On Translation Functors for General Linear and Symmetric Groups^{∗}

Jonathan Brundan and Alexander Kleshchev^{†}

1 Introduction

Let 𝔽 be an algebraically closed field of characteristic p. A central problem in the representation theory of the general linear group GL(n) = GL_{n}(𝔽) is to understand the structure of the tensor space V^{⊗r}, where V is the natural GL(n)-module. To do this (inductively) we would like information about the structure of tensor products of the form M ⊗ V, where M is an irreducible (or Weyl or tilting) module.

Given α ∈ ℤ/pℤ, we will define functors Tr^{α} and Tr_{α}, which roughly speaking are given by tensoring with the natural GL(n)-module V and its dual V^{∗} respectively, then projecting onto certain blocks determined by the residue α. In particular, for any rational GL(n)-module M,

M ⊗ V ≅ ⨁_{α∈ℤ/pℤ}Tr^{α} M and M ⊗ V^{∗} ≅ ⨁_{α∈ℤ/pℤ}Tr_{α} M.

In fact, these functors can be viewed as special cases of Jantzen’s translation functors. Our main results prove the following facts about Tr^{α} and Tr_{α}:

(1) Fix an irreducible rational GL(n)-module L. We give a precise combinatorial criterion for N := Tr^{α} L (resp. Tr_{α} L) to be irreducible; whenever this occurs, the inverse decomposition numbers for the irreducible module N can be computed from knowledge of the ones for L.

(2) Even when N is not irreducible, we show that it is always a (contravariantly) self-dual indecomposable module, and describe its (simple) socle and head, as well as the space of high weight vectors in N, precisely.

(3) We construct a natural filtration of N by precisely b high weight modules, for some combinatorially defined constant b (depending on L and α), compute certain composition multiplicities in N and describe the endomorphism ring of N using Casimir-type operators.

(4) In particular, we show that the dimension of the endomorphism ring of N is precisely b, and that the Loewy length of N is at least 2b − 1.

∗1991 subject classification: 20G05, 20C05.

Special cases of these results, with b ≤ 2, follow from [J, II.7]. In general, the natural number b can take any value. We also obtain various related results describing the action of Tr^{α} and Tr_{α} on Weyl modules, and obtain combinatorial criteria for L ⊗ V and L ⊗ V^{∗} to be completely reducible.

We also consider applications to the representation theory of the symmetric group Σ_{n} over 𝔽. The functor Tr^{α} corresponds under the Schur functor to a certain ‘α-induction’ functor Ind^{α} : 𝔽Σ_{n} -mod → 𝔽Σ_{n+1} -mod. Roughly speaking, the functor Ind^{α} can be defined as ordinary induction from Σ_{n} to Σ_{n+1} followed by projection onto certain blocks. This goes back to G. Robinson [R]. As consequences of the above results (1)–(4) for Tr^{α}, we will obtain analogous results for the functor Ind^{α}, which are the counterparts of the results proved in [K2]–[K6], [B1], [B2] for the ‘α-restriction’ functor Res_{α}.

In section 2 and section 3, we state our results precisely, for general linear and symmetric groups respectively. In the remaining sections 4–8, we prove the results, usually by relating the problems under consideration to certain branching problems studied in [K2]–[K6].

2 Statement of results for general linear groups

To describe our main results for GL(n), we need a little notation. Let X(n) denote the set of all n-tuples λ = (λ_{1}, …, λ_{n}) ∈ ℤ^{n} and X^{+}(n) ⊂ X(n) denote all λ ∈ X(n) satisfying λ_{1} ≥ ⋯ ≥ λ_{n}. As we explain in section 4, X^{+}(n) can be identified with the dominant weights for the root system of GL(n). So for λ ∈ X^{+}(n), we have the (rational) GL(n)-modules L_{n}(λ), Δ_{n}(λ) and ∇_{n}(λ), which are the irreducible, standard (or Weyl) and costandard modules of highest weight λ, respectively. For 1 ≤ i ≤ n, let ε_{i} denote the n-tuple (0, …, 0, 1, 0, …, 0) where 1 is in the ith position. We also denote the natural GL(n)-module by V_{n} and its dual by V_{n}^{∗}.

Given (a, b) ∈ ℤ × ℤ, define the corresponding p-residue res(a, b) to be (b − a) regarded as an element of the ring ℤ/pℤ. For α ∈ ℤ/pℤ and λ ∈ X(n), define the α-content of λ to be the integer:

cont_{α}(λ) := |\{(a, b) | \begin{matrix}1 ≤ a ≤ n, 0 < b ≤ λ_{a}, \\ res(a, b) = α \end{matrix} \}| − |\{(a, b) | \begin{matrix}1 ≤ a ≤ n, λ_{a} ≤ b < 0, \\ res(a, b) = α \end{matrix} \}| .

Say λ, μ ∈ X(n) are linked, written λ ∼ μ, if cont_{α}(λ) = cont_{α}(μ) for all α ∈ ℤ/pℤ. The linkage principle proved in [CL] implies that if Ext_{GL(n)}^{1}(L_{n}(λ), L_{n}(μ)) ≠ 0, for λ, μ ∈ X^{+}(n), then λ ∼ μ. We remark that in fact the blocks of GL(n) (which can in general be smaller than the linkage classes) are explicitly known, see [J, II.7.2(3)].

Let C_{n} denote the category of all rational GL(n)-modules. For any λ ∈ X(n), let C_{n}(λ) denote the full subcategory of C_{n} consisting of all M ∈ C_{n} such that all composition factors of M are of the form L_{n}(μ) for μ ∼ λ. By the linkage principle,

C_{n} ≅ ⨁_{λ} C_{n}(λ)

where λ runs over the set of ∼-equivalence classes in X(n).

Fix a residue α ∈ ℤ/pℤ. We can now define the functors

Tr^{α} : C_{n} → C_{n}  and  Tr_{α} : C_{n} → C_{n}.

It suffices to define the restriction of Tr^{α} (resp. Tr_{α}) to C_{n}(λ), for any λ ∈ X(n) (then we may extend the functors additively to all of C_{n}). Given M ∈ C_{n}(λ), we let Tr^{α} M (resp. Tr_{α} M) denote the largest submodule of M ⊗ V_{n} (resp. M ⊗ V_{n}^{∗}) all of whose composition factors are of the form L_{n}(μ) with

cont_{α}(μ) = cont_{α}(λ) + 1 (resp. cont_{α}(μ) = cont_{α}(λ) − 1),

and cont_{β}(μ) = cont_{β}(λ) for all α ≠ β ∈ ℤ/pℤ. By the linkage principle, Tr^{α} M (resp. Tr_{α} M) is a direct summand of M ⊗ V_{n} (resp. M ⊗ V_{n}^{∗}). Given a morphism θ : M → N, Tr^{α} θ is just the restriction to Tr^{α} M of the natural map θ ⊗ 1 : M ⊗ V_{n} → N ⊗ V_{n}, and similarly for Tr_{α}.

On any fixed block C_{n}(λ), the functor Tr^{α} (resp. Tr_{α}), for a suitable choice of α, coincides with the translation functor T_{λ}^{μ} defined in [J, II.7.6], for a weight μ ∈ X(n) such that the dominant conjugate of (μ − λ) is equal to the highest weight of V_{n} (resp. V_{n}^{∗}). We note initially that the argument of [J, II.7.6] shows easily that the functors Tr^{α} and Tr_{α} are (left and right) adjoint to one another, and both are exact.

In the next combinatorial definitions, the notions of normal and good first appeared in [K3]; the dual notions of conormal and cogood are new. The reader may be more familiar with normal and good nodes; we reserve this terminology for the symmetric group setting when definitions are ‘transposed’, see section 3. In the definitions, we call a map ψ from a set M ⊆ ℤ to a set N ⊆ ℤ increasing (resp. decreasing) if ψ(m) > m (resp. ψ(m) < m) for all m ∈ M.

Fix λ ∈ X^{+}(n) and 1 ≤ i ≤ n. We say i is λ-removable if either i = n or 1 ≤ i < n and λ_{i} > λ_{i+1}; equivalently, i is λ-removable if λ − ε_{i} ∈ X^{+}(n). We say i is λ-addable if either i = 1 or 1 < i ≤ n and λ_{i} < λ_{i−1}; equivalently, i is λ-addable if λ + ε_{i} ∈ X^{+}(n).

Say i is normal for λ if i is λ-removable and there is a decreasing injection from the set of

λ-addable j with i < j ≤ n and res(i, λ_{i}) = res(j, λ_{j} + 1)

into the set of

λ-removable j′ with i < j′ ≤ n and res(i, λ_{i}) = res(j′, λ_{j′}).

Say i is good for λ if i is normal for λ and there is no j that is normal for λ with 1 ≤ j < i and res(j, λ_{j}) = res(i, λ_{i}).

Say i is conormal for λ if i is λ-addable and there is an increasing injection from the set of

λ-removable j with 1 ≤ j < i and res(j, λ_{j}) = res(i, λ_{i} + 1)

into the set of

λ-addable j′ with 1 ≤ j′ < i and res(j′, λ_{j′} + 1) = res(i, λ_{i} + 1).

Say i is cogood for λ if i is conormal for λ and there is no j that is conormal for λ with i < j ≤ n and res(j, λ_{j} + 1) = res(i, λ_{i} + 1).

We refer the reader to the combinatorial Lemma 5.9 for an explanation of the duality between these definitions. For a fixed α ∈ ℤ/pℤ, there is at most one i that is good for λ with res(i, λ_{i}) = α. Moreover, such an i exists if and only if there is at least one j that is normal for λ with res(j, λ_{j}) = α. A similar statement is true for conormal and cogood.

Our first result describes the effect of Tr^{α} on standard modules (the analogous result for costandard modules follows easily since Tr^{α} commutes with contravariant duality):

Theorem A. Fix λ ∈ X^{+}(n) and a residue α ∈ ℤ/pℤ. Then, Tr^{α} Δ_{n}(λ) is zero unless there is at least one λ-addable i with 1 ≤ i ≤ n and res(i, λ_{i} + 1) = α. In that case,

(i) Tr^{α} Δ_{n}(λ) has a filtration with factors Δ_{n}(λ + ε_{j}) for all λ-addable j with 1 ≤ j ≤ n and res(j, λ_{j} + 1) = α, each appearing with multiplicity one;

(ii) the head of Tr^{α} Δ_{n}(λ) is ⨁_{j} L_{n}(λ + ε_{j}) where the sum is over all j with 1 ≤ j ≤ n such that j is normal for λ + ε_{j} and res(j, λ_{j} + 1) = α;

Next we consider Tr^{α} applied to an irreducible module. Here and later [M : L] denotes the multiplicity of an irreducible module L as a composition factor of a module M.

Theorem B. Fix λ ∈ X^{+}(n) and a residue α ∈ ℤ/pℤ. Then, Tr^{α} L_{n}(λ) is zero unless there is a (necessarily unique) i that is cogood for λ with res(i, λ_{i} + 1) = α. In that case,

(i) Tr^{α} L_{n}(λ) is an indecomposable, contravariantly self-dual module, with simple socle and head isomorphic to L_{n}(λ + ε_{i});

(ii) Tr^{α} L_{n}(λ) is irreducible if and only if there is a unique j with 1 ≤ j ≤ n such that j is conormal for λ and res(j, λ_{j} + 1) = α; in particular, L_{n}(λ) ⊗ V_{n} is completely reducible if and only if for every α ∈ ℤ/pℤ, there is at most one j such that j is conormal for λ and res(j, λ_{j} + 1) = α;

(iii) for any μ ∈ X^{+}(n),

Hom_{GL(n)}(Δ_{n}(μ), Tr^{α} L_{n}(λ)) = \{ \begin{matrix}𝔽 & if μ = λ + ε_{j} for some j that is conormal for λ, \\ & with res(j, λ_{j} + 1) = α; \\ 0 & otherwise; \end{matrix}

(iv) for any λ-addable j with 1 ≤ j ≤ n,

[Tr^{α} L_{n}(λ) : L_{n}(λ + ε_{j})] = \{ \begin{matrix}b_{j} & if j is conormal for λ and res(j, λ_{j} + 1) = α, \\ 0 & otherwise \end{matrix}

where b_{j} denotes the number of k with 1 ≤ k ≤ j such that k is conormal for λ and res(k, λ_{k} + 1) = α;

(v) the endomorphism ring End_{GL(n)}(Tr^{α} L_{n}(λ)) is isomorphic to the truncated polynomial ring 𝔽[T]/(T^{b}), of dimension b, where b is the number of j with 1 ≤ j ≤ n such that j is conormal for λ and res(j, λ_{j} + 1) = α.

If, in the language of [J, II.6.2], λ + ε_{i} lies in the upper closure of the facet containing λ, where i is as in Theorem B, then one can easily deduce that Tr^{α} L_{n}(λ) is irreducible directly from [J, II.7.15]. Theorem B(ii) gives a necessary and sufficient condition for Tr^{α} L_{n}(λ) to be irreducible, and shows that there are many other more general circumstances when Tr^{α} L_{n}(λ) is irreducible. The significance of this is the following corollary, which follows immediately from Theorem B(i) and (ii), by exactness of Tr^{α}:

Corollary 1. Fix λ ∈ X^{+}(n) and α ∈ ℤ/pℤ. Suppose that there is a unique i with 1 ≤ i ≤ n such that i is conormal for λ and res(i, λ_{i} + 1) = α. If

ch L_{n}(λ) = ∑_{μ∈X^{+}(n)} c_{λ,μ} ch Δ_{n}(μ)

with almost all coefficients c_{λ,μ} equal to zero, then

ch L_{n}(λ + ε_{i}) = ∑_{μ∈X^{+}(n)} c_{λ,μ} ch Tr^{α} Δ_{n}(μ),

which is known by Theorem A(i).

Our next result gives further information about the structure of Tr^{α} L_{n}(λ):

Theorem C. Fix λ ∈ X^{+}(n) and a residue α ∈ ℤ/pℤ. Let 1 = s_{1} < s_{2} < ⋯ < s_{b} denote the set of all j with 1 ≤ j ≤ n such that j is conormal for λ and res(j, λ_{j} + 1) = α. Then, N := Tr^{α} L_{n}(λ) has a filtration 0 = N_{0} < N_{1} < ⋯ < N_{b} = N such that:

(i) for 1 ≤ i ≤ b, N_{i}/N_{i−1} is a non-zero quotient of Δ_{n}(λ + ε_{s_{i}});

(ii) for 1 ≤ i ≤ j ≤ b, dim Hom_{GL(n)}(N_{j}/N_{j−1}, N_{i}/N_{i−1}) = [N_{i}/N_{i−1} : L_{n}(λ + ε_{s_{j}})] = 1; (iii) for 1 ≤ i < b, the extension 0 → N_{i}/N_{i−1} → N_{i+1}/N_{i−1} → N_{i+1}/N_{i} → 0 does not

split; (iv) the Loewy length of N_{i}/N_{i−1} is at least b − i + 1;

(v) the Loewy length of N is at least 2b − 1.

We state now the dual results to Theorems A, B and C, for the adjoint functor Tr_{α}. The dual statement to Corollary 1 is easily deduced from Theorem B′, and we leave the details to the reader.

Theorem A′. Fix λ ∈ X^{+}(n) and a residue α ∈ ℤ/pℤ. Then, Tr_{α} Δ_{n}(λ) is zero unless there is at least one λ-removable i with 1 ≤ i ≤ n and res(i, λ_{i}) = α. In that case,

(i) Tr_{α} Δ_{n}(λ) has a filtration with factors Δ_{n}(λ−ε_{j}) for all λ-removable j with 1 ≤ j ≤ n and res(j, λ_{j}) = α, each appearing with multiplicity one;

(ii) the head of Tr_{α} Δ_{n}(λ) is ⨁_{j} L_{n}(λ − ε_{j}) where the sum is over all j with 1 ≤ j ≤ n such that j is conormal for λ − ε_{j} and res(j, λ_{j}) = α;

Theorem B′. Fix λ ∈ X^{+}(n) and a residue α ∈ ℤ/pℤ. Then, Tr_{α} L_{n}(λ) is zero unless there is a (necessarily unique) i that is good for λ with res(i, λ_{i}) = α. In that case,

(i) Tr_{α} L_{n}(λ) is an indecomposable, contravariantly self-dual module, with simple socle and head isomorphic to L_{n}(λ − ε_{i});

(ii) Tr_{α} L_{n}(λ) is irreducible if and only if there is a unique j with 1 ≤ j ≤ n such that j is normal for λ and res(j, λ_{j}) = α; in particular, L_{n}(λ) ⊗ V_{n}^{∗} is completely reducible if and only if for every α ∈ ℤ/pℤ, there is at most one j such that j is normal for λ and res(j, λ_{j}) = α;

(iii) for any μ ∈ X^{+}(n),

Hom_{GL(n)}(Δ_{n}(μ), Tr_{α} L_{n}(λ)) = \{ \begin{matrix}𝔽 & if μ = λ − ε_{j} for some j that is normal for λ, \\ & with res(j, λ_{j}) = α; \\ 0 & otherwise; \end{matrix}

(iv) for any λ-removable j with 1 ≤ j ≤ n,

[Tr_{α} L_{n}(λ) : L_{n}(λ − ε_{j})] = \{ \begin{matrix}b_{j} & if j is normal for λ and res(j, λ_{j}) = α, \\ 0 & otherwise \end{matrix}

where b_{j} is the number of k with j ≤ k ≤ n such that k is normal for λ and res(k, λ_{k}) = α;

(v) the endomorphism ring End_{GL(n)}(Tr_{α} L_{n}(λ)) is isomorphic to the truncated polynomial ring 𝔽[T]/(T^{b}), of dimension b, where b is the number of j with 1 ≤ j ≤ n such that j is normal for λ and res(j, λ_{j}) = α.

Theorem C′. Fix λ ∈ X^{+}(n) and a residue α ∈ ℤ/pℤ. Let n = s_{1} > s_{2} > ⋯ > s_{b} denote the set of all j with 1 ≤ j ≤ n such that j is normal for λ and res(j, λ_{j}) = α. Then, N := Tr_{α} L_{n}(λ) has a filtration 0 = N_{0} < N_{1} < ⋯ < N_{b} = N such that:

(i) for 1 ≤ i ≤ b, N_{i}/N_{i−1} is a non-zero quotient of Δ_{n}(λ − ε_{s_{i}});

(ii) for 1 ≤ i ≤ j ≤ b, dim Hom_{GL(n)}(N_{j}/N_{j−1}, N_{i}/N_{i−1}) = [N_{i}/N_{i−1} : L_{n}(λ + ε_{s_{j}})] = 1; (iii) for 1 ≤ i < b, the extension 0 → N_{i}/N_{i−1} → N_{i+1}/N_{i−1} → N_{i+1}/N_{i} → 0 does not

split.

(iv) the Loewy length of N_{i}/N_{i−1} is at least b − i + 1;

(v) the Loewy length of N is at least 2b − 1.

In the remainder of the paper, we will not mention the functors Tr^{α} and Tr_{α} again, preferring simply to work with the functors ? ⊗ V_{n} and ? ⊗ V_{n}^{∗}.

We now indicate where proofs of the above theorems can be found in the main body of the paper. Using the linkage principle and contravariant duality, Theorems A(i) and A′(i) are special cases of Lemma 4.8, and Theorems A(ii) and A′(ii) follow from Theorems 5.12(ii) and 5.10(ii) respectively. Theorems B(iii) and B′(iii) follow from the linkage principle and Theorems 5.12(i) and 5.10(i) respectively, while the description of the socles in Theorems B(i) and B′(i) follow from Theorems 5.12(iii) and 5.10(iii). In particular, since these socles are either simple (or zero), the modules Tr^{α} L_{n}(λ) and Tr_{α} L_{n}(λ) are indecomposable (or zero), and they are obviously self-dual as Tr^{α} and Tr_{α} commute with contravariant duality, giving the remaining parts of Theorems B(i) and B′(i). Theorem B(ii) follows from parts (i) and (iii) together with Proposition 4.7, applied to N = Tr^{α} L_{n}(λ) and M = Tr^{α} ∇_{n}(λ), and Theorem B′(ii) follows similarly. Theorem B′(iv) and (v) are Theorem 7.7 and Theorem 8.10 respectively; Theorem B(iv) and (v) follow from these using the duality argument of the proof of Corollary 6.5 together with the combinatorial Lemma 5.9. Finally, Theorem C′(i) follows from Theorem 6.3 together with the linkage principle, as explained in (8.5). The remaining parts of Theorem C′ are proved in Theorem 8.14 and Corollary 8.15, and Theorem C follows from Theorem C′ by the duality argument again.

3 Statement of results for symmetric groups

Let Σ_{r} be the symmetric group on r letters. If λ is a partition of r we write λ ⊢ r. We denote by S^{λ} the Specht module over 𝔽Σ_{r} corresponding to a partition λ ⊢ r, and by D^{λ} the irreducible 𝔽Σ_{r}-module, corresponding to a p-regular partition λ ⊢ r (the reader is referred to [J1], [J2] or [JK] for these and other standard notions in the representation theory of symmetric groups).

Fix a partition λ = (λ_{1}, λ_{2}, …) ⊢ r. We identify λ with its Young diagram

λ = {(i, j) ∈ ℕ × ℕ | j ≤ λ_{i}}.

The elements of ℕ×ℕ are called nodes. A node of the form (i, λ_{i}) is called a removable node (of λ) if λ_{i} > λ_{i+1}; a node of the form (i, λ_{i} + 1) is called an addable node (for λ) if i = 1 or i > 1 and λ_{i} < λ_{i−1}. If A = (i, λ_{i}) is a removable node, we let

λ_{A} := λ \ {A} = (λ_{1}, …, λ_{i−1}, λ_{i} − 1, λ_{i+1}, …),

the partition of (r − 1) obtained by removing A from λ. If B = (i, λ_{i} + 1) is an addable node, we let

λ^{B} := λ ∪ {B} = (λ_{1}, …, λ_{i−1}, λ_{i} + 1, λ_{i+1}, …),

the partition of (r + 1) obtained by adding B to λ. The p-residue of a node A = (i, j) is defined as in section 2: res A := (j − i) ∈ ℤ/pℤ. We say a node A = (i, j) is to the right (resp. left) of B = (k, l) if j > l (resp. j < l).

A removable node A (of λ) is called a normal node if for every addable node B to the right of A with res B = res A there exists a removable node C(B) strictly between A and B with res C(B) = res A, and moreover B ≠ B′ implies C(B) ≠ C(B′). A removable node is called a good node if it is the leftmost among the normal nodes of a fixed residue.

An addable node B (for λ) is called a conormal node if for every removable node A to the left of B with res A = res B there exists an addable node C(A) strictly between B and A with res C(A) = res B, and moreover A ≠ A′ implies C(A) ≠ C(A′). An addable node is called a cogood node if it is the rightmost among the conormal nodes of a fixed residue.

In the next theorem, the last two equalities are new, while the first two equalities can be deduced from [K3, 0.4,0.5] and Lemma 5.9(iv) using Frobenius reciprocity.

Theorem D. Fix p-regular partitions λ ⊢ r, μ ⊢ (r + 1). Then,

Hom_{Σ_{r+1}}(D^{μ}, D^{λ} ↑^{Σ_{r+1}}) = \{ \begin{matrix}𝔽 & if μ = λ^{B} for some cogood node B for λ, \\ 0 & otherwise; \end{matrix}
Hom_{Σ_{r+1}}(S^{λ} ↑^{Σ_{r+1}}, D^{μ}) = \{ \begin{matrix}𝔽 & if λ = μ_{A} for some normal node A for μ, \\ 0 & otherwise; \end{matrix}
Hom_{Σ_{r+1}}(S^{μ}, D^{λ} ↑^{Σ_{r+1}}) = \{ \begin{matrix}𝔽 & if μ = λ^{B} for some conormal node B for λ, \\ 0 & otherwise; \end{matrix}
Hom_{Σ_{r+1}}(S^{λ} ↑^{Σ_{r+1}}, (S^{μ})^{∗}) = \{ \begin{matrix}𝔽 & if μ = λ^{B} for some addable node B for λ, \\ 0 & otherwise. \end{matrix}

For α ∈ ℤ/pℤ and a partition λ, define the α-content of λ to be the integer

cont_{α}(λ) := |{A ∈ λ | res A = α}|

(which is a special case of the definition in section 2). For two partitions λ and μ we write λ ∼ μ, if cont_{α}(λ) = cont_{α}(μ) for all α ∈ ℤ/pℤ. The ‘Nakayama Conjecture’ (proved e.g. in [JK]) claims that 𝔽Σ_{r}-modules D^{λ} and D^{μ} are in the same block if and only if λ ∼ μ.

Fix a residue α ∈ ℤ/pℤ. We define the functors

Ind^{α} : 𝔽Σ_{r}-mod → 𝔽Σ_{r+1}-mod  and  Res_{α} : 𝔽Σ_{r}-mod → 𝔽Σ_{r−1}-mod,

by defining them first on a module M in any fixed block, and then extending additively to all of 𝔽Σ_{r}-mod. Assume M belongs to the block corresponding to the residue contents c_{0}, c_{1}, …, c_{p−1}, that is, every composition factor of M is of the form D^{λ} with cont_{β}(λ) = c_{β} for all β ∈ ℤ/pℤ. We let Ind^{α} M (resp. Res_{α} M) denote the largest submodule of M ↑^{Σ_{r+1}} (resp. M ↓_{Σ_{r−1}}) all of whose composition factors are of the form D^{μ} with cont_{α}(μ) = c_{α} + 1 (resp. cont_{α}(μ) = c_{α} − 1), and cont_{β}(μ) = c_{β} for all α ≠ β ∈ ℤ/pℤ. Given a morphism θ : M → N, Ind^{α} θ is just the restriction to Ind^{α} M of the natural map θ̂: M ↑^{Σ_{r+1}}→ N ↑^{Σ_{r+1}} induced by θ, and similarly for Res_{α}. We have

M ↑^{Σ_{r+1}} ≅ ⨁_{α∈ℤ/pℤ}Ind^{α} M and M ↓_{Σ_{r−1}} ≅ ⨁_{α∈ℤ/pℤ}Res_{α} M.

The functors just defined are called Robinson’s α-induction and α-restriction functors (cf. [JK, 6.3.16]).

Our next result describes the effect of Ind^{α} on irreducible modules, and is the symmetric group analogue of Theorem B.

Theorem E. Fix a p-regular partition λ ⊢ r and a residue α ∈ ℤ/pℤ. Then, Ind^{α} D^{λ} is zero unless λ has at least one conormal node of residue α. In that case,

(i) Ind^{α} D^{λ} is an indecomposable, self-dual module, with simple socle and head isomorphic to D^{λ^{B}} where B is the (unique) cogood node of residue α;

(ii) Ind^{α} D^{λ} is irreducible if and only if there is a unique conormal node of residue α; in particular, the induced module D^{λ} ↑^{Σ_{r+1}} is completely reducible if and only if all conormal nodes have different residues.

(iii) for any p-regular μ ⊢ (r + 1),

Hom_{Σ_{r+1}}(S^{μ}, Ind^{α} D^{λ}) = \{ \begin{matrix}𝔽 & if μ = λ^{B} for some conormal node B with res B = α, \\ 0 & otherwise; \end{matrix}

(iv) for any addable node B such that λ^{B} is p-regular,

[Ind^{α} D^{λ} : D^{λ^{B}}] = \{ \begin{matrix}d_{B} & if B is conormal for λ and res B = α, \\ 0 & otherwise \end{matrix}

where d_{B} denotes the number of conormal nodes C to the left of B (counting B itself) such that res C = α;

(v) the endomorphism ring End_{Σ_{r+1}}(Ind^{α} D^{λ}) is isomorphic to the truncated polynomial ring 𝔽[T]/(T^{d}), of dimension d, where d is the number of conormal nodes B with res B = α.

We note that a criterion for complete reducibility of D^{λ} ↑^{Σ_{r+1}} different from the one in Theorem E(ii) was found in [K5, Theorem C]. It is difficult to see directly that the two combinatorial conditions are equivalent (see Corollary 8.12(iii)). Also, Theorem E(i) is immediate from [K4, Theorem 3.2], using the combinatorial Lemma 5.9.

The next result is parallel to Corollary 1, and is useful in calculating inverse decomposition matrices. It follows immediately from Theorem E(ii) together with exactness of the functor Ind^{α}; in the second part of the statement, we have used the (known) fact that for any μ ⊢ r, Ind^{α} S^{μ} has a filtration with factors S^{μ^{C}} for all addable nodes C for μ with res C = α.

Corollary 2. Let λ ⊢ r be p-regular, and α ∈ ℤ/pℤ. Suppose that λ has a unique conormal node B of residue α. If (in the Grothendieck group)

D^{λ} = ∑_{μ⊢r} c_{λ,μ}S^{μ}  then  D^{λ^{B}} = ∑_{μ⊢r} c_{λ,μ} Ind^{α} S^{μ} = ∑_{μ⊢r} c_{λ,μ} (∑_{C} S^{μ^{C}}) ,

where the last summation is over all addable nodes C for μ with res C = α.

We state now for completeness the dual result to Theorem E, for the functor Res_{α}. These results are known, see [K3, 0.4,0.5], [K4, 3.1], [K5, Theorem B], [K6, 1.4].

Theorem E′. Fix a p-regular partition λ ⊢ r and a residue α ∈ ℤ/pℤ. Then, Res_{α} D^{λ} is zero unless λ has at least one normal node of residue α. In that case,

(i) Res_{α} D^{λ} is an indecomposable, self-dual module, with simple socle and head isomorphic to D^{λ_{A}} where A is the (unique) good node of residue α;

(ii) Res_{α} D^{λ} is irreducible if and only if there is a unique normal node of residue α; in particular, the restriction D^{λ} ↓_{Σ_{r−1}} is completely reducible if and only if all normal nodes have different residues.

(iii) for any p-regular μ ⊢ (r − 1),

Hom_{Σ_{r−1}}(S^{μ}, Res_{α} D^{λ}) = \{ \begin{matrix}𝔽 & if μ = λ_{A} for some normal node A with res A = α, \\ 0 & otherwise; \end{matrix}

(iv) for any removable node A such that λ_{A} is p-regular,

[Res_{α} D^{λ} : D^{λ_{A}}] = \{ \begin{matrix}d_{A} & if A is normal for λ and res A = α, \\ 0 & otherwise \end{matrix}

where d_{A} denotes the number of normal nodes C to the right of A (counting A itself) such that res C = α;

(v) the endomorphism ring End_{Σ_{r−1}}(Res_{α} D^{λ}) is isomorphic to the truncated polynomial ring 𝔽[T]/(T^{d}), of dimension d, where d is the number of normal nodes A with res A = α.

Remark. We remark that an analogue of Theorem C can also be proved for the functor Ind^{α}. More precisely, fix a p-regular partition λ ⊢ r and a residue α ∈ ℤ/pℤ. Let B_{1}, B_{2} … B_{d} be all conormal nodes of residue α counted from left to right. Then, one can show that I := Ind^{α} D^{λ} has a filtration 0 = I_{0} < I_{1} < ⋯ < I_{d} = I such that:

(i) for 1 ≤ j ≤ d, I_{j}/I_{j−1} is a non-zero quotient of S^{λ^{B_{j}}};

(ii) for 1 ≤ j ≤ k ≤ d such that λ^{B_{k}} is p-regular, [I_{j}/I_{j−1} : D^{λ^{B_{k}}}] = 1;

(iii) the extension 0 → I_{j}/I_{j−1} → I_{j+1}/I_{j−1} → I_{j+1}/I_{j} → 0 does not split;

(iv) if every λ^{B_{j}} is p-regular, then the Loewy length of I is at least 2d − 1.

We now indicate briefly how to deduce Theorems D and E from the results for GL(n) stated in section 2, using Schur functors. The key point is that a Schur functor applied to a GL(n)-module of the form M ⊗ V_{n} (where M is polynomial of degree r < n) gives the 𝔽Σ_{r+1}-module induced from the 𝔽Σ_{r}-module obtained by applying the Schur functor to M; see [BK, Theorem 4.17] for the precise statement. Now fix p-regular partitions λ ⊢ r and μ ⊢ (r + 1), and choose n > r. We can regard the transpose partitions λ^{t} and μ^{t} as elements of X^{+}(n). In [BK, Corollary 4.18] (cf. [DE, Lemma 2.5]), we used the Schur functor to show that:

Hom_{Σ_{r+1}}(D^{μ}, D^{λ} ↑^{Σ_{r+1}}) ≅ Hom_{GL(n)}(L_{n}(μ^{t}), L_{n}(λ^{t}) ⊗ V_{n}),
Hom_{Σ_{r+1}}(S^{μ}, D^{λ} ↑^{Σ_{r+1}}) ≅ Hom_{GL(n)}(Δ_{n}(μ^{t}), L_{n}(λ^{t}) ⊗ V_{n}),
Hom_{Σ_{r+1}}(S^{λ} ↑^{Σ_{r+1}}, D^{μ}) ≅ Hom_{GL(n)}(L_{n}(μ^{t}), ∇_{n}(λ^{t}) ⊗ V_{n}),
Hom_{Σ_{r+1}}(S^{λ} ↑^{Σ_{r+1}}, (S^{μ})^{∗}) ≅ Hom_{GL(n)}(Δ_{n}(μ^{t}), ∇_{n}(λ^{t}) ⊗ V_{n}).

Theorem D follows from these isomorphisms, (5.1) and Theorem 5.12 (note for symmetric groups we have ‘transposed’ the combinatorial definitions of normal, conormal and cogood from section 2). Now Theorem E(iii) and the fact that Ind^{α} D^{λ} has the socle as claimed in Theorem E(i) follow immediately from Theorem D on taking blocks. Since the socle is simple (whenever it is non-zero), it follows that Ind^{α} D^{λ} is indecomposable, and it is selfdual as D^{λ} is and Ind^{α} commutes with duality, proving the remaining parts of Theorem E(i). Theorem E(iv) follows from Theorem B(iv) together with the following fact proved in [BK, Theorem 4.16] (where λ ⊢ r and μ ⊢ (r + 1) are p-regular and n > r):

[D^{λ} ↑^{Σ_{r+1}}: D^{μ}] = [L_{n}(λ^{t}) ⊗ V_{n} : L_{n}(μ^{t})].

To prove Theorem E(ii), we note that as Ind^{α} D^{λ} has simple socle and head isomorphic to D^{λ^{B}} (where B is as in (i)), Ind^{α} D^{λ} is irreducible if and only if [Ind^{α} D^{λ} : D^{λ^{B}}] = 1. So Theorem E(ii) follows from Theorem E(iv) (alternatively, one can deduce Theorem E(ii) directly from Theorem B(ii) using [JS, 2.13] and a block argument). Finally, we observe that if λ ⊢ r < n is p-regular, then L_{n}(λ^{t})⊗V_{n} has p-restricted socle and head, which follows from [BK, Corollary 2.12]. So by [BK, Theorem 4.17],

End_{GL(n)}(L_{n}(λ^{t}) ⊗ V_{n}) ≅ End_{Σ_{r+1}}(D^{λ} ↑^{Σ_{r+1}}).

This isomorphism is even an isomorphism of 𝔽-algebras, as follows from the proof of [BK, Theorem 4.17] (see [BK, Lemma 2.17(ii)]), since the isomorphism is given by a natural restriction map. Theorem E(v) follows from Theorem B(v) by the same argument at the level of blocks.

4 A complete reducibility criterion

In the remainder of the paper, we prove the results stated in section 2 and section 3. We begin in this preliminary section with some general results in the setting of rational representations of an arbitrary reductive algebraic group as in [J]. Actually the results in this section (except for Lemma 4.8) are true for an arbitrary quasi-hereditary algebra with a duality fixing the simple modules. In later sections, we will only be concerned with GL(n).

Fix a (connected) reductive algebraic group G over our algebraically closed field 𝔽. A G-module always means a rational 𝔽G-module as in [J]. Fix a maximal torus T < G and a Borel subgroup B^{+} > T. Let R denote the root system of G relative to T, and R^{+} ⊂ R denote the positive roots determined by the choice of positive Borel subgroup B^{+}. Let X(T) denote the character group of T, and call elements of X(T) weights. The Weyl group W = N_{G}(T)/T acts on X(T), and we let ⟨., .⟩ denote some fixed W -invariant inner product on X(T) ⊗_{ℤ} ℝ. For μ ∈ X(T) ⊗_{ℤ} ℝ, μ^{∨} denotes \frac{2μ}{⟨μ,μ⟩}. Let w_{0} denote the longest element of W. We have the usual dominance order on X(T), defined by λ > μ if (λ − μ) is a sum of positive roots. The weight λ ∈ X(T) is dominant (relative to B^{+}) if ⟨λ, α^{∨}⟩ ≥ 0 for all α ∈ R^{+}; we let X^{+}(T) denote the set of all dominant weights λ ∈ X(T). For λ ∈ X^{+}(T), we have the G-modules Δ(λ) (denoted V (λ) in [J]), ∇(λ) (denoted H^{0}(λ) in [J]) and L(λ), which are the standard, costandard and irreducible G-modules of highest weight λ respectively.

In the special case that G = GL(n), we always take T = T (n) to be all diagonal invertible matrices and B^{+} = B^{+}(n) to be all upper triangular invertible matrices. The character group of T (n) is the free abelian group with generators ε_{1}, …, ε_{n}, where ε_{i} denotes the standard character defined by diag(t_{1}, …, t_{n}) ↦ t_{i} for all t_{1}, …, t_{n} ∈ 𝔽^{×}, and we identify this with the set X(n) of all n-tuples λ = (λ_{1}, …, λ_{n}) ∈ ℤ^{n}, by letting (λ_{1}, …, λ_{n}) ∈ X(n) correspond to the character ∑_{i} λ_{i}ε_{i}. For λ = (λ_{1}, …, λ_{n}) ∈ X(n), we write:

|λ| for λ_{1} + ⋯ + λ_{n} ∈ ℤ;
λ̄ for (λ_{1}, …, λ_{n−1}) ∈ X(n − 1);
λ̃ for (λ_{1}, …, λ_{n}, 0) ∈ X(n + 1).

The root system of GL(n) is the set {ε_{i} − ε_{j} | 1 ≤ i, j ≤ n, i ≠ j}, and the root ε_{i} − ε_{j} is positive if i < j. A weight (λ_{1}, …, λ_{n}) ∈ X(n) is dominant precisely when λ_{1} ≥ ⋯ ≥ λ_{n}, and we let X^{+}(n) ⊂ X(n) denote the set of all such dominant weights. For λ ∈ X^{+}(n), we then denote the irreducible, standard and costandard for GL(n) of highest weight λ by L_{n}(λ), Δ_{n}(λ) and ∇_{n}(λ) respectively, as in section 2.

If M is a finite dimensional G-module, the dual M^{∗} is defined as usual to be the dual vector space with action (gf)(m) = f(g^{−1}m) for g ∈ G, f ∈ M^{∗} and m ∈ M. Recall that L(λ)^{∗} ≅ L(λ^{∗}) and Δ(λ)^{∗} = ∇(λ^{∗}), where λ^{∗} := −w_{0}λ. We also define the contravariant dual M^{τ} of M. This is the dual vector space M^{∗} with new action defined by (gf)(m) = f(τ(g)m) for g ∈ G, f ∈ M^{∗} and m ∈ M, where τ is the Chevalley antiautomorphism of G as in [J, II.1.16]. For λ ∈ X^{+}(T), we have L(λ)^{τ} ≅ L(λ) and Δ(λ)^{τ} ≅ ∇(λ). In the case G = GL(n), τ is just matrix transposition, while for λ ∈ X(n), λ^{∗} = −w_{0}λ = (−λ_{n}, …, −λ_{1}).

The following fact proved in [J, II.4.13] is of central importance:

4.1. For λ, μ ∈ X^{+}(T), Ext_{G}^{i}(Δ(λ), ∇(μ)) is zero unless i = 0 and λ = μ, when it is one dimensional.

We say that a G-module M has a ∇-filtration (resp. a Δ-filtration) if it has an ascending filtration 0 = M_{(0)} < M_{(1)} < … with ⋃_{i≥0} M_{(i)} = M such that each factor M_{(i)}/M_{(i−1)} is isomorphic to a (possibly infinite) direct sum of copies of ∇(λ^{(i)}) (resp Δ(λ^{(i)})) for some λ^{(i)} ∈ X^{+}(n). We recall the Donkin-Mathieu theorem [D], [M]:

4.2. If M, N are G-modules with ∇-filtrations, then M ⊗ N has a ∇-filtration.

For the remainder of the section, fix G-modules W and N such that:

(P1) W has a ∇-filtration

0 = W_{(0)} < W_{(1)} < ⋯ < W_{(s)} = W

where W_{(t)}/W_{(t−1)} ≅ ∇(λ^{(t)}) for some λ^{(t)} ∈ X^{+}(T), t = 1, 2, …, s.

(P2) N is a submodule of W satisfying N ≅ N^{τ}. Define N_{(t)} to be the intersection N ∩ W_{(t)}; then N has the filtration

0 = N_{(0)} ≤ N_{(1)} ≤ ⋯ ≤ N_{(s)} = N,

and N_{(t)}/N_{(t−1)} is a (possibly zero) submodule of ∇(λ^{(t)}) for 1 ≤ t ≤ s.

We have in mind the following example for the modules W and N. Take W = ∇(λ) ⊗ ∇(ν) and N = L(λ) ⊗ L(ν) (or any fixed block of these); then, (P1) is satisfied by (4.2), while (P2) is obvious. Another example is if W = ∇(λ) ↓_{H}, N = L(λ) ↓_{H} where H is a Levi subgroup of G.

We prove some general facts about modules W and N for use later in the paper. We will need the following known lemma:

4.3. Lemma. For λ ∈ X^{+}(T) and any finite dimensional G-module V all of whose composition factors are of the form L(ν) for ν ≯ λ, Ext_{G}^{1}(Δ(λ), V) = 0.

Proof. We argue by induction on dimension. Pick a maximal submodule W of V such that V/W ≅ L(ν) for ν ≯ λ. The long exact sequence of cohomology yields the exact sequence Ext_{G}^{1}(Δ(λ), W) → Ext_{G}^{1}(Δ(λ), V) → Ext_{G}^{1}(Δ(λ), L(ν)). By induction, the left hand Ext is zero, so it suffices to show that Ext_{G}^{1}(Δ(λ), L(ν)) = 0 which is done in the proof of [J, II.2.14].

Applying the lemma to V = Δ(μ) and dualizing with [J, II.2.12(3)], we deduce the well-known:

4.4. For λ, μ ∈ X^{+}(T) with μ ≯ λ, Ext_{G}^{1}(Δ(λ), Δ(μ)) = Ext_{G}^{1}(∇(μ), ∇(λ)) = 0.

We can reorder the ∇-filtration of W in (P1) if necessary using (4.4), to assume from now on that λ^{(i)} > λ^{(j)} implies i > j for all 1 ≤ i, j ≤ s.

4.5. Lemma. For 1 ≤ t ≤ s and λ ∈ X^{+}(T), there is an exact sequence

0 → Hom_{G}(Δ(λ), N_{(t−1)}) → Hom_{G}(Δ(λ), N_{(t)}) → Hom_{G}(Δ(λ), N_{(t)}/N_{(t−1)}) → 0.

Proof. We have the long exact sequence

0 → Hom_{G}(Δ(λ), N_{(t−1)}) → Hom_{G}(Δ(λ), N_{(t)}) → Hom_{G}(Δ(λ), N_{(t)}/N_{(t−1)})

→ Ext_{G}^{1}(Δ(λ), N_{(t−1)}) → ….

Note that Hom_{G}(Δ(λ), N_{(t)}/N_{(t−1)}) ⊆ Hom_{G}(Δ(λ), ∇(λ^{(t)})), and the latter is zero by (4.1) unless λ = λ^{(t)}. If Hom_{G}(Δ(λ), N_{(t)}/N_{(t−1)}) = 0 the result is obvious, so we may assume that λ = λ^{(t)}. This means by the choice of ordering of the ∇-filtration of W that for u < t, λ^{(u)} ≯ λ. All composition factors of N_{(t−1)} are of the form L(ν) for ν ≤ λ^{(u)} and some 1 ≤ u < t, hence also satisfy ν ≯ λ. So Ext_{G}^{1}(Δ(λ), N_{(t−1)}) = 0 by Lemma 4.3, and the result follows from the long exact sequence.

4.6. Corollary. For λ ∈ X^{+}(T), dim Hom_{G}(Δ(λ), N) is equal to the number of t with 1 ≤ t ≤ s such that λ = λ^{(t)} and N_{(t)} ≠ N_{(t−1)}.

Proof. Note that dim Hom_{G}(Δ(λ), N_{(t)}) = ∑_{u=1}^{t} dim Hom_{G}(Δ(λ), N_{(u)}/N_{(u−1)}), which follows from lemma by induction on t. Now take t = s to deduce that

dim Hom_{G}(Δ(λ), N) = ∑_{t=1}^{s} dim Hom_{G}(Δ(λ), N_{(t)}/N_{(t−1)}).

Since N_{(t)}/N_{(t−1)} is a submodule of ∇(λ^{(t)}), Hom_{G}(Δ(λ), N_{(t)}/N_{(t−1)}) is zero unless λ = λ^{(t)} and N_{(t)} ≠ N_{(t−1)}, when it is one dimensional. The result follows.

Now we obtain our main result in the general setting:

4.7. Proposition. The G-module N is completely reducible if and only if

Hom_{G}(L(λ), N) ≅ Hom_{G}(Δ(λ), N)

for all λ ∈ X^{+}(T).

Proof. Obviously, if N is completely reducible then Hom_{G}(L(λ), N) ≅ Hom_{G}(Δ(λ), N) for all λ. Conversely, suppose Hom_{G}(L(λ), N) ≅ Hom_{G}(Δ(λ), N) for all λ ∈ X^{+}(T). Let d := dim Hom_{G}(Δ(λ), N); by Corollary 4.6, d is equal to the number of t with 1 ≤ t ≤ s such that λ^{(t)} = λ and N_{(t)} ≠ N_{(t−1)}.

We claim that N_{(t)}/N_{(t−1)} ≅ L(λ^{(t)}) whenever it is non-zero, for all 1 ≤ t ≤ s. Suppose for a contradiction that this is false, and take u ≤ s maximal such that N_{(u)}/N_{(u−1)} is reducible. This implies that Hom_{G}(N_{(u)}/N_{(u−1)}, L(λ)) = 0 as N_{(u)}/N_{(u−1)} is a submodule of ∇(λ), where λ := λ^{(u)}. So, using contravariant duality,

d = dim Hom_{G}(Δ(λ), N) = dim Hom_{G}(L(λ), N) = dim Hom_{G}(N, L(λ))

≤ ∑_{t=1}^{s} dim Hom_{G}(N_{(t)}/N_{(t−1)}, L(λ))

= ∑_{t=1}^{u−1} dim Hom_{G}(N_{(t)}/N_{(t−1)}, L(λ)) + ∑_{t=u+1}^{s} dim Hom_{G}(N_{(t)}/N_{(t−1)}, L(λ)).

For t < u such that N_{(t)} ≠ N_{(t−1)}, the module N_{(t)}/N_{(t−1)} is a submodule of ∇(λ^{(t)}), so has highest weight λ^{(t)} ≯ λ. Hence in this case, Hom_{G}(N_{(t)}/N_{(t−1)}, L(λ)) is zero unless λ^{(t)} = λ, when it is at most one dimensional. For t > u, N_{(t)}/N_{(t−1)} is either zero or L(λ^{(t)}) by maximality of u, so Hom_{G}(N_{(t)}/N_{(t−1)}, L(λ)) is zero unless λ^{(t)} = λ and N_{(t)} ≠ N_{(t−1)}, when it is exactly one dimensional. This shows that

∑_{t=1}^{u−1} dim Hom_{G}(N_{(t)}/N_{(t−1)}, L(λ)) + ∑_{t=u+1}^{s} dim Hom_{G}(N_{(t)}/N_{(t−1)}, L(λ)) ≤ d − 1.

Comparing this with our previous inequality gives the desired contadiction.

We have now shown that N_{(t)}/N_{(t−1)} ≅ L(λ^{(t)}) whenever it is non-zero. Hence by Corollary 4.6 again, dim Hom_{G}(L(λ), N) = [N : L(λ)] for all λ ∈ X^{+}(T). Hence, N coincides with its socle, so is completely reducible.

Now, let ε ∈ X^{+}(n) be a miniscule weight (see [H, §13 ex. 13] where such weights are called minimal). Then L(ε) = Δ(ε) = ∇(ε). It is of importance that the ∇-filtration of W = ∇(λ) ⊗ ∇(ε) in (P1) is in fact multiplicity-free:

4.8. Lemma. Let ε ∈ X^{+}(T) be a miniscule weight. For any λ ∈ X^{+}(T), the G-module ∇(λ) ⊗ L(ε) has a ∇-filtration with factors

{∇(λ + wε) | for all w ∈ W such that λ + wε ∈ X^{+}(T)},

each appearing with multiplicity one. In particular by (4.1), for any μ ∈ X^{+}(T)

Hom_{G}(Δ(μ), ∇(λ) ⊗ L(ε)) = \{ \begin{matrix}𝔽 & if μ ∈ λ + W ε, \\ 0 & otherwise. \end{matrix}

Proof. The fact that ∇(λ) ⊗ L(ε) has a ∇-filtration follows from (4.2) (or [D, Proposition 4.2.1] or [W]). To compute the factors appearing in a ∇-filtration, it suffices to work at the level of characters. Consequently, the multiplicities are the same as in characteristic 0. Now use [H, §26 ex. 9]. We note that the lemma can also be deduced as a special case of [J, II.7.13].

5 Primitive vectors and the socle of L_{n}(λ) ⊗ V_{n}^{∗}

As observed in section 4, the GL(n)-modules W = ∇_{n}(λ) ⊗ V_{n} (resp. ∇_{n}(λ) ⊗ V_{n}^{∗}) and N = L_{n}(λ) ⊗ V_{n} (resp. L_{n}(λ) ⊗ V_{n}^{∗}) satisfy the hypotheses (P1) and (P2) of section 4, for any λ ∈ X^{+}(n). In this section, we will relate the combinatorial notions from section 2 to the general properties of the modules W and N considered in section 4.

We point out that the weights ε_{1} and −ε_{n} are miniscule for G = GL(n), so we have the following special cases of Lemma 4.8:

5.1. Fix λ, μ ∈ X^{+}(n).

(i) The space Hom_{GL(n)}(Δ_{n}(μ), ∇_{n}(λ)⊗V_{n}) is zero unless μ = λ+ε_{i} for some λ-addable i, when it is one dimensional.

(ii) The space Hom_{GL(n)}(Δ_{n}(μ), ∇_{n}(λ) ⊗ V_{n}^{∗}) is zero unless μ = λ − ε_{i} for some λremovable i, when it is one dimensional.

We work now with the hyperalgebra U(n) of GL(n) (denoted Dist(GL(n)) in [J]), taking notation from [BKS, Section 2]. The hyperalgebra can be defined by base change starting from the universal enveloping algebra U(n, ℂ) of the Lie algebra 𝔤𝔩(n, ℂ). For 1 ≤ i, j ≤ n, let x_{i,j} denote the element of 𝔤𝔩(n, ℂ) corresponding to the n × n matrix with 1 in the ij-entry and zeros elsewhere. Let U(n, ℤ) be the ℤ-subalgebra of U(n, ℂ) generated by

\{x_{i,j}^{(r)}, (\begin{matrix}x_{i,i} \\ r \end{matrix}) | 1 ≤ i, j ≤ n, i ≠ j, r ≥ 0\} ,

where x^{(r)} denotes the divided power \frac{x^{r}}{r!} and (\begin{matrix}x \\ r \end{matrix}) denotes \frac{x(x−1)...(x−r+1)}{r!}. The hyperalgebra

U(n) = U(n, 𝔽) is then U(n, ℤ) ⊗_{ℤ} 𝔽. For 1 ≤ i < j ≤ n, we denote the image of x_{i,j}^{(r)} and

x_{j,i}^{(r)} in U(n) by E_{i,j}^{(r)} and F_{i,j}^{(r)} respectively, and the image of (\begin{matrix}x_{i,i} \\ r \end{matrix}) by (\begin{matrix}H_{i} \\ r \end{matrix}). Letting

U^{−}(n), U^{0}(n) and U^{+}(n) denote the subalgebras of U(n) generated by

\{F_{i,j}^{(r)}\}_{1≤i<j≤n,r≥0} , \{(\begin{matrix}H_{i} \\ r \end{matrix})\}_{1≤i≤n,r≥0} and \{E_{i,j}^{(r)}\}_{1≤i<j≤n,r≥0}

respectively, we have the usual ‘triangular decomposition’ U(n) = U^{−}(n)U^{0}(n)U^{+}(n).

Any rational GL(n)-module M is naturally a U(n)-module (see [J, I.8]). We say that a vector v ∈ M is a primitive vector if it is annihilated by E_{i,j}^{(k)} for all 1 ≤ i < j ≤ n and k ≥ 1. We say M is a high weight module of high weight λ if it is generated by a primitive vector of weight λ. By [J, II.2.13], the standard module Δ_{n}(λ) is universal amongst all high weight modules of high weight λ. This implies:

5.2. For any GL(n)-module M and λ ∈ X^{+}(T), the dimension of the space of all primitive vectors in M of weight λ is equal to dim Hom_{GL(n)}(Δ_{n}(λ), M).

Let f_{1}, …, f_{n} denote the basis for V_{n}^{∗} that is dual to the natural basis for V_{n}. The action of the generators of U(n) on V_{n}^{∗} is given by:

E_{i,j}^{(l)}f_{k} = −δ_{l,1}δ_{i,k}f_{j}, F_{i,j}^{(l)}f_{k} = −δ_{l,1}δ_{j,k}f_{i}, (\begin{matrix}H_{i} \\ l \end{matrix}) f_{k} = −δ_{l,1}δ_{i,k}f_{k}  (5.3)

for 1 ≤ i, j, k ≤ n and l ≥ 1.

For the remainder of the section, let P = LY denote the standard parabolic subgroup of GL(n), where L ≅ GL(n − 1) × G(1) (embedded diagonally) and Y is the unipotent radical generated by the root subgroups corresponding to the roots ε_{i} − ε_{n}, for i = 1, 2, … n − 1. Note that for any GL(n)-module M, the Y -fixed points M^{Y} of M are L-invariant, so we can regard M^{Y} as a GL(n − 1)-module in a natural way.

5.4. Lemma. For any GL(n)-module M, the linear map e : M → M ⊗ V_{n}^{∗} defined for v ∈ M by

e(v) := v ⊗ f_{n} + ∑_{h=1}^{n−1} E_{h,n}v ⊗ f_{h}

is an injective GL(n − 1)-homomorphism such that e(M) ⊇ (M ⊗ V_{n}^{∗})^{Y} .

Proof. The map e is obviously injective as f_{1}, …, f_{n} are linearly independent. To prove that it is a GL(n − 1)-homomorphism, we work with the naturally embedded hyperalgebra U(n − 1) < U(n). Note that for any vector v ∈ M that is a weight vector for the maximal torus T (n − 1) of GL(n − 1), the weights of v and e(v) relative to T (n − 1) are equal. So, to prove that e is a GL(n − 1)-homomorphism, it suffices to show that e(Xv) = Xe(v) for any v ∈ M and any X ∈ U(n − 1) of the form E_{i,j}^{(k)} or F_{i,j}^{(k)} with 1 ≤ i < j < n and k > 0. We have (interpreting E_{n,n} as 1):

E_{i,j}^{(k)}e(v) = E_{i,j}^{(k)} (∑_{h=1}^{n} E_{h,n}v ⊗ f_{h}) = ∑_{h=1}^{n}(E_{i,j}^{(k)}E_{h,n}v) ⊗ f_{h} + (E_{i,j}^{(k−1)}E_{i,n}v) ⊗ (E_{i,j}f_{i})

= ∑_{h=1}^{n} E_{h,n}E_{i,j}^{(k)}v ⊗ f_{h} + E_{i,n}E_{i,j}^{(k−1)}v ⊗ f_{j} + E_{i,n}E_{i,j}^{(k−1)}v ⊗ (−f_{j}) = e(E_{i,j}^{(k)}v),

and

F_{i,j}^{(k)}e(v) = F_{i,j}^{(k)} (∑_{h=1}^{n} E_{h,n}v ⊗ f_{h}) = ∑_{h=1}^{n}(F_{i,j}^{(k)}E_{h,n}v) ⊗ f_{h} + (F_{i,j}^{(k−1)}E_{j,n}v) ⊗ (F_{i,j}f_{j})

= ∑_{h=1}^{n} E_{h,n}F_{i,j}^{(k)}v ⊗ f_{h} + E_{j,n}F_{i,j}^{(k−1)}v ⊗ f_{i} + E_{j,n}F_{i,j}^{(k−1)}v ⊗ (−f_{i}) = e(F_{i,j}^{(k)}v).

For the last part, take any element w = ∑_{h=1}^{n} v_{h} ⊗ f_{h} of (M ⊗ V_{n}^{∗})^{Y}. Then E_{i,n}w = 0 for any 1 ≤ i < n. That is,

0 = E_{i,n}(∑_{h=1}^{n} v_{h} ⊗ f_{h}) = ∑_{h=1}^{n} E_{i,n}v_{h} ⊗ f_{h} + v_{i} ⊗ E_{i,n}f_{i} = ∑_{h=1}^{n} E_{i,n}v_{h} ⊗ f_{h} − v_{i} ⊗ f_{n}.

Since f_{1}, …, f_{n} are linearly independent, we conclude that v_{i} = E_{i,n}v_{n} for i = 1, …, n − 1. So w = e(v_{n}) and we have shown that (M ⊗ V_{n}^{∗})^{Y} ⊆ e(M).

We recall the notion of level introduced by Seitz. For λ ∈ X^{+}(n), we define the ℓth level M^{(ℓ)} of any submodule M of ∇_{n}(λ) to be the sum of the weight spaces M_{ν} for all ν ∈ X(n) satisfying ν_{n} = λ_{n} + ℓ. Observe that M^{(ℓ)} is a GL(n − 1)-submodule of M and M = ⨁_{ℓ≥0} M^{(ℓ)}.

5.5. Proposition. For any λ ∈ X^{+}(n) and any submodule M of ∇_{n}(λ), the restriction of the map e from Lemma 5.4 to M^{(0)} ⊕ M^{(1)} gives an isomorphism

ē: M^{(0)} ⊕ M^{(1)} → (M ⊗ V_{n}^{∗})^{Y}

as GL(n − 1)-modules.

Proof. We first check that for v ∈ M^{(0)} ⊕ M^{(1)}, e(v) lies in (M ⊗ V_{n}^{∗})^{Y}. We have to prove that E_{i,n}^{(k)}e(v) = 0 for any 1 ≤ i < n and k > 0. We have

E_{i,n}^{(k)} (∑_{h=1}^{n} E_{h,n}v ⊗ f_{h}) = ∑_{h=1}^{n} E_{i,n}^{(k)}E_{h,n}v ⊗ f_{h} + E_{i,n}^{(k−1)}E_{i,n}v ⊗ E_{i,n}f_{i},

which is clearly zero by weights if k > 1. For k = 1, it equals E_{i,n}v ⊗ f_{n} + E_{i,n}v ⊗ (−f_{n}) which is again 0.

Applying Lemma 5.4, this shows that ē is an injective GL(n − 1)-homomorphism from M^{(0)}⊕M^{(1)} to (M ⊗V_{n}^{∗})^{Y}. To show that ē is surjective take w = ∑_{h=1}^{n} v_{h}⊗f_{h} ∈ (M ⊗V_{n}^{∗})^{Y}. We may assume that w is a weight vector of weight ν ∈ X(n). By Lemma 5.4, we can write w = e(v) for some weight vector v ∈ M of weight ν + ε_{n}. We now prove by downward induction on the dominance order on ν that v lies in M^{(0)} ⊕ M^{(1)}, to complete the proof.

If v is a GL(n − 1)-primitive vector in M, then e(v) = w is a GL(n)-primitive vector, since we have already shown that it is Y -invariant. So (5.1) and (5.2) imply that the weight ν of w is λ − ε_{j} for some 1 ≤ j ≤ n. But then, the weight of v is λ − ε_{j} + ε_{n} so v lies in M^{(0)} ⊕ M^{(1)} as required.

Otherwise, if v is not GL(n − 1)-primitive, then we can find 1 ≤ i < j < n and k > 0 such that E_{i,j}^{(k)}v ≠ 0. By induction E_{i,j}^{(k)}v lies in M^{(0)} ⊕ M^{(1)} so v does too by weights.

Recall the notation λ̄ from section 4.

5.6. Corollary. For λ, μ ∈ X^{+}(n) and a submodule M of ∇_{n}(λ), the restriction of the map ē from Proposition 5.5 gives a bijection between the GL(n − 1)-primitive vectors in M^{(0)} ⊕ M^{(1)} of weight μ + ε_{n} and the GL(n)-primitive vectors in M ⊗ V_{n}^{∗} of weight μ. In particular, if μ_{n} = λ_{n},

Hom_{GL(n)}(Δ_{n}(μ), M ⊗ V_{n}^{∗}) ≅ Hom_{GL(n−1)}(Δ_{n−1}(μ̄), M^{(1)}).

Proof. The first statement follows immediately from Proposition 5.5, since a vector v ∈ M ⊗ V_{n}^{∗} is GL(n)-primitive if and only if it is GL(n − 1)-primitive and lies in (M ⊗ V_{n}^{∗})^{Y}. If μ_{n} = λ_{n}, a vector v ∈ M^{(0)} ⊕ M^{(1)} has GL(n)-weight μ + ε_{n} if and only if it lies in M^{(1)} and has GL(n − 1)-weight μ̄. So the second statement now follows using (5.2).

5.7. Remark. A functorial proof of the Hom-space isomorphism in Corollary 5.6 is given in [BK, Corollary 2.10]. The elementary argument given here provides more information.

Now we can prove the main results of the section. In fact, using Corollary 5.6, these will follow from the following theorem first proved in [K3, Theorem 4.2] (see also [B1, Theorem 5.3]) concerning the branching rule from GL(n) to GL(n − 1):

5.8. Suppose λ ∈ X^{+}(n), μ ∈ X^{+}(n − 1). Then, the space Hom_{GL(n−1)}(Δ_{n−1}(μ), L_{n}(λ)^{(1)}) is zero unless μ = λ̄ − ε_{i} for some i with 1 ≤ i < n that is normal for λ, in which case it is one dimensional.

Recall the combinatorial notions from section 2 and the notation λ^{∗} = (−λ_{n}, …, −λ_{1}) from section 4. For 1 ≤ i ≤ n, we let i^{∗} := n + 1 − i. Obviously, given λ ∈ X^{+}(n), i is λ-removable if and only if i^{∗} is λ^{∗}-addable, and dually i is λ-addable if and only if i^{∗} is λ^{∗}-removable. We have the following combinatorial lemma:

5.9. Lemma. (i) i is normal for λ if and only if i^{∗} is conormal for λ^{∗}; (ii) i is good for λ if and only if i^{∗} is cogood for λ^{∗}; (iii) i is good for λ if and only if i is normal for λ and conormal for λ − ε_{i}; (iii)′ i is cogood for λ if and only if i is conormal for λ and normal for λ + ε_{i}. (iv) i is cogood for λ if and only if i is λ-addable and i is good for λ + ε_{i}; (iv)′ i is good for λ if and only if i is λ-removable and i is cogood for λ − ε_{i}.

Proof. Note that the primed statements are equivalent to the unprimed statements, applying ∗. Also, parts (i) and (ii) follow immediately from the definitions.

Now consider (iii). We first prove that if i is both normal and conormal for λ then i is good for λ. Assume that i is normal for λ but not good for λ. Then, there is a j with 1 ≤ j < i, j normal for λ and res(i, λ_{i}) = res(j, λ_{j}). Since j is normal, there is a decreasing injection from the set of

λ-addable k with j < k < i and res(j, λ_{j}) = res(k, λ_{k} + 1)

into the set of λ-removable k′ with j < k′ < i, res(j, λ_{j}) = res(k′, λ_{k′}). So, there are strictly more

λ-removable h with j ≤ h < i and res(i, λ_{i}) = res(h, λ_{h})

than there are

λ-addable h′ with j < h′ < i and res(i, λ_{i}) = res(h′, λ_{h′} + 1).

This means that there cannot be an increasing injection from the set of

λ-removable h with 1 ≤ h < i and res(i, λ_{i}) = res(h, λ_{h})

into the set of

λ-addable h′ with 1 ≤ h′ < i and res(i, λ_{i}) = res(h′, λ_{h′} + 1).

Hence, i is not conormal for λ − ε_{i}.

Conversely, if i is not normal for λ then it is not good, so suppose that i is normal for λ but not conormal for λ − ε_{i}. Then, we can find some λ-removable j < i such that there is no increasing injection from the set of

λ-removable k with j ≤ k < i and res(i, λ_{i}) = res(k, λ_{k})

into the set of

λ-addable k′ with j < k′ < i and res(i, λ_{i}) = res(k′, λ_{k′} + 1).

If we take the largest such j, then there must be an increasing bijection θ_{1} from the set of

λ-removable k with j < k < i and res(i, λ_{i}) = res(k, λ_{k})

into the set of

λ-addable k′ with j < k′ < i and res(i, λ_{i}) = res(k′, λ_{k′} + 1).

Since i is normal, there is a decreasing injection θ_{2} from the set of

λ-addable k with i < k ≤ n and res(i, λ_{i}) = res(k, λ_{k} + 1)

into the set of

λ-removable k′ with i < k′ ≤ n and res(i, λ_{i}) = res(k′, λ_{k′}).

Now combining θ_{1}^{−1} and θ_{2} gives a decreasing injection θ from the set of

λ-addable k with j < k ≤ n and res(i, λ_{i}) = res(k, λ_{k} + 1)

into the set of

λ-removable k′ with j < k′ ≤ n and res(i, λ_{i}) = res(k′, λ_{k′}).

Noting res(j, λ_{j}) = res(i, λ_{i}), this shows that j is also normal for λ. Therefore i is not good for λ. This proves (iii) (hence (iii)′).

Now we deduce (iv). By (iii), i is λ-addable and good for λ + ε_{i} if and only if i is conormal for λ and normal for λ + ε_{i}. By (iii)′, this is if and only if i is cogood for λ, as required for (iv).

5.10. Theorem. Fix λ, μ ∈ X^{+}(n).

(i) The space Hom_{GL(n)}(Δ_{n}(μ), L_{n}(λ)⊗V_{n}^{∗}) is zero unless μ = λ−ε_{i} for some 1 ≤ i ≤ n that is normal for λ, in which case it is one dimensional.

(ii) The space Hom_{GL(n)}(L_{n}(μ), ∇_{n}(λ)⊗V_{n}^{∗}) is zero unless μ = λ−ε_{i} for some 1 ≤ i ≤ n that is conormal for λ − ε_{i}, in which case it is one dimensional.

(iii) The space Hom_{GL(n)}(L_{n}(μ), L_{n}(λ)⊗V_{n}^{∗}) is zero unless μ = λ−ε_{i} for some 1 ≤ i ≤ n that is good for λ, in which case it is one dimensional.

Proof. We note initially that by (5.1), we may assume that μ = λ − ε_{i} for some 1 ≤ i ≤ n that is λ-removable. Moreover, all three Hom spaces are certainly either zero or one dimensional. Part (i) is now immediate from (5.8) and Corollary 5.6 providing i < n. If i = n, note that Hom_{GL(n)}(Δ_{n}(λ − ε_{n}), L_{n}(λ) ⊗ V_{n}^{∗}) is one dimensional as λ − ε_{n} is the highest weight of L_{n}(λ) ⊗ V_{n}^{∗}, while n is always normal for λ according to the definition in section 2. Hence, (i) is true for all i. Part (ii) follows easily from (i) and Lemma 5.9(i), because Hom_{GL(n)}(L_{n}(λ − ε_{i}), ∇_{n}(λ) ⊗ V_{n}^{∗}) ≅ Hom_{GL(n)}(Δ_{n}(λ^{∗}), L_{n}(λ^{∗} + ε_{i^{∗}}) ⊗ V_{n}^{∗}).

Finally, we prove (iii). By Lemma 5.9(iii), it suffices to show that

Hom_{GL(n)}(L_{n}(λ − ε_{i}), L_{n}(λ) ⊗ V_{n}^{∗})

is non-zero if and only if i is normal for λ and conormal for λ−ε_{i}. The forward implication is obvious because of (i) and (ii). For the converse, assume that i is normal for λ and conormal for λ−ε_{i}. Take a generator θ of the one dimensional space Hom_{GL(n)}(Δ_{n}(λ−ε_{i}), ∇_{n}(λ)⊗V_{n}^{∗}). Observe that by (ii), θ factors through the quotient L_{n}(λ − ε_{i}) of Δ_{n}(λ − ε_{i}). Also, by (i), the image of θ lies in the subspace L_{n}(λ) ⊗ V_{n}^{∗} of ∇_{n}(λ) ⊗ V_{n}^{∗}. So θ induces a non-zero homomorphism from L_{n}(λ − ε_{i}) to L_{n}(λ) ⊗ V_{n}^{∗}, as required.

Applying Proposition 4.7, we obtain the following complete reducibility criterion:

5.11. Corollary. For λ ∈ X^{+}(n), the module L_{n}(λ)⊗V_{n}^{∗} is completely reducible if and only if every i with 1 ≤ i ≤ n that is normal for λ is good for λ.

We also have the dual statements:

5.12. Theorem. Fix λ, μ ∈ X^{+}(n).

(i) The space Hom_{GL(n)}(Δ_{n}(μ), L_{n}(λ) ⊗ V_{n}) is zero unless μ = λ + ε_{i} for some 1 ≤ i ≤ n that is conormal for λ, in which case it is one dimensional.

(ii) The space Hom_{GL(n)}(L_{n}(μ), ∇_{n}(λ)⊗V_{n}) is zero unless μ = λ+ε_{i} for some 1 ≤ i ≤ n that is normal for λ + ε_{i}, in which case it is one dimensional.

(iii) The space Hom_{GL(n)}(L_{n}(μ), L_{n}(λ)⊗V_{n}) is zero unless μ = λ+ε_{i} for some 1 ≤ i ≤ n that is cogood for λ, in which case it is one dimensional.

Proof. Note that by contravariant duality,

Hom_{GL(n)}(L_{n}(μ), L_{n}(λ) ⊗ V_{n}) ≅ Hom_{GL(n)}(L_{n}(λ) ⊗ V_{n}, L_{n}(μ))
≅ Hom_{GL(n)}(L_{n}(λ), L_{n}(μ) ⊗ V_{n}^{∗}).

So (iii) follows easily from Theorem 5.10(iii) and Lemma 5.9. Parts (i) and (ii) follow similarly from Theorem 5.10(ii) and (i) respectively.

Applying Proposition 4.7 again we deduce:

5.13. Corollary. For λ ∈ X^{+}(n), the module L_{n}(λ)⊗V_{n} is completely reducible if and only if every i with 1 ≤ i ≤ n that is conormal for λ is cogood for λ.

6 A filtration of L_{n}(λ) ⊗ V_{n}^{∗}

We now construct a filtration of the module L_{n}(λ) ⊗ V_{n}^{∗} by high weight modules. This will allow us in the first place to prove that certain composition multiplicities are zero.

Fix λ ∈ X^{+}(n) and let w_{λ} (resp. v_{λ}) denote a non-zero high weight vector in Δ_{n}(λ) (resp. L_{n}(λ)) throughout the section. Let r_{a} < ⋯ < r_{1} = n denote the set of all r with 1 ≤ r ≤ n which are λ-removable. Let s_{b} < ⋯ < s_{1} = n denote the set of all s with 1 ≤ s ≤ n which are normal for λ.

6.1. Lemma. The module Δ_{n}(λ) ⊗ V_{n}^{∗} is generated as a U^{−}(n)-module by the vectors

{w_{λ} ⊗ f_{r_{i}} | 1 ≤ i ≤ a}.

Proof. We first prove by induction on i that any vector w ⊗ f_{i} ∈ Δ_{n}(λ) ⊗ V_{n}^{∗} lies in the U^{−}(n)-module generated by w_{λ} ⊗ f_{1}, …, w_{λ} ⊗ f_{i}. Since w_{λ} generates Δ_{n}(λ) as a U^{−}(n)- module, we can find y ∈ U^{−}(n) such that yw_{λ} = w. Then,

y(w_{λ} ⊗ f_{i}) = (yw_{λ}) ⊗ f_{i} + X

where X is a linear combination of terms of the form w′ ⊗ f_{i′} for w′ ∈ Δ_{n}(λ) and i′ < i. By induction, all such terms lie in the U^{−}(n)-submodule generated by w_{λ} ⊗ f_{1}, …, w_{λ} ⊗ f_{i−1}, hence w ⊗ f_{i} lies in the submodule generated by w_{λ} ⊗ f_{1}, …, w_{λ} ⊗ f_{i} as required.

This shows that Δ_{n}(λ) ⊗ V_{n}^{∗} is generated by {w_{λ} ⊗ f_{i} | 1 ≤ i ≤ n}. Now suppose that i is not λ-removable. Let j be minimal such that j is λ-removable with j > i. Since λ_{i} = λ_{j}, the weight λ − ε_{i} + ε_{j} is not a weight of Δ_{n}(λ), so F_{i,j}w_{λ} = 0. Hence,

F_{i,j}(w_{λ} ⊗ f_{j}) = w_{λ} ⊗ (F_{i,j}f_{j}) = −w_{λ} ⊗ f_{i}.

Hence, w_{λ} ⊗ f_{i} lies in the U^{−}(n)-submodule generated by w_{λ} ⊗ f_{j}.

6.2. Lemma. For 1 ≤ i ≤ a, let M_{(i)} be the U^{−}(n)-submodule of Δ_{n}(λ) ⊗ V_{n}^{∗} generated by M_{(i−1)} and w_{λ} ⊗ f_{r_{i}}. Then, each M_{(i)} is GL(n)-stable and

0 = M_{(0)} < M_{(1)} < ⋯ < M_{(a)} = M

is a Δ-filtration of M := Δ_{n}(λ) ⊗ V_{n}^{∗} with M_{(i)}/M_{(i−1)} ≅ Δ_{n}(λ − ε_{r_{i}}) for each i. Moreover, the image of w_{λ} ⊗ f_{r_{i}} is a non-zero primitive vector in M_{(i)}/M_{(i−1)}.

Proof. We first prove by induction on i ≤ a that w_{λ} ⊗ f_{r_{i}} + M_{(i−1)} is either zero or a primitive vector in M_{(i)}/M_{(i−1)}, hence that each M_{(i)} is a GL(n)-module. The induction starts trivially with i = 0. So take i > 0 and assume the claim for all smaller i. Moreover, if M_{(i−1)} = M then M_{(i)} = M_{(i−1)} = M and the result is trivial, so we can assume that M_{(i−1)} ≠ M. Observe then that by Lemma 6.1, M/M_{(i−1)} is generated as a U^{−}(n)-module by the vectors {w_{λ} ⊗f_{r_{j}} +M_{(i−1)} |j ≥ i}. If w_{λ} ⊗f_{r_{i}} +M_{(i−1)} is zero then M_{(i)} = M_{(i−1)} and the conclusion is immediate. So we may assume that w_{λ} ⊗ f_{r_{i}} ∉ M_{(i−1)}. But that means that w_{λ} ⊗ f_{r_{i}} + M_{(i−1)} is a non-zero vector of maximal weight λ − ε_{r_{i}} in M/M_{(i−1)}. Hence, w_{λ} ⊗ f_{r_{i}} is a primitive vector as required.

Noting that M_{(a)} = M by Lemma 6.1, we have now constructed a filtration

0 = M_{(0)} ≤ ⋯ ≤ M_{(a)} = M

with each factor M_{(i)}/M_{(i−1)} isomorphic to a (possible zero) quotient of Δ_{n}(λ − ε_{r_{i}}). It just remains to show that in fact M_{(i)}/M_{(i−1)} ≅ Δ_{n}(λ−ε_{r_{i}}) for each i. This follows immediately by dimension from Lemma 4.8 or (5.1).

6.3. Theorem. For 1 ≤ i ≤ b, let N_{(i)} be the U^{−}(n)-submodule of L_{n}(λ) ⊗ V_{n}^{∗} generated by N_{(i−1)} and v_{λ} ⊗ f_{s_{i}}. Then, each N_{(i)} is GL(n)-stable and

0 = N_{(0)} < N_{(1)} < ⋯ < N_{(b)} = N

is a filtration of N := L_{n}(λ)⊗V_{n}^{∗} such that N_{(i)}/N_{(i−1)} is a non-zero quotient of Δ_{n}(λ−ε_{s_{i}}) for each i. Moreover, the image of v_{λ} ⊗ f_{r_{i}} is a non-zero primitive vector in N_{(i)}/N_{(i−1)}.

Proof. For 1 ≤ i ≤ a, let M̄_{(i)} be the image of M_{(i)} in the filtration of Lemma 6.2 under the quotient Δ_{n}(λ) ⊗ V_{n}^{∗} → L_{n}(λ) ⊗ V_{n}^{∗}. This gives a filtration

0 = M̄_{(0)} ≤ M̄_{(1)} ≤ ⋯ ≤ M̄_{(a)} = N.

By contravariant duality, Corollary 4.6 and Theorem 5.10(i), M̄_{(i)} ≠ M̄_{(i−1)} if and only if r_{i} is normal for λ. The theorem follows immediately on deleting all trivial factors M̄_{(i)}/M̄_{(i−1)} with r_{i} not normal for λ from this filtration.

Our first application of this high weight filtration is the following result, which is the analogue for tensor products of [K6, Lemma 7.4]:

6.4. Lemma. Let μ ∈ X^{+}(n) be any weight of the form μ = λ − ε_{j} − ∑_{i=1}^{j−2} a_{i}(ε_{i} − ε_{i+1}) for some 1 ≤ j ≤ n and non-negative coefficients a_{i} not all of which are zero. Then, [L_{n}(λ) ⊗ V_{n}^{∗} : L_{n}(μ)] = 0.

Proof. Let κ = ∑_{i=1}^{j−2} a_{i}(ε_{i} − ε_{i+1}). Suppose that L_{n}(μ) is a composition factor of L_{n}(λ) ⊗ V_{n}^{∗}. Then, it is a composition factor of some factor V := N_{(i)}/N_{(i−1)} of the high weight filtration constructed in Theorem 6.3, for some 1 ≤ i ≤ b. Note that V is a high weight module of high weight λ − ε_{s_{i}}, generated by the primitive vector v_{λ} ⊗ f_{s_{i}} + N_{(i−1)}. We can find a filtration 0 ≤ V′′ < V′ ≤ V such that V′/V′′ ≅ L_{n}(μ). Let v ∈ V′ be a vector of weight μ such that v + V′′ is a non-zero primitive vector in L_{n}(μ).

Since μ = λ−ε_{j} −κ and V is generated as a U^{−}(n)-module by the vector v_{λ}⊗f_{s_{i}} +N_{(i−1)}, which has weight λ − ε_{s_{i}}, we must have that λ − ε_{j} − κ ≤ λ − ε_{s_{i}}. Hence, 0 ≤ κ + ε_{j} − ε_{s_{i}} so that j ≤ s_{i} by the definition of κ. We can find operators y_{1} and y_{2} lying in the −(ε_{j} − ε_{s_{i}})- and −κ-weight spaces of U^{−}(n) respectively, such that

v = y_{2}y_{1}(v_{λ} ⊗ f_{s_{i}}) + N_{(i−1)}.

Observe that y_{1} and y_{2} commute and so

v = y_{2}y_{1}(v_{λ} ⊗ f_{s_{i}}) + N_{(i−1)} = y_{1}(y_{2}(v_{λ} ⊗ f_{s_{i}})) + N_{(i−1)} = y_{1}((y_{2}v_{λ}) ⊗ f_{s_{i}}) + N_{(i−1)}.

Since this is non-zero, y_{2}v_{λ} must be a non-zero vector in L_{n}(λ), so we can find x_{2} lying in the κ-weight space of U^{+}(n) such that x_{2}y_{2}v_{λ} = v_{λ}. Since v is a primitive vector modulo V′′ and κ ≠ 0, x_{2}v ∈ V′′. But x_{2} and y_{1} commute, so

x_{2}v = x_{2}y_{1}((y_{2}v_{λ}) ⊗ f_{s_{i}}) + N_{(i−1)} = y_{1}((x_{2}y_{2}v_{λ}) ⊗ f_{s_{i}}) + N_{(i−1)} = y_{1}(v_{λ} ⊗ f_{s_{i}}) + N_{(i−1)}.

But then y_{2}x_{2}v = v, which contradicts the fact that x_{2}v ∈ V′′.

We formulate the dual statement for completeness:

6.5. Corollary. Let μ ∈ X^{+}(n) be any weight of the form μ = λ + ε_{j} − ∑_{i=j+1}^{n−1} a_{i}(ε_{i} − ε_{i+1}) for some 1 ≤ j ≤ n and non-negative coefficients a_{i} not all of which are zero. Then, [L_{n}(λ) ⊗ V_{n} : L_{n}(μ)] = 0.

Proof. Given a GL(n)-module M, let d(M) denote the GL(n)-module M but with action twisted by the automorphism g ↦ (g^{−1})^{t} of GL(n). Note that d(L_{n}(λ)) = L_{n}(λ^{∗}) and d(V_{n}) = V_{n}^{∗}. Now

[L_{n}(λ) ⊗ V_{n} : L_{n}(μ)] = [d(L_{n}(λ) ⊗ V_{n}) : d(L_{n}(μ))] = [L_{n}(λ^{∗}) ⊗ V_{n}^{∗} : L_{n}(μ^{∗})],

and the corollary follows immediately from Lemma 6.4.

7 Some composition multiplicities in L_{n}(λ) ⊗ V_{n}^{∗}

We calculate the multiplicity [L_{n}(λ)⊗V_{n}^{∗} : L_{n}(λ−ε_{i})] for any λ ∈ X^{+}(n) and any 1 ≤ i ≤ n. To do this, we will relate the latter multiplicities to branching multiplicities in the first level L_{n}(λ)^{(1)} of the restriction of L_{n}(λ) to GL(n − 1), as defined in section 5. The latter multiplicities are known from [K6, Corollary 9.4(ii)]:

7.1. Fix λ ∈ X^{+}(n) and any λ-removable i with 1 ≤ i < n. Then, [L_{n}(λ)^{(1)} : L_{n−1}(λ̄ − ε_{i})] is zero unless i is normal for λ, in which case it equals the number of j with i ≤ j < n such that j is normal for λ and res(i, λ_{i}) = res(j, λ_{j}).

We let Y denote the subgroup of GL(n) generated by root subgroups corresponding to the roots ε_{1} − ε_{n}, …, ε_{n−1} − ε_{n} as in section 5. As noted earlier, taking Y -fixed points gives a functor from the category of GL(n)-modules to the category of GL(n − 1)-modules. We state [J, II.2.11]:

7.2. For λ ∈ X^{+}(n), L_{n−1}(λ̄) ≅ L_{n}(λ)^{Y} = ⨁_{ν} L_{n}(λ)_{ν} where the sum is over all ν ∈ X(n) with ν_{n} = λ_{n}.

We say a GL(n)-module M is polynomial of degree r if all non-zero weight spaces of M are of weight μ ∈ X(n) with |μ| = r and μ_{i} ≥ 0 for all 1 ≤ i ≤ n.

7.3. Lemma. Fix λ ∈ X^{+}(n) with λ_{n} = 0. Let M be a finite dimensional GL(n)-module that is polynomial of degree r := |λ|. Then,

[M : L_{n}(λ)] = [M^{Y} : L_{n−1}(λ̄)].

Proof. Let 0 = M_{(0)} < ⋯ < M_{(s)} = M be a composition series of M. We prove by induction on i that [M_{(i)} : L_{n}(λ)] = [M_{(i)}^{Y} : L_{n−1}(λ̄)], starting from i = 0. Assume the result is true for i − 1. Let M_{(i)}/M_{(i−1)} ≅ L_{n}(μ) for μ ∈ X^{+}(n). Since |μ| = r = |λ|, we note that μ = λ if and only if μ̄ = λ̄. We have the natural exact sequence

0 → M_{(i−1)}^{Y} → M_{(i)}^{Y} → (M_{(i)}/M_{(i−1)})^{Y} .

By (7.2), (M_{(i)}/M_{(i−1)})^{Y} ≅ L_{n}(μ)^{Y} ≅ L_{n−1}(μ̄), so the exact sequence implies

[M_{(i−1)}^{Y} : L_{n−1}(λ̄)] ≤ [M_{(i)}^{Y} : L_{n−1}(λ̄)] ≤ [M_{(i−1)}^{Y} : L_{n−1}(λ̄)] + [L_{n−1}(μ̄) : L_{n−1}(λ̄)].

To prove the induction step, we need to show that right hand inequality is always equality. This is obvious if μ̄ ≠ λ̄, or equivalently if μ ≠ λ. So suppose that μ = λ. Then we need to show that the right hand map in the above exact sequence is surjective. Take v + M_{(i−1)} ∈ (M_{(i)}/M_{(i−1)})^{Y}. By (7.2)

L_{n}(λ)^{Y} = ⨁_{ν} L_{n}(λ)_{ν},

where the sum is over ν ∈ X(n) with ν_{n} = 0. So we may assume that v is a weight vector of weight ν with ν_{n} = 0. But then v ∈ M_{(i)} is Y -invariant by weights. So v ∈ M_{(i)}^{Y}, as required for surjectivity.

Recall the equivalence relation ∼ from section 2. Now we can relate [L_{n}(λ)⊗V_{n}^{∗} : L_{n}(μ)] to branching multiplicities in the first level in almost all cases:

7.4. Lemma. Fix λ, μ ∈ X^{+}(n) with μ_{n} = λ_{n} = 0 and |μ| = |λ|−1. Suppose that μ ≁ λ−ε_{n}. Then,

[L_{n}(λ) ⊗ V_{n}^{∗} : L_{n}(μ)] = [L_{n}(λ)^{(1)} : L_{n−1}(μ̄)].

Proof. Let N = L_{n}(λ) ⊗ V_{n}^{∗}. By the linkage principle, we may write N = N′ ⊕ N′′ for unique submodules N′, N′′ such that all composition factors of N′ (resp. N′′) are of the form L_{n}(ν) for ν ∼ λ − ε_{n} (resp. ν ≁ λ − ε_{n}). Note that [N : L_{n}(μ)] = [N′′ : L_{n}(μ)] since μ ≁ λ − ε_{n}. By Proposition 5.5, N^{Y} ≅ L_{n}(λ)^{(0)} ⊕ L_{n}(λ)^{(1)}. But L_{n−1}(μ̄) is not a composition factor of L_{n}(λ)^{(0)} by degree. So,

[L_{n}(λ)^{(1)} : L_{n−1}(μ̄)] = [N^{Y} : L_{n−1}(μ̄)] = [(N′)^{Y} : L_{n−1}(μ̄)] + [(N′′)^{Y} : L_{n−1}(μ̄)].

As L_{n}(μ) is not a composition factor of N′ and |ν| = |λ| − 1 = |μ| for any composition factor L_{n}(ν) of N′, (7.2) implies that [(N′)^{Y} : L_{n−1}(μ̄)] = 0. It just remains to show that [N′′ : L_{n}(μ)] = [(N′′)^{Y} : L_{n−1}(μ̄)]. This will follow immediately from Lemma 7.3 once we have shown that N′′ is polynomial.

Note that the submodule N_{(1)} of N in the filtration of Theorem 6.3 is a quotient of Δ_{n}(λ − ε_{n}), so N_{(1)} ≤ N′ and N′′ is a quotient of N/N_{(1)}. By Theorem 6.3, N/N_{(1)} has a filtration by high weight modules all of whose high weights are of the form λ − ε_{i} for λ-removable i < n. Hence, N/N_{(1)} is polynomial, so N′′ is too.

To give a similar result in the case that μ ∼ λ − ε_{n}, we need to allow n to vary. Recall the notation λ̃ from section 4.

7.5. Lemma. Fix λ, μ ∈ X^{+}(n). Then,

[L_{n}(λ) ⊗ V_{n}^{∗} : L_{n}(μ)] = [L_{n+1}(λ̃) ⊗ V_{n+1}^{∗} : L_{n+1}(μ̃)]

+ [L_{n+1}(λ̃ − ε_{n+1})^{(1)} : L_{n}(μ)] − [L_{n+1}(λ̃)^{(1)} : L_{n}(μ)].

Proof. Let S denote the exact functor from the category of GL(n+1)-modules to the category of GL(n)-modules defined by sending a GL(n + 1)-module N to the GL(n)-submodule of N consisting of the sum of all weight spaces N_{γ} with γ_{n+1} = 0, and by restriction on morphisms. Let N = L_{n+1}(λ̃) ⊗ V_{n+1}^{∗}. We compute [S(N) : L_{n}(μ)] in two different ways.

First, take a composition series 0 = N^{(0)} < N^{(1)} < ⋯ < N^{(s)} = N for N. As S is exact,

[S(N) : L_{n}(μ)] = ∑_{i=1}^{s}[S(N_{(i)}/N_{(i−1)}) : L_{n}(μ)].

Observe that L_{n+1}(λ̃ − ε_{n+1}) appears in the composition series precisely once as λ̃ − ε_{n+1} is the highest weight of N. Moreover, S(L_{n+1}(λ̃ − ε_{n+1})) is the first level L_{n+1}(λ̃ − ε_{n+1})^{(1)} of L_{n+1}(λ̃ − ε_{n+1}). So, this unique composition factor L_{n+1}(λ̃ − ε_{n+1}) of N contributes [L_{n+1}(λ̃ − ε_{n+1})^{(1)} : L_{n}(μ)] to the right hand side of this expression. By Lemma 6.4, all other composition factors of N are of the form L_{n+1}(ν) for ν ∈ X^{+}(n + 1) with ν_{n+1} ≥ 0. For such ν, S(L_{n+1}(ν)) is nonzero only if ν_{n+1} = 0, in which case it is L_{n}(ν̄) by (7.2). Hence [S(L_{n+1}(ν)) : L_{n}(μ)] is zero unless ν = μ̃, when it is one. So we obtain a total contribution of [N : L_{n+1}(μ̃)] to [S(N) : L_{n}(μ̄)] from such composition factors. We have shown:

[S(N) : L_{n}(μ)] = [L_{n+1}(λ̃) ⊗ V_{n+1}^{∗} : L_{n+1}(μ̃)] + [L_{n+1}(λ̃ − ε_{n+1})^{(1)} : L_{n}(μ)].

Second, we note N = L_{n+1}(λ̃)⊗V_{n+1}^{∗} ≅ L_{n+1}(λ̃)⊗V_{n}^{∗}⊕L_{n+1}(λ̃)⊗V_{1}^{∗} as a GL(n)-module, where V_{n+1}^{∗} = V_{n}^{∗} ⊕ V_{1}^{∗} is a decomposition of V_{n+1}^{∗} with respect to GL(n). So, S(N), which just amounts to picking certain weight spaces, is equal to (L_{n+1}(λ̃)^{(0)} ⊗ V_{n}^{∗}) ⊕ L_{n+1}(λ̃)^{(1)}. So using (7.2) again,

[S(N) : L_{n}(μ)] = [L_{n}(λ) ⊗ V_{n}^{∗} : L_{n}(μ)] + [L_{n+1}(λ̃)^{(1)} : L_{n}(μ)].

Comparing these two different expressions for [S(N) : L_{n}(μ)] gives the result.

7.6. Proposition. Fix λ, μ ∈ X^{+}(n) with λ_{n} = μ_{n} = 0. Then,

[L_{n}(λ) ⊗ V_{n}^{∗} : L_{n}(μ)] = \{ \begin{matrix}[L_{n+1}(λ̃ − ε_{n+1})^{(1)} : L_{n}(μ)] & if μ ∼ λ − ε_{n}. \\ [L_{n}(λ)^{(1)} : L_{n−1}(μ̄)] & if μ ≁ λ − ε_{n} \end{matrix}

Proof. If μ ≁ λ − ε_{n}, this is just Lemma 7.4. So suppose that μ ∼ λ − ε_{n}. By Lemma 7.5,

[L_{n}(λ) ⊗ V_{n}^{∗} : L_{n}(μ)] = [L_{n+1}(λ̃) ⊗ V_{n+1}^{∗} : L_{n+1}(μ̃)]

+ [L_{n+1}(λ̃ − ε_{n+1})^{(1)} : L_{n}(μ)] − [L_{n+1}(λ̃)^{(1)} : L_{n}(μ)].

We can compute the first term on the right hand side using Lemma 7.4, as res(n, λ̃_{n}) ≠ res(n + 1, λ̃_{n+1}). Hence, [L_{n+1}(λ̃) ⊗ V_{n+1}^{∗} : L_{n+1}(μ̃)] = [L_{n+1}(λ̃)^{(1)} : L_{n}(μ)]. Now the right hand side simplifies to give the required result.

Now we can deduce the main result of the section:

7.7. Theorem. Fix λ ∈ X^{+}(n) and a λ-removable i with 1 ≤ i ≤ n. Then,

[L_{n}(λ) ⊗ V_{n}^{∗} : L_{n}(λ − ε_{i})]

is zero unless i is normal for λ, when it equals the number of j with i ≤ j ≤ n such that j is normal for λ and res(i, λ_{i}) = res(j, λ_{j}).

Proof. By tensoring with a power of determinant if necessary, we may assume that λ_{n} = 0. If i = n, the composition multiplicity is always 1, and the statement is true. So we also assume 1 ≤ i < n. Now use Proposition 7.6 together with (7.1) and the combinatorial definition of normal.

8 The endomorphism ring of L_{n}(λ) ⊗ V_{n}^{∗}

Now we describe the endomorphism ring of L_{n}(λ) ⊗ V_{n}^{∗} using Casimir-type operators. This gives further information about the structure of the high weight filtration of Lemma 6.2.

Throughout the section, fix λ ∈ X^{+}(n) and let α_{i} ∈ ℤ/pℤ denote the residue res(i, λ_{i}), for 1 ≤ i ≤ n. Let

Z_{n}(λ) := X_{n} + ∑_{1≤i<j≤n} [(H_{i} − i)(H_{j} − j) − (λ_{i} − i)(λ_{j} − j)] + ∑_{1≤i≤n}(λ_{i} − i)  (8.1)

where

X_{n} := − ∑_{1≤i<j≤n} F_{i,j}E_{i,j}.

This element Z_{n}(λ) differs from the central element C_{2} constructed in [CL, 2.2] by a scalar, so Z_{n}(λ) lies in the centre of U(n).

8.2. Lemma. Z_{n}(λ)v = (X_{n} + α_{i})v, for any vector v of weight λ − ε_{i} (in some GL(n)- module) and 1 ≤ i ≤ n.

Proof. It suffices to check that

∑_{1≤h<k≤n} [(λ_{h} − δ_{hi} − h)(λ_{k} − δ_{ki} − k) − (λ_{h} − h)(λ_{k} − k)] + ∑_{1≤h≤n}(λ_{h} − h) = λ_{i} − i

which is routine.

8.3. Lemma. Fix α ∈ ℤ/pℤ. Suppose M is a GL(n)-module with a filtration

0 = M_{(0)} < M_{(1)} < ⋯ < M_{(a)} = M

such that each factor M_{(i)}/M_{(i−1)} is a quotient of Δ_{n}(λ−ε_{k}) for some 1 ≤ k ≤ n with α_{k} = α. Then, for any 1 ≤ j ≤ i ≤ a, (Z_{n}(λ) − α)^{j}M_{(i)} ⊆ M_{(i−j)}. In particular, (Z_{n}(λ) − α)^{a}M = 0.

Proof. The general case follows by induction from the special case i = j = 1. So suppose i = j = 1 and let v ∈ M_{(1)} be a non-zero primitive vector of weight λ − ε_{k}. Since α_{k} = α, Lemma 8.2 implies that (Z_{n}(λ) − α)v = X_{n}v, which is zero as v is primitive. Since v generates M_{(1)} and (Z_{n}(λ)−α) lies in the centre of U(n), it follows that (Z_{n}(λ)−α)M_{(1)} = 0 as required.

Now let n = s_{1} > ⋯ > s_{b} denote the set of all j which are normal for λ with 1 ≤ j ≤ n. Let N = L_{n}(λ) ⊗ V_{n}^{∗} and fix a non-zero primitive vector v_{λ} in L_{n}(λ). Let

0 = N_{(0)} < N_{(1)} < ⋯ < N_{(b)} = N  (8.4)

be the high weight filtration of Theorem 6.3, where N_{(i)}/N_{(i−1)} is a (nonzero) high weight module generated by the image of v_{λ} ⊗ f_{s_{i}}. For α ∈ ℤ/pℤ, let

s1^{α} > ⋯ > sb_{α}^{α}

denote the set of all s_{j} with 1 ≤ j ≤ b and α_{s_{j}} = α. Note that given any 1 ≤ i, j ≤ n, λ − ε_{i} ∼ λ − ε_{j} if and only if α_{i} = α_{j}. So by the linkage principle, we can decompose N as

N = ⨁_{α∈ℤ/pℤ}N^{α}

where N^{α} is the largest submodule of N all of whose composition factors are of the form L_{n}(ν) with cont_{α}(ν) = cont_{α}(λ) − 1 and cont_{γ}(ν) = cont_{γ}(λ) for γ ≠ α. The high weight filtration of N in (8.4) yields a high weight filtration of N^{α},

0 = N_{(0)}^{α} < N_{(1)}^{α} < ⋯ < N_{(b_{α})}^{α} = N^{α}  (8.5)

such that N_{(i)}^{α}/N_{(i−1)}^{α} is a non-zero quotient of Δ_{n}(λ − ε_{s_{i}^{α}}) and N_{(i)}^{α}/N_{(i−1)}^{α} ≅ N_{(j)}/N_{(j−1)}, where j is uniquely determined by s_{j} = si^{α}. Indeed, if π^{α} is the projection from N onto N^{α} then N_{(i)}^{α} is just π^{α}(N_{(j)}).

8.6. Lemma. For any α ∈ ℤ/pℤ, (i) (Z_{n}(λ) − α)^{b_{α}}N^{α} = 0; (ii) (Z_{n}(λ) − γ)N^{α} = N^{α} for any α ≠ γ ∈ ℤ/pℤ; (iii) N^{α} = ∏_{γ≠α}(Z_{n}(λ) − γ)^{b_{γ}}N. (iv) N_{(i)}^{α}/N_{(i−1)}^{α} is generated as a high weight by the primitive vector

∏_{γ≠α}(Z_{n}(λ) − γ)^{b_{γ}}(v_{λ} ⊗ f_{s_{i}^{α}}) + N_{(i−1)}^{α}.

Proof. Part (i) follows immediately from Lemma 8.3. For (ii), suppose γ ≠ α. Then (Z_{n}(λ) − γ) = (Z_{n}(λ) − α) + (α − γ). Since (Z_{n}(λ) − α) induces a nilpotent endomorphism of N^{α} by (i) and α −γ ≠ 0, it follows that (Z_{n}(λ)−γ) induces an automorphism of N^{α}. For (iii), note that by (i), ∏_{γ≠α}(Z_{n}(λ) − γ)^{b_{γ}} annihilates all blocks N_{γ} of N for γ ≠ α, while by (ii) it induces an automorphism of N^{α}. Finally, (iv) follows from (iii) and the description of N_{(i)}^{α} in the paragraph preceding this lemma.

The same argument applies to the GL(n − 1)-module Q := L_{n}(λ)^{(1)}. Let t_{1} > ⋯ > t_{c} denote the set of all j which are normal for λ with 1 ≤ j < n (actually, c = b − 1, t_{i} = s_{i+1}, i = 1, …, b−1). Then, by [BKS, Lemma 3.7, Proposition A.2], Q has a high weight filtration

0 = Q_{(0)} < Q_{(1)} < ⋯ < Q_{(c)} = Q  (8.7)

where Q_{(i)}/Q_{(i−1)} is a (non-zero) high weight module generated by the image of F_{t_{i},n}v_{λ}, which is a primitive vector in Q_{(i)}/Q_{(i−1)} of weight λ − ε_{t_{i}} + ε_{n}. For α ∈ ℤ/pℤ, let

t1^{α} > ⋯ > tc_{α}^{α}

denote the set of all t_{j} with 1 ≤ j ≤ c and α_{t_{j}} = α. By the linkage principle for GL(n − 1), we can decompose Q as

Q = ⨁_{α∈ℤ/pℤ}Q^{α}

where Q^{α} is the largest submodule of Q all of whose composition factors are of the form L_{n−1}(ν) with cont_{α}(ν) = cont_{α}(λ) − 1 and cont_{γ}(ν) = cont_{γ}(λ) for γ ≠ α. Now we can state the following result proved in [K5, Theorem 2.14] (the element ϕ_{α} defined here is precisely the element φ_{Γ} defined in [K5, Definition 2.9], as follows from the definitions using Lemma 8.2):

8.8. For any α ∈ ℤ/pℤ, let ϕ_{α} denote the image of (Z_{n−1}(λ̄) − α) in End_{𝔽}(Q^{α}). Then, the endomorphisms

{1, ϕ_{α}, …, (ϕ_{α})^{c_{α}−1}}

form a basis for End_{GL(n−1)}(Q^{α}).

To deduce the analogous result for N^{α}, we need one computational lemma:

8.9. Lemma. Let e : L_{n}(λ) → L_{n}(λ) ⊗ V_{n}^{∗} be the map defined in Lemma 5.4.

(i) e(Z_{n−1}(λ̄)v) = Z_{n}(λ)e(v), for any v ∈ L_{n}(λ)^{(0)} ⊕ L_{n}(λ)^{(1)} of weight λ − ε_{i} + ε_{n} and 1 ≤ i < n.

(ii) e(F_{i,n}v_{λ}) = (Z_{n}(λ) − α_{n})(v_{λ} ⊗ f_{i}), for 1 ≤ i < n.

Proof. (i) Note that X_{n} = X_{n−1} − ∑_{1≤i<n} F_{i,n}E_{i,n}. Now e is a GL(n − 1)-map, so X_{n−1} commutes with e, while every E_{i,n} annihilates e(v) as e(v) is fixed by the unipotent radical Y from Proposition 5.5. Hence, e(X_{n−1}v) = X_{n}e(v). Now, to prove (i) note that if v has GL(n)-weight λ − ε_{i} + ε_{n}, then e(v) has weight λ − ε_{i}. So, applying Lemma 8.2 twice,

Z_{n}(λ)e(v) = X_{n}e(v) + α_{i}e(v) = e((X_{n−1} + α_{i})v) = e(Z_{n−1}(λ̄)v),

as required.

(ii) We calculate

e(F_{i,n}v_{λ}) = ∑_{j=i}^{n}(E_{j,n}F_{i,n}v_{λ}) ⊗ f_{j} = (λ_{i} − λ_{n})v_{λ} ⊗ f_{j} + ∑_{j=i+1}^{n}(F_{i,j}v_{λ}) ⊗ f_{j},

and using Lemma 8.2,

(Z_{n}(λ) − α_{n})(v_{λ} ⊗ f_{i}) = (α_{i} − α_{n})(v_{λ} ⊗ f_{i}) + X_{n}(v_{λ} ⊗ f_{i})

= (α_{i} − α_{n})(v_{λ} ⊗ f_{i}) + ∑_{j=i+1}^{n} F_{i,j}(v_{λ} ⊗ f_{j})

= (α_{i} − α_{n})(v_{λ} ⊗ f_{i}) + ∑_{j=i+1}^{n}((F_{i,j}v_{λ}) ⊗ f_{j} − v_{λ} ⊗ f_{i})

= (α_{i} − α_{n} − n + i)(v_{λ} ⊗ f_{i}) + ∑_{j=i+1}^{n}(F_{i,j}v_{λ}) ⊗ f_{j}.

The result follows since α_{i} + i − α_{n} − n = λ_{i} − λ_{n} in 𝔽.

8.10. Theorem. For any α ∈ ℤ/pℤ, let ψ_{α} denote the image of (Z_{n}(λ) − α) in End_{𝔽}(N^{α}). Then, the endomorphisms

{1, ψ_{α}, …, (ψ_{α})^{b_{α}−1}}

form a basis for End_{GL(n)}(N^{α}), and (ψ_{α})^{b_{α}} = 0.

Proof. We first show that {1, ψ_{α}, …, (ψ_{α})^{b_{α}−1}} are linearly independent endomorphisms. This is obvious if b_{α} ≤ 1, so we assume that b_{α} > 1. Note first that the argument of Lemma 8.6(iii) applies equally well to the GL(n − 1)-module Q, to show that:

Q^{α} = ∏_{γ≠α}(Z_{n−1}(λ̄) − γ)^{c_{γ}}Q.

Consequently, the high weight filtration of Q in (8.7) yields a high weight filtration of Q^{α},

0 = Q_{(0)}^{α} < Q_{(1)}^{α} < ⋯ < Q_{(c_{α})}^{α} = Q^{α}  (8.11)

where Q_{(i)}^{α}/Q_{(i−1)}^{α} is a high weight module generated by the image of

∏_{γ≠α}(Z_{n−1}(λ̄) − γ)^{c_{γ}}F_{t_{i}^{α},n}v_{λ},

precisely as in Lemma 8.6(iv). By (8.8), (Z_{n−1}(λ̄) − α)^{c_{α}−1}Q^{α} ≠ 0. However, (Z_{n−1}(λ̄) − α)^{c_{α}−1} annihilates the submodule Q_{(c_{α}−1)}^{α} of Q^{α} by Lemma 8.3. Hence,

(Z_{n−1}(λ̄) − α)^{c_{α}−1} ∏_{γ≠α}(Z_{n−1}(λ̄) − γ)^{c_{γ}}F_{t,n}v_{λ} ≠ 0

where t = tc_{α}^{α}. By definition, b_{α} = c_{α} unless α = α_{n}, when b_{α} = c_{α} + 1. So, on applying the injective map e to this non-zero vector, Lemma 8.9 implies:

(Z_{n}(λ) − α)^{b_{α}−1} ∏_{γ≠α}(Z_{n}(λ) − γ)^{b_{γ}}v_{λ} ⊗ f_{t} ≠ 0.

By Lemma 8.6(iii), this shows that (ψ_{α})^{b_{α}−1} ≠ 0. But by Lemma 8.6(i), (ψ_{α})^{b_{α}} = 0, so 1, ψ_{α}, …, ψ_{α}^{b_{α}−1} are certainly linearly independent endomorphisms.

It now just remains to check that dim End_{GL(n)}(N^{α}) = b_{α} for all α ∈ ℤ/pℤ. Since

End_{GL(n)}(N) ≅ ⨁_{α∈ℤ/pℤ}End_{GL(n)}(N^{α}),

it suffices to show that dim End_{GL(n)}(N) ≤ b. Let M denote Δ_{n}(λ) ⊗ V_{n}^{∗}, with the notation of Lemma 6.2 for the Δ-filtration of M constructed there. Then,

dim End_{GL(n)}(N) ≤ dim Hom_{GL(n)}(M, N) ≤ ∑_{i=1}^{a} dim Hom_{G}(Δ_{n}(λ − ε_{r_{i}}), N).

The right hand side is precisely the number b of j with 1 ≤ j ≤ n that are normal for λ, thanks to Theorem 5.10(i). This completes the proof.

8.12. Corollary. (i) The dimension of End_{GL(n)}(L_{n}(λ) ⊗ V_{n}^{∗}) is equal to the number of j with 1 ≤ j ≤ n which are normal for λ.

(ii) The dimension of End_{GL(n)}(L_{n}(λ) ⊗ V_{n}) is equal to the number of j with 1 ≤ j ≤ n which are conormal for λ.

(iii) The number of j with 1 ≤ j ≤ n that are normal for λ is equal to the number of k with 1 ≤ k ≤ n which are conormal for λ.

Proof. Part (i) follows immediately Theorem 8.10. Note that

End_{GL(n)}(L_{n}(λ) ⊗ V_{n}) ≅ End_{GL(n)}(L_{n}(λ^{∗}) ⊗ V_{n}^{∗}).

Now apply (i) and Lemma 5.9(i) to deduce (ii). For (iii) use the isomorphism

End_{GL(n)}(L_{n}(λ) ⊗ V_{n}^{∗}) ≅ End_{GL(n)}(L_{n}(λ) ⊗ V_{n})

together with (i) and (ii).

8.13. Remark. We do not know a combinatorial proof of the purely combinatorial statement in Corollary 8.12(iii).

In (iv) of the next theorem, we refer to the Loewy length of a finite dimensional GL(n)- module M. We recall briefly the definition. We let 0 = S_{0}(M) ≤ S_{1}(M) ≤ … denote the socle series of M, and M = J^{0}(M) ≥ J^{1}(M) ≥ … denote the radical series of M as in [L, section I.8]. Then, the Loewy length of M is the smallest ℓ such that S_{ℓ}(M) = M; by [L, I.8.2(i)], this is also the smallest ℓ such that J^{ℓ}(M) = 0.

8.14. Theorem. Fix α ∈ ℤ/pℤ and let 0 = N_{(0)}^{α} < N_{(1)}^{α} < ⋯ < N_{(b_{α})}^{α} = N^{α} be the high weight filtration of (8.5).

(i) For 1 ≤ i ≤ k ≤ b_{α}, the vector (Z_{n}(λ) − α)^{k−i} ∏_{γ≠α}(Z_{n}(λ) − γ)^{b_{γ}}(v_{λ} ⊗ f_{s_{k}^{α}}) + N_{(i−1)}^{α} is a non-zero primitive vector in N_{(i)}^{α}/N_{(i−1)}^{α} of weight λ − ε_{s_{k}^{α}}.

(ii) for 1 ≤ i ≤ k ≤ b_{α},

dim Hom_{GL(n)}(N_{(k)}^{α}/N_{(k−1)}^{α}, N_{(i)}^{α}/N_{(i−1)}^{α}) = [N_{(i)}^{α}/N_{(i−1)}^{α} : L_{n}(λ + ε_{s_{k}^{α}})] = 1;

(iii) For 1 ≤ i < b_{α}, the extension 0 → N_{(i)}^{α}/N_{(i−1)}^{α} → N_{(i+1)}^{α}/N_{(i−1)}^{α} → N_{(i+1)}^{α}/N_{(i)}^{α} → 0 does not split.

(iv) The Loewy length of N_{(i)}^{α}/N_{(i−1)}^{α} is at least b_{α} − i + 1.

Proof. Throughout the proof, write Z = Z_{n}(λ) for short.

(i) By Lemma 8.6(iv), the module N_{(k)}^{α} is generated by N_{(k−1)}^{α} and the vector ∏_{γ≠α}(Z − γ)^{b_{γ}}(v_{λ}⊗f_{s_{k}^{α}}), which is primitive modulo N_{(k−1)}^{α}. Therefore, since (Z−α)^{k−i}N_{(k−1)}^{α} ⊂ N_{(i−1)}^{α} by Lemma 8.3, it follows that

v = (Z − α)^{k−i} ∏_{γ≠α}(Z − γ)^{b_{γ}}(v_{λ} ⊗ f_{s_{k}^{α}}) + N_{(i−1)}^{α}

is either primitive or zero. Also by Lemma 8.3, v lies in N_{(i)}^{α}/N_{(i−1)}^{α}, so it just remains to show that v is non-zero. Suppose now for a contradiction that v = 0. Then

(Z − α)^{k−i}N_{(k)}^{α} ⊆ N_{(i−1)}^{α}.

Since (Z − α)^{b_{α}−k}N^{α} ⊆ N_{(k)}^{α} and (Z − α)^{i−1}N_{(i−1)}^{α} = 0 by Lemma 8.3, it follows that (Z − α)^{b_{α}−1}N^{α} = 0, which contradicts Theorem 8.10.

(ii) By (i), left multiplication by (Z − α)^{k−i} induces a non-zero homomorphism between N_{(k)}^{α}/N_{(k−1)}^{α} and N_{(i)}^{α}/N_{(i−1)}^{α}. Hence both the Hom-dimension and the composition multiplicity are at least one. So, since [N_{(i)}^{α}/N_{(i−1)}^{α} : L_{n}(λ + ε_{s_{k}^{α}})] ≥ 1 for all 1 ≤ i ≤ k, [N^{α} : L_{n}(λ + ε_{s_{k}^{α}})] ≥ k with equality only if [N_{(i)}^{α}/N_{(i−1)}^{α} : L_{n}(λ + ε_{s_{k}^{α}})] = 1 for all i ≤ k. But Theorem 7.7 shows that equality does hold. So [N_{(i)}^{α}/N_{(i−1)}^{α} : L_{n}(λ + ε_{s_{k}^{α}})] = 1 and this also implies that the Hom-dimension is at most one.

(iii) If the extension is split, Lemma 8.3 implies that (Z − α) annihilates N_{(i+1)}^{α}/N_{(i−1)}^{α}. But by Lemma 8.6(iv), the vector w := ∏_{γ≠α}(Z − γ)^{b_{γ}}(v_{λ} ⊗ f_{s_{i+1}^{α}}) belongs to N_{(i+1)}^{α}, and by (i), (Z − α)w + N_{(i−1)}^{α} is a non-zero vector of N_{(i+1)}^{α}/N_{(i−1)}^{α}. Hence (Z − α) does not annihilate N_{(i+1)}^{α}/N_{(i−1)}^{α}.

(iv) Let L = N_{(i)}^{α}/N_{(i−1)}^{α}. For i ≤ k ≤ b_{α}, let

v_{k} = (Z − α)^{k−i} ∏_{γ≠α}(Z − γ)^{b_{γ}}(v_{λ} ⊗ f_{s_{k}^{α}}) + N_{(i−1)}^{α}

which by (i) is a non-zero primitive vector in L.

We claim first that for k > i, v_{k} lies in the GL(n)-submodule of L generated by v_{k−1}. By Lemma 8.6(iv), we can write

(Z − α) ∏_{γ≠α}(Z − γ)^{b_{γ}}(v_{λ} ⊗ f_{s_{k}^{α}}) + N_{(k−2)}^{α} = y ∏_{γ≠α}(Z − γ)^{b_{γ}}(v_{λ} ⊗ f_{s_{k−1}^{α}}) + N_{(k−2)}^{α}

for some y ∈ U^{−}(n). Applying the central element (Z − α)^{k−i−1}, which sends N_{(k−2)}^{α} into N_{(i−1)}^{α}, this shows that v_{k} = yv_{k−1} as required.

Now let 0 = S_{0} ≤ S_{1} ≤ … denote the socle series of L. We show that for i ≤ k ≤ b_{α}, v_{k} ∉ S_{b_{α}−k}, by downward induction on k. In particular, this will show that v_{i} ∉ S_{b_{α}−i}, hence that the socle series of L has length at least b_{α} − i + 1. The induction starts trivially with k = b_{α}, since v_{b_{α}} ≠ 0. So now take k < b_{α} and assume by induction that v_{k+1} ∉ S_{b_{α}−k−1}. Suppose for a contradiction that v_{k} ∈ S_{b_{α}−j}. Let v̄_{k} = v_{k} + S_{b_{α}−k−1} ∈ S_{b_{α}−k}/S_{b_{α}−k−1}. Since v_{k+1} ∈ U(n)v_{k} and v_{k+1} ∉ S_{b_{α}−k−1}, we conclude that v̄_{k} and v̄_{k+1} = v_{k+1} + S_{b_{α}−k−1} are two non-zero primitive vectors in S_{b_{α}−k}/S_{b_{α}−k−1}, which have different weights. Since v̄_{k+1} ∈ U(n)v̄_{k}, this contradicts the semisimplicity of S_{b_{α}−k}/S_{b_{α}−k−1}.

8.15. Corollary. For α ∈ ℤ/pℤ, the Loewy length of N^{α} is at least 2b_{α} − 1, where b_{α} is the number of j with 1 ≤ j ≤ n which are normal for λ and α_{j} = α.

Proof. Let ℓ denote the Loewy length of N^{α}. Consider the submodule N_{(1)}^{α} ⊂ N^{α} generated by ∏_{γ≠α}(Z_{n}(λ) − γ)^{b_{γ}}(v_{λ} ⊗ f_{s_{1}^{α}}) as in Theorem 8.14. Note that N_{(1)}^{α} has simple head L_{n}(μ) where μ is the highest weight λ − ε_{s_{1}^{α}} of N, hence that [N^{α} : L_{n}(μ)] = 1.

By Theorem 8.14(iv), the Loewy length of N_{(1)}^{α} is at least b_{α}, so as N_{(1)}^{α} has simple head L_{n}(μ), we conclude that [S_{b_{α}−1}(N_{(1)}^{α}) : L_{n}(μ)] = 0, whence L_{n}(μ) appears with multiplicity 1 in N_{(1)}^{α}/S_{b_{α}−1}(N_{(1)}^{α}). By [L, Lemma 8.5(i)], S_{b_{α}−1}(N_{(1)}^{α}) = N_{(1)}^{α} ∩ S_{b_{α}−1}(N^{α}). So N_{(1)}^{α}/S_{b_{α}−1}(N_{(1)}^{α}) embeds into N^{α}/S_{b_{α}−1}(N^{α}), which implies [N^{α}/S_{b_{α}−1}(N^{α}) : L_{n}(μ)] = 1. Therefore

[S_{b_{α}−1}(N^{α}) : L_{n}(μ)] = 0.

Note by contravariant duality that N^{α}/J^{i}(N^{α}) ≅ (S_{i}(N^{α}))^{τ} . Hence, as N^{α} contains L_{n}(μ) with multiplicity one,

[J^{b_{α}−1}(N^{α}) : L_{n}(μ)] = 1.

Since J^{b_{α}−1}(N^{α}) ⊆ S_{ℓ−(b_{α}−1)}(N^{α}) by [L, Lemma I.8.2(ii)], this shows that

[S_{ℓ−b_{α}+1}(N^{α}) : L_{n}(μ)] = 1.

Hence, S_{ℓ−b_{α}+1}(N^{α}) strictly contains S_{b_{α}−1}(N^{α}). So, ℓ − b_{α} + 1 > b_{α} − 1, as required.

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brundan@darkwing.uoregon.edu, klesh@math.uoregon.edu

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