .

Write ν[j] for the composition obtained from ν by replacing the first part equal to zero with j. Then, a calculation from the definitions shows that for ν ⊩ k − j,

S_{(d^{k−j})}(q)R_{ν}(q^{d}) ∏_{i=1}^{dj−1}(q^{n−i} − 1) = S_{(d^{k})}(q)R_{ν[j]}(q^{d}).

Also, Lemma 5.5a shows that

∑_{μ⊢k−j} b_{λ,μ}m_{μ,ν} = m_{λ,ν[j]}

Using these formulae, we can replace the two summations over j and ν with one summation over γ ⊩ k (so γ = ν[j]). We obtain:

dim L(σ, λ′) = ∑_{γ⊩k}S_{(d^{k})}(q)R_{γ}(q^{d})m_{λ,γ}

which verifies the induction step.

5.5e. Remarks. (i) For the unipotent case σ = 1, it is interesting to “set q = 1” in the right hand side of the dimension formula in Theorem 5.5d. Observe that S_{λ}(1) is 0 unless λ = (1^{n}), when it is 1. So the right hand side of the expression in the theorem is equal simply to m_{λ,(1^{n})}, the weight multiplicity of the (1^{n})-weight space of the irreducible module L_{n}(λ) for the classical Schur algebra S_{1}(n, n). This is well-known to be same as the dimension dim D^{λ′} of the irreducible F Σ_{n}-module parametrized by the partition λ′, or zero if λ is not p-restricted.

(ii) The idea in this section is extended in [Br] to give a similar result to Theorem 5.5d concerning the Brauer character values of the irreducible F G_{n}-modules at unipotent elements. In particular, it is shown there that

S_{λ}(q) = ∑_{μ⊢n} K_{λ,μ}^{−1}K̃_{μ′,(1^{n})}(q)

where K^{−1} = (K_{λ,μ}^{−1}) is the inverse of the matrix of Kostka numbers of [M, I, (6.4)] and K̃ = (K̃_{λ,μ}(q)) is the matrix of renormalized Kostka-Foulkes polynomials as in [M, III, (7.11)].

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