A CHARACTER FORMULA FOR A FAMILY OF

SIMPLE MODULAR REPRESENTATIONS OF GL_{n}

Olivier Mathieu and Georges Papadopoulo

Abstract. Let K be an algebraically closed field of finite characteristic p, and let n ≥ 1 be an integer. In the paper, we give a character formula for all simple rational representations of GL_{n}(K) with highest weight any multiple of any fundamental weight. Our formula is slightly more general: say that a restricted dominant weight λ is special if there are integers i ≤ j such that λ = ∑_{i≤k≤j} a_{k}ω_{k} and ∑_{i≤k≤j} a_{k} ≤ p − (j − i). Indeed, we compute the character of any simple module L(λ) where λ = λ_{0} + pλ_{1} + ... + p^{r}λ_{r} with all λ_{i} are special. By stabilization, we get a character formula for a family of irreducible rational GL_{∞}(K)-modules.

Introduction. In the paper, we will prove a character formula for a stable family of simple rational representations of GL_{n}(K)-modules. In order to explain the result, we first need to give a few definitions about the combinatorics of polynomial weights and Young diagrams. Then we will state the main result and we will briefly compare it with previous results. At the end of the introduction, we will describe the main ingredients of its proof.

From now on, fix an integer n ≥ 1 and an algebraically closed field K of characteristic p > 0. Let H be the Cartan subgroup of GL_{n}(K) consisting of diagonal matrices. Denote by ϵ_{1}, ϵ_{2}, … the weights of the natural ndimensional representation V of GL_{n}(K). Recall that the highest weight of ⋀^{i} V is the weight ω_{i} = ∑_{k≤i} ϵ_{k}. A weight μ = ∑_{1≤i≤n} l_{i}ϵ_{i} is called polynomial if l_{i} ≥ 0 for all i. Its degree is |μ| = ∑_{1≤i≤n} l_{i}. Any integer l ≥ 0 admits a unique finite p-adic expansion l = ∑_{0≤j} l(j)p^{j}, where 0 ≤ l(j) < p and l(j) = 0 for j >> 0. Similarly, any polynomial weight μ admits a unique finite p-adic expansion μ = ∑_{0≤j} p^{j}μ(j), where μ(j) = ∑_{1≤i≤n} l_{i}(j)ϵ_{i}.

Let m < p be an integer. For a Young diagram Y, set c_{1}(Y), c_{2}(Y), … (respectively l_{1}(Y), l_{2}(Y), …) the number of boxes in the first column, in the second column (respectively in the first line, in the second line) and so on.... Its size |Y | is the total number of boxes. We say that Y is m-special if the number of columns is ≤ m and if c_{1}(Y) − c_{m}(Y) ≤ p − m. A tableau of

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 20C20, 20G05

shape Y is a labeling of the boxes of Y by the integers 1, 2, …, n. As usual, a tableau is called semi-standard if the filling is non decreasing from left to right and increasing from top to bottom. For a tableau T, denote by T [i] the subset of boxes with labels ≤ i (when T is semi-standard, T [i] is again a Young diagram). Set η_{i}(T) the number of times the label i occurs in T (i.e. the cardinality of T [i] \ T [i − 1]). The weight of T is w(T) = ∑_{i≥1} η_{i}(T)ϵ_{i}. By definition, a tableau is m-semi-standard if all Young diagrams T [i] are m-special.

A polynomial weight λ = ∑_{i≤k≤j} a_{k}ω_{k} is called special if it is restricted, dominant and if m(λ) ≤ p − (j − i), where m(λ) = ∑_{i≤k≤j} a_{k}. Note that we have m(λ) < p for any special weight λ (in the special case i = j, this follows from the hypothesis that λ is restricted). Recall that the Young diagram Y (λ) associated with a dominant polynomial weight λ is defined by the requirement: λ = ∑_{1≤i≤n} l_{i}(Y)ϵ_{i}. Indeed a restricted polynomial dominant weight λ is special if and only if the associated Young diagram Y (λ) is m(λ)-special.

Let λ be a special weight, let μ be a polynomial weight. Set N(λ, μ) the number of m(λ)-semi-standard tableaux of shape Y (λ) and with weight μ. By definition, we have N(0, 0) = 1 and N(λ, μ) = 0 if the degrees of λ and μ are distinct.

Theorem 0.1. Let λ be a dominant weight of the form λ = ∑_{0≤k≤r} p^{k}λ_{k}, where all weights λ_{k} are special. Let L(λ) be the simple GL_{n}(K)-module with highest weight λ. Any weight of L(λ) is polynomial, and for any polynomial weight μ, we have dim L(λ)_{μ} = ∏_{0≤k≤r} N(λ_{k}, μ(k)).

Remark: It is obvious that any weight μ of L(λ) is polynomial, but this property is necessary to define its p-adic expansion (μ(k))_{k≥0}.

Remark: By definition, the fundamental weights of GL_{n} are the weights ω_{i} for some 1 ≤ i < n. For any s with 0 ≤ s ≤ p−1, the weight sω_{i} is special. Hence the theorem gives a character formula for the modules L(N.ω_{i}) with N arbitrary.

Remark: The definition of special weights is indeed independent of n, i.e. if λ is a special weight for GL_{n}(K) its natural extension to GL_{N}(K) is again special for any N ≥ n. Thus the previous theorem gives rise to character formula for all GL_{∞}(K)-modules with highest weights of the form λ = ∑_{0≤k≤r} p^{k}λ_{k}, where all weights λ_{k} are special (i.e. the family of such λ is stable).

There is a general conjecture, due to Lusztig [Lu1,Lu2], about the character of simple rational GL_{n}(K)-modules. The experts believe that the conjecture holds for p ≥ n (see e.g. the introduction of [So]), and this conjecture has been proved for p >> 0 by Andersen, Jantzen and Soergel [AJS]. Anyhow, most character formulas treated in Theorem 0.1 are outside the validity domain of Lusztig’s Conjecture.

The proof is based on the following three ingredients:

(i) First one uses Steinberg’s tensor product formula [St] to reduce the statement to the case where λ is special. It turns out that the Steinberg’s formula is especially simple in our setting, because any weight of L(λ) is a unique combination of weights of the modules L(p^{k}λ_{k}).

(ii) We strongly use an idea of Donkin [D2]: Donkin proved that 𝕄 = ⋀(V ⊗ W) is a dual pair under GL_{n}(K) × GL(W) (here V = K^{n} and W is any vector space). This dual pair is called Howe’s skew dual pair, because it has been found by Howe in the context of fields of characteristic zero [H]. Then Donkin showed that the character of simple modules can be deduced from the character of tilting modules, and conversely. However, the character of all tilting modules is still unknown. Our approach is based on the same dual pair. We show that the character of simple GL(V)-modules can be also deduced from tensor produts mutiplicities of tilting GL(W)-modules.

(iii) Similarly, the general tensor product multiplicities of tilting modules are unknown. However the main result of [GM1,GM2] (Verlinde’s formula for algebraic groups) describes some of them. More precisely, we use Verlinde’s formula for GL(W) with W of dimension 1, 2, …, p − 1, and the computable tilting multiplicities in 𝕄 correspond exactly with the special weights.

Acknowledgments: We thank S. Donkin, A. Kleshchev and P. Littelmann for helpful conversations. The research of the first author is supported by UA1 du CNRS. The research of the second author is supported at Paris 7 by UA748 du CNRS.

§1 General results about tilting modules. Let K be an algebraically closed field of characteristic p, let G be a reductive group over K, let B be a Borel subgroup, and let H ⊂ B be a Cartan subgroup. We will set by U the unipotent radical of B and by U^{−} the unipotent radical of the opposed Borel subgroup. Denote by P^{+} the set of dominant weights. For λ ∈ P^{+}, denote by L(λ) (respectively Δ(λ), ∇(λ)) the simple module (respectively the Weyl module, the dual of the Weyl module) with highest weight λ.

By G-module, we mean rational G-module of finite dimension. A good filtration of a G-module M is a filtration whose subquotients are dual of Weyl modules. A G-module M is tilting if M and M^{∗} have a good filtration. Recall the following known result:

Theorem 1.1.

(i ) For each λ ∈ P^{+}, there exists a unique indecomposable tilting module T (λ) which admits λ as highest weight. Moreover, dim(T (λ)_{λ}) = 1.

(ii ) Any tilting module is the direct sum of indecomposable tilting modules of type T (λ). The tilting modules T (λ) and T (μ) are isomorphic if and only if λ = μ.

(iii) The tensor product of two tilting modules is a tilting module.

References for the Theorem are as follows: the general notion of tilting modules for any quasi-hereditary algebra is due to Ringel [R]. The assertions (i), (ii) as they are quoted are due to Donkin [D2] (Theorem 1.1). Assertion (iii) follows from the fact that the tensor product of two G-modules with a good filtration has a good filtration: for groups of type A (which are indeed the only groups used here), it has been established in [W], for the general case see [D1,M].

Let M be a G-module. Denote by T^{G}(M) the image of the composite map M^{U} → M → M_{U^{−}} where M^{U} is the space of U-invariants of M and M_{U^{−}} is the space of U^{−}-coinvariants of M. Since T^{G}(M) is an H-module, there is a weight decomposition T^{G}(M) = ⊕_{λ∈P^{+}}T_{λ}^{G}(M).

Lemma 1.2. Let M be an indecomposable tilting module.

(i) T^{G}(M) has dimension one.

(ii) Let λ be the unique weight of T^{G}(M). Then we have M ≃ T (λ).

Proof: It is clear that T^{G}(N) ≠ 0 for any non-zero G-module N, because any maximal weight of N is a weight of T^{G}(N). Let λ be any weight of T^{G}(M) and choose v ∈ M_{λ}^{U} such that its image in T^{G}(M) is not zero. Denote by the same notation v_{λ} the highest weight vector in Δ(λ), in ∇(λ) and in T (λ). By the universal property of Weyl modules, there is a map ψ_{1} : Δ(λ) → M sending v_{λ} to v. Similarly, there is a map ψ_{2} : M → ∇(λ) sending v to v_{λ}.

Now, there is a canonical injection Δ(λ) ↪ T (λ) whose quotient has a filtration by Weyl modules. We have Ext_{G}^{1}(Δ(μ), ∇(μ′)) = 0, for any μ, μ′ ∈ P^{+} ([CPSV], corollary 3.3). Since M has a good filtration, we have Ext_{G}^{1}(T (λ)/Δ(λ), M) = 0. Thus, the map ψ_{1} can be extended to a map ϕ_{1} : T (λ) → M. In the same way, there is a canonical surjection T (λ) ↠ ∇(λ), and the map ψ_{2} can be lifted to a map ϕ_{2} : M → T (λ). So we get the following commutative diagram:

T (λ)
ϕ_{2} ↗ ↓

Δ(λ) ⟶^{ψ_{1}} M ⟶^{ψ_{2}} ∇(λ)

↓ ↗ ϕ_{1}

T (λ)

By definition, we have ψ_{2} ∘ ψ_{1}(v_{λ}) = v_{λ}. Therefore, ϕ_{2} ∘ ϕ_{1} is a non nilpotent endomorphism of the indecomposable module M. By Fitting’s Lemma, ϕ_{2} ∘ ϕ_{1} is an invertible map. Thus, T (λ) is a direct factor of M and so we have M ≃ T (λ). □

Corollary 1.3. Let M be a tilting G-module and let C be its commutant. (i) We have M ≃ ⊕_{λ∈P^{+}}T_{λ}^{G}(M) ⊗ T (λ) as a G-module. (ii) For any λ ∈ P^{+}, the C-module T_{λ}^{G}(M) is zero or simple.

Proof: For N ≥ 0, let Mat_{N}(K) be the K-algebra of N × N matrices. By Theorem 1.1, there exists an isomorphism of G-modules M ≃ ⊕_{λ∈P^{+}}P (λ)^{N_{λ}}. Hence C contains the subalgebra C^{0} = ⊕_{λ∈P^{+}}Mat_{N_{λ}}(K). By Lemma 1.2, we have M = ⊕_{λ∈P^{+}}P (λ)^{N_{λ}} as G × C^{0}-modules, hence Assertion (i) holds. Moreover the T_{λ}^{G}(M) is zero or is a simple C^{0}-module, and Assertion (ii) follows. □

Lemma 1.4. Let M, N be two G-modules. If M is indecomposable of dimension divisible by p, then the dimension of any direct summand of M ⊗ N is divisible by p.

Proof: See [GM1] (Lemma 2.7.). □

§2 Howe’s skew duality for the pair (GL(V) × GL(W)). From now on, fix an integer n ≥ 1 and set V = K^{n}. Denote by (v_{i})_{1≤i≤n} its natural basis and by H the torus of GL(V) consisting of the diagonal matrices. Let us call ϵ_{1}, ...ϵ_{n} the characters of the torus H defined by ϵ_{i}(diag(t_{1}, ..., t_{n})) = t_{i}, where diag(t_{1}, ..., t_{n}) is the diagonal matrix with entries t_{1}, t_{2}, . . .. The root system Δ of GL(V) with respect to H is the set of all characters ϵ_{i} − ϵ_{j} for 1 ≤ i ≠ j ≤ n. As usual, we will choose the subset Π = {α_{i}, 1 ≤ i ≤ n − 1} (where α_{i} = ϵ_{i} − ϵ_{i+1}) as a basis of the root system Δ. Denote by U (respectively: U^{−}) the unipotent radical of the Borel subgroup associated with the basis Π (respectively the unipotent radical of the opposed Borel subgroup). Also let P, P^{+} be the lattice of weights and the set of dominant weights with respect to Π. The simple module with highest weight λ ∈ P^{+} is denoted by L(λ).

These notations are those used in the introduction. In what follows, we will use another vector space W of dimension m, with basis (w_{i})_{1≤i≤m}. In order to avoid conflicts of notations, corresponding objects associated with GL(W) will be denoted slightly differently. The torus of GL(W) and the unipotent radicals of its Borel subgroup and its opposed Borel subgroup will be denoted by H(W), U(W) and U^{−}(W). The basis of characters of H(W) will be denoted by ϵ_{1}^{W} , …, ϵ_{m}^{W}. Similarly, the set of dominant weights will be denoted by P_{W}^{+} and the tilting GL(W)-module with highest weight λ ∈ P_{W}^{+} is denoted by T^{W} (λ).

For any Young diagram Y contained in the n×m rectangle (i.e. such that c_{1}(Y) ≤ n and l_{1}(Y) ≤ m), we set λ(Y) = ∑_{1≤i≤n} l_{i}(Y)ϵ_{i} and λ^{T} (Y) = ∑_{1≤i≤m} c_{i}(Y)ϵ_{i}^{W}. By definition, λ(Y) belongs to P^{+} and λ^{T} (Y) belongs to P_{W}^{+} .

Set 𝕄 = ⋀(V ⊗ W) and let K[GL(V)] and K[GL(W)] be the group algebras of GL(V) and GL(W). Let ρ_{V} : K[GL(V)] → End_{K}(𝕄) and ρ_{W} : K[GL(W)] → End_{K}(𝕄) the maps induced by the actions of GL(V) and GL(W) on 𝕄.

Theorem 2.1 (Donkin). We have ρ_{V} ((K[GL(V)]) = End_{GL(W)}(𝕄) and dually, ρ_{W} ((K[GL(W)]) = End_{GL(V)}(𝕄).

Theorem 2.1 is proved in [D], proposition (3.11) (see also [AR] for a generalization to other classical groups). For fields of characteristic zero, this duality is due to Howe [H]. In that setting, Howe showed that 𝕄 is isomorphic to ⊕_{Y} L(λ(Y)) ⊗ L^{W} (λ^{T} (Y)), where Y runs over all Young diagrams contained in the n × m rectangle. There is an analogous property in finite characteristics, as it is shown in the next lemma:

Lemma 2.2. Let Y be a Young diagram of size d contained in the n × m rectangle.

(i) As GL(V)-modules, T_{λ^{T} (Y)}^{GL(W)}(𝕄) is isomorphic to L(λ(Y)).

(ii) Let μ = ∑_{1≤k≤n} k_{i}ϵ_{i} be a polynomial weight of degree d. We have

L(λ(Y))_{μ} ≃ T_{λ^{T} (Y)}^{GL(W)}(⋀^{k_{1}} W ⊗ ⋀^{k_{2}} W ⊗ …).

Proof: For 1 ≤ i ≤ n and 1 ≤ j ≤ m, let b_{i,j} the box of the n × m rectangle located at the intersection of the i^{th} line and the j^{th} column and set x_{i,j} = v_{i} ⊗ w_{j}. When d > 0, drawn the Young diagram Y inside the rectangle in a such way that it contains the upper left box b_{1,1}. For example:

Let X (respectively X^{−}) be the subspace of V ⊗ W generated by all x_{i,j} with b_{i,j} ∈ Y (respectively with b_{i,j} ∉ Y). Note that N is a d-dimensional U × U(W)-submodule of V ⊗ W and X^{−} is a (nm − d)-dimensional U^{−} × U^{−}(W)-submodule of V ⊗ W. Choose x, x^{−} be non-zero vectors in ⋀^{d} X and in ⋀^{nm−d} X^{−}.

The vector x is U × U(W)-invariant of weight (λ(T), λ^{T} (V)). Hence it defines an element \overline{x} ∈ T_{λ^{T} (Y)}^{GL(W)}(𝕄). Moreover, x ∧ x^{−} is a non-zero vector of the trivial U^{−}(W)-module ⋀^{nm}(V ⊗ W). Hence the image of x in 𝕄_{U^{−}(W)} is not zero, and we have \overline{x} ≠ 0. Hence \overline{x} is a non-zero U-invariant vector of weight λ(Y). By Corollary 1.3, the non-zero GL(V)-module T_{λ^{T} (Y)}^{GL(W)}(𝕄) is

simple. Hence we have T_{λ^{T} (Y)}^{GL(W)}(𝕄) ≃ L(λ(Y)) and Assertion (i) is proved.

Identify V ⊗ W ≃ W ⊕ W ⊕ …, where the i^{th}-factor W is v_{i} ⊗ W. Thus we get 𝕄 ≃ ⋀ W ⊗ ⋀ W … (n times). Thus the eigenspace of weight μ of the GL(V)-module T^{GL(W)}(𝕄) is T^{GL(W)}(⋀^{k_{1}} W ⊗ ⋀^{k_{2}} W ⊗ …), and Assertion (ii) follows. □

Corollary 2.3. Let Y be a Young diagram of size d contained in the n × m rectangle, and let μ = ∑_{1≤k≤n} k_{i}ϵ_{i} be a polynomial weight of degree d. The dimension of L(λ(Y))_{μ} is the multiplicity (as a direct summand) of the indecomposable GL(W)-module T (λ^{T} (Y)) in ⋀^{k_{1}} W ⊗ ⋀^{k_{2}} W ⊗ . . ..

Proof: The assertion follows from Corollary 1.3 and Lemma 2.2. □

Remark: By Corollary 2.3, the knowledge of tensor product multiplicities of tilting modules determines the character formula of simple modules. Indeed, Donkin showed how to deduce the character formula of simple modules from the character of tilting modules. With his notations, Donkin’s formula is (see [D2], Lemma 3.1):

[T (λ) : ∇(μ)] = [∇(μ^{T}) : L(λ^{T})].

§3 Some multiplicities of tilting GL(W)-modules in ⋀(V ⊗W). Let W be a vector space of dimension m < p. We will use the notations of Section 2 together with the next notations. For 1 ≤ j < m, set h_{j}^{W} = (ϵ_{j}^{W})^{∗} − (ϵ_{j+1}^{W})^{∗} (i.e. h_{j}^{W} is the j^{th} simple coroot) and set h_{0}^{W} = (ϵ_{1}^{W})^{∗} − (ϵ_{m}^{W})^{∗} (i.e. h_{0}^{W} is the highest coroot). Also set ω_{j}^{W} = ∑_{1≤k≤j} ϵ_{j}^{W} for all 0 ≤ j ≤ m and denote by Ω_{j}^{W} the set of all weights of the form ϵ_{k_{1}}^{W} + ϵ_{k_{2}}^{W} + ⋯ + ϵ_{k_{j}}^{W} with k_{1} < k_{2} < ⋯ < k_{j}. Thus Ω_{j}^{W} is the set of weights of ⋀^{j} W and ω_{j}^{W} is its highest weight. Let ρ^{W} ∈ P_{W} be any weight with ρ^{W} (h_{j}^{W}) = 1 for all 1 ≤ j < m. By definition, the interior of the fundamental alcove is the set C_{W} of all λ ∈ P_{W}^{+} such that (λ + ρ^{W})(h_{0}^{W}) < p.

Lemma 3.1. Let λ ∈ P_{W}^{+} . Then p divides dim T (λ) if and only if λ ∉ C_{W}.

Lemma 3.2. Let j be an integer with 0 ≤ j ≤ m. We have T (ω_{j}^{W}) ≃ ⋀^{j} W. In particular, the set of weights of T (ω_{j}^{W}) is Ω_{j}^{W} and each weight appears with multiplicity one.

Lemma 3.3. Let λ ∈ C_{W} and let j with 0 ≤ j ≤ m. We have T (λ)⊗T (ω_{i}) = ⊕_{ν}T (λ + ν), where the sum runs over all ν ∈ Ω_{j}^{W} such that (λ + ν) ∈ P^{+}.

References for the previous three lemmas are as follows: Lemma 3.2 follows from the fact that ⋀^{j} W is simple and is a direct summand of the tilting module V^{⊗j} (see [D2] for details). Lemma 3.1 and 3.3 follows from the main result of [GM2] (announced in [GM1]). More precisely, the ”only if” part of Lemma 3.1 is a consequence of Andersen’s linkage principle [A], and the ”if” part follows from Lemma 1.4. For the peculiar case considered here, there is a quick proof of Lemmas 3.1 and 3.3 in [M2] (see Proposition 10 and Lemma 12).

Lemma 3.4. Let k_{1}, …, k_{n} be integers with 0 ≤ k_{i} ≤ m. We have:

⋀^{k_{1}} W ⊗ ... ⊗ ⋀^{k_{n}} W = T ⊕ [⊕_{(ν_{1},...,ν_{n})}T (ν_{1} + ... + ν_{n})], where T is a sum of indecomposable tilting modules of dimension divisible

by p and where the sum runs over all n-tuples (ν_{1}, ..., ν_{n}) ∈ Ω_{k_{1}}^{W} × ⋯ × Ω_{k_{n}}^{W} such that ν_{1} + ν_{2} + ... + ν_{i} belongs to C_{W}, for any i with 1 ≤ i ≤ n.

Proof: Let λ ∈ P_{W}^{+} and let k be an integer with 0 ≤ k ≤ m. Assume first that λ ∈ C_{W}. It follows from Lemmas 3.1, 3.2 and 3.3 that we have:

(3.4.1) T (λ) ⊗ ⋀^{k} W ≃ T ⊕ [⊕_{ν}T (λ + ν)], where T is a sum of indecomposable tilting modules of dimension divisible by p and where the sum runs over all ν ∈ Ω_{k}^{W} such that λ + ν belongs to C_{W}. Next, assume that λ ∉ C_{W}. From Lemmas 1.4 and 3.1, we get:

(3.4.2) T (λ) ⊗ ⋀^{k} W ≃ T, where T is a sum of indecomposable tilting modules of dimension divisible by p. Note that for n = 1, the assertion of Lemma 3.4 is obvious: indeed the conditions ν_{1} ∈ C_{W} and ν_{1} ∈ Ω_{k_{1}}^{W} simply mean ν_{1} = ω_{k_{1}}^{W}. Thus, Lemma 3.4 follows, by induction over n, from the assertions (3.4.1) and (3.4.2). □

Example 3.5 For this example, we will consider the case m = p − 1. For any k ∈ ℤ, set θ_{k}^{W} = ω_{a}^{W} + b.ω_{m}^{W}, where k = a + mb and 0 ≤ a < m. It is clear that C_{W} = {θ_{k}^{W} |k ∈ ℤ}. Moreover θ_{k}^{W} is the highest weight of δ^{⊗b} ⊗ ⋀^{a} W, where δ is the determinant character of GL(W). Thus Lemma 3.4 can be stated as follows:

⋀^{k_{1}} W ⊗ ... ⊗ ⋀^{k_{n}} W = T ⊕ T (θ_{k_{1}+···+k_{n}}^{W}),

where T is a sum of indecomposable tilting modules of dimension divisible by p. Using Corollary 2.3, we get that for any special weight λ with m(λ) = p − 1, the module L(λ) is multiplicity free. See Example 4.5 for an explicit realization of these modules.

§4 Character of L(λ), λ being a special weight. We will use the notations of the previous two sections. In particular, the dimensions of V and W are n and m. Moreover, we will assume that m < p. Also denote by C_{W}^{pol} the set of all polynomial weights of C_{W}. A restricted dominant polynomial weight λ ∈ P is called m-special if there are integers i, j with λ = ∑_{i≤k≤j} a_{k}ω_{k}, ∑_{i≤k≤j} a_{k} ≤ m and j − i ≤ p − m. So by definition, 0 is special. To check that a non zero weight is special, we can choose i, j such that a_{i} and a_{j} are not zero. Any m-special weight is m(λ)-special, i.e. it is special. Let Y oung(n, m) the set of all m-special Young diagrams of height ≤ n.

Lemma 4.1. (i) The map Y ↦ λ(Y) is a bijection from the set Y oung(n, m) to the set of all m-special weights.

(ii) The map Y ↦ λ^{T} (Y) is a bijection from the set Y oung(n, m) to C_{W}^{pol}.

Proof of Assertion (i): The map Y ↦ λ(Y) is a bijection between the set of all Young diagrams of height ≤ n and the set of dominant polynomial weights of GL(V). More explicitly, this map is given by: Y ↦ ∑_{k≥1} ω_{c_{k}(Y)}. We have m(Y (λ)) = l_{1}(Y), hence Young diagrams with at most m columns correspond with weights λ with m(λ) ≤ m. Moreover if l_{1}(Y) ≤ m, we have λ(Y) = ∑_{i≤k≤j} a_{k}ω_{k}, where i = c_{m}(Y), j = c_{1}(Y). Thus λ(Y) is m-special if and only if Y is m-special.

Proof of Assertion (ii): The map Y ↦ λ^{T} (Y) is a bijection between Young diagrams of height ≤ m and dominant polynomial weight of GL(W). We have λ^{T} (Y) = ∑_{k≥1} c_{k}(Y)ϵ_{k}^{W}. We have ρ^{W} (h_{0}^{W}) = m − 1, and λ^{T} (Y)(h_{0}^{W}) = c_{1}(Y) − c_{m}(Y). Hence Y is m-special, if and only if:

c_{1}(Y) − c_{m}(Y) ≤ p − m, if and only if

λ^{T} (Y)(h_{0}^{W}) ≤ p + 1 − ρ^{W} (h_{0}^{W}), if and only if

(λ^{T} (Y) + ρ^{W})(h_{0}^{W}) < p, if and only if

λ^{T} (Y) belongs to C_{W}^{pol}. □

Lemma 4.2. Let Y ∈ Y oung(n, m) and let μ = ∑_{1≤i≤n} k_{i}ϵ_{i} be a polynomial weight such that |Y | = |μ|. There is a natural bijection Ψ_{m} from

(i) the set of all m-semi-standard tableaux of shape Y and weight μ, to (ii) the set of all n-tuples (ν_{1}, ..., ν_{n}) ∈ Ω_{k_{1}}^{W} ×⋯×Ω_{k_{n}}^{W} such that ∑_{1≤i≤n} ν_{i}

= λ^{T} (Y) and such that ν_{1} + ν_{2} + ... + ν_{i} belongs to C_{W}, for any i with 1 ≤ i ≤ n.

Proof: Let T be a tableau of shape Y and weight μ. Define n weights ν_{1}, …, ν_{n} ∈ P_{W} by the requirement:

ν_{1} + ⋯ + ν_{i} = λ^{T} (T [i]), for all i with 1 ≤ i ≤ n.

Note that T [i] \ T [i − 1] contains exactly k_{i} boxes. Assume that T is semi-standard. Then any two boxes of T [i] \ T [i − 1] are located on different columns. Denote by j_{1} < ⋯ < j_{k_{i}} the numbering of the non empty columns of T [i]\T [i−1]. We have ν_{i} = ϵ_{j_{1}}^{W} +ϵ_{j_{2}}^{W} +…, hence ν_{i} belongs to Ω_{k_{i}}^{W}. Thus it is clear that the map Ψ : T ↦ (ν_{1}, ..., ν_{n}) is a bijection from

(i) the set of all semi-standard tableaux of shape Y and weight μ, to (ii) the set of all n-tuples (ν_{1}, ..., ν_{n}) ∈ Ω_{k_{1}}^{W} ×⋯×Ω_{k_{n}}^{W} such that ∑_{1≤i≤n} ν_{i}

= λ^{T} (Y) and such that ν_{1} + ν_{2} + ... + ν_{i} belongs to P_{W}^{+} , for any i with 1 ≤ i ≤ n.

Indeed, this bijection Ψ is equivalent to the rule of Richardson and Littlewood ([LR], see also [L]). Denote by Ψ_{m} the restriction of Ψ to the set of all m-semi-standard tableaux of shape Y and weight μ. As the weights ν_{1} + ν_{2} + ... + ν_{i} are polynomial, it follows from Lemma 4.1 (ii) that Ψ_{m} is the bijection required by Lemma 4.2. □

Theorem 4.3. Let λ be a special weight. Any weight of L(λ) is polynomial and for any polynomial weight μ, the dimension of L(λ)_{μ} is the number of m-semi-standard tableaux of shape Y (λ) and of weight μ.

Proof: It is obvious and well known that the weights of L(λ) are polynomial ([G]). Let μ = ∑_{1≤i≤n} k_{i}ϵ_{i} be a polynomial weight, set Y = Y (λ) and m = m(λ). By Corollary 2.3, the dimension of L(λ(Y))_{μ} is the multiplicity of the indecomposable GL(W)-module T (λ^{T} (Y)) in ⋀^{k_{1}} W ⊗ ⋀^{k_{2}} W ⊗ . . .. By Lemma 4.1, λ^{T} (Y) belongs to the fundamental alcove C_{W}. By Lemma 3.1, the dimension of the tilting GL(W)-module T (λ^{T} (Y)) is not divisible by p. Hence by Lemma 3.4, dim(L(λ(Y))_{μ}) is the number of all n-tuples (ν_{1}, ..., ν_{n}) ∈ Ω_{k_{1}}^{W} × ⋯ × Ω_{k_{n}}^{W} such that ∑_{1≤i≤n} ν_{i} = λ^{T} (Y) and such that ν_{1} + ν_{2} + ... + ν_{i} belongs to C_{W}, for any i with 1 ≤ i ≤ n. Hence by Lemma 4.2, dim(L(λ(Y))_{μ}) is also the number of m-semi-standard tableaux of shape Y (λ) and of weight μ. □

Example 4.4 Consider the polynomial dominant weight λ = 2ω_{1} + ω_{3} of GL_{4}(K), and set μ = ϵ_{1} + ϵ_{2} + ϵ_{3} + 2ϵ_{4}. There are three semi-standard tableaux of shape Y (λ) and weight μ, namely:

1 2 4 3 4 and 1 3 4 2 4 and 1 4 4 2 3

Assume that K is a field of characteristic 5. Then , the semi-standard Young tableau

T = 1 4 4 2 3

is not 3-semi-standard, because T [3] is not 3-special thus we have dim L(λ)_{μ} = 2. In characteristic 0, the corresponding dimension is 3.

Example 4.5 Let λ be a special weight with m(λ) = p − 1. It follows from Example 3.5 that all the weights of L(λ) have mutiplicity 1. Indeed, it is easy to find an explicit description of the module L(λ). As m(λ) = p − 1, there are integers a, b, i such that λ = aω_{i} +bω_{i+1} and a+b = p−1. Here we will denote by x_{1}, …, x_{n} the basis of V = K^{n}. Denote by \overline{S}V the quotient of the algebra SV by the ideal generated by the x_{i}^{p}. Set N = (p − 1)i + b. We have \overline{S}^{N}V ≃ L(λ), where \overline{S}^{N}V is the degree N component of \overline{S}V. Indeed it is easy to check that the GL(V)-module \overline{S}^{N}V is simple and that (x_{1}.x_{2} … x_{i})^{p−1}.x_{i+1}^{b} is its highest weight vector.

§5 Proof of the Theorem. Say that a polynomial weight μ = ∑_{1≤i≤n} k_{i}ϵ_{i} is reduced if all k_{i} are ≤ p − 1. Note that a reduced dominant polynomial weight is necessarily restricted. For any polynomial weight μ, the weights μ(k) occurring in its p-adic decomposition are reduced.

Lemma 5.1. (i) Any special weight is reduced.

(ii) Let r ≥ 0 and let μ = ∑_{0≤k≤r} p^{k}.μ_{k} be a polynomial weight where are all μ_{k} are reduced. Then μ_{k} = μ(k) for all 0 ≤ k ≤ r.

(iii) Let λ be a reduced dominant polynomial weight. Then any weight of L(λ) is reduced.

Proof: Let λ be a special weight. We have λ = ∑_{1≤i≤n} k_{i}ϵ_{i}, with k_{1} = m(λ) < p. As λ is dominant, we have k_{i} ≤ k_{1}, and λ is reduced. Thus Assertion (i) holds. Assertion (ii) is obvious. Let λ be a reduced dominant polynomial weight. Let X be the set of all linear combinations ∑_{1≤k≤n} x_{i}ϵ_{i}, where the x_{i} are real numbers with 0 ≤ x_{i} ≤ p − 1. Then X is a convex set which is stable by Σ_{n} (the Weyl group of GL_{n}(K)). Hence any weight μ of L(λ) belongs to X, and μ is reduced. Thus Assertion (iii) holds. □

Lemma 5.2. Let r ≥ 0 and let λ be a polynomial dominant weight of the form λ = ∑_{k≥0} p^{k} λ_{k}, where are all λ_{k} are reduced and dominant. Let μ be a polynomial weight. Then we have:

L(λ)_{μ} ≃ L(λ_{0})_{μ(0)} ⊗ L(λ_{1})_{μ(1)} ⊗ . . ..

Proof: For g = (g_{i,j})_{1≤i,j≤n} ∈ GL_{n}(K), set F r(g) = (g_{i,j}^{p})_{1≤i,j≤n}. The map F r : GL_{n}(K) → GL_{n}(K), called the Frobenius map, is a morphism of groups. By Steinberg’s product formula (see [St], Theorem 41), there is an isomorphism L(λ) ≃ L(λ_{0}) ⊗ L(λ_{1}) ⊗ …, where the action of GL_{n}(K) on the k^{th}-factor is shifted by F r^{k}. Thus we have L(λ)_{μ} = Σ_{(μ_{0},μ_{1},...)}L(λ_{0})_{μ_{0}} ⊗ L(λ_{1})_{μ_{1}} ⊗ …, where the sum runs over all (r + 1)-uples (μ_{k})_{0≤k≤r} such that μ = ∑_{k≥0} p^{k}μ_{k} and each μ_{k} is a weight of L(λ_{k}). By Assertion (iii) of Lemma 5.1, the weights μ_{k} are reduced. Then, by Assertion (ii) of Lemma 5.1, we have μ_{k} = μ(k). Thus Lemma 5.2 holds. □

Let S_{n} the set of all weights of the form λ = ∑_{0≤k≤r} p^{k} λ_{k}, where all weights λ_{k} are special. For such a λ, we have λ_{k} = λ(k) (by Lemma 5.1 (ii)). Any polynomial weight for GL_{n}(K) can be viewed as a polynomial weight for GL_{N}(K) for any N ≥ n. Note that the condition defining special weights does not involves n. Hence we have S_{n} ⊂ S_{n+1} ⊂ …, i.e. the families S_{n} are stable. For λ being a dominant polynomial weight, the simple GL_{n}(K)- module is defined as well for n infinite, and the following theorem holds as well for n infinite (with the convention that S_{∞} = ∪_{n≥1}S_{n}). Also recall that for any special weight λ and any polynomial weight μ, we denote by N(λ, μ) the number of m(λ)-semi-standard tableaux of shape Y (λ) and with weight μ.

Theorem 5.3. (n being finite or infinite) Let λ ∈ S_{n}. Any weight of the GL_{n}(K)-module L(λ) is polynomial, and for any polynomial weight μ, we have dim L(λ)_{μ} = ∏_{0≤k} N(λ(k), μ(k)).

Proof: First assume that n is finite. By Assertion (i) of Lemma 5.1, any special weights is reduced. Hence Theorem 5.3 follows from Lemma 5.2 and Theorem 4.3. The case n infinite follows by inductive limit. □

§6 Restrictions to Young subgroups. Let us consider GL_{n−1}(K) as the subgroup of GL_{n}(K) as usual. For any GL_{n−1}-module L, denotes by ch(L) its character. For any dominant weight μ of GL_{n−1}(K), we will denote by L_{n−1}(μ) the simple GL_{n−1}(K)-module with highest weight μ (we keep the notation L(λ) for the simple GL_{n}(K)-module with highest weight λ).

Lemma 6.1. Let λ be a dominant weight of GL_{n}(K) and let A be a finite set of dominant weight of GL_{n−1}(K). Assume that ch(L(λ)|_{GL_{n−1}(K)}) = ∑_{μ∈A} ch(L_{n−1}(μ)). Then we have L(λ)|_{GL_{n−1}(K)} = ⊕_{μ∈A}L_{n−1}(μ). In particular L(λ)|_{GL_{n−1}(K)} is semi-simple.

Proof: As the characters of simple modules are linearly independent, the module L(λ)|_{GL_{n−1}(K)} has a composition series in which each L_{n−1}(μ), for all μ ∈ A, occurs exactly once. Note that L(λ) carries a non degenerate contravariant form. Let S be a simple submodule of L(λ)|_{GL_{n−1}(K)}. Thus L(λ)|_{GL_{n−1}(K)}/S^{⊥} is isomorphic to S. As S does not occur as a quotient of L(λ)|_{GL_{n−1}(K)}/S, we have S ∩ S^{⊥} = 0. Hence S is a direct summand and L(λ)|_{GL_{n−1}(K)} is semi-simple. Thus, we have L(λ)|_{GL_{n−1}(K)} = ⊕_{μ∈A}L_{n−1}(μ). □

Let m < p. For any Young diagrams Y ∈ Y oung(n, m), let V (I) the set of all Young diagrams Y′ such that:

(i) c_{k}(Y′) ≤ c_{k}(Y) ≤ c_{k}(Y′) + 1, for all k ≥ 1,

(ii) Y′ ∈ Y oung(n − 1, m).

These conditions are indeed equivalent to the fact that there is a m-semi-standard tableau T of shape Y such that T [n − 1] = Y′, T [n] = Y. Also the definition of V (Y) is independent of m. Namely if Y is m-special and is m′-special for m ≠ m′, then condition (ii) is automatically satisfied.

Theorem 6.2. Let λ ∈ S_{n}. Then we have

L(λ)|_{GL_{n−1}(K)} = ⊕_{(Y_{0},Y_{1},...)}L_{n−1}(λ(Y_{0}) + pλ(Y_{1}) + …), where the direct sum runs over all uples (Y_{0}, Y_{1}, …) ∈ V (Y (λ(0))) ×

V (Y (λ(1))) × . . .. In particular, L(λ)|_{GL_{n−1}(K)} is semi-simple, multiplicity free, and the highest weights of any simple summand is in S_{n−1}.

Proof: Let Y ∈ Y oung(n, m). There is a natural bijection from

(i) the set of all m-semi-standard tableaux T of shape Y with labels ≤ n, to

(ii) the set of pairs (Y′, T′), where Y′ ∈ V (Y) and T′ is a m-semi-standard tableaux T of shape Y with labels ≤ n − 1.

Explicitly, Y′ = T [n − 1] and T′ is the tableau induced by T. It follows from Theorem 5.3 that we have

ch(L(λ)|_{GL_{n−1}(K)}) = ∑_{(Y_{0},Y_{1},...)} ch(L_{n−1}(λ(Y_{0}) + pλ(Y_{1}) + …)), where the sum runs over all uples (Y_{0}, Y_{1}, …) ∈ V (Y (λ(0))) × V (Y (λ(1))) × . . .. Let (Y_{0}, Y_{1}, …) be such an uple and set μ = ∑_{0≤k}(p^{k}λ(Y_{k})). It follows from Assertions (i) and (ii) of Lemma 5.1 that (λ(Y_{k}))_{0≤k} are the terms of the p-adic expansion of the polynomial weight μ. Hence the decomposition μ = ∑_{0≤k}(p^{k}λ(Y_{k})) with (Y_{0}, Y_{1}, …) ∈ V (Y (λ(0))) × V (Y (λ(1))) × … is unique. Thus the theorem follows from Lemma 6.1. □

Lemma 6.3. Let G_{1} …, G_{k} be k groups and let L be a finite dimensional G_{1} × ⋯ × G_{k}-module. Assume that L is semi-simple as G_{i}-module for all 1 ≤ i ≤ k. Then L is semi-simple as G_{1} × ⋯ × G_{k}-module.

A proof of this lemma can be found in [K] (lemma 1.6). □

For any k non-negative integers a_{1}, … a_{k} with n = a_{1} + a_{2} + …, there is a natural embedding of GL_{a_{1}}(K) × ⋯ × GL_{a_{k}}(K) inside GL_{n}(K).

Corollary 6.4. Let λ ∈ S_{n}. As a GL_{a_{1}}(K) × … GL_{a_{k}}(K)-module, L(λ) is semi-simple.

Proof: Using Theorem 6.2, we prove by induction over b that L(λ)|_{GL_{n−b}(K)} is semi-simple for all b ≤ n. Thus the corollary follows from Lemma 6.3. □

Remark: Let m < p and let λ be a special weight of degree n. The weight space L(λ)_{ϵ_{1}+···+ϵ_{n}} is the simple representation of Σ_{n} associated with the Young diagram transposed of Y (λ) (see [J]). Its dimension is computed by Theorem 4.3, and its restriction to subgroups Σ_{a_{1}} × ⋯ × Σ_{a_{k}} is semi-simple by Corollary 6.4. These two results for the symmetric groups were already established: we recover the main results of [M2] and [K]. Indeed the proof of the semi-simplicity of L(λ)_{GL_{a_{1}} (K)×···×GL_{a_{k}} (K)} is similar to the proof of Kleshchev for representations of the symmetric groups.

Bibliography:

[AM] A.M. Adamovitch, G.L. Rybnikov, Tilting modules for classical groups and Howe duality in positive characteristic. Transform. Groups, 1 (1996) 1-34.

[A] H.H. Andersen: The strong linkage principle. J.Reine Angew Math., 315 (1980) 53-55.

[AJS] H.H Andersen, J.C. Jantzen, W. Soergel, Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p. Astérisque, 220 (1994).

[CPSV] E. Cline, B. Parshall, L. Scott, W. van der Kallen, Rational and generic

cohomology. Invent. Math., 39 (1977) 143-163.

[D1] S. Donkin: Rational representations of algebraic groups. Springer Verlag, Lect.Notes in Math. 1140 (1985).

[D2] S. Donkin, On tilting modules for algebraic groups. Math. Z. 212 (1993),

39-60.

[GM1] G. Georgiev, O. Mathieu, Catégorie de fusion pour les groupes de Chevalley. C. R. Acad. Sci. Paris, 315 (1992), no. 6, 659–662.

[GM2] G. Georgiev, O. Mathieu, Fusion rings for modular representations of Chevalley groups. Mathematical aspects of conformal and topological field theories and quantum groups (South Hadley, 1992), Contemp. Math., 175 (1994) 89-100.

[G] J.A. Green,Polynomial representations of GL_{n}. Lecture Notes in Mathematics, 830. Springer-Verlag, Berlin-New York, 1980.

[J] J.C. Jantzen: Representations of algebraic groups, Academic Press, Orlando(1987).

[K] A. Kleshchev, Completely splittable representations of symmetric groups. J. Algebra, 181 (1996) 584-592.

[H] R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., 8 (1995) 1-182.

[Li] P. Littelmann,A generalization of the Littlewood-Richardson rule. J. Algebra, 130 (1990) 328-368.

[LR] D.E. Littlewood, A.R. Richardson,Group characters and algebra, Philos. Trans. Roy. Soc. London Ser. A, 233 (1934), 49-141.

[Lu1] G. Lusztig, Some problems in the representation theory of finite Chevalley groups. Classical groups and related topics (Santa Cruz, 1979) Proc. Symp. Pure Math., 37 (19780) 313-317.

[Lu2] G. Lusztig, Hecke algebras and Jantzen’s generic decomposition patterns. Adv. Math., 37 (1980) 121-164.

[M1] O. Mathieu, Filtrations of G-modules. Ann. Sci. Ecole Norm. Sup., 23 (1990) 625-644.

[M2] O. Mathieu, On the dimension of some modular irreducible representations of the symmetric group. Lett. Math. Phys., 38 (1996), 23-32.

[R] C.M. Ringel, The category of modules with good filtrations over a quasihereditary algebra has almost split sequences. Math. Z., 208 (1991) 209-223.

[So] W. Soergel, Conjecture de Lusztig., Séminaire Bourbaki (1994), Astérisque, 237 (1996) 75-86.

[St] R. Steinberg, Lectures on Chevalley groups. Notes, Yale University, (1967).

[W] Wang Jian-Pian: Sheaf cohomology on G/B and tensor products of Weyl modules. J.of Algebra, 77 (1982) 162-185.

Authors Addresses: O. M.: Université Louis Pasteur, IRMA, 7 rue René Descartes, 67000 F-Strasbourg.

G. P.:Université Louis Pasteur, IRMA, 7 rue René Descartes, 67000 F-Strasbourg.