Semisimple restrictions from GL(n) to GL(n − 1)^{1}

Jonathan Brundan  Alexander Kleshchev
Department of Pure Mathematics  Department of Mathematics
Cambridge University  University of Oregon
16, Mill Lane, Cambridge, CB2 1SB  Eugene, Oregon
England  U.S.A.

     E-mail: brundan@pmms.cam.ac.uk  E-mail: klesh@math.uoregon.edu

Irina Suprunenko

Institute of Mathematics Academy of Sciences of Belarus Minsk, 220072 Belarus

              E-mail: imanb%imanb.belpak.minsk.by@demos.su

Abstract

We obtain a criterion for the restriction of an irreducible rational GL(n)- module to the naturally embedded subgroup GL(n − 1) to be semisimple, over an arbitrary algebraically closed field. In that case, we describe the composition factors of the restriction explicitly. As an application, we classify the completely splittable representations of general linear groups and give an exact character formula for these modules.

Introduction

In this paper, we study rational representations of the algebraic group GL(n) = GL(n, 𝔽) defined over an algebraically closed field 𝔽 of characteristic p ≥ 0. Our main result gives a combinatorial criterion for the restriction of an irreducible GL(n)- module to the naturally embedded subgroup GL(n−1) to be semisimple. In that case, we describe the composition factors of the restriction explicitly. This extends earlier work of Kleshchev [K3] and Brundan [B1] where analogous results for symmetric groups and Hecke algebras were proved.

As an application – in section 6 – we will classify all completely splittable representations of general linear groups, extending [K4]. By definition, a completely splittable representation is an irreducible module which is semisimple on restriction to every Levi subgroup of GL(n). For example, all irreducible GL(n)-modules of high weight a multiple of a fundamental dominant weight are completely splittable, by Theorem 6.2. Our results allow the dimensions of all weight spaces of a completely splittable representation to be determined exactly. Character formulae for these representations have recently also been obtained (independently) by Mathieu and Papadopoulo [MP], using completely different methods.

11991 subject classification. Primary 20G05.

We now describe the main result of the paper in detail. Recall that rational irreducible 𝔽GL(n)-modules are parametrised by dominant weights, which can be identified with n-tuples of integers λ = (λ_{1}, …, λ_{n}) where λ_{1} ≥ ⋯ ≥ λ_{n} and λ_{n} is arbitrary. Let L_{n}(λ) denote the corresponding irreducible GL(n)-module.

The dominant weight λ is p-restricted if p = 0, or p ≠ 0 and λ_{i} − λ_{i+1} < p for all 1 ≤ i < n. If λ = (λ_{1}, …, λ_{n}) and μ = (μ_{1}, …, μ_{n}) are dominant weights, we denote by λ+μ the weight whose ith part is λ_{i}+μ_{i} for i = 1, 2, …, n, and for a positive integer m we write mλ for the weight whose ith part is mλ_{i}. Given this notation, we may write an arbitrary dominant weight λ (non-uniquely) as λ = λ(0) + pλ(1) + ⋯ + p^{d}λ(d) for some d ≥ 0, with each λ(i) p-restricted. We call any such expansion a p-adic expansion of λ.

Fix now a dominant weight λ = (λ_{1}, …, λ_{n}). If λ_{n} ≥ 0, λ should be regarded as a partition, with corresponding diagram [λ]. This is the set

{(i, j) ∈ ℕ × ℕ | λ_{i} ≠ 0, 1 ≤ j ≤ λ_{i}}

which we always identify with an array of boxes in the plane. For example, if λ = (3, 2), then [λ] is

We say i is a removable row (for λ) if 1 ≤ i < n and λ_{i} ≠ λ_{i+1}. Notice that if λ_{n} ≥ 0 and i is a removable row, then the node (i, λ_{i}) at the end of the ith row of [λ] can be removed to leave the diagram of a proper partition – that is, removable rows contain removable nodes.

Given (i, j) ∈ ℤ × ℤ (not necessarily ℕ × ℕ), define the corresponding p-residue res_{p}(i, j) to be i − j regarded as an element of the ring ℤ/pℤ. In the above example, the 3-residues of [λ] are:

0 2 1 1 0

For 1 ≤ i ≤ j < n, define B^{λ}(i, j) := j − i + λ_{i} − λ_{j+1} ∈ ℤ/pℤ. We shall use the fact that this is just the difference between the p-residues of (j + 1, λ_{j+1} + 1) and (i, λ_{i}). Given i, j with 1 ≤ i < j < n, a B-chain from i to j is a chain i = i_{0} < ⋯ < i_{r} = j such that B^{λ}(i_{s−1}, i_{s}) = 0 (in ℤ/pℤ) for all 1 ≤ s ≤ r. Finally, let R_{i}(λ) := {res_{p}(i, j) | λ_{i+1} < j ≤ λ_{i}}, i = 1, 2, …, n − 1.

We can now give the two main combinatorial definitions of the paper. If λ is a p-restricted dominant weight, we say that λ is a generalized Jantzen-Seitz weight (GJS for short) if whenever there are 1 ≤ i < j < n with R_{i}(λ)∩R_{j}(λ) ≠ ∅, then there is a B-chain from i to j. More generally, if λ is an arbitrary weight, we say that λ is a generalized Jantzen-Seitz weight if λ is dominant and each λ(i) in a p-adic expansion of λ is GJS.

Next, suppose that λ = (λ_{1}, …, λ_{n}) is a p-restricted GJS weight. The weight μ = (μ_{1}, …, μ_{n−1}) is allowable (for λ) providing

(1) λ_{i+1} ≤ μ_{i} ≤ λ_{i} for all 1 ≤ i < n;

(2) if μ_{i} ≠ λ_{i} and there is a B-chain from i to j for some 1 ≤ i < j < n, then μ_{j} = λ_{j+1}.

The definitions ensure that such weights are p-restricted and dominant. More generally, if λ is an arbitrary GJS weight with p-adic expansion λ = λ(0) + pλ(1) + ⋯ + p^{d}λ(d), we say that the weight μ = (μ_{1}, …, μ_{n−1}) is allowable (for λ) if μ = μ(0)+pμ(1)+⋯+ p^{d}μ(d) for weights μ(0), …, μ(d) such that μ(i) is allowable for λ(i) for i = 0, 1, …, d.

Note that these two definitions are independent of the choice of p-adic expansion of λ, in the case that λ is not p-restricted. We shall shortly give some examples to illustrate the definitions, but first, we state the main result of the paper.

Main Theorem Let λ be a dominant weight. Then, res_{GL(n−1)}^{GL(n)} L_{n}(λ) is semisimple if and only if λ is GJS. In that case,

res_{GL(n−1)}^{GL(n)} L_{n}(λ) ≅ ⨁_{μ allowable} L_{n−1}(μ).

Examples The following are examples of p-restricted GJS weights if p = 3 and n ≥ 7:

Example one: 0 2 1 1 0 2 1 0 2 1 2 0  Example two: 0 2 1 0 2 1 1 0 2 1 2 1 0 2 0 2 1 0 1 0 2 2 1 0

In example one, the following are the allowable weights if n > 7: (3, 2^{3}, 1^{3}), (3, 2^{2}, 1^{4}), (3, 2^{3}, 1^{2}), (3, 2^{2}, 1^{3}), (2^{4}, 1^{2}) and (2^{3}, 1^{3}). In the case n = 7, omit the first two weights from this list (as they contain more than (n−1) non-zero parts). In example two, allowable weights are obtained by removing nodes ‘from the bottom up’ – the possibilities for n > 7 are (6, 4^{3}, 3, 2, 1),(6, 4^{3}, 3, 2),(6, 4^{3}, 3, 1),(6, 4^{3}, 2, 1),(6, 4^{2}, 3, 2, 1),(5, 4^{2}, 3, 2, 1) and (4^{3}, 3, 2, 1). Again, if n = 7, omit the first weight from this list.

All the results of the paper are expected to generalize (without significant alterations to the proofs) to quantum GL(n) – by which we mean the quantum algebra obtained by base change from Lusztig’s integral form for the Drinfeld-Jimbo quantized enveloping algebra U_{q}(𝔤𝔩_{n}) [L]. The results in the quantum case will be valid over an arbitrary field 𝔽, and at an arbitrary root of unity v ∈ 𝔽^{×}. To obtain the correct statements of the results for quantum GL(n) – in the p-restricted case only – replace the integer p in all the above definitions with the integer ℓ, which by definition is the smallest positive integer such that v^{−ℓ+1} + v^{−ℓ+3} + ⋯ + v^{ℓ−1} = 0 in 𝔽, or 0 if no such integer exists. More complicated modifications are needed in the non-p-restricted case, using the quantum version of Steinberg’s tensor product theorem. The necessary technical theory to generalize the proofs here to the quantum case can be found in [B1] (see also Remark 2.7).

The paper is organised as follows. In section 1, we will give some equivalent statements of the above definitions and results. In particular, we restate results in terms of dominant weights for the algebraic group SL(n, 𝔽), since this may be more familiar to some readers. In section 2, we set up notation and recall well known basis theorems for standard and costandard modules for GL(n). In section 3, we use these basis theorems to obtain a useful theoretical criterion for res_{GL(n−1)}^{GL(n)} L_{n}(λ) to be semisimple. The central tool in the proof of the Main Theorem is introduced in section 4, where we define Kleshchev’s lowering operators and prove some technical properties. We then prove the Main Theorem in section 5. Finally, in section 6, we give the application to classifying completely splittable representations. There is also an appendix containing a short proof of the standard basis theorem and straightening rule.

Acknowledgements The first author would like to thank G. Seitz for making this collaboration possible and the University of Oregon for its hospitality whilst this work was completed. We also thank J. C. Jantzen for pointing out [J2], Satz II.6. The second author was partially supported by the NSF. The third author was partially supported by the Basic Research Foundation of Belarus.

1 Some equivalent statements of the main results

We wish first to reformulate the definition of GJS. Recall the two examples of GJS weights given in the introduction. The first of these examples is ‘trivially’ a GJS weight as in fact R_{i}(λ) ∩ R_{j}(λ) = ∅ for all 1 ≤ i < j < n. The second example has the (stronger) property that there is a B-chain between all pairs of removable rows 1 ≤ i < j < n. Weights with this latter property are the usual Jantzen-Seitz weights of [JS]. We want to show that these are essentially the only two ways prestricted generalized Jantzen-Seitz weights can occur.

Given removable rows 1 ≤ i < j < n, a B-chain i = i_{0} < ⋯ < i_{r} = j is proper if every i_{s} is a removable row. Note that if i < k < j and k is not a removable row then B^{λ}(i, k) + B^{λ}(k, j) = B^{λ}(i, j). So if a B-chain from i to j exists, then a proper B-chain from i to j exists.

1.1 Lemma Let λ be GJS. Suppose i_{0} < i_{1} < ⋯ < i_{r} and j_{0} < j_{1} < ⋯ < j_{s} are two proper B-chains and that either i_{0} = j_{0} or i_{r} = j_{s}. Then there is a proper Bchain min(i_{0}, j_{0}) = k_{0} < k_{1} < ⋯ < k_{t} = max(i_{r}, j_{s}) such that {i_{0}, …, i_{r}, j_{0}, …, j_{s}} ⊆ {k_{0}, …, k_{t}}.

Proof. We prove this in the case that i_{0} = j_{0}, leaving the similar case i_{r} = j_{s} to the reader. Suppose first that i_{1} < j_{1}. Then, B^{λ}(i_{0}, i_{1}) = 0 = B^{λ}(j_{0}, j_{1}). Hence, as i_{0} = j_{0}, the nodes (i_{1}, λ_{i_{1}+1} + 1) and (j_{1}, λ_{j_{1}+1} + 1) have the same residue. These residues are elements of R_{i_{1}}(λ) and R_{j_{1}}(λ) by definition, so R_{i_{1}}(λ) meets R_{j_{1}}(λ). So there is a proper B-chain from i_{1} to j_{1} by definition of GJS. So, we may refine the chain j_{0} < j_{1} < ⋯ < j_{s} to assume that j_{1} = i_{1}. Similarly, if j_{1} < i_{1}, we may refine the chain i_{0} < i_{1} < ⋯ < i_{r} to assume that i_{1} = j_{1}. Now, the proof is easily completed by induction.

The lemma implies in particular that if i < j are removable rows and there is a B-chain from i to j for a GJS weight λ, then there is a unique maximal proper B-chain from i to j.

1.2 Lemma Let λ = (λ_{1}, …, λ_{n}) be a p-restricted dominant weight. Then, λ is GJS if and only if one of the following two conditions holds:

(JS1) R_{i}(λ) ∩ R_{j}(λ) = ∅ for all 1 ≤ i < j < n;

(JS2) whenever 1 ≤ i < j < n are consecutive removable rows, then B^{λ}(i, j) = 0.

Proof. (⇐) is obvious.

(⇒) Let λ be GJS. Suppose that (JS1) is false, so that we can find 1 ≤ i < j < n with R_{i}(λ) ∩ R_{j}(λ) ≠ ∅. Let i = i_{0} < i_{1} < ⋯ < i_{r} = j be a maximal proper B-chain from i to j.

We first claim that R_{i_{0}}(λ) ∪ R_{i_{1}}(λ) ∪ ⋯ ∪ R_{i_{r}}(λ) = ℤ/pℤ. Let a_{s} = res_{p}(i_{s}, λ_{i_{s}}) for all 0 ≤ s ≤ r. Take any a ∈ R_{i}(λ) ∩ R_{j}(λ). Then, R_{i_{0}}(λ) contains a, a − 1, …, a_{0}, R_{i_{1}}(λ) equals a_{0} − 1, a_{0} − 2, …, a_{1}, R_{i_{2}}(λ) equals a_{1} − 1, a_{1} − 2, …, a_{2} and so on. Finally, R_{i_{r}}(λ) contains a_{r−1} − 1, …, a. This proves the claim.

Next, we claim that in fact {i_{0}, i_{1}, …, i_{r}} are all of the removable rows between i and j. Well, suppose not. Then, there is some i < k < j with k removable and equal to no i_{s}. By the previous claim, R_{k}(λ) meets some R_{i_{s}}(λ) for some 0 ≤ s ≤ r. If i_{s} < k, then there is a B-chain from i_{s} to k and from i_{s} to j, so that the B-chain from i_{s} to j can be refined to a strictly longer proper B-chain from i_{s} to j by Lemma 1.1, contradicting maximality of the original B-chain from i to j. So, i_{s} > k and there is a B-chain from k to i_{s}. This time, the B-chain from i to i_{s} can be refined to a strictly longer proper B-chain by Lemma 1.1, giving the required contradiction in this case.

Hence, B^{λ}(a, b) = 0 for all consecutive removable rows i ≤ a < b ≤ j. Now, let i^{−}, j^{+} be the smallest and largest removable rows respectively, so i^{−} ≤ i and j^{+} ≥ j. By the first claim, R_{i^{−}}(λ) meets R_{i^{+}}(λ) for some i ≤ i^{+} ≤ j. If i^{−} = i, we take i^{+} = j so that i^{−} < i^{+} always. Now repeating the previous argument shows that B^{λ}(a, b) = 0 for all consecutive removable rows i^{−} ≤ a < b ≤ i^{+}. Similarly, there is i ≤ j^{−} ≤ j with j^{−} < j^{+} such that B^{λ}(a, b) = 0 for all consecutive removable rows j^{−} ≤ a < b < j^{+}. Since the intervals {i^{−}, …, i^{+}}, {i, …, j}, {j^{−}, …, j^{+}} overlap, this shows that (JS2) holds, completing the proof.

As a corollary of the Main Theorem and Lemma 1.2, we now describe the semisimple restrictions from SL(n) to SL(n − 1) (embedded as in the GL case). Let ℓ = n − 1, and ω_{1}, …, ω_{ℓ} be the fundamental dominant weights for the root system A_{ℓ} (as in [Bou]). If Λ = b_{1}ω_{1} + ⋯ + b_{l−1}ω_{l−1} + b_{ℓ}ω_{ℓ} is a weight for A_{ℓ} we define res(Λ) to be b_{1}ω_{1} + ⋯ + b_{ℓ−1}ω_{ℓ−1} considered as a weight for A_{ℓ−1}.

A dominant weight Λ = b_{1}ω_{1}+⋯+b_{ℓ}ω_{ℓ} is called p-restricted if p = 0 or b_{1}, …, b_{ℓ} < p. Every dominant weight can be represented in the form

Λ = a_{1}ω_{i_{1}} + ⋯ + a_{k}ω_{i_{k}}, a_{1}, …, a_{k} > 0. (1.3)

By convention, we represent the weight Λ = 0 in (1.3) by taking k = 0. For such Λ and 1 ≤ r ≤ s ≤ k define

B_{r,s}^{Λ} := i_{s} − i_{r} + ∑_{v=r}^{s} a_{v} ∈ ℤ/pℤ.

We say that Λ satisfies (JS1′) if, for every 1 ≤ r < s ≤ k, one has B_{r,s}^{Λ} − v ≠ 0 (in ℤ/pℤ) for all 0 < v < a_{r} + a_{s}. We say that Λ satisfies (JS2′) if B_{r,r+1}^{Λ} = 0 for all 1 ≤ r < k. Finally, we say that a p-restricted dominant weight Λ is GJS if Λ satisfies (JS1′) or (JS2′). The zero weight is GJS by convention.

Let α_{1}, …, α_{ℓ} be simple roots in A_{ℓ} corresponding to ω_{1}, …, ω_{ℓ}, respectively. Denote the root α_{i} + α_{i+1} + ⋯ + α_{ℓ} by α(i, ℓ). Fix Λ written in the form (1.3). An A_{ℓ}-weight M is called Λ-allowable if M = Λ − n_{1}α(i_{1}, ℓ) − ⋯ − n_{k}α(i_{k}, ℓ) for some integers n_{i} such that

(1) 0 ≤ n_{i} ≤ a_{i}, i = 1, 2, …, k;

(2) if n_{i} ≠ 0 and B_{i,j}^{Λ} = 0 for some 1 ≤ i < j ≤ k then n_{j} = a_{j}.

Recall Steinberg’s tensor product theorem [St], Theorem 41. Let Λ = Λ_{0} + pΛ_{1} + ⋯ + p^{d}Λ_{d} be the (coordinate-wise) p-adic expansion. Denote the irreducible SL(n)- module with high weight Λ by L(Λ). Steinberg’s theorem implies that for p > 0, L(Λ) can be represented as the tensor product L(Λ) = L(Λ_{0}) ⊗ L(Λ_{1})^{[1]} ⊗ ⋯ ⊗ L(Λ_{d})^{[d]} where ^{[j]} means the j-th Frobenius twist.

1.4 Theorem (SL-version of Main Theorem) Let Λ be an arbitrary dominant weight with p-adic expansion Λ = Λ_{0} + pΛ_{1} + ⋯ + p^{d}Λ_{d}, where d = 0 if p = 0. The restriction res_{SL(n−1)}^{SL(n)} L(Λ) is semisimple if and only if every Λ_{j} is GJS. In that case

res_{SL(n−1)}^{SL(n)} L(Λ) = ⨁ L(res(M_{0})) ⊗ L(res(M_{1}))^{[1]} ⊗ ⋯ ⊗ L(res(M_{d}))^{[d]}

where the sum is over all tuples (M_{0}, M_{1}, …, M_{d}) such that M_{j} is Λ_{j}-allowable, j = 0, 1, …, d.

2 The standard and costandard basis theorems

To prove the Main Theorem, we will work with the hyperalgebra U(n) corresponding to the algebraic group GL(n), which we may do as the categories of finite dimensional rational GL(n)-modules and finite dimensional U(n)-modules are equivalent [J1], I.7. We begin by recalling the definition of this hyperalgebra, and then introduce notation for the standard and costandard bases of standard and costandard modules.

2.1 The hyperalgebra. The hyperalgebra U(n) = U(n, 𝔽) corresponding to the algebraic group GL(n) = GL(n, 𝔽), can be defined by base change from a Kostant ℤ-form U(n, ℤ) for the universal enveloping algebra U(n, ℂ) of the Lie algebra 𝔤𝔩(n, ℂ), as in [CL]. For 1 ≤ i, j ≤ n, let X_{i,j} denote the element of 𝔤𝔩(n, ℂ) corresponding to the n × n matrix with a 1 in the ij-entry and zeros elsewhere. Let

X_{i,j}^{(r)} := \frac{(X_{i,j})^{r}}{r!} , \binom{X_{i,j}}{r} := \frac{X_{i,j}(X_{i,j} − 1) … (X_{i,j} − r + 1)}{r!}

in U(n, ℂ). Then, U(n, ℤ) is the ℤ-subalgebra of U(n, ℂ) generated by

\{1, X_{i,j}^{(r)}, \binom{X_{i,i}}{r} | 1 ≤ i, j ≤ n, i ≠ j, r ≥ 1\}

and the hyperalgebra U(n) over 𝔽 is 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 \binom{X_{i,i}}{r}

by \binom{H_{i}}{r}. We use E_{i}^{(r)} and F_{i}^{(r)} as shorthands for E_{i,i+1}^{(r)} and F_{i,i+1}^{(r)}.

Let U^{+}(n) and U^{−}(n) respectively be the subalgebras of U(n) generated by the elements {1, E_{i}^{(r)} | 1 ≤ i < n, r ≥ 1} and {1, F_{i}^{(r)} | 1 ≤ i < n, r ≥ 1}. Let U^{0}(n) denote

the ‘diagonal’ subalgebra generated by \{1, \binom{H_{i}}{r} | 1 ≤ i ≤ n, r ≥ 1\}. Recall that U^{−}(n)

has a PBW basis which we always order as follows:

F_{1,2}^{(N_{1,2})}F_{1,3}^{(N_{1,3})}F_{2,3}^{(N_{2,3})} … F_{1,n}^{(N_{1,n})} … F_{n−1,n}^{(N_{n−1,n})}

as N runs over all upper triangular matrices with entries in ℤ_{≥0} and zeros on the diagonal.

We identify the n-tuple λ = (λ_{1}, …, λ_{n}) ∈ ℤ^{n} with the weight λ : U^{0}(n) → 𝔽 which maps \binom{H_{i}}{r} to \binom{λ_{i}}{r} for all 1 ≤ i ≤ n, r ≥ 1. Let ε_{i} denote the U(n)-weight (0, …, 0, 1, 0, …, 0) (where the 1 is in the ith position). As usual, there is a dominance order defined on the set X(n) := ℤ^{n} of all weights: for μ, λ ∈ X(n), μ ≤ λ if λ − μ can be written as ∑_{i=1}^{n−1} a_{i}(ε_{i} − ε_{i+1}) with each a_{i} a non-negative integer. The weight λ ∈ X(n) is dominant if λ_{1} ≥ ⋯ ≥ λ_{n} with λ_{n} ∈ ℤ arbitrary. Let X^{+}(n) denote the set of all dominant weights.

For λ ∈ X^{+}(n), let L_{n}(λ), Δ_{n}(λ) and ∇_{n}(λ) denote the irreducible, standard (or Weyl) and costandard (or co-Weyl) U(n)-modules of high weight λ respectively. By definition, a high weight vector in a U(n)-module is a weight vector annihilated by E_{i}^{(r)} for all 1 ≤ i < n, r ≥ 1. So the standard module Δ_{n}(λ) is generated by a high weight vector e_{λ} (unique up to scalars), and U^{0}(n) acts on e_{λ} by the weight λ. Recall that L_{n}(λ) = Δ_{n}(λ)/ rad Δ_{n}(λ) where rad Δ_{n}(λ) denotes the unique maximal proper submodule, and let f_{λ} be the image of e_{λ} in this quotient. Also, L_{n}(λ) is the simple socle of ∇_{n}(λ).

Let δ denote the n-tuple (1, 1, …, 1) which is the weight of the one dimensional determinant module det. Then, for any c ∈ ℤ, Δ_{n}(λ) ⊗ det^{c} ≅ Δ_{n}(λ + cδ) , and similarly for L_{n}(λ). Using this observation, it is easy to reduce all the results in the introduction to the case that λ_{n} ≥ 0, when we may identify λ with a partition. We will do this from now on in the paper, and let Λ^{+}(n) denote the set of all partitions (λ_{1}, …, λ_{n}) ∈ ℤ^{n} with λ_{1} ≥ ⋯ ≥ λ_{n} ≥ 0. For convenience, let Λ(n) ⊂ X(n) denote the set of all n-tuples of non-negative integers, so Λ^{+}(n) = Λ(n) ∩ X^{+}(n). Elements of Λ(n) are compositions.

Let U(n − 1), U^{−}(n − 1), U^{0}(n − 1), U^{+}(n − 1) denote the naturally embedded subalgebras of U(n), U^{−}(n), U^{0}(n), U^{+}(n) respectively corresponding to the subgroup GL(n − 1) < GL(n) (embedded into the top left hand corner of the matrices). We will talk about U(n − 1)-weights and U(n − 1)-high weight vectors when it is necessary to distinguish between these notions and the corresponding notions for U(n).

For λ ∈ X(n) and μ in either X(n − 1) or X(n), and 1 ≤ i ≤ j < n, define

C^{λ}(i, j) := j − i + λ_{i} − λ_{j},
B^{λ}(i, j) := j − i + λ_{i} − λ_{j+1},
B^{μ,λ}(i, j) := j − i + μ_{i} − λ_{j+1},

all regarded as elements of ℤ/pℤ. Next, given 1 ≤ i < j ≤ n, define

ℭ^{λ}(i, j) := {a | i < a < j, C^{λ}(i, a) = 0},
𝔅^{λ}(i, j) := {a | i ≤ a < j, B^{λ}(i, a) = 0},
𝔅^{μ,λ}(i, j) := {a | i ≤ a < j, B^{μ,λ}(i, a) = 0}.

Observe that B^{λ}(i, j) = B^{λ,λ}(i, j) and 𝔅^{λ}(i, j) = 𝔅^{λ,λ}(i, j).

2.2 Tableaux. Suppose that λ ∈ Λ(n) is a composition, where λ = (λ_{1}, …, λ_{n}). Recall the definition of the diagram [λ] of λ from the introduction. A λ-tableau is a function t : [λ] → {1, …, n}, which we regard just as the diagram [λ] with boxes filled with integers in {1, …, n}. A tableau t is simply a λ-tableau for some λ ∈ Λ(n), and in which case λ is the shape of t. If 1 appears μ_{1} times, 2 appears μ_{2} times, …, n appears μ_{n} times in t then we say that the weight of t is (μ_{1}, μ_{2}, …, μ_{n}).

A tableau is row standard if the entries increase weakly along the rows. If λ ∈ Λ^{+}(n) is a partition, we say a λ-tableau is standard if the entries increase weakly along rows and strictly down columns. The λ-tableau with every entry on the ith row equal to i is denoted 1_{λ}. It is the unique standard λ-tableau of weight λ.

Given a row standard tableau t and 1 ≤ m ≤ n, let t[m] denote the tableau obtained by removing all nodes with entry greater than m. As t is row standard, t[m] is well-defined and has the shape of some composition. Note also that t equals t[n] as the entries in t are at most n. If in addition every row on the ith row is at least i for all i, then t[m] has at most m non-empty rows, and we can define shape(t[m]) to be the shape of the tableau t[m] regarded as an element of Λ(m).

We can now define a partial order on the set of all row standard tableaux, called the dominance order on tableaux, as follows. Given λ-tableaux s and t, write s ≤ t if either s = t, or there is some 1 ≤ m ≤ n such that s[i] has the same shape as t[i] for i = m + 1, …, n but the shape of s[m] is strictly lower than the shape of t[m] in the dominance order on Λ(n). For example,

1 3 4 2 2 < 1 2 4 2 3 < 1 2 2 3 4 .

Say that μ belongs to λ, written μ ⟵ λ, if λ = (λ_{1}, …, λ_{m}) ∈ Λ^{+}(m) and μ = (μ_{1}, …, μ_{m−1}) ∈ Λ^{+}(m − 1) for some m > 1, satisfying λ_{i+1} ≤ μ_{i} ≤ λ_{i} for all 1 ≤ i < m. So, [μ] is the diagram of a partition, obtained from [λ] by removing nodes from the bottoms of columns. Given μ ⟵ λ ∈ Λ^{+}(n), we define the level of μ to be the integer ∑_{i=1}^{n−1}(λ_{i} − μ_{i}), and let 1_{μ,λ} denote the standard λ-tableau with (i, j)-entry equal to i if (i, j) ∈ [μ] or n otherwise. For example, if μ = (2, 1, 0) and λ = (3, 2, 0, 0) (so n = 4) then

1_{μ,λ} = 1 1 4 2 4 .

2.3 The standard basis theorem. Fix now a partition λ = (λ_{1}, …, λ_{n}) ∈ Λ^{+}(n). Given a row standard λ-tableau t such that every entry on row i of t is at least i, let

F_{t} := F_{1,2}^{(N_{1,2})}F_{1,3}^{(N_{1,3})}F_{2,3}^{(N_{2,3})} … F_{1,n}^{(N_{1,n})} … F_{n−1,n}^{(N_{n−1,n})}

where N_{i,j} equals the number of j’s on row i of t. This is just a monomial in the PBW basis for U^{−}(n). For example, F_{1_{λ}} is the identity in U^{−}(n). Given μ ⟵ λ, let F_{μ,λ} denote F_{1_{μ,λ}}; explicitly, F_{μ,λ} = F_{1,n}^{(λ_{1}−μ_{1})} … F_{n−1,n}^{(λ_{n−1}−μ_{n−1})}.

The standard basis theorem [CL], 3.5 says that

{F_{t}e_{λ} | t a standard λ-tableau}

is a basis for Δ_{n}(λ). We will also need the straightening rule proved in Theorem A.4 in the appendix (also note the remarks after the proof of Theorem A.4): if t is any row standard but not standard λ-tableau such that every entry on row i is at least i, then F_{t}e_{λ} can be written as a linear combination of standard basis elements F_{s}e_{λ}, s standard, where s > t in the dominance order on tableaux. Moreover, as the weight of the vector F_{s}e_{λ} is precisely the weight of s as defined in (2.2), the standard tableaux s occurring in this expansion of F_{t}e_{λ} have the same weight as t.

2.4 Lemma Given μ ⟵ λ and an element S ∈ U^{−}(n), the F_{μ,λ}e_{λ}-coefficient of Se_{λ} when expanded in terms of the standard basis for Δ_{n}(λ) is equal to the F_{μ,λ}-coefficient of S when written in terms of the PBW basis for U^{−}(n).

Proof. Take a monomial X = F_{1,2}^{(N_{1,2})}F_{1,3}^{(N_{1,3})}F_{2,3}^{(N_{2,3})} … F_{1,n}^{(N_{1,n})} … F_{n−1,n}^{(N_{n−1,n})} in the PBW basis for U^{−}(n). Let M_{i} = N_{i,i+1} + N_{i,i+2} + ⋯ + N_{i,n}. Suppose first that X cannot be written in the form F_{t} for some row standard λ-tableau t with every entry on row i at least i. This means that for some 1 ≤ i < n, M_{i} > λ_{i}. Now note that X can be rearranged to equal

X = F_{1,2}^{(N_{1,2})}F_{1,3}^{(N_{1,3})} … F_{1,n}^{(N_{1,n})} … F_{i,i+1}^{(N_{i,i+1})} … F_{i,n}^{(N_{i,n})} … F_{n−1,n}^{(N_{n−1,n})}.

All weights of Δ_{n}(λ) lie in Λ(n). So, the assumption that M_{i} > λ_{i} means that F_{i,i+1}^{(N_{i,i+1})} … F_{i,n}^{(N_{i,n})} … F_{n−1,n}^{(N_{n−1,n})}e_{λ} is zero as its weight is not in Λ(n). Hence, Xe_{λ} = 0.

So, we may assume that X = F_{t} for some row standard λ-tableau t with every entry on row i at least i. In that case, if t is standard, it contributes to the F_{μ,λ}e_{λ} coefficient of Se_{λ} precisely when t = 1_{μ,λ}. If t is not standard, then by the straightening rule, F_{t}e_{λ} expands as a sum of standard basis elements F_{s}e_{λ} for s > t and s of the same weight as t. But the tableau 1_{μ,λ} is minimal amongst row standard λ-tableaux of the same weight as 1_{μ,λ} for which every entry on row i is at least i, in the dominance order on tableaux. So s ≠ 1_{μ,λ} for each s. Hence, no term F_{s}e_{λ} in the expansion contributes to the F_{μ,λ}e_{λ}-coefficient of Se_{λ}.

2.5 The costandard basis theorem. The basic reference here is [G]. Let A(n) = 𝔽[c_{i,j} | 1 ≤ i, j ≤ n], a free polynomial ring. Regarding elements of A(n) as functions GL(n) → 𝔽, A(n) is a (left) rational GL(n)-module with action

(g.f)(g′) = f(g′g)

for all g, g′ ∈ GL(n), f ∈ A(n). Hence, A(n) is a U(n)-module.

Suppose first that λ is a partition with diagram consisting of a single column of length r. Let t be a λ-tableau with entries t_{1}, …, t_{r} reading down this column. Define f_{t} ∈ A(n) to be the bideterminant

det ( \begin{matrix}c_{1,t_{1}} & ⋯ & c_{1,t_{r}} \\ ⋮ & ... & ⋮ \\ c_{r,t_{1}} & ⋯ & c_{r,t_{r}} \end{matrix} ) .

More generally, given any λ ∈ Λ^{+}(n) and a λ-tableau t with columns t^{(1)}, …, t^{(s)}, define f_{t} ∈ A(n) to be the product f_{t} := f_{t^{(1)}} … f_{t^{(s)}}. By [G], 4.4, ∇_{n}(λ) can be defined as the submodule of A(n) spanned by the f_{t} for all λ-tableaux t. [Note that our f_{t} corresponds to Green’s bideterminant (T_{l} : T_{i}) where l and i are the row sequences of the λ-tableaux 1_{λ} and t, respectively.] The following elementary facts about the f_{t} are known [G]. (One can use the map φ defined in [G], 4.4 to prove (1) and (2). For (3) see [G], 4.6b.)

(1) F_{i,j}^{(r)}.f_{t} = ∑_{s} f_{s} and E_{j,i}^{(r)}.f_{t} = ∑_{s} f_{s}, where the sums are over all λ-tableaux s obtained from t by replacing r entries equal to i by j’s in all possible ways.

(2) \binom{H_{i}}{r}.f_{t} = \binom{N_{i}}{r}f_{t} where N_{i} is the number of entries equal to i in t.

(3) f_{t} = 0 if t contains repeated entries in some column.

By [G], 4.5, we have the costandard basis theorem:

{f_{t} | t a standard λ-tableau}

is a basis for ∇_{n}(λ). By (1) and (3), f_{1_{λ}} is a U(n)-high weight vector, of weight λ by (2), so we may choose the high weight vector f_{λ} defined in (2.1) to be f_{1_{λ}}. Given μ ⟵ λ, let f_{μ,λ} denote f_{1_{μ,λ}}.

2.6 Lemma Let μ ⟵ λ.

(i) The vector f_{μ,λ} is a U(n − 1)-high weight vector in ∇_{n}(λ) of U(n − 1)-weight μ.

(ii) For i = 1, …, n − 1, let a_{i} = ∑_{s=1}^{i}(λ_{s} − μ_{s}). Then, E_{1}^{(a_{1})} … E_{n−1}^{(a_{n−1})}f_{μ,λ} = f_{λ}.

Proof. Use (1)–(3) and the definition of the tableau 1_{μ,λ}.

2.7 Remark Part (ii) of this lemma will allow us to raise U(n−1)-high weight vectors in L_{n}(λ) to f_{λ}. In the quantum case – where we do not know of an analogue of the costandard basis theorem in the literature – one would need to argue more carefully here, using induction together with the explicit construction of f_{μ,λ} given below in Theorem 5.5.

3 Branching rules

We now review some of the results on branching rules proved in [K1, K3, B1, B2], and prove some other important preliminary results. Throughout the section, λ ∈ Λ^{+}(n) denotes a fixed partition with λ = (λ_{1}, …, λ_{n}).

The following result is proved in [B1], Theorem 3.19, as a consequence of the standard basis theorem. It is also proved in Proposition A.2 in the appendix.

3.1 Proposition Let μ(1), μ(2), …, μ(N) be all partitions μ ⟵ λ ordered so that μ(i) > μ(j) in the usual dominance order implies i < j. Define Δ^{i} to be the U(n − 1)- submodule of Δ = Δ_{n}(λ) generated by {F_{μ(1),λ}e_{λ}, …, F_{μ(i),λ}e_{λ}}. Then

(0) = Δ^{0} ⊂ Δ^{1} ⊂ ⋯ ⊂ Δ^{N} = Δ_{n}(λ) (3.2)

is a U(n − 1)-stable filtration, with Δ^{i}/Δ^{i−1} ≅ Δ_{n−1}(μ(i)) for i = 1, 2, …, N. Moreover, the image of F_{μ(i),λ}e_{λ} in Δ^{i}/Δ^{i−1} is a U(n − 1)-high weight vector.

The result shows that Δ_{n}(λ) has a Weyl filtration as a U(n − 1)-module, with factors {Δ_{n−1}(μ)|μ ⟵ λ} each occurring with multiplicity one. This well known fact seems to have been noticed first by Jantzen in the proof of [J2], Satz II.6.

3.3 Corollary For μ ∈ X^{+}(n − 1),

dim Hom_{U(n−1)}(Δ_{n−1}(μ), ∇_{n}(λ)) = \{ \begin{matrix}1 & if μ ⟵ λ, \\ 0 & otherwise. \end{matrix}

If μ ⟵ λ, the U(n − 1)-high weight vector in the image of Δ_{n−1}(μ) under any such non-zero homomorphism is a scalar multiple of f_{μ,λ}.

Proof. The first statement follows from the proposition by contravariant duality and [J1], II.4.16(a). For the second, use Lemma 2.6(i).

Following [B1], we say that μ is normal for λ, written μ ⟵^{norm} λ, if μ ∈ Λ^{+}(n − 1) and

dim Hom_{U(n−1)}(Δ_{n−1}(μ), L_{n}(λ)) ≠ 0.

We say that μ is good for λ, written μ ⟵^{good} λ, if μ ∈ Λ^{+}(n − 1) and

dim Hom_{U(n−1)}(L_{n−1}(μ), L_{n}(λ)) ≠ 0.

Since L_{n−1}(λ) is a quotient of Δ_{n−1}(μ), we know that μ ⟵^{good} λ implies μ ⟵^{norm} λ. Moreover, since L_{n}(λ) ⊆ ∇_{n}(λ), we may use Corollary 3.3 to conclude that μ ⟵^{norm} λ implies μ ⟵ λ.

We will use the following criteria for normality; the first follows from Corollary 3.3, the second is proved in [B2], Theorem 5.2.

3.4 Criterion Fix μ ⟵ λ. Then, μ ⟵^{norm} λ if and only if f_{μ,λ} ∈ L_{n}(λ).

3.5 Criterion Fix μ ⟵ λ. Then, μ ⟵^{norm} λ if and only if every vector e ∈ rad Δ_{n}(λ) has zero F_{μ,λ}e_{λ}-coefficient when written in terms of the standard basis for Δ_{n}(λ).

By Criterion 3.5 and Lemma 2.4, we obtain:

3.6 Lemma If μ ⟵ λ and there exists S ∈ U^{−}(n) such that

(i) S has non-zero F_{μ,λ}-coefficient when written in terms of the PBW basis for U^{−}(n),

(ii) Sf_{λ} = 0 (in L_{n}(λ)) then μ is not normal for λ.

In this paper, we are concerned with the property that res_{U(n−1)}^{U(n)} L_{n}(λ) is semisimple. Our aim now is to show that this holds if and only if every μ ⟵^{norm} λ is in fact good. As in Proposition 3.1, let μ(1), μ(2), …, μ(N) be all partitions μ ⟵ λ ordered so that μ(i) > μ(j) in the usual dominance order implies i < j. Let

(0) = Δ^{0} ⊂ Δ^{1} ⊂ ⋯ ⊂ Δ^{N} = Δ_{n}(λ)

be the filtration of (3.2). Denote the image of Δ^{i} under the natural map Δ → L = Δ/ rad Δ by L^{i}. We get a filtration

(0) = L^{0} ⊆ L^{1} ⊆ ⋯ ⊆ L^{N} = L.

With this notation, we obtain a third criterion for normality:

3.7 Lemma Given 1 ≤ i ≤ N as above, μ(i) ⟵^{norm} λ if and only if L^{i} ≠ L^{i−1}.

Proof. First, consider the weights of Δ/Δ^{i−1}. In Δ^{i}/Δ^{i−1} ≅ Δ_{n−1}(μ(i)), the weight μ(i) occurs with multiplicity one, all other weights being lower in the dominance order. The remaining weights of Δ/Δ^{i−1} are of the form {γ|there exists j > i with γ ≤ μ(j)}. But μ(j) ≱ μ(i) if j > i, so μ(i) only occurs with multiplicity one in Δ/Δ^{i−1}, and moreover, other weights of Δ/Δ^{i−1} do not dominate μ(i). Hence the span of all weight spaces in Δ/Δ^{i−1} different from μ(i) is stable under the action of U^{−}(n − 1)U^{0}(n − 1). So there exists a non-zero U^{−}(n − 1)U^{0}(n − 1)-homomorphism Δ/Δ^{i−1} → 𝔽_{μ(i)}. By Frobenius reciprocity for algebraic groups [J1], I.3.4, we obtain a non-zero U(n − 1)- homomorphism

θ : Δ/Δ^{i−1} → ∇_{n−1}(μ(i)).

We claim that θ is non-zero on Δ^{i}/Δ^{i−1}. Indeed, soc ∇_{n−1}(μ(i)) = L_{n−1}(μ(i)), hence im θ ⊇ L_{n−1}(μ(i)). So the μ(i)-weight space of im θ is non-zero. Now we can use the fact that the μ(i)-weight space of Δ/Δ^{i−1} is contained in Δ^{i}/Δ^{i−1}.

Extend θ to the map θ̂: Δ → ∇_{n−1}(μ(i)), zero on Δ^{i−1}, and non-zero on Δ^{i}.

Suppose that μ(i) ⟵^{norm} λ. By contravariant duality, this is equivalent to the fact that there exists a non-zero U(n − 1)-homomorphism φ : L → ∇_{n−1}(μ(i)). This homomorphism can be extended to a (non-zero) homomorphism φ̂ : Δ → ∇_{n−1}(μ(i)). Since dim Hom_{U(n−1)}(Δ, ∇_{n−1}(μ(i))) = 1 (which follows from Corollary 3.3), we conclude that φ̂ is proportional to θ̂. This implies rad Δ ⊆ ker θ̂, i.e. θ̂ factors through L = Δ/ rad Δ. Now, θ̂ is zero on Δ^{i−1} and non-zero on Δ^{i}, so it factors to give a map which is zero on L^{i−1} and non-zero on L^{i}. Hence L^{i} ≠ L^{i−1}.

Conversely, suppose L^{i} ≠ L^{i−1}. Then F_{μ(i),λ}e_{λ} + L^{i−1} is a non-zero high weight vector of L/L^{i−1}. So there is a non-zero U^{−}(n − 1)U^{0}(n − 1)-homomorphism L → 𝔽_{μ(i)}. By Frobenius reciprocity again, this gives a non-zero U(n − 1)-homomorphism L → ∇_{n−1}(μ(i)), and consequently by contravariant duality we obtain a non-zero homomorphism Δ_{n−1}(μ(i)) → L.

3.8 Corollary L_{n}(λ) is generated as a U^{−}(n − 1)-module by {F_{μ,λ}f_{λ} | μ ⟵^{norm} λ}.

Proof. Note that L^{i}/L^{i−1} is a homomorphic image of Δ^{i}/Δ^{i−1}, and hence is a module generated by a high weight vector F_{μ(i),λ}f_{λ} + L_{i−1}. Now apply Lemma 3.7.

Now we are ready to prove the main result of this section.

3.9 Proposition Let λ ∈ Λ^{+}(n). The following statements are equivalent:

(i) res_{U(n−1)}^{U(n)} L_{n}(λ) is semisimple;

(ii) every μ ⟵^{norm} λ is good;

(iii) for each μ ⟵^{norm} λ, f_{μ,λ} is a scalar multiple of

F_{μ,λ}f_{λ} + ∑_{{ν>μ, \atop ν ⟵^{norm}λ}} S_{ν}F_{ν,λ}f_{λ}

for certain coefficients S_{ν} ∈ U^{−}(n − 1);

(iv) for each μ ⟵^{norm} λ and every S ∈ U^{−}(n) such that f_{μ,λ} is a scalar multiple of Sf_{λ}, the element S has non-zero F_{μ,λ}-coefficient when written in terms of the PBW basis.

Proof. (i)⇒(ii). If μ ⟵^{norm} λ and the restriction is semisimple, then f_{μ,λ} must generate an irreducible U(n − 1)-module, so that μ is good as required.

Assume (ii) holds. For every 1 ≤ i ≤ N, put

M_{i} := {μ(j) | 1 ≤ j ≤ i, μ(j) ⟵^{norm} λ}.

Also put M_{0} = ∅.

We prove by induction on i = 0, 1, …, N that L^{i} is semisimple with composition factors {L_{n−1}(μ) | μ ∈ M_{i}}. Since L^{0} = (0), the induction base is clear. Let i > 0. If μ(i) is not normal then L^{i} = L^{i−1} by virtue of Lemma 3.7, and we are done. Assume that μ(i) is normal, hence good by assumption. Then f_{μ(i),λ} ∈ L by Criterion 3.4, and f_{μ(i),λ} generates a copy of L_{n−1}(μ(i)). By Corollary 3.8, we may write

f_{μ(i),λ} = ∑_{ν ⟵^{norm}λ} S_{ν}F_{ν,λ}f_{λ}

for certain S_{ν} ∈ U^{−}(n − 1). By considering weights, we can assume that S_{ν} = 0 unless μ(i) ≤ ν and that S_{μ(i)} ∈ 𝔽. So,

f_{μ(i),λ} − a_{i}F_{μ(i),λ}f_{λ} = ∑_{{ν>μ(i), \atop ν ⟵^{norm}λ}} S_{ν}F_{ν,λ}f_{λ} (3.10)

for a scalar a_{i} = S_{μ(i)} ∈ 𝔽. The right hand side of this expression lies in L^{i−1} which, by the inductive hypothesis, is semisimple with composition factors L_{n−1}(ν) for ν ∈ M_{i−1}. Since f_{μ(i),λ} generates a copy of L_{n−1}(μ(i)), we have f_{μ(i),λ} ∉ L^{i−1}. Hence, a_{i} ≠ 0. Moreover, by definition, L^{i} is generated by L^{i−1} and F_{μ(i),λ}f_{λ}, hence L^{i} is also generated by L^{i−1} and f_{μ(i),λ}, which implies that L^{i} is semisimple with composition factors L_{n−1}(μ) where μ ∈ M_{i−1} ∪ {μ(i)} = M_{i}.

Now (ii) ⇒ (i) follows immediately by taking i = N. Also (ii) ⇒ (iii) follows from (3.10) since we proved that a_{i} ≠ 0.

(iv)⇒(iii). This is obvious using Corollary 3.8, as in the proof of (3.10).

(iv)⇒(ii). Suppose (iv) holds but there is some μ ⟵^{norm} λ that is not good. Then, for some ν < μ with ν ⟵^{norm} λ, f_{ν,λ} = T f_{μ,λ} for some T ∈ U^{−}(n − 1). By (iii), we may write

f_{ν,λ} = ( ∑_{{γ≥μ, \atop γ ⟵^{norm}λ}} T S_{γ}F_{γ,λ}) f_{λ}.

Each T S_{γ} lies in U^{−}(n − 1), and F_{γ,λ} ≠ F_{ν,λ} as γ ≥ μ > ν. So, by the choice of ordering of the PBW basis, the element in the brackets on the right hand side of this expression has zero F_{ν,λ}-coefficient when written in terms of the PBW basis. This contradicts (iv).

(iii)⇒(iv). Suppose that (iii) holds but that we can find μ ⟵^{norm} λ and S ∈ U^{−}(n) with zero F_{μ,λ}-coefficient such that f_{μ,λ} = Sf_{λ}. By (iii), we can find an element

S′ = a_{μ}F_{μ,λ} + ∑_{{ν>μ, \atop ν ⟵^{norm}λ}} S_{ν}F_{ν,λ}

for coefficients a_{μ} ∈ 𝔽^{×}, S_{ν} ∈ U^{−}(n − 1) such that S′f_{λ} = f_{μ,λ} = Sf_{λ}. Hence, (S′ − S)f_{λ} = 0. But the F_{μ,λ}-coefficient of (S′ − S) is a_{μ} which is non-zero, so this contradicts normality of μ by Lemma 3.6.

4 Lowering operators

We next review the definition of Kleshchev’s lowering operators, following the reformulation described in [B1]. We then prove two technical properties of the lowering operators that are central to the proof of the Main Theorem. Throughout the section, λ = (λ_{1}, …, λ_{n}) denotes a fixed element of Λ^{+}(n).

We use the non-standard notation (i..j) to denote the open interval {i+1, …, j−1} of ℕ. Given 1 ≤ i < j ≤ n and B ⊆ (i..j), let F_{i,j}^{B} denote F_{i,b_{1}}F_{b_{1},b_{2}} … F_{b_{r},j} if B = {b_{1} < ⋯ < b_{r}} (we interpret F_{i,j}^{∅} as F_{i,j}). Let μ be an element of either Λ^{+}(n − 1) or Λ^{+}(n), and take A ⊆ (i..j). The lowering operator S_{i,j}^{μ}(A) can be defined as

S_{i,j}^{μ}(A) := ∑_{B⊆(i..j)} H_{i,j}^{μ}(A, B)F_{i,j}^{B}

for certain scalars H_{i,j}^{μ}(A, B). These scalars are defined by evaluating a certain polynomial H_{i,j}(A, B) ∈ ℤ[x_{i}, …, x_{j−1}] (defined in the next paragraph) at x_{k} := res_{p}(k, μ_{k}) for all k. In particular, note that (x_{k} − x_{i}) evaluates to C^{μ}(i, k) (see (2.1)).

The polynomial H_{i,j}(A, B), for arbitrary subsets A, B of (i..j), is defined to be the following rational expression in ℚ(x_{i}, …, x_{j−1}):

H_{i,j}(A, B) := ∑_{D⊆B\A}(−1)^{|D|} \frac{∏_{t∈A}(x_{t} − x_{D_{i}(t)})}{∏_{t∈B}(x_{t} − x_{D_{i}(t)})}.

Here, given any set D ⊆ (i..j) and any t ∈ (i..j), D_{i}(t) denotes the greatest element of D that is strictly smaller than t, or i if no such element exists. It is not immediate that this rational expression is a polynomial, but this is proved in [B1], Lemma 4.6. The first lemma follows easily from these definitions.

4.1 Lemma If B ⊆ A ⊆ (i..j), then H_{i,j}^{μ}(A, B) = ∏_{a∈A\B} C^{μ}(i, a).

We now prove two technical facts about the operators S_{i,j}^{μ}(A). The first of these follows easily from the commutator relations proved in [B1] or [K3]. This will be used to construct U(n − 1)-high weight vectors.

4.2 Proposition (First technical fact) Let λ ∈ Λ^{+}(n) be p-restricted and μ ⟵^{norm} λ, so that f_{μ,λ} ∈ L_{n}(λ) (see Criterion 3.4). Suppose A ⊆ (i..n) satisfies

S_{j,n}^{μ}(A ∩ (j..n))f_{μ,λ} = 0

for all i < j < n with j ∉ A. Then S_{i,n}^{μ}(A)f_{μ,λ} is a U(n − 1)-high weight vector (possibly zero) in L_{n}(λ).

Proof. As λ is p-restricted, it suffices by [K2], Theorem B to show that

E_{j−1}S_{i,n}^{μ}(A)f_{μ,λ} = 0

for all 1 < j < n. Now note that f_{μ,λ} is a weight vector of weight μ. So S_{i,n}^{μ}(A)f_{μ,λ} = S_{i,n}(A)f_{μ,λ}, where S_{i,n}(A) is (the classical analogue of) the operator S_{i,n}(A) defined in [B1] or [B2]. Also, E_{j−1}f_{μ,λ} = 0 for 1 < j < n, so by [B1], Lemma 4.11(i), E_{j−1}S_{i,n}^{μ}(A)f_{μ,λ} = 0 unless j − 1 ∈ {i} ∪ A and j ∉ A. In this remaining case, [B1], Lemma 4.11(ii) shows that

E_{j−1}S_{i,n}^{μ}(A)f_{μ,λ} = −S_{i,j−1}^{μ}(A ∩ (i..j − 1))S_{j,n}^{μ}(A ∩ (j..n))f_{μ,λ}

which is zero by assumption.

We still need to be able to show that the high weight vectors in Proposition 4.2 are non-zero under suitable circumstances. For this, we need a second technical fact, which is rather harder to prove. First, a preliminary lemma, which is easily verified working in U(n, ℤ):

4.3 Lemma Fix 1 ≤ i < j ≤ n and a ≥ 0. The following commutator relation holds in U(n):

[E_{j−1}^{(a+1)}, F_{i,j}] = \{ \begin{matrix}E_{j−1}^{(a)}F_{i,j−1} & if i < j − 1, \\ E_{j−1}^{(a)} (H_{j−1} − H_{j} + a) & if i = j − 1. \end{matrix}

Next, for 1 ≤ i < j ≤ n and A ⊆ (i..j), we define the polynomial K_{i,j}(A) ∈ ℤ[x_{i}, …, x_{j−1}; y_{i+1}, …, y_{j}] as in [B1]. By definition,

K_{i,j}(A) := ∑_{B⊆(i..j)} (H_{i,j}(A, B) ∏_{t∈B∪{i}}(y_{t+1} − x_{t})) .

The key property of K_{i,j}(A) is as follows:

4.4 Lemma Let B, C ⊆ (i..j) be any sets such that there is a bijection θ : C → B with θ(c) ≥ c for all c ∈ C. Let A = (i..j) \ C. Then,

K_{i,j}(A) ≡ ∏_{t∈{i,...,j−1}\B}(y_{t+1} − x_{i}) (modulo J),

where J is the ideal of ℤ[x_{i}, …, x_{j−1}; y_{i+1}, …, y_{j}] generated by {y_{θ(c)+1} − x_{c} | c ∈ C}.

Proof. This is immediate from [B1], Lemma 4.13(ii).

Define K_{i,j}^{μ,λ}(A) by evaluating the indeterminates in the polynomial K_{i,j}(A) at

x_{k} := res_{p}(k, μ_{k}); y_{k+1} := res_{p}(k + 1, λ_{k+1} + 1).

Note (x_{k} − x_{i}) evaluates to C^{μ}(i, k) and (y_{k+1} − x_{i}) evaluates to B^{μ,λ}(i, k). So,

K_{i,j}^{μ,λ}(A) = ∑_{B⊆(i..j)} (H_{i,j}^{μ}(A, B) ∏_{t∈B∪{i}} B^{μ,λ}(t, t)) .

4.5 Proposition (Second technical fact) Let μ ⟵^{norm} λ, so that f_{μ,λ} ∈ L_{n}(λ). Let 1 ≤ i < j ≤ n and C ⊆ ℭ^{μ}(i, j). Set B = 𝔅^{μ,λ}(i, j). Suppose that there exists a bijection θ : C → B such that θ(c) ≥ c for all c ∈ C. Let A = (i..j) \ C. Then, S_{i,j}^{μ}(A)f_{μ,λ} ≠ 0.

Proof. Suppose μ is in level l (see (2.2)), and let μ̄ = (μ_{1}, …, μ_{n−1}, λ_{n} + l) be the U(n)-weight of f_{μ,λ}. We may write μ̄ = λ − ∑_{s=1}^{n−1} a_{s}(ε_{i} − ε_{i+1}) for unique nonnegative integers a_{s}. For notational convenience, we let X_{1} = E_{1}^{(a_{1})} … E_{i−1}^{(a_{i−1})} and X_{2} = E_{j}^{(a_{j})} … E_{n−1}^{(a_{n−1})}.

Step one. We first show that for any h < j,

E_{j−1}^{(a_{j−1}+1)}X_{2}F_{h,j}f_{μ,λ} = \{ \begin{matrix}E_{j−1}^{(a_{j−1})}X_{2}F_{h,j−1}f_{μ,λ} & if h < j − 1, \\ B^{μ,λ}(j − 1, j − 1)E_{j−1}^{(a_{j−1})}X_{2}f_{μ,λ} & if h = j − 1. \end{matrix}

For this, we observe that X_{2} commutes with F_{h,j−1} and F_{h,j}, and that (by considering weights), E_{j−1}^{(a_{j−1}+1)}X_{2}f_{μ,λ} = 0. The first case follows immediately from these two observations and the first case of Lemma 4.3. For the second case, where h = j − 1, Lemma 4.3 implies that E_{j−1}^{(a_{j−1}+1)}X_{2}F_{h,j}f_{μ,λ} = E_{j−1}^{(a_{j−1})} (H_{j−1} − H_{j} + a_{j−1}) X_{2}f_{μ,λ}. So, we need to evaluate (H_{j−1} − H_{j} + a_{j−1}) on the weight λ − ∑_{t=1}^{j−1} a_{t}(ε_{t} − ε_{t+1}); H_{j} evaluates to λ_{j} +a_{j−1} on this. On the other hand, this weight also equals μ̄+∑_{t=j}^{n−1} a_{t}(ε_{t} − ε_{t+1}) and H_{j−1} evaluates to μ_{j−1} on this. Consequently, (H_{j−1} − H_{j} + a_{j−1}) evaluates to μ_{j−1} − λ_{j} = B^{μ,λ}(j − 1, j − 1) on X_{2}f_{μ,λ}, and step one follows.

Step two. We next show that

X_{1}E_{i}^{(a_{i}+1)} … E_{j−1}^{(a_{j−1}+1)}X_{2}S_{i,j}^{μ}(A)f_{μ,λ} = K_{i,j}^{μ,λ}(A)f_{λ}.

Note that E_{1}^{(a_{1})} … E_{n−1}^{(a_{n−1})}f_{μ,λ} = f_{λ} by Lemma 2.6(ii). So, it suffices to show by definition of S_{i,j}^{μ}(A) and K_{i,j}^{μ,λ}(A) that

X_{1}E_{i}^{(a_{i}+1)} … E_{j−1}^{(a_{j−1}+1)}X_{2}F_{i,j}^{D}f_{μ,λ} = ( ∏_{t∈D∪{i}} B^{μ,λ}(t, t)) E_{1}^{(a_{1})} … E_{n−1}^{(a_{n−1})}f_{μ,λ}

for all subsets D ⊆ (i..j). This follows from step one by induction on (j − i).

Step three. We finally show that the right hand side of the expression in step two is non-zero, to prove the lemma. To see this, just note that B^{μ,λ}(c, θ(c)) = B^{μ,λ}(i, θ(c)) − C^{μ}(i, c) = 0 for all c ∈ C, so by Lemma 4.4,

K_{i,j}^{μ,λ}(A) = ∏_{t∈{i,...,j−1}\B} B^{μ,λ}(i, t)

which is non-zero by definition of B.

5 Proof of the Main Theorem

We are now in a position to prove the Main Theorem.

We begin by reducing the proof to the case that λ is p-restricted. So suppose that p > 0 and take an arbitrary λ ∈ Λ^{+}(n) and let λ = λ(0) + pλ(1) + ⋯ + p^{d}λ(d) be a p-adic expansion of λ, where each λ(i) is p-restricted. Let F : GL(n) → GL(n) denote the Frobenius pth power map, and given a GL(n)-module V, let V^{[j]} denote the jth Frobenius twist of V, with new action g.v = F^{j}(g)v for v ∈ V, g ∈ GL(n). Then, Steinberg’s tensor product theorem [St], Theorem 41 tells us that L_{n}(λ) ≅ L_{n}(λ(0))⊗L_{n}(λ(1))^{[1]} ⊗⋯⊗L_{n}(λ(d))^{[d]}. Let soc(λ) denote the socle of the restriction res_{GL(n−1)}^{GL(n)} L_{n}(λ). The following result is proved in [K2]:

5.1 Theorem Let λ = λ(0)+pλ(1)+⋯+p^{d}λ(d) be a p-adic expansion of λ ∈ Λ^{+}(n). For all i, each composition factor of soc(λ(i)) is p-restricted, and

soc(λ) ≅ soc(λ(0)) ⊗ soc(λ(1))^{[1]} ⊗ ⋯ ⊗ soc(λ(d))^{[d]}.

Now, observe that res_{GL(n−1)}^{GL(n)} L_{n}(λ) is semisimple if and only if dim soc(λ) =

dim L_{n}(λ). Consequently, by the theorem, we see that res_{GL(n−1)}^{GL(n)} L_{n}(λ) is semisimple if and only if dim soc(λ(i)) = dim L_{n}(λ(i)) for each i, which is if and only if each res_{GL(n−1)}^{GL(n)} L_{n}(λ(i)) is semisimple. Moreover, in that case, the theorem shows that the composition factors of the restriction are the corresponding twisted tensor products of the composition factors of each res_{GL(n−1)}^{GL(n)} L_{n}(λ(i)).

Hence, to prove the Main Theorem, it suffices to consider the case that λ is prestricted. So, for the remainder of the section, we assume that λ = (λ_{1}, …, λ_{n}) is a p-restricted dominant weight. Our first aim is to show that if λ is GJS, then res_{U(n−1)}^{U(n)} L_{n}(λ) is semisimple. We need a preliminary lemma:

5.2 Lemma Fix i with 1 ≤ i < n together with integers x_{i} > 0 and x_{k} ≥ 0 for all k ∈ (i..n). Let A be any subset of (i..n) such that k ∈ (i..n) and x_{k} ≠ 0 imply k ∈ A. Then, the F_{i,n}^{(x_{i})}F_{i+1,n}^{(x_{i+1})} … F_{n−1,n}^{(x_{n−1})}-coefficient of F_{i,n}^{(x_{i}−1)}F_{i+1,n}^{(x_{i+1})} … F_{n−1,n}^{(x_{n−1})}S_{i,n}^{λ}(A) when

written in terms of the PBW basis is x_{i} ∏_{a∈A} (C^{λ}(i, a) + x_{a}) .

Proof. A routine induction on (n − i) shows that for any subset B of (i..n), the F_{i,n}^{(x_{i})} … F_{n−1,n}^{(x_{n−1})}-coefficient of F_{i,n}^{(x_{i}−1)}F_{i+1,n}^{(x_{i+1})} … F_{n−1,n}^{(x_{n−1})}F_{i,n}^{B} is x_{i} ∏_{b∈B} x_{b}. Consequently, by definition of S_{i,n}^{λ}(A), the corresponding coefficient in F_{i,n}^{(x_{i}−1)}F_{i+1,n}^{(x_{i+1})} … F_{n−1,n}^{(x_{n−1})}S_{i,n}^{λ}(A) is

x_{i} ∑_{B⊆(i..n)} H_{i,n}^{λ}(A, B) ∏_{b∈B} x_{b}.

Now note that if B ⊆ A, then H_{i,n}^{λ}(A, B) is given by Lemma 4.1. If B ⊈ A, then x_{b} = 0 for some b ∈ B by assumption. Consequently, the required coefficient equals

x_{i} ∑_{B⊆A} ∏_{a∈A\B} C^{λ}(i, a) ∏_{b∈B} x_{b}

which factorizes to give the conclusion.

5.3 Proposition Suppose that λ ∈ Λ^{+}(n) is a p-restricted GJS weight and μ ⟵^{norm} λ. Then, μ is allowable.

Proof. Suppose that μ is not allowable, so that there are removable rows 1 ≤ i < j < n such that μ_{i} < λ_{i}, μ_{j} > λ_{j+1}, with B^{λ}(i, j) = 0. Choose i, j so that the B-chain i < j cannot be refined to a longer proper B-chain. The strategy is as follows. Let y_{k} = λ_{k} − μ_{k} for all 1 ≤ k < n and A = {i < a < j + 1 | C^{λ}(i, a) ≠ 0}. We will consider the element

S := \underbrace{F_{1,n}^{(y_{1})} … F_{i−1,n}^{(y_{i−1})}}_{S_{1}} \underbrace{F_{i,n}^{(y_{i}−1)}F_{i+1,n}^{(y_{i+1})} … F_{j,n}^{(y_{j})}}_{S_{2}} \underbrace{F_{j+1,n}^{(y_{j+1}+1)}F_{j+2,n}^{(y_{j+2})} … F_{n−1,n}^{(y_{n−1})}}_{S_{3}} S_{i,j+1}^{λ}(A).

of U^{−}(n). We wish to show that Sf_{λ} = 0 but that S has non-zero F_{μ,λ}-coefficient when expanded in terms of the PBW basis for U^{−}(n). This will show that μ is not normal by Lemma 3.6. For notational convenience we split S as a product S_{1}S_{2}S_{3}S_{i,j+1}^{λ}(A), where the S_{i} are the terms indicated above. If j + 1 = n, then we take S_{3} = 1.

We first show that 𝔅^{λ}(i, j + 1) is the disjoint union of ℭ^{λ}(i, j + 1) and {j}, and that every removable row k with i < k < j is an element of A. First, note that B^{λ}(i, j) = 0 ≠ C^{λ}(i, j) as λ is p-restricted. Also, if k is not a removable row, then C^{λ}(i, k) = B^{λ}(i, k). Thus, it suffices to show given any removable row k with i < k < j, that k is not an element of ℭ^{λ}(i, j + 1) or 𝔅^{λ}(i, j + 1).

If C^{λ}(i, k) = 0 then B^{λ}(k, j) = B^{λ}(i, j) − C^{λ}(i, k) = 0. So the B-chain i < j can be refined by Lemma 1.1, contradicting maximality. If B^{λ}(i, k) = 0, the B-chain i < j can be refined, again by Lemma 1.1, to give a contradiction.

We have shown in particular that there is no injection from 𝔅^{λ}(i, j +1) to ℭ^{λ}(i, j + 1). So, S_{i,j+1}^{λ}(A)f_{λ} = 0 by [B1], Lemma 5.2. Hence, Sf_{λ} = 0.

Now we consider the F_{μ,λ}-coefficient of S when expanded in terms of the PBW basis. Suppose first that j +1 ≠ n. By definition, S_{i,j+1}^{λ}(A) is a sum of monomials of the form

F_{i,j+1}^{B}H_{i,j+1}^{λ}(A, B). Now, S_{3}F_{i,j+1}^{B} = F_{i,n}^{B}F_{j+1,n}^{(y_{j+1})} … F_{n−1,n}^{(y_{n−1})} together with the term F_{i,j+1}^{B}S_{3} which will not contribute to the F_{μ,λ}-coefficient of S. Moreover, H_{i,n}^{λ}(A, B) equals 0 if B ⊈ (i..j + 1) (by [B1], Lemma 4.6(ii)) and H_{i,j+1}^{λ}(A, B) otherwise. Hence, the F_{μ,λ}-coefficient of S equals the F_{μ,λ}-coefficient of

S_{1}S_{2}S_{i,n}^{λ}(A)F_{j+1,n}^{(y_{j+1})} … F_{n−1,n}^{(y_{n−1})}

if j + 1 ≠ n. If j + 1 = n, then by definition the F_{μ,λ}-coefficient of S is the F_{μ,λ}- coefficient of S_{1}S_{2}S_{i,n}^{λ}(A).

In both cases, this coefficient is equal to the F_{i,n}^{(y_{i})} … F_{j,n}^{(y_{j})}-coefficient of S_{2}S_{i,n}^{λ}(A), which we computed in Lemma 5.2. It equals

y_{i} ∏_{a∈A} (C^{λ}(i, a) + y_{a}) .

It just remains to show that this expression is non-zero. First note that 0 < y_{i} ≤ λ_{i} − λ_{i+1} by assumption. So, y_{i} ≢ 0 (p), as λ is p-restricted. Also, C^{λ}(i, j) + y_{j} = B^{λ}(i, j) + y_{j} − (λ_{j} − λ_{j+1}) ≠ 0 as B^{λ}(i, j) = 0 and y_{j} ≢ λ_{j} − λ_{j+1} (p) by assumption. So, if the expression vanishes, then some C^{λ}(i, a) + y_{a} = 0 for some i < a < j. Now, if y_{a} = 0, this is not the case by definition of A.

So, we may assume that C^{λ}(i, k) + y_{k} = 0 for some i < k < j with y_{k} > 0, so that in particular k is a removable row. We now show that there is a B-chain from i to k, so that Lemma 1.1 gives a contradiction to maximality of the B-chain i < j. If y_{k} < λ_{k} − λ_{k+1}, C^{λ}(i, k) + y_{k} = 0 implies that R_{i}(λ) meets R_{k}(λ), giving a B-chain from i to k. Finally, if y_{k} = λ_{k} − λ_{k+1}, then C^{λ}(i, k) + y_{k} = B^{λ}(i, k) = 0, so again there is a B-chain from i to k.

5.4 Corollary If λ ∈ Λ^{+}(n) is a p-restricted GJS weight, then res_{U(n−1)}^{U(n)} L_{n}(λ) is semisimple.

Proof. We are going to use a consequence of the linkage principle, which says that L_{n−1}(ν) is a composition factor of Δ_{n−1}(μ) only if ν and μ have the same residue content (that is, the p-residues in the diagrams [ν], [μ] are in 1–1 correspondence). Indeed, it is proved in [J1], II.6.13 that L_{n−1}(ν) is a composition factor of Δ_{n−1}(μ) only if ν and μ are conjugate under the action of the affine Weyl group W_{a} corresponding to U(n − 1). However it is well known (see e.g. [CL], section 4) that μ and ν are W_{a}-conjugate if and only if they have the same residue content.

Now, by Proposition 3.9, to prove the semisimplicity of the restriction, it suffices to show that every f_{μ,λ} for μ ⟵^{norm} λ generates an irreducible U(n−1)-module. Suppose not. Then we can find weights μ and ν, both normal for λ, such that L_{n−1}(ν) is a composition factor of U(n − 1)f_{μ,λ}. That is, by Proposition 5.3 and the universality of standard modules, there are allowable weights μ and ν such that L_{n−1}(ν) is a composition factor of Δ_{n−1}(μ).

¿From the linkage principle, ν and μ have the same residue content. By Lemma 1.2, λ either satisfies (JS1), when clearly no two distinct ν, μ ⟵ λ have the same residue content, or (JS2). In the latter case, no two allowable weights even lie in the same level, so again cannot have the same residue content.

We now want to prove the converse to Proposition 5.3 – that is, if λ is GJS and μ is allowable, then μ is normal.

5.5 Theorem If λ ∈ Λ^{+}(n) is a p-restricted GJS weight, then

res_{U(n−1)}^{U(n)} L_{n}(λ) ≅ ⨁ ; μ allowable L_{n−1}(μ).

Proof. By Corollary 5.4 and Proposition 5.3, it suffices to show that if ν is allowable then ν ⟵^{norm} λ. We prove this by induction on level, the result being clear for level 0 when ν = (λ_{1}, …, λ_{n−1}) [which is always normal as f_{ν,λ} = f_{λ}]. So let ν be allowable in level l and suppose that the result has been proved for all smaller levels. Choose i minimal such that ν = (μ_{1}, …, μ_{i−1}, μ_{i} − 1, μ_{i+1}, …, μ_{n−1}) for some μ ⟵ λ. By definition of allowable, μ is also allowable, so by induction, μ ⟵^{norm} λ.

Now, let B = 𝔅^{μ,λ}(i, n), C = ℭ^{μ}(i, n). We claim that B = C. For this, note first that i ∉ B as λ is p-restricted. Also, for k > i, if k is not a removable row for λ, then B^{μ,λ}(i, k) = C^{μ}(i, k). So, it suffices to consider a removable row (for λ) k > i. Suppose first that k > i is an element of C. Then, as ν is allowable and λ is GJS, C^{μ}(i, k) = 0 implies that μ_{k} = λ_{k+1}, for otherwise R_{i}(λ) meets R_{k}(λ) and hence μ_{k} = λ_{k+1}, giving a contradiction. So B^{μ,λ}(i, k) = k − i + μ_{i} − λ_{k+1} = k − i + μ_{i} − μ_{k} = C^{μ}(i, k) = 0, and k ∈ B. Conversely, suppose that k > i is an element of B. We show that there is a B-chain from i to k. This is immediate if λ_{i} = μ_{i}, since then 0 = B^{μ,λ}(i, k) = B^{λ}(i, k). And if λ_{i} ≠ μ_{i}, then B^{μ,λ}(i, k) = 0 implies that R_{i}(λ) meets R_{k}(λ), so again there is a B-chain from i to k by definition of GJS. Consequently, as ν is allowable, μ_{k} = λ_{k+1}, and k ∈ C as required.

So, B = C as claimed. Let A = (i..n) \ C. Proposition 4.5 (taking θ : C → B to be the identity) implies that S_{i,n}^{μ}(A)f_{μ,λ} is a non-zero element of L_{n}(λ). We now show that it is a U(n − 1)-high weight vector, so that ν ⟵^{norm} λ as required. To prove this, we use Proposition 4.2 to see that it suffices to show that

S_{j,n}^{μ}(A ∩ (j..n))f_{μ,λ} = 0

for all i < j < n with j ∉ A.

Suppose for a contradiction that there is some j ∈ (i..n) \ A with S_{j,n}^{μ}(A′)f_{μ,λ} ≠ 0 where A′ = A ∩ (j..n). Choose the greatest such j, so that Proposition 4.2 applied to A′ shows that S_{j,n}^{μ}(A′)f_{μ,λ} is a non-zero U(n − 1)-high weight vector of weight ν′ = (μ_{1}, …, μ_{j} − 1, …, μ_{n−1}). Then μ_{j} > λ_{j+1}. On the other hand, j ∉ A implies C^{μ}(i, j) = 0. So R_{i}(λ) meets R_{j}(λ). Hence, there is a B-chain from i to j which forces μ_{j} = λ_{j+1} since ν is allowable. The contradiction obtained completes the proof.

Finally, we need to prove the converse to the Main Theorem.

5.6 Theorem Let λ ∈ Λ^{+}(n) be a p-restricted weight that is not GJS. Then, the restriction of L_{n}(λ) to U(n − 1) is not semisimple.

Proof. The idea here is as follows: using the argument in the proof of Theorem 5.5 we will construct an operator T ∈ U^{−}(n) such that T f_{λ} is a non-zero U(n − 1)-high weight vector of weight μ, where the F_{μ,λ}-coefficient of T when written in terms of the PBW basis is zero. This will show that L_{n}(λ) does not have semisimple restriction to U(n − 1) because of Proposition 3.9(iv). The argument is quite long, so we break it into a number of steps.

Step one. We can find rows i < j such that R_{i}(λ) meets R_{j}(λ) but there is no B-chain from i to j. That is, we can find nodes (i, a) and (j, b) in [λ] such that

(1) 1 ≤ i < j < n, λ_{i+1} < a ≤ λ_{i} and λ_{j+1} < b ≤ λ_{j};

(2) res_{p}(i, a) = res_{p}(j, b);

(3) there is no B-chain from i to j. We wish to choose (i, a) to be maximal with respect to the lexicographic ordering on ℕ × ℕ such that (1)–(3) hold for some (j, b). That is, for every node (i′, a′) with either i′ > i or i′ = i, a′ > a, there is no node (j′, b′) such that (1)–(3) hold for (i′, a′) and (j′, b′).

We now define μ ⟵ λ as follows. First, let μ_{1} = λ_{1}, …, μ_{i−1} = λ_{i−1} and μ_{i} = a − 1, μ_{j} = b. The remaining μ_{k} for i < k < n, k ≠ j are defined by

μ_{k} = \{ \begin{matrix}λ_{k+1} & if there is a B-chain from i to k, \\ λ_{k+1} & if k > j and there is a B-chain from j to k, \\ λ_{k} & otherwise. \end{matrix}

Suppose μ is in level l. Let μ(0) = (λ_{1}, …, λ_{n−1}), and define μ(s) for l ≥ s > 0 inductively by choosing k to be maximal such that the kth row of μ is different from the kth row of μ(s − 1) and letting μ(s) equal μ(s − 1) with one extra node removed from this kth row.

Step two. We wish to prove by induction on s that μ(s) ⟵^{norm} λ. This is clear for s = 0, so consider μ(s + 1) assuming by induction that μ(s) ⟵^{norm} λ. The argument proceeds as in the proof of Theorem 5.5. For convenience, let ν = μ(s + 1), γ = μ(s) and suppose ν is obtained from γ by removing one node from the kth row. Note that by the construction in step one, if k < j then either k = i or there is a B-chain from i to k, and if k > j then there is a B-chain from either i or j to k.

Let B = 𝔅^{γ,λ}(k, n). We claim that B ⊆ ℭ^{γ}(k, n). Take any h ∈ B. Then h > k since λ is p-restricted. If h is not a removable row for λ, then B^{γ,λ}(k, h) = C^{γ}(k, h) and h ∈ ℭ^{γ}(k, n). So we may assume that h is a removable row for λ. We show first that there is a B-chain from k to h; this is immediate if λ_{k} = γ_{k}, since then 0 = B^{γ,λ}(k, h) = B^{λ}(k, h). If λ_{k} ≠ γ_{k}, B^{γ,λ}(k, h) = 0 implies that (k, γ_{k} + 1) has the same p-residue as (h, λ_{h+1} + 1). In other words, R_{k}(λ) meets R_{h}(λ), so the required B-chain from k to h exists by the maximality assumption on (i, a). In particular, this shows that h ≠ j – for if h = j, then k < j so there is a B-chain from i to k (or i = k) and from k to j, but by assumption there is no B-chain from i to j. Consequently, there is a B-chain from either i or j to k (or k = i or j) and from k to h, so by the construction in step one, γ_{h} = λ_{h+1}. So, B^{γ,λ}(k, h) = h−k +γ_{k} −λ_{h+1} = h−k +γ_{k} −γ_{h} = C^{γ}(k, h), and h ∈ ℭ^{γ}(k, n). This proves the claim.

Set A = (k..n) \ B. Then, as B ⊆ ℭ^{γ}(k, n), Proposition 4.5 (taking θ to be the identity and C := B) implies that S_{k,n}^{γ}(A)f_{γ,λ} is a non-zero element of L_{n}(λ). We now show that it is a U(n − 1)-high weight vector, so that ν ⟵^{norm} λ as required. To prove this, we use Proposition 4.2 to see that it suffices to show that S_{h,n}^{γ}(A ∩ (h..n))f_{γ,λ} = 0 for all k < h < n with h ∉ A. Well, take the largest h with k < h < n, h ∉ A such that S_{h,n}^{γ}(A′)f_{γ,λ} ≠ 0 where A′ = A ∩ (h..n). Then, Proposition 4.2 applied to A′ shows that S_{h,n}^{γ}(A′)f_{γ,λ} is a non-zero U(n − 1)-high weight vector of weight ν′ = (γ_{1}, …, γ_{h−1}, γ_{h} − 1, γ_{h+1}, …, γ_{n−1}), for some h > k. In particular, this means h ∈ B must be a removable row for λ. But we proved in the previous paragraph that for any removable row h ∈ B we have γ_{h} = λ_{h+1} so ν′ ⟵ λ is false, giving a contradiction.

Step three. Now let γ = (μ_{1}, …, μ_{i−1}, μ_{i} + 1, μ_{i+1}, …, μ_{n−1}) [that is, γ = μ(l − 1)]. By step two, f_{γ,λ} ∈ L_{n}(λ). So there is an operator S ∈ U^{−}(n) such that Sf_{λ} = f_{γ,λ}.

Moreover, we showed in step two that if B = 𝔅^{γ,λ}(i, n), A = (i..n) \ B, then j ∈ A and S_{i,n}^{γ}(A)Sf_{λ} is a non-zero high weight vector, so a non-zero multiple of f_{μ,λ}.

We now claim that the F_{μ,λ}-coefficient of T := S_{i,n}^{γ}(A)S when written in terms of the PBW basis is zero. This suffices to complete the proof of the theorem by Proposition 3.9(iv).

For the claim, we may write S as a linear combination of monomials in the PBW basis of the form F_{i,i+1}^{(N_{i,i+1})}F_{i,i+2}^{(N_{i,i+2})}F_{i+1,i+2}^{(N_{i+1,i+2})} … F_{i,n}^{(N_{i,n})} … F_{n−1,n}^{(N_{n−1,n})} for non-negative integers N_{h,k}. By definition, S_{i,n}^{γ}(A) equals ∑_{B⊆(i..n)} H_{i,n}^{γ}(A, B)F_{i,n}^{B}. So, no F_{μ,λ}-coefficient

arises from the PBW-expansion of each monomial in the product S_{i,n}^{γ}(A)S except possibly in the case B = ∅. But by Lemma 4.1, H_{i,n}^{γ}(A, ∅) = ∏_{d∈A} C^{γ}(i, d). As j ∈ A and

C^{γ}(i, j) = j − i + γ_{i} − γ_{j} = j − i + a − b = res_{p}(j, b) − res_{p}(i, a) = 0, this product is zero, proving the claim.

Theorem 5.5 and Theorem 5.6, together with the remarks after Theorem 5.1, complete the proof of the Main Theorem.

6 Application: completely splittable representations

We define the notion of a completely splittable representation of GL(n) by analogy with the definition for symmetric groups – see [K4]. Recall that a standard Levi subgroup of GL(n) is a subgroup of the form GL(n_{1}) × GL(n_{2}) × ⋯ × GL(n_{k}), n = n_{1} + n_{2} + ⋯ + n_{k}, embedded diagonally. A subgroup of GL(n) is called a Levi subgroup if it is conjugate to a standard Levi subgroup. A rational irreducible GL(n)-module is called completely splittable if its restriction to any Levi subgroup is semisimple.

In this section, we will give a classification of all completely splittable representations. Of course, if p = 0, every rational irreducible GL(n)-module is completely splittable, so we exclude the case p = 0 for the remainder of this section. We begin with an elementary lemma.

6.1 Lemma Let L be a rational irreducible GL(n)-module. Then L is completely splittable if and only if res_{GL(m)}^{GL(n)} L is semisimple for all m < n (the subgroup GL(m) being embedded into the top left corner).

Proof. We consider a standard Levi subgroup H = GL(m) × \overbrace{GL(1) × ⋯ × GL(1)}^{(n−m) times}. Any (rational) irreducible H-module remains irreducible on restriction to GL(m) < H. So if L is completely splittable then res_{GL(m)}^{GL(n)} L is semisimple.

Conversely, assume this restriction is semisimple for all m < n. It suffices to prove that res_{H}^{GL(n)} L is semisimple for all standard Levi subgroups H = GL(n_{1}) × ⋯ × GL(n_{k}). Any subgroup GL(n_{j}) appearing in this direct product is conjugate to the subgroup GL(n_{j}) embedded into the top left corner. So by assumption the restriction to any such a subgroup is semisimple. Now the result follows from the following general fact, proved for example in [K4], 1.6: let G_{1} and G_{2} be groups, G = G_{1} × G_{2}, 𝔽 be an algebraically closed field, and M be a finite dimensional 𝔽G-module such that res_{G_{i}}^{G} M is semisimple for i = 1, 2. Then M is semisimple. (The assumption that the G_{i} are finite made in [K4], 1.6 was never used in the proof).

Now let λ = (λ_{1}, …, λ_{n}) be an arbitrary dominant weight for GL(n). Let i be the smallest removable row for λ and let j be the largest removable row for λ – so 1 ≤ i ≤ j < n. We define

ψ_{n}(λ) := j − i + λ_{i} − λ_{j+1}.

Note this is just B^{λ}(i, j), but regarded now as an integer not as an element of ℤ/pℤ. If λ = cδ = (c, c, …, c) then there are no removable rows and we put ψ_{n}(λ) = 0.

The easiest way to compute ψ_{n}(λ) in practise is to first tensor with a power of det (which does not affect ψ_{n}) to ensure that λ_{n} = 0. In that case, we regard λ as a partition, and the number ψ_{n}(λ) is the length of a particular hook in the diagram [λ]. In fact, providing λ_{n} = 0, then ψ_{n}(λ) = χ(λ′) where χ is defined in [K4] and λ′ is the transpose of the partition λ. For example, let λ = (9, 9, 5, 5, 4, 4, 2, 0, …, 0) ∈ Λ^{+}(n) with n ≥ 8. The boxes of the corresponding hook are marked with × in the following picture, and ψ_{n}(λ) = 14.

× × × × × × × × ×

×

×

×

×

×

On the other hand, if λ = (9, 9, 5, 5, 4, 4, 2) is a weight for GL(7), one should first replace λ by (7, 7, 3, 3, 2, 2, 0) and then see that ψ_{7}(λ) = 11.

The following theorem classifies all completely splittable representations:

6.2 Theorem Let λ ∈ X^{+}(n) be an arbitrary dominant weight with p-adic expansion λ = λ(0) + pλ(1) + ⋯ + p^{d}λ(d). Then L_{n}(λ) is completely splittable if and only if ψ_{n}(λ(i)) ≤ p for all i = 0, 1, …, d.

Proof. We note that Theorem 5.1 implies that a Steinberg tensor product is completely splittable if and only if all the terms involved are completely splittable. So to prove the theorem we need only consider the case that λ = (λ_{1}, …, λ_{n}) is p-restricted.

We now proceed by induction on n. For n = 1 the result is trivial. Let n > 1. By tensoring with det if necessary, we assume that λ_{n} = 0. The result is obvious if λ = 0, so suppose also that λ ≠ 0. Let i be the smallest removable row of λ, and let j be the largest removable row, so that ψ_{n}(λ) = j − i + λ_{i} − λ_{j+1}. Our assumption that λ_{n} = 0 implies that λ_{j+1} = 0.

Assume ψ_{n}(λ) ≤ p. Then λ is GJS (as (JS1) holds). So, by the Main Theorem, res_{GL(n−1)}^{GL(n)} L_{n}(λ) is semisimple with composition factors {L_{n−1}(μ) | μ allowable for λ}. If μ is allowable for λ, ψ_{n−1}(μ) ≤ ψ_{n}(λ) + 1 with equality if and only if μ_{i} ≠ λ_{i} and μ_{j} ≠ 0. If ψ_{n}(λ) = p, then there is a B-chain from i to j, so such a μ would not be allowable. Consequently, ψ_{n−1}(μ) ≤ p for all allowable μ. So, by the induction hypothesis, each L_{n−1}(μ) is completely splittable. Now it suffices to apply Lemma 6.1 to prove that L_{n}(λ) is completely splittable as required.

In the other direction, assume that ψ_{n}(λ) > p. If res_{GL(n−1)}^{GL(n)} L_{n}(λ) is not semisimple, we are done. Otherwise λ is GJS, and

res_{GL(n−1)}^{GL(n)} L_{n}(λ) ≅ ⨁ ; μ allowable L_{n−1}(μ)

(by the Main Theorem). So, by the inductive hypothesis and Lemma 6.1, it suffices to show that ψ_{n−1}(μ) > p for some μ allowable for λ. If i = j, then p < ψ_{n}(λ) = λ_{i} which contradicts the p-restrictedness of λ. Otherwise, note that μ = (λ_{1}, …, λ_{j−1}, 0, …, 0) ∈ Λ^{+}(n−1) is allowable, and ψ_{n−1}(μ) = ψ_{n}(λ)−1. If ψ_{n−1}(μ) ≤ p we must have ψ_{n−1}(μ) = p and ψ_{n}(λ) = p + 1. But then R_{i}(λ) meets R_{j}(λ), and there is no B-chain from i to j, which contradicts the fact that λ is GJS.

We call the weight λ ∈ X^{+}(n) completely splittable if it satisfies the combinatorial condition in Theorem 6.2. Of course, the theorem shows that λ is completely splittable if and only if L_{n}(λ) is completely splittable.

In [MP], Mathieu and Papadopoulo have given a character formula for modules L_{n}(λ) with λ completely splittable. As a consequence of their character formula, they were able to show that these modules are completely splittable, giving an alternative proof of the implication (⇐) in Theorem 6.2. Our methods also give the explicit character formula for completely splittable representations obtained in [MP]. In our notation, the result is as follows:

6.3 Theorem Let λ ∈ Λ^{+}(n) be completely splittable. For μ ∈ Λ(n), the dimension of the μ-weight space of L_{n}(λ) is equal to the number of standard λ-tableaux t of weight μ, satisfying the extra condition (†) that shape(t[m]) is allowable for shape(t[m + 1]) for m = n − 1, n − 2, …, 1.

Proof. Note that if t is a standard λ-tableau with shape(t[m + 1]) completely splittable and shape(t[m]) is allowable for shape(t[m + 1]) for some m, then shape(t[m]) is also completely splittable by the argument of Theorem 6.2 – so the condition (†) makes sense.

The argument now is the same as the proof of the standard basis theorem given in the appendix. Arguing as there, using the Main Theorem and Lemma 3.7, one can conclude that if μ(i) is normal for λ, then L^{i}/L^{i−1} ≅ L_{n−1}(μ(i)) (notation as in Lemma 3.7). Then, induction on n yields that the set of F_{t}f_{λ} for all standard λ-tableaux t satisfying (†) is a basis for L_{n}(λ). This stronger statement immediately implies the character formula as stated.

We conclude by describing an equivalent condition to (†) in the p-restricted case. Fix λ ∈ Λ^{+}(n) with λ both p-restricted and completely splittable. We define a p-hook to be a pair of nodes (i, j) and (i′, j′) in [λ] with i − j + p − 1 = i′ − j′. We refer to (i, j) as the top-right node in the p-hook, and (i′, j′) as the bottom-left node. Given a standard λ-tableau t, we say a p-hook is bad (in t) if

(1) the entry in the top-right node of the hook is strictly larger than the entry in the bottom-left node;

(2) no entry in the column containing the bottom-left node is equal to the entry in the top-right node. We claim that if t is a standard λ-tableau, with λ p-restricted and completely splittable, then the condition (†) of Theorem 6.3 is equivalent to the condition that there are no bad p-hooks in t. To prove this, first observe, by Theorem 6.2 and the definition of ψ_{n}(λ), that shape(t[n − 1]) is allowable for λ if and only if there is no bad p-hook in t with top right entry equal to n. Now by induction, the condition (†) holds for t[n − 1] if and only if there is no bad p-hook in t[n − 1], which is if and only if there is no bad p-hook in t with top right entry at most n − 1. The claim follows immediately from these two statements.

So, by the claim, the dimension of the μ-weight space of L_{n}(λ) is equal to the number of standard λ-tableaux t of weight μ, containing no bad p-hooks.

6.4 Example Let p = 3, n = 4, λ = (3, 2, 1, 1), μ = (2, 1, 2, 2). Note that ψ_{4}(λ) = 3, so λ is completely splittable. The standard λ-tableaux of weight μ are just the following:

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

Of these, the first contains a bad p-hook (namely, the nodes (1, 3), (2, 2)), and the second does not. Thus, the μ-weight space of L_{n}(λ) is one dimensional.

Appendix: a short proof of the straightening rule

We were unable to find a suitable reference for the straightening rule used in section 2, so include here a short proof based on [J2], Satz II.6. We take our notation from (2.1). The standard basis theorem and the straightening rule will follow easily from the following lemma.

A.1 Lemma For λ ∈ X^{+}(n), the module Δ_{n}(λ) is generated as a U^{−}(n − 1)-module by

𝔊 := {F_{1,n}^{(a_{1})} … F_{n−1,n}^{(a_{n−1})}e_{λ} | 0 ≤ a_{i} ≤ λ_{i} − λ_{i+1} for i = 1, …, n − 1} .

Proof. By the PBW basis,

Δ_{n}(λ) = U^{−}(n)e_{λ} = ∑_{b_{1},...,b_{n−1}≥0} U^{−}(n − 1)F_{1,n}^{(b_{1})} … F_{n−1,n}^{(b_{n−1})}e_{λ}.

So we need to show that F_{1,n}^{(b_{1})} … F_{n−1,n}^{(b_{n−1})}e_{λ} lies in the U^{−}(n − 1)-submodule generated by 𝔊, for any integers b_{1}, …, b_{n−1} ≥ 0. To prove this, we use induction on the tuple (b_{1}, …, b_{n−1}), ordered lexicographically, the conclusion being immediate if b_{i} ≤ λ_{i} − λ_{i+1} for all i. So take a tuple (b_{1}, …, b_{n−1}) with b_{i} > λ_{i} − λ_{i+1} for some i.

We need the commutator formula

F_{i,n}^{(b_{i})}F_{i+1,n}^{(b_{i+1})} = F_{i+1,n}^{(b_{i}+b_{i+1})}F_{i,i+1}^{(b_{i})} − ∑_{j=1}^{b_{i}} F_{i,i+1}^{(j)}F_{i,n}^{(b_{i}−j)}F_{i+1,n}^{(b_{i+1}+j)}

derived in the proof of [J2], Satz II.6. To prove this, observe that the subalgebra generated by {F_{i,i+1}^{(j)}, F_{i+1,n}^{(j)} | j ≥ 0} is a copy of U^{−}(3), so it suffices to prove the following identity in U^{−}(3, ℤ):

F_{2,3}^{(r+s)}F_{1,2}^{(r)} = ∑_{j=0}^{r} F_{1,2}^{(j)}F_{1,3}^{(r−j)}F_{2,3}^{(s+j)}.

For this, we work in the ring of formal power series U^{−}(3, ℤ)[[u, v]]. If t = u, v or uv and i > j, y_{i,j}(t) denotes 1 + tX_{j,i} + ⋯ + t^{r} \frac{X_{j,i}^{r}}{r!} + … in U^{−}(3, ℤ)[[u, v]]. By [St], Lemma 15, we have the identity y_{2,3}(v)y_{1,2}(u) = y_{1,2}(u)y_{1,3}(uv)y_{2,3}(v) and the required formula follows by equating u^{r}v^{r+s}-coefficients.

Next note, by considering the action of SL(2), that F_{i,i+1}^{(b_{i})}e_{λ} = 0 as, by assumption, b_{i} > λ_{i} − λ_{i+1}. Combining this with the commutator formula from the previous paragraph, we deduce

F_{1,n}^{(b_{1})} … F_{n−1,n}^{(b_{n−1})}e_{λ} = − ∑_{j=1}^{b_{i}} F_{i,i+1}^{(j)}F_{1,n}^{(b_{1})} … F_{i,n}^{(b_{i}−j)}F_{i+1,n}^{(b_{i+1}+j)} … F_{n−1,n}^{(b_{n−1})}e_{λ}.

All the terms on the right hand side involve tuples strictly lower than (b_{1}, …, b_{n−1}) in the lexicographic order, so lie in the U^{−}(n − 1)-submodule generated by G by the induction hypothesis.

As in (2.2), we write μ ⟵ λ if λ = (λ_{1}, …, λ_{n}) ∈ X^{+}(n) and μ = (μ_{1}, …, μ_{n−1}) ∈ X^{+}(n − 1) for some n > 1, satisfying the condition λ_{i+1} ≤ μ_{i} ≤ λ_{i} for i = 1, …, n − 1. Recall given μ ⟵ λ that F_{μ,λ} := F_{1,n}^{(λ_{1}−μ_{1})} … F_{n−1,n}^{(λ_{n−1}−μ_{n−1})}. In this notation, Lemma A.1 says that Δ_{n}(λ) is generated by {F_{μ,λ}e_{λ} | μ ⟵ λ}.

A.2 Proposition Take λ ∈ X^{+}(n) and fix an ordering μ(1), …, μ(N) of the weights {μ ⟵ λ} such that μ(i) > μ(j) implies i < j. For 1 ≤ i ≤ N, let Δ^{i} be the U^{−}(n − 1)-module of Δ = Δ_{n}(λ) generated by {F_{μ(j),λ}e_{λ} | j ≤ i}. Then,

(0) = Δ^{0} ⊂ Δ^{1} ⊂ ⋯ ⊂ Δ^{N} = Δ

is a filtration of Δ as a U(n − 1)-module with Δ^{i}/Δ^{i−1} ≅ Δ_{n−1}(μ(i)). Moreover, the image of F_{μ(i),λ}e_{λ} in Δ^{i}/Δ^{i−1} is a U(n − 1)-high weight vector.

Proof. We first claim that each Δ^{i} is U(n−1)-stable and that the image of F_{μ(i),λ}e_{λ} in Δ^{i}/Δ^{i−1} is a (possibly zero) U(n−1)-high weight vector. We prove this by induction on i, so take i with 1 ≤ i ≤ N and assume by induction that Δ^{i−1} is U(n − 1)-stable (which is obvious in the case i = 1). Our assumptions imply that μ(i) is maximal with respect to the dominance ordering amongst {μ(j) | j ≥ i}. But Δ/Δ^{i−1} is generated as a U^{−}(n − 1)-module by {F_{μ(j),λ}e_{λ} + Δ^{i−1} | j ≥ i}, so the image of F_{μ(i),λ}e_{λ} is either zero or a vector of maximal weight in Δ/Δ^{i−1}. The claim follows.

Now note that Δ^{N} = Δ by Lemma A.1. So, we have constructed a filtration of Δ as a U(n − 1)-module, where the factors are high weight modules. In particular, the dimension of Δ is at most ∑_{μ⟵λ} dim Δ_{n−1}(μ), with equality if and only if Δ_{i}/Δ_{i−1} equals the universal high weight module Δ_{n−1}(μ(i)) for all i, which is what we want to prove. But now a calculation using Weyl’s dimension formula shows that equality does indeed hold.

We introduce some new notation, which is a variant on the Gelfand-Zetlin patterns of [GZ]. We write μ ⊂ λ if λ = (λ_{1}, …, λ_{m}) ∈ X(m) and μ = (μ_{1}, …, μ_{m−1}) ∈ X(m− 1) for some m > 1, satisfying the condition μ_{i} ≤ λ_{i} for i = 1, …, m − 1. For λ ∈ X(n), a λ-chain is a chain C = (C(1) ⊂ C(2) ⊂ ⋯ ⊂ C(n)) of weights with C(n) = λ [so C(i) ∈ X(i) for each i]. A weight chain is simply a λ-chain for some λ. The weight chain (C(1) ⊂ ⋯ ⊂ C(n)) is standard if in fact C(1) ⟵ C(2) ⟵ … ⟵ C(n) [so C(i) ∈ X^{+}(i) for each i].

Given μ ⊂ λ, with λ ∈ X(m), define F_{μ,λ} := F_{1,m}^{(λ_{1}−μ_{1})} … F_{m−1,m}^{(λ_{m−1}−μ_{m−1})}. Given a weight chain C = (C(1) ⊂ ⋯ ⊂ C(n)), let F_{C} := F_{C(1),C(2)}F_{C(2),C(3)} … F_{C(n−1),C(n)}. Observe that {F_{C} | for fixed λ ∈ X(n) and all λ-chains C} is precisely our usual PBW basis for U^{−}(n) in this new notation.

A.3 Theorem (The standard basis theorem) Fix λ ∈ X^{+}(n). The set

{F_{C}e_{λ} | for all standard λ-chains C}

is a basis for Δ_{n}(λ).

Proof. Induction on n using Proposition A.2.

We now want to prove the straightening rule, which describes how F_{C}e_{λ} for an arbitrary monomial F_{C} in the PBW basis for U^{−}(n) expands in terms of the standard basis for Δ_{n}(λ). There is a natural partial order on weight chains. Given chains C = (C(1) ⊂ ⋯ ⊂ C(n)) and D = (D(1) ⊂ ⋯ ⊂ D(n)), we write C ≤ D if either C(i) = D(i) for all i, or C(n) = D(n), …, C(i + 1) = D(i + 1) and C(i) < D(i) for some i. We call this the dominance order on weight chains.

A.4 Theorem (The straightening rule) Fix λ ∈ X^{+}(n) and let Δ = Δ_{n}(λ). Given an arbitrary λ-chain C, the vector F_{C}e_{λ} can be expanded uniquely as a linear combination of basis elements F_{D}e_{λ} for standard λ-chains D satisfying D ≥ C in the dominance order on λ-chains.

Proof. Given μ ⟵ λ, let Δ_{μ} be the U^{−}(n − 1)-submodule of Δ generated by {F_{ν,λ}e_{λ} | ν ⟵ λ, ν ≥ μ} and let Δ′_{μ} be the U^{−}(n − 1)-submodule of Δ generated by {F_{ν,λ}e_{λ} | ν ⟵ λ, ν > μ}. Choosing the ordering of {μ ⟵ λ} suitably in Proposition A.2, we see that Δ_{μ} is spanned by basis elements F_{D}e_{λ} for all standard λ-chains D = (D(1) ⟵ … ⟵ D(n)) with D(n − 1) ≥ μ.

As Δ is generated as a U^{−}(n − 1)-module by {F_{ν,λ}e_{λ} | ν ⟵ λ} by Proposition A.2, F_{C(n−1),C(n)}e_{λ} lies in the sum of the modules Δ_{ν} for ν ⟵ λ, ν ≥ C(n − 1). So, we can expand F_{C}e_{λ} as a linear combination of terms F_{D}e_{λ} for standard λ-chains D with D(n − 1) ≥ C(n − 1). If D(n − 1) ≠ C(n − 1), then D > C as required.

So we need to consider terms in the expansion with D(n − 1) = C(n − 1). This implies in particular that C(n − 1) ⟵ λ, so letting μ = C(n − 1), F_{C}e_{λ} lies in Δ_{μ} and its image in Δ_{μ}/Δ′_{μ} ≅ Δ_{n−1}(μ) is the vector F_{C′}e_{μ} where e_{μ} is the image of F_{μ,λ}e_{λ} and C′ is the μ-chain (C(1) ⊂ ⋯ ⊂ C(n − 1)). By induction on n, we can expand F_{C′}e_{μ} as a linear combination of terms F_{D′}e_{μ} for μ-chains D′ satisfying D′ ≥ C′. Hence, F_{C}e_{λ} can be expanded as required modulo Δ′_{μ}. But Δ′_{μ} is spanned by basis elements of the form F_{D}e_{λ} where the chain D = (D(1) ⟵ … ⟵ D(n)) satisfies D(n−1) > C(n−1), so again D > C, and the result follows.

To obtain the standard basis theorem and straightening rule as described in (2.3), it just remains to translate between the notation of weight chains and the tableaux notation of (2.2). If λ ∈ Λ^{+}(n), the map t ↦ (shape(t[1]) ⊂ shape(t[2]) ⊂ ⋯ ⊂ shape(t[n])) gives a injection from the set of row standard λ-tableau with entries in the ith row at least i into the set of λ-chains. It induces a bijection between standard λtableaux and standard λ-chains. Given this, it is straightforward to translate between the two notations.

References

[Bou] N. Bourbaki, Groupes et algebres de Lie, chapitres IV-VI, Hermann, Paris, 1968.

[B1] J. W. Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type A, 1996, submitted to Proc. London Math. Soc.

[B2] J. W. Brundan, Lowering operators for GL(n) and quantum GL(n), to appear in Proceedings Seattle conference on representations, cohomology and actions of finite groups.

[CL] R. W. Carter and G. Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242.

[GZ] I. M. Gelfand and M. L. Zetlin, Finite dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk. USSR 71 (1950), 825–828. [in Russian]

[G] J. A. Green, Polynomial representations of GL_{n}, Lecture Notes in Math., vol. 830, Springer-Verlag, 1980.

[J1] J. C. Jantzen, Representations of algebraic groups, Academic Press, Orlando, 1987.

[J2] J. C. Jantzen, Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen, Bonner Math. Schriften, No. 67, 1973.

[JS] J. C. Jantzen and G. M. Seitz, On the representation theory of the symmetric groups, Proc. London Math. Soc. 65 (1992), 475–504.

[K1] A. S. Kleshchev, On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, I, Proc. London Math. Soc. 69 (1994), 515–540.

[K2] A. S. Kleshchev, On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups, II, Comm. Alg. 22 (1994), 6175–6208.

[K3] A. S. Kleshchev, Branching rules for modular representations of symmetric groups, II, J. reine angew. Math. 459 (1995), 163–212.

[K4] A. S. Kleshchev, Completely splittable representations of symmetric groups, J. Alg. 181 (1996), 584–592.

[L] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Am. Math. Soc. 3 (1990), 257–297.

[MP] O. Mathieu and G. Papadopoulo, A character formula for a family of simple modular representations of GL_{n}, preprint, 1997.

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