Some Remarks on Branching Rules and Tensor Products for Algebraic Groups^{∗}

Jonathan Brundan^{†} and Alexander Kleshchev^{†}

1 Introduction and Preliminaries

Let 𝔽 be an algebraically closed field of characteristic p > 0. In [BK1, BK2], we have revealed and exploited various relations between the branching rules from GL_{n}(𝔽) to its Levi subgroups on one hand, and decompositions of tensor products over GL_{n}(𝔽) itself on the other. For example, if L is some irreducible rational GL_{n}(𝔽)-module and V is the natural GL_{n}(𝔽)-module, there is a close relationship between the highest weight vectors (relative to GL_{n−1}(𝔽)) in the restriction L ↓_{GL_{n−1}(𝔽)} and the highest weight vectors (relative to GL_{n}(𝔽)) in the tensor product L ⊗ V^{∗}. In this paper we obtain more results in this direction, some of which are valid for an arbitrary type.

To describe our main results, we adopt standard Lie theoretic notation. Let G be a (connected) reductive algebraic group over 𝔽. As in [J], R denotes the root system of G with respect to a fixed maximal torus T, R^{+} ⊂ R denotes the set of positive roots determined by a choice of Borel subgroup B^{+} containing T, and {α_{1}, …, α_{ℓ}} ⊂ R^{+} is the corresponding base for R. Denote the highest (long) root of R by α_{0} and the longest element of the Weyl group W = N_{G}(T)/T by w_{0}. We write X(T) for the character group Hom(T, 𝔽^{×}), Y (T) for the cocharacter group Hom(𝔽^{×}, T) and let ⟨·, ·⟩ be the natural pairing X(T) × Y (T) → ℤ. For α ∈ R, α^{∨} denotes the corresponding coroot in Y (T), and X^{+}(T) denotes the set {λ ∈ X(T) | ⟨λ, α_{i}^{∨}⟩ ≥ 0, i = 1, …, ℓ} of dominant weights.

All G-modules are assumed to be rational. For λ ∈ X^{+}(T), we have the G-modules L(λ), Δ(λ) and ∇(λ), which are the irreducible, the standard (or Weyl), and the costandard Gmodules with highest weight λ. Let Dist(G) be the algebra of distributions of G as in [J, I.7], which is generated by Dist(T) and the ‘divided power’ root generators X_{α}^{(n)}, Y_{α}^{(n)} for α ∈ R^{+}, n ≥ 1. Write X_{i}^{(n)} = X_{α_{i}}^{(n)}, Y_{i} = Y_{α_{i}}^{(n)} for i = 1, …, ℓ. If G is semisimple and simply connected (which we may assume for the proofs), Dist(G) coincides with the hyperalgebra of G arising from the Chevalley construction. We note that any G-module is a Dist(G)-module in a natural way; see [J, I.7.11, II.1.20].

Given a weight ν ∈ X(T) and a G-module M, M_{ν} will denote the ν-weight space of M. If in addition μ ∈ X^{+}(T) is a dominant weight, we define

M^{μ} := {v ∈ M | X_{i}^{(b_{i})}v = 0 for all b_{i} > ⟨μ, α_{i}^{∨}⟩ and i = 1, 2, …, ℓ}

∗1991 subject classification: 20G05.

†Both authors partially supported by the NSF.

and let M_{ν}^{μ} := M^{μ} ∩ M_{ν} denote its ν-weight space. Our first result generalizes a well known fact in characteristic 0.

Theorem A. Let λ, μ ∈ X^{+}(T), and M be any G-module. Then

Hom_{G}(Δ(λ), M ⊗ ∇(μ)) ≅ M_{λ−μ}^{μ}.

To explain our interest in the theorem, suppose that M = L(ν) is an irreducible module for some fixed ν ∈ X^{+}(T). Then, for μ large relative to ν, we see that M_{λ−μ}^{μ} = M_{λ−μ}, so by the theorem, L(ν)_{λ−μ} ≅ Hom_{G}(Δ(λ), L(ν) ⊗ ∇(μ)). So to compute the formal character of L(ν) it suffices to describe the Hom space in Theorem A for λ, μ large. In view of the universality of standard modules, this is equivalent to describing the highest weight vectors of weight λ in L(ν) ⊗ ∇(μ).

We note that Hom_{G}(Δ(λ), L(ν)⊗∇(μ)) ≅ Hom_{G}(L(ν^{∗}), ∇(λ^{∗})⊗∇(μ)) where ν^{∗}, λ^{∗} are the dual dominant weights; its dimension is precisely the multiplicity of L(ν^{∗}) in the socle of ∇(λ^{∗}) ⊗ ∇(μ). Our next result reveals some extra structure related to restricted weights of the socle of such tensor products. Recall that a dominant weight λ is called p^{r}-restricted if ⟨λ, α_{i}^{∨}⟩ < p^{r} for all i = 1, 2, …, ℓ. A semisimple module will be called p^{r}-restricted if all of its composition factors have p^{r}-restricted highest weights.

Theorem B. Let μ, ν ∈ X^{+}(T) and α_{0} ∈ R be the highest root. If μ is p^{r}-restricted and ⟨ν, α_{0}^{∨}⟩ < p^{r} then the socle of ∇(μ) ⊗ ∇(ν) is p^{r}-restricted.

In particular, we note that any miniscule weight ν satisfies the condition in Theorem B for all r. Theorem B is false if we weaken the assumption ⟨ν, α_{0}^{∨}⟩ < p^{r} to assume only that ν is p^{r}-restricted; see Remark 3.5 for a counterexample in this case.

Now we specialize to the case that G = GL(n) = GL_{n}(𝔽). As usual, take T to be all diagonal matrices in GL(n) and B^{+} to be all upper triangular matrices. We identify the weight lattice X(T) with the set X(n) of all n-tuples λ = (λ_{1}, λ_{2}, …, λ_{n}) of integers, λ corresponding to the character diag(t_{1}, …, t_{n}) ↦ t_{1}^{λ_{1}} … t_{n}^{λ_{n}}, and X^{+}(T) with the set X^{+}(n) = {λ ∈ X(n) | λ_{1} ≥ ⋯ ≥ λ_{n}}. We write L_{n}(λ), Δ_{n}(λ), ∇_{n}(λ) for the irreducible, standard and costandard modules, and ε_{i} denotes the weight (0, …, 0, 1, 0, …, 0) with 1 in the ith position.

The connection between Theorem A and our earlier results [BK1, BK2] arises as follows. Embed GL(n − 1) into the top left hand corner of GL(n). If μ = −ℓε_{n} for ℓ ≥ 0, the space M_{λ−μ}^{μ} appearing in Theorem A is precisely the space of vectors in M_{λ−μ} which are highest

weight vectors with respect to the subgroup GL(n−1), satisfying in addition X_{n−1}^{(b)}v = 0 for any b > ℓ. By directly constructing the isomorphism appearing in Theorem A, we obtain the following extension of Theorem A to irreducible modules in one important special case.

Theorem C. Fix λ, μ ∈ X^{+}(n) with λ_{n} = μ_{n}. For any submodule M of ∇_{n}(λ),

Hom_{GL(n)}(L_{n}(μ), M ⊗ ∇_{n}(−ℓε_{n})) ≅ Hom_{GL(n−1)}(L_{n−1}(μ̄), M ↓_{GL(n−1)})

where ℓ = ∑_{i=1}^{n}(λ_{i}−μ_{i}) and μ̄ = (μ_{1}, …, μ_{n−1}) denotes the restriction of μ to T ∩GL(n−1).

We believe it is an important problem to describe the socle of L_{n}(λ) ↓_{GL(n−1)} (which appears in Theorem C if M = L_{n}(λ)), for any λ ∈ X^{+}(n). We refer to this problem as the modular branching problem for the general linear group. A complete answer only exists in some special cases, namely, the ‘first level’ and when L_{n}(λ) ↓_{GL(n−1)} is semisimple; see [K4, B1, BKS]. By the known characteristic 0 branching rule, together with basic properties of good filtrations, the space Hom_{GL(n−1)}(Δ_{n−1}(μ), ∇_{n}(λ) ↓_{GL(n−1)}) is 0 unless λ_{i+1} ≤ μ_{i} ≤ λ_{i} for i = 1, …, n−1, when it is 1-dimensional. Hence, each of the three spaces

Hom_{GL(n−1)}(Δ_{n−1}(μ),L_{n}(λ) ↓_{GL(n−1)}),
Hom_{GL(n−1)}(L_{n−1}(μ),∇_{n}(λ) ↓_{GL(n−1)}),
Hom_{GL(n−1)}(L_{n−1}(μ),L_{n}(λ) ↓_{GL(n−1)})

are at most 1-dimensional, the last of which computes the socle. Moreover, the last Hom space is non-zero if and only if both of the first two are non-zero.

Our final result, which is a consequence of Theorem C, reduces the problem of calculating any of the three Hom spaces to just the first. We are not aware of a direct proof of Theorem D working only with branching rules.

Theorem D. Fix λ ∈ X^{+}(n) and μ ∈ X^{+}(n−1) such that λ_{i+1} ≤ μ_{i} ≤ λ_{i} for 1 ≤ i ≤ n−1. Let \overline{λ^{∗}} = (−λ_{n}, −λ_{n−1}, …, −λ_{2}) ∈ X^{+}(n − 1) and μ̃^{∗} = (−λ_{n}, −μ_{n−1}, …, −μ_{1}) ∈ X^{+}(n). Then,

Hom_{GL(n−1)}(L_{n−1}(μ), ∇_{n}(λ) ↓_{GL(n−1)}) ≅ Hom_{GL(n−1)}(Δ_{n−1}(\overline{λ^{∗}}), L_{n}(μ̃^{∗}) ↓_{GL(n−1)}).

Consequently, L_{n−1}(μ) lies in the socle of L_{n}(λ) ↓_{GL(n−1)} if and only if both

Hom_{GL(n−1)}(Δ_{n−1}(μ), L_{n}(λ) ↓_{GL(n−1)}) and Hom_{GL(n−1)}(Δ_{n−1}(\overline{λ^{∗}}), L_{n}(μ̃^{∗}) ↓_{GL(n−1)})

are non-zero.

In particular, Theorem D means that to calculate the socle of L_{n}(λ) ↓_{GL(n−1)} for all λ, it is sufficient to calculate the space of GL(n − 1)-highest weight vectors in L_{n}(λ) for all λ, or equivalently, the socle of L_{n}(λ) ↓_{B^{+}∩GL(n−1)}. In [B2, §5.3], the first author described an algorithm for calculating the space of highest weight vectors in L_{n}(λ) ↓_{GL(n−1)}. This is computationally intensive, depending on first calculating the Gram matrix for the contravariant form on certain weight spaces of Weyl modules, so is viable only for partitions of size |λ| < 12. Combining this with Theorem D means it is now possible to compute explicitly the socle of L_{n}(λ) ↓_{GL(n−1)} for small λ.

Finally, we remark that there is an analogue of Theorem B for the branching problem: if λ ∈ X^{+}(n) is p^{r}-restricted, the socle of ∇_{n}(λ) ↓_{GL(n−1)} is also p^{r}-restricted. This is a generalization of [K1, Theorem B] (for type A), where this was proved with ∇_{n}(λ) replaced by L_{n}(λ). In fact, the proof of the more general version is identical to the proof in [K1], combined with Lemma 3.2 from this paper.

2 Proof of Theorem A

We will assume throughout the section that G is semisimple and simply connected. Theorem A (and Theorem B) as stated in the introduction reduce to this case by standard arguments. The point is that then, the algebra of distributions Dist(G) can be identified with the hyperalgebra U of G, so can be constructed explicitly by first choosing a Chevalley system (x_{α})_{α∈R}, (h_{i})_{1≤i≤ℓ} in the corresponding semisimple Lie algebra 𝔤 over ℂ, then taking the ℤ-subalgebra U_{ℤ} of the universal enveloping algebra of 𝔤 generated by all x_{α}^{k}/k!, and finally setting U = U_{ℤ} ⊗ F; see [J, II.1.12] and [S]. The elements X_{α}^{(n)}, Y_{α}^{(n)} ∈ Dist(G) coincide with (x_{α}^{n}/n!) ⊗ 1, (x_{−α}^{n}/n!) ⊗ 1 ∈ U respectively, for α ∈ R^{+}.

By [J, II.1.20], there is an equivalence of categories between the category of all G-modules and the category of locally finite U-modules. We denote by U^{+} (resp. U^{−}) the subalgebra of U generated by all X_{α}^{(k)} (resp. Y_{α}^{(k)}) for α ∈ R^{+}, k ≥ 0. Also, let U^{0} be the subalgebra generated by all

(\begin{matrix}H_{i} \\ k \end{matrix}) := \frac{h_{i}(h_{i} − 1) … (h_{i} − k + 1)}{k!} ⊗ 1

for 1 ≤ i ≤ ℓ and k ≥ 0. Kostant’s ℤ-form for U_{ℤ} [S, Theorem 2] gives a PBW type basis for each of U, U^{−}, U^{0} and U^{+}, on tensoring with 𝔽.

We call a weight vector v in a G-module a highest weight vector if it is annihilated by all X_{α}^{(k)} for α ∈ R^{+}, k ≥ 1. The following fundamental result can be found in [J, II.2.13b)].

2.1. (Universality of standard modules) The module Δ(μ) is generated by any highest weight vector v_{μ} of weight μ, and, moreover, any G-module generated by a highest weight vector of weight μ is a quotient of Δ(μ).

We will often regard elements of X(T) as homomorphisms U^{0} → 𝔽. For a dominant weight μ let

X(μ) := {X_{i}^{(b_{i})} | 1 ≤ i ≤ ℓ, b_{i} > ⟨μ, α_{i}^{∨}⟩},
Y (μ) := {Y_{i}^{(b_{i})} | 1 ≤ i ≤ ℓ, b_{i} > ⟨μ, α_{i}^{∨}⟩},
Ω(μ) := {X_{α}^{(b_{α})}, H − μ(H) | α ∈ R^{+}, b_{α} ≥ 1, H ∈ U^{0}}.

The next lemma is well known. We prove it for completeness as we could not find a proof in the literature.

2.2. Lemma. For μ ∈ X^{+}(T), let I(μ) be the left ideal of U generated by Y (μ) ∪ Ω(μ). Then, Δ(μ) ≅ U/I(μ).

Proof. Let v_{μ} be a highest weight vector in Δ(μ) of weight μ. Consider the U-module homomorphism U → Δ(μ), u ↦ uv_{μ}. As Uv_{μ} = Δ(μ) and I(μ)v_{μ} = 0, this homomorphism yields a surjection U/I(μ) → Δ(μ). By the universality of standard modules and the equivalence of categories between locally finite U-modules and G-modules, it suffices to prove that V (μ) := U/I(μ) is finite dimensional.

We prove this as in [H2, 21.4] by showing that the weights of V (μ) are permuted by the Weyl group W associated to the root system R. Let s_{i} ∈ W be the simple reflection corresponding to α_{i}. Since W is generated by its simple reflections, we just need to prove that s_{i}ν is a weight of V (μ) whenever ν is a weight of V (μ).

Take 0 ≠ v ∈ V (μ)_{ν}. Our goal is to establish that X_{i}^{(k)}v = Y_{i}^{(k)}v = 0 for k ≫ 0. Then the vector exp(X_{i}) exp(−Y_{i}) exp(X_{i})v will be a well-defined non-zero vector of weight s_{i}ν. Note that ν + kα_{i} is not a weight of V (μ) for k large enough, so X_{i}^{(k)}v = 0 for such k. To prove the claim for Y_{i} we may assume, using the PBW type basis for U^{−}, that v = Y_{β_{1}}^{(b_{1})} … Y_{β_{m}}^{(b_{m})} + I(μ) where {β_{1}, …, β_{m}} are the positive roots.

By induction on b_{1} + ⋯ + b_{m} we now show that Y_{i}^{(k)}Y_{β_{1}}^{(b_{1})} … Y_{β_{m}}^{(b_{m})} ∈ I(μ) if k >

3(b_{1} + ⋯ + b_{m}) + μ_{i}. If b_{1} + ⋯ + b_{m} = 0 this is clear as Y_{i}^{(k)} ∈ I(μ) for k > μ_{i}. For the inductive step, let r = min{s | b_{s} > 0}. To apply the inductive hypothesis it suffices to note that Y_{i}^{(k)}Y_{β_{r}}^{(b_{r})} is a linear combination of elements of the form u_{j}Y_{i}^{(k−j)} where j ≤ 3b_{r}, which follows for example by [K1, 1.8(ii)].

Lemma 2.2 gives generators and relations for Δ(μ) as a U-module. However, to prove Theorem A, we need generators and relations for Δ(μ) as a U^{+}-module.

2.3. Lemma. For μ ∈ X^{+}(T), let I^{−}(μ) be the left ideal of U^{−} generated by Y (μ). Then, Δ(μ) ↓_{U^{−}} ≅ U^{−}/I^{−}(μ).

Proof. Let J(μ) denote the left ideal of U generated by Ω(μ). Then, Z(μ) := U/J(μ) is the Verma module of highest weight μ. Using the PBW type bases, the map θ : U^{−} → Z(μ), Y ↦ Y + J(μ) is an isomorphism of U^{−}-modules. Lemma 2.2 implies that Δ(μ) ≅ Z(μ)/F (μ) where F (μ) is the image of I(μ) in Z(μ). So it suffices to show that θ maps I^{−}(μ) onto F (μ), or equivalently, that UY (μ) ⊆ U^{−}Y (μ) + J(μ).

We can write U = U^{−}U^{+}U^{0} by [S, Theorem 2]. Clearly, elements of U^{0} applied to the elements of Y (μ) change them to proportional ones. So we just need to prove that for any X ∈ U^{+}, the element XY_{i}^{(b)} belongs to U^{−}Y (μ) + J(μ) providing b > ⟨μ, α_{i}^{∨}⟩. We may assume that X = X_{α}^{(a)} for some α ∈ R^{+}, a ≥ 1. If α ≠ α_{i}, the weight aα − bα_{i} is not a sum of negative roots, so X_{α}^{(a)}Y_{i}^{(b)} lies in J(μ). So we may assume that α = α_{i} is a simple root, and moreover, by weights, that a ≤ b. Then, using [S, Lemma 5], we get

X_{i}^{(a)}Y_{i}^{(b)} + J(μ) = Y_{i}^{(b−a)} (\begin{matrix}H_{i}−b−a+2a \\ a \end{matrix}) + J(μ) = Y_{i}^{(b−a)} (\begin{matrix}⟨μ,α_{i}^{∨}⟩−(b−a) \\ a \end{matrix}) + J(μ).

If b − a > ⟨μ, α_{i}^{∨}⟩ we have Y_{i}^{(b−a)} ∈ Y (μ). Otherwise ⟨μ, α_{i}^{∨}⟩ − (b − a) is a non-negative

integer strictly less than a, so (\begin{matrix}⟨μ,α_{i}^{∨}⟩−(b−a) \\ a \end{matrix}) = 0.

2.4. Corollary. For μ ∈ X^{+}(T), let I^{+}(μ) be the left ideal of U^{+} generated by X(μ). Then, Δ(μ) ↓_{U^{+}} ≅ U^{+}/I^{+}(−w_{0}μ).

Proof. Let n_{0} ∈ N_{G}(T) be any representative of w_{0} ∈ W = N_{G}(T)/T. This acts on U by the adjoint action Ad. Moreover, Ad n_{0} sends U^{−} isomorphically onto U^{+} and I^{−}(μ) isomorphically onto I^{+}(−w_{0}μ). Using these observations, the result follows immediately from Lemma 2.3.

Recall the definition of M_{λ−μ}^{μ} from the introduction. Now we can prove Theorem A.

2.5. Theorem. Let λ, μ ∈ X^{+}(T), and M be any G-module. Then

Hom_{G}(Δ(λ), M ⊗ ∇(μ)) ≅ M_{λ−μ}^{μ}.

Proof. Let 𝔽_{λ} be the 1-dimensional B^{+}-module of weight λ, and let A⊲B^{+} be the unipotent radical of B^{+}. Using the universality of standard modules we get

Hom_{G}(Δ(λ), M ⊗ ∇(μ)) ≅ Hom_{B^{+}}(𝔽_{λ}, M ⊗ ∇(μ))
≅ ((M ⊗ ∇(μ))^{A})_{λ}
≅ Hom_{A}(∇(μ)^{∗}, M)_{λ}

where the last λ-weight space is taken with respect to the action (t · φ)(f) = tφ(t^{−1}f) for φ ∈ Hom_{A}(∇(μ)^{∗}, M), f ∈ ∇(μ)^{∗}. Moreover, since ∇(μ)^{∗} ≅ Δ(−w_{0}μ) and U^{+} ≅ Dist(A), [J, I.7.16] implies

Hom_{A}(∇(μ)^{∗}, M)_{λ} ≅ Hom_{U^{+}}(Δ(−w_{0}μ), M)_{λ}.

The natural isomorphism Hom_{U^{+}}(U^{+}, M) → M combined with Corollary 2.4 induces an isomorphism

F : Hom_{U^{+}}(Δ(−w_{0}μ), M) → M^{μ}, φ ↦ φ(v_{−μ})

where v_{−μ} is a lowest weight vector in Δ(−w_{0}μ) of weight −μ. For t ∈ T and a weight vector φ ∈ Hom_{U^{+}}(Δ(−w_{0}μ), M)_{λ},

t(φ(v_{−μ})) = t(φ(t^{−1}tv_{−μ})) = (t · φ)(tv_{−μ}) = (λ − μ)(t)φ(v_{−μ}).

Hence, F sends the λ-weight space of Hom_{U^{+}}(Δ(−w_{0}μ), M) isomorphically onto the (λ−μ)- weight space of M^{μ}.

3 Proof of Theorem B

Now we turn to the proof of Theorem B, which will ultimately be deduced as a consequence of Steinberg’s tensor product theorem. We continue with the notation and assumptions from section 2; in particular, G is semisimple and simply connected.

3.1. Lemma. For ν ∈ X^{+}(T) and m ≥ 0, ν − w_{0}ν ≱ mα_{0} if and only if ⟨ν, α_{0}^{∨}⟩ < m.

Proof. If ⟨ν, α_{0}^{∨}⟩ ≥ m then ν − mα_{0} is a weight of Δ(ν), hence ν − mα_{0} ≥ w_{0}ν, which is the lowest weight of Δ(ν). Hence, ν − w_{0}ν ≥ mα_{0}. Conversely, suppose ν − w_{0}ν ≥ mα_{0}. Then ν − w_{0}ν = mα_{0} + κ where κ is a sum of positive roots. Now, ⟨ν − w_{0}ν, α_{0}^{∨}⟩ = m⟨α_{0}, α_{0}^{∨}⟩ + ⟨κ, α_{0}^{∨}⟩ ≥ m⟨α_{0}, α_{0}^{∨}⟩ = 2m. On the other hand, ⟨ν − w_{0}ν, α_{0}^{∨}⟩ = ⟨ν, α_{0}^{∨}⟩ − ⟨w_{0}ν, α_{0}^{∨}⟩ = 2⟨ν, α_{0}^{∨}⟩, since ⟨w_{0}ν, α_{0}^{∨}⟩ = ⟨ν, (w_{0}^{−1}α_{0})^{∨}⟩ = ⟨ν, (w_{0}α_{0})^{∨}⟩ and w_{0}α_{0} = −α_{0}. Thus, ⟨ν, α_{0}^{∨}⟩ ≥ m.

3.2. Lemma. Let λ ∈ X^{+}(T) be p^{r}-restricted,and v_{μ} ∈ ∇_{n}(λ) be a non-zero weight vector of weight μ. If v_{μ} is annihilated by all X_{α}^{(k)} for all 1 ≤ k < p^{r}and all α ∈ R^{+}, then μ = λ.

Proof. We let U_{r} denote the subalgebra of U generated by {X_{α}^{(k)}, Y_{α}^{(k)}}_{α∈R^{+},k<p^{r}}, which is the algebra of distributions of G_{r}, the rth Frobenius kernel of G (see [J, II.3]). The assumptions imply that the U_{r}-module M generated by v_{μ} is non-zero and has all weights less than or equal to μ. Pick L(ν) lying in the socle of M, so that ν is p^{r}-restricted with ν ≤ μ. Certainly, μ ≤ λ, so the result will follow if we can show that ν = λ.

For this, we claim that ∇(λ) has simple socle L(λ) as a U_{r}-module. By the argument of [H1, Proposition 1.1] (which proves the special case r = 1), Δ(λ) is generated as a U_{r}- module by any highest weight vector of weight λ. This easily implies that Δ(λ) has simple head as a U_{r}-module, hence proving the claim on dualizing.

3.3. Theorem. Fix μ, ν ∈ X^{+}(T) where μ is p^{r}-restricted and ⟨ν, α_{0}^{∨}⟩ < p^{r}. The socle of ∇(μ) ⊗ ∇(ν) is p^{r}-restricted.

Proof. We say a vector v ∈ ∇(μ) ⊗ ∇(ν) is weakly primitive if X_{α}^{(k)}v = 0 for all α ∈ R^{+} and all k with 0 < k < p^{r}. Fix a weakly primitive weight vector v ∈ ∇(μ) ⊗ ∇(ν) of weight δ. Write δ = μ + ν − κ for some κ ∈ X(T). We first claim that κ ≤ ν − w_{0}ν. Write

v = ∑_{γ,i,j} x_{δ−γ}^{i} ⊗ y_{γ}^{j}

summing over γ ∈ X(T) and i, j over index sets I_{γ}, J_{γ} respectively. In this expression, {x_{β}^{i}}_{i∈I_{β}} and {y_{γ}^{j}}_{j∈J_{γ}} denote linearly independent vectors of the weight spaces ∇(μ)_{β} and ∇(ν)_{γ} respectively. Let γ_{0} be a minimal weight such that J_{γ_{0}} is non-empty. Then for any α ∈ R^{+} and any k with 0 < k < p^{r} we have

0 = X_{α}^{(k)}v = ∑_{i,j} (X_{α}^{(k)}x_{δ−γ_{0}}^{i}) ⊗ y_{γ_{0}}^{j}

+ [a linear combination of vectors of the form x_{β}^{i} ⊗ y_{γ}^{j} with γ ≰ γ_{0}].

We conclude by linear independence of {y_{γ}^{j}}_{j∈J_{γ}} that X_{α}^{(k)}x_{δ−γ_{0}}^{i} = 0 for any α ∈ R^{+} and 0 < k < p^{r}. Since μ is p^{r}-restricted, it follows from Lemma 3.2 that x_{δ−γ_{0}}^{i} is a high weight vector in ∇(μ). Thus, δ − γ_{0} = μ, hence γ_{0} = δ − μ = ν − κ. This shows that ν − κ is a weight of ∇(ν), so ν − κ ≥ w_{0}ν, which implies the claim.

Now, assume for a contradiction that the Steinberg tensor product L(λ) ⊗ L(λ′)^{[r]} is a submodule of ∇(μ) ⊗ ∇(ν) for some p^{r}-restricted λ and some λ′ ≠ 0. Let v_{λ} and v_{λ′}^{+} be high weight vectors of L(λ) and L(λ′)^{[r]} = L(p^{r}λ′), respectively. Also let v_{λ′}^{−} be the lowest weight vector of L(λ′)^{[r]}. Then both v_{λ} ⊗ v_{λ′}^{+} and v_{λ} ⊗ v_{λ′}^{−} are weakly primitive (in the latter case, this follows by the definition of the action of U on Frobenius twists). The weights of these two vectors are λ + p^{r}λ′ and λ + p^{r}w_{0}λ′ respectively. Set

λ + p^{r}λ′ = μ + ν − κ_{1}, λ + p^{r}w_{0}λ′ = μ + ν − κ_{2}.

By the claim, we have κ_{2} ≤ ν − w_{0}ν. On the other hand, κ_{2} − κ_{1} = p^{r}(λ′ − w_{0}λ′) ≥ p^{r}α_{0}, the last inequality being true by [K3, Lemma 1.5]. It follows that κ_{2} ≥ κ_{1} + p^{r}α_{0} ≥ p^{r}α_{0}, whence ν − w_{0}ν ≥ p^{r}α̃. This contradicts the assumption on ν because of Lemma 3.1.

3.4. Corollary. Let μ be a dominant p^{r}-restricted weight, and ν be any miniscule weight. If M is any submodule of ∇(μ) then the socle of M ⊗ L(ν) is p^{r}-restricted. In particular, the socle of L(μ) ⊗ L(ν) is p^{r}-restricted.

Proof. This follows immediately from Theorem 3.3, since if ν is miniscule then ⟨ν, α^{∨}⟩ is 0 or 1 for all α ∈ R^{+}.

3.5. Remark. One might ask whether Theorem 3.3 is true more generally, namely, is it true that the socle of ∇(μ) ⊗ ∇(ν) is p-restricted as long as both μ, ν ∈ X^{+}(n) are p-restricted. We give a counterexample which shows that this is false in general. Consider the 2-restricted dominant weights μ = ν = 3ε_{1} + 2ε_{2} + ε_{3} for GL(4). Put λ = 6ε_{1} + 3ε_{2} + 2ε_{3} + ε_{4}. By the Littlewood-Richardson rule and [W], the module ∇(μ) ⊗ ∇(ν) has a ∇-filtration, with ∇(λ) as one of its quotients. Hence there exists a non-zero homomorphism from Δ(λ) to ∇(μ) ⊗ ∇(ν). However, ∇(λ) is irreducible in characteristic 2, as follows e.g. from [K2, 2.2(iv)]. So we get a non-2-restricted irreducible module in the socle of ∇(μ) ⊗ ∇(ν).

4 Proof of Theorems C and D

From now on, we assume that G = GL(n). In the notation from the introduction, the root system R ⊂ X(T) is the set {ε_{i} − ε_{j} | 1 ≤ i, j ≤ n}. For i < j, we denote the root ε_{i} − ε_{j} by α(i, j). Write E_{i,j}^{(k)} for X_{α(i,j)}^{(k)} and F_{i,j}^{(k)} for Y_{α(i,j)}^{(k)}. We fix an integer ℓ ≥ 0 throughout the section.

Let P = LY be the standard parabolic subgroup of GL(n), where L ≅ GL(n − 1)GL(1) (embedded diagonally) and Y is the unipotent radical generated by the root subgroups corresponding to the roots α(i, n), for i = 1, 2, … n − 1. Note that for any GL(n)-module N, the Y -fixed points N^{Y} of N are L-invariant, so we can regard N^{Y} as a GL(n−1)-module in a natural way. Also, for λ = (λ_{1}, …, λ_{n}) ∈ X^{+}(n), j ∈ ℤ and any submodule M of ∇_{n}(λ), the jth level of M is defined by

M^{j} := ⨁_{{ν∈X(n), \atop ν_{n}=λ_{n}+j}} M_{ν}.

This is a weight space for the 1-dimensional torus GL(1) that centralizes GL(n − 1) in GL(n), so

M ↓_{GL(n−1)}≅ ∑_{j≥0} M^{j}.

4.1. Lemma. Let λ, μ ∈ X^{+}(n). The dimension of Hom_{GL(n)}(Δ_{n}(μ), ∇_{n}(λ) ⊗ ∇(−ℓε_{n})) is 1 if μ_{n} ≤ λ_{n} and λ_{i+1} ≤ μ_{i} ≤ λ_{i} for i = 1, …, n − 1, and is 0 otherwise.

Proof. By [W], ∇_{n}(λ) ⊗ ∇(−ℓε_{n}) has a good filtration, so [J, II.4.16a)] implies that the Hom dimension is equal to the multiplicity of ∇_{n}(μ) in a good filtration of ∇_{n}(λ)⊗∇(−ℓε_{n}). Now the result follows for example from the Littlewood-Richardson rule.

Let V be the natural GL(n)-module, and let {f_{1}, …, f_{n}} be the basis of V^{∗} dual to the canonical basis of V. By [J, I.2.16(4)], the module ∇(−ℓε_{n}) is precisely the ℓth symmetric power S^{ℓ}(V^{∗}). Let Λ(n, ℓ) be the set of all n-tuples (λ_{1}, …, λ_{n}) of non-negative integers with λ_{1} + ⋯ + λ_{n} = ℓ. For β = (β_{1}, …, β_{n}) ∈ Λ(n, ℓ), we set

f_{β} = f_{1}^{β_{1}} … f_{n}^{β_{n}}  ∈ S^{ℓ}(V^{∗}),
E(β) = E_{1,n}^{(β_{1})}E_{2,n}^{(β_{2})} … E_{n−1,n}^{(β_{n−1})} ∈ Dist(GL(n)).

Then {f_{β} | β ∈ Λ(n, ℓ)} is a basis for S^{ℓ}(V^{∗}), and in particular, the set of weights of S^{ℓ}(V^{∗}) is precisely the set −Λ(n, ℓ), all with multiplicity one. Also if β = (β_{1}, …, β_{n}), γ = (γ_{1}, …, γ_{n}) ∈ Λ(n, ℓ), we write

|β̄| = β_{1} + ⋯ + β_{n−1}, (γ̄ ; β̄) = (\begin{matrix}γ_{1} \\ β_{1} \end{matrix}) … (\begin{matrix}γ_{n−1} \\ β_{n−1} \end{matrix}) .

Then for 1 ≤ r < s ≤ n, t ≥ 1, we have

E_{r,s}^{(t)}f_{β} = (−1)^{t} (\begin{matrix}β_{r} \\ t \end{matrix}) f_{β−tα(r,s)} and F_{r,s}^{(t)}f_{β} = (−1)^{t} (\begin{matrix}β_{s} \\ t \end{matrix}) f_{β+tα(r,s)}

(note that (\begin{matrix}β_{i} \\ t \end{matrix}) = 0 if β_{i} < t, so the right hand sides above are interpreted as 0 if β_{r} < t or

β_{s} < t, respectively).

Let M be an arbitrary GL(n)-module. As {f_{β} | β ∈ Λ(n, ℓ)} is a basis of S^{ℓ}(V^{∗}), any element w ∈ M ⊗ S^{ℓ}(V^{∗}) can be written uniquely in the form

w = ∑_{β∈Λ(n,r)} w_{β} ⊗ f_{β}.

We refer to w_{β} as the β-component of w. Define a linear map

e : M → M ⊗ S^{ℓ}(V^{∗}), v ↦ ∑_{β∈Λ(n,ℓ)}(E(β)v) ⊗ f_{β}.

4.2. Lemma. For any GL(n)-module M, the map e is an injective GL(n − 1)-module homomorphism.

Proof. Clearly, e linear. It is injective since the ℓε_{n}-component of e(v) is v. Let B′ be the subgroup B^{+} ∩ GL(n − 1) of all upper triangular matrices in GL(n − 1), and W′ be the subgroup of all permutation matrices in GL(n − 1). As B′ and W′ generate GL(n − 1), it suffices to prove that e is both a B′-homomorphism and a W′-homomorphism.

To prove that e is a W′-homomorphism take σ to be a permutation of {1, …, n−1} and denote by the same letter σ the corresponding permutation matrix. Then

σe(v) = σ ∑_{β∈Λ(n,ℓ)} E(β)v ⊗ f_{β} = ∑_{β∈Λ(n,ℓ)}(σE(β)v) ⊗ (σf_{β})

= ∑_{β∈Λ(n,ℓ)}(σE(β)σ^{−1}σv) ⊗ f_{σβ} = ∑_{β∈Λ(n,ℓ)}(E(σβ)σv) ⊗ f_{σβ} = e(σv).

To prove that e is a B′-homomorphism, we first note that it is a T -homomorphism, which follows from the fact that for v ∈ M_{μ}, the restriction of the weight of any (E(β)v) ⊗ f_{β} to T is the same as the restriction of μ. So, it suffices to prove that e(E_{r,s}^{(t)}v) = E_{r,s}^{(t)}e(v) for any v ∈ M, 1 ≤ r < s ≤ n − 1 and t > 0. First, we note that for m ≥ 0, the commutator formula from [S, Lemma 15] implies that

E_{r,s}^{(m)}E(β) = E_{r,s}^{(m)}E_{s,n}^{(β_{s})}E(β − β_{s}α(s, n)) = ∑_{i=0}^{min(m,β_{s})} E_{r,n}^{(i)}E_{s,n}^{(β_{s}−i)}E_{r,s}^{(m−i)}E(β − β_{s}α(s, n))

= ∑_{i=0}^{min(m,β_{s})} E_{r,n}^{(i)}E(β − iα(s, n))E_{r,s}^{(m−i)} = ∑_{i=0}^{min(m,β_{s})} (\begin{matrix}i+β_{r} \\ i \end{matrix}) E(β + iα(r, s))E_{r,s}^{(m−i)}.

Hence,

E_{r,s}^{(t)}e(v) = E_{r,s}^{(t)} ∑_{β∈Λ(n,ℓ)}(E(β)v) ⊗ f_{β} = ∑_{{β∈Λ(n,ℓ), \atop 0≤m≤t}} (E_{r,s}^{(m)}E(β)v) ⊗ (E_{r,s}^{(t−m)}f_{β})

= ∑_{{β∈Λ(n,ℓ), \atop 0≤m≤t}} ∑_{i=0}^{min(m,β_{s})}(−1)^{t−m} (\begin{matrix}β_{r} \\ t−m \end{matrix}) (\begin{matrix}i+β_{r} \\ i \end{matrix}) (E(β + iα(r, s))E_{r,s}^{(m−i)}v) ⊗ f_{β−(t−m)α(r,s)}.

Fix any γ ∈ Λ(n, ℓ). Then the γ-component of the above expression is

∑_{m=max(0,t−γ_{s})}^{t} ∑_{i=0}^{min(m,γ_{s}−(t−m))} (−1)^{t−m} (\begin{matrix}γ_{r}+t−m \\ t−m \end{matrix}) (\begin{matrix}i+γ_{r}+t−m \\ i \end{matrix}) E(γ + (t − m + i)α(r, s))E_{r,s}^{(m−i)}v

which equals

∑_{j=max(0,t−γ_{s})}^{t}c_{j}E(γ + (t − j)α(r, s))E_{rs}^{(j)}v

where c_{j} = ∑_{m=j}^{t}(−1)^{t−m} (\begin{matrix}γ_{r}+t−m \\ t−m \end{matrix}) (\begin{matrix}γ_{r}+t−j \\ m−j \end{matrix}). An elementary substitution now shows that

c_{j} = 0 if j < t. Hence, the γ-component is E(γ)E_{r,s}^{(t)}v, proving that E_{r,s}^{(t)}e(v) = e(E_{r,s}^{(t)}v).

4.3. Lemma. For any GL(n)-module M we have (M ⊗ S^{ℓ}(V^{∗}))^{Y} ⊆ e(M).

Proof. Let w = ∑_{β∈Λ(n,ℓ)} w_{β} ⊗ f_{β} ∈ (M ⊗ S^{ℓ}(V^{∗}))^{Y}. We have to show that

w_{γ} = E(γ)w_{ℓε_{n}}

for any γ ∈ Λ(n, ℓ). We prove this by induction on |γ̄|. If |γ̄| = 0, the result is clear. Let |γ̄| > 0. Considering the ℓε_{n}-component of the equation E(γ)w = 0 gives

∑_{β⪯γ}(−1)^{|β̄|}E(γ − β)w_{β} = 0

where β ⪯ γ means β_{1} ≤ γ_{1}, …, β_{n−1} ≤ α_{n−1}. Now the induction hypothesis gives us

0 = ∑_{β≺γ}(−1)^{|β̄|}E(γ − β)E(β)w_{ℓε_{n}} + (−1)^{|γ̄|}w_{γ} = ∑_{β≺γ}(−1)^{|β̄|} (γ̄ ; β̄) E(γ)w_{ℓε_{n}} + (−1)^{|γ̄|}w_{γ}.

The lemma now follows from the identity ∑_{β⪯γ}(−1)^{|β̄|} (γ̄ ; β̄) = 0.

4.4. Lemma. Let M be a submodule of ∇_{n}(λ). Let v ∈ M^{0} ⊕ ⋯ ⊕ M^{ℓ}. Then e(v) ∈ (M ⊗ S^{ℓ}(V^{∗}))^{Y}.

Proof. Let 1 ≤ r < n and t > 0. Then

E_{r,n}^{(t)}e(v) = E_{r,n}^{(t)} ∑_{β∈Λ(n,ℓ)}(E(β)v) ⊗ f_{β} = ∑_{β∈Λ(n,ℓ)} ∑_{m=0}^{t} (E_{r,n}^{(t−m)}E(β)v) ⊗ (E_{r,n}^{(m)}f_{β})

= ∑_{β∈Λ(n,ℓ)} ∑_{m=0}^{min(t,β_{r})}((\begin{matrix}t−m+β_{r} \\ t−m \end{matrix}) E(β + (t − m)α(r, n))v) ⊗ ((−1)^{m} (\begin{matrix}β_{r} \\ m \end{matrix}) f_{β−mα(r,n)}) .

Let γ ∈ Λ(n, ℓ). The γ-component of the expression above is

∑_{m=0}^{min(t,γ_{n})}(−1)^{m} (\begin{matrix}t+γ_{r} \\ t−m \end{matrix}) (\begin{matrix}γ_{r}+m \\ m \end{matrix}) E(γ + tα(r, n))v.

If t > γ_{n} then γ_{1} + ⋯ + γ_{n−1} + t > ℓ, so E(γ + tα(r, n))v = 0 as v belongs to the top ℓ levels. Otherwise,

∑_{m=0}^{t}(−1)^{m} (\begin{matrix}t+γ_{r} \\ t−m \end{matrix}) (\begin{matrix}γ_{r}+m \\ m \end{matrix}) = 0

which completes the proof.

4.5. Theorem. Let M be any submodule of ∇_{n}(λ) and let ē denote the restriction of e to M^{0} ⊕ ⋯ ⊕ M^{ℓ}. Then ē is an isomorphism between M^{0} ⊕ ⋯ ⊕ M^{ℓ} and (M ⊗ S^{ℓ}(V^{∗}))^{Y} as GL(n − 1)-modules.

Proof. Lemma 4.2 and Lemma 4.4 imply that ē is a well-defined injective homomorphism of GL(n − 1)-modules. To prove that it is surjective, take v ∈ (M ⊗ S^{ℓ}(V^{∗}))^{Y} of weight ν = (ν_{1}, …, ν_{n}). By Lemma 4.3, v = e(v_{ℓε_{n}}), so it remains to show that v_{ℓε_{n}} lies in the first ℓ levels of M. Suppose for a contradiction this is false, and choose v, ν so that ν is maximal in the dominance order subject to the condition v_{ℓε_{n}} ∉ M^{0} ⊕ ⋯ ⊕ M^{ℓ}.

If v_{ℓε_{n}} is a GL(n − 1)-highest weight vector, then by Lemma 4.2, v = e(v_{ℓε_{n}}) is a GL(n − 1)-highest weight vector, which by assumption is Y -invariant. Hence, v is a GL(n)- highest weight vector so, using the universality of standard modules, Lemma 4.1 implies that ν_{n} ≤ λ_{n}. But the weight of v_{ℓε_{n}} is ν + ℓε_{n}, so this contradicts the assumption v_{ℓε_{n}} ∉ M^{0} ⊕ ⋯ ⊕ M^{ℓ}.

So, v_{ℓε_{n}} is not a GL(n−1)-highest weight vector, so we can find some 1 ≤ i < j < n, k > 0 such that E_{i,j}^{(k)}v_{ℓε_{n}} ≠ 0. Applying the injective map e, this implies that E_{i,j}^{(k)}v ≠ 0. As E_{i,j}^{(k)}v

is Y -invariant, the maximality of ν now implies that E_{i,j}^{(k)}v_{ℓε_{n}} ∈ M^{0} ⊕ ⋯ ⊕ M^{ℓ}. But again this implies that v_{ℓε_{n}} ∈ M^{0} ⊕ ⋯ ⊕ M^{ℓ}, giving the desired contradiction.

4.6. Corollary. Fix λ, μ ∈ X^{+}(n) and a submodule M ≤ ∇_{n}(λ). The restriction of the map ē from Theorem 4.5 gives a bijection between the GL(n − 1)-highest weight vectors in M^{0} ⊕ ⋯ ⊕ M^{ℓ} of weight μ + ℓε_{n} and the GL(n)-highest weight vectors in M ⊗ S^{ℓ}(V^{∗}) of weight μ.

Proof. This follows from Theorem 4.5 since a vector v ∈ M ⊗ S^{ℓ}(V^{∗}) is GL(n)-primitive if and only if it is GL(n − 1)-primitive and lies in (M ⊗ S^{ℓ}(V^{∗}))^{Y}.

4.7. Theorem. Fix λ, μ ∈ X^{+}(n) and a submodule M ≤ ∇_{n}(λ). Let μ̄ = (μ_{1}, …, μ_{n−1}). (i) If μ_{n} ≤ λ_{n} then Hom_{GL(n)}(Δ_{n}(μ), M ⊗S^{ℓ}(V^{∗})) ≅ Hom_{GL(n−1)}(Δ_{n−1}(μ̄), M^{ℓ+μ_{n}−λ_{n}}). (ii) If μ_{n} = λ_{n} then Hom_{GL(n)}(L_{n}(μ), M ⊗ S^{ℓ}(V^{∗})) ≅ Hom_{GL(n−1)}(L_{n−1}(μ̄), M^{ℓ}).

Proof. (i) A vector v ∈ M has GL(n)-weight μ + ℓε_{n} if and only if it has GL(n − 1)-weight μ̄ and lies in M^{ℓ+μ_{n}−λ_{n}}. Since μ_{n} ≤ λ_{n}, such vectors lie in the first ℓ levels of M. So Corollary 4.6 now implies that there is a bijection between the GL(n − 1)-highest weight vectors of (T ∩ GL(n − 1))-weight μ̄ in M^{ℓ+μ_{n}−λ_{n}} and the GL(n)-highest weight vectors in M ⊗ S^{ℓ}(V^{∗}) of weight μ. The result now follows using the universality of standard modules.

(ii) Suppose now that μ_{n} = λ_{n}. In view of (i), it suffices to show that a GL(n − 1)- highest weight vector v ∈ M^{ℓ} of weight μ + ℓε_{n} generates an irreducible GL(n − 1)-module if and only if e(v) ∈ M_{n}(λ) ⊗ S^{ℓ}(V^{∗}) generates an irreducible GL(n)-module. Equivalently, applying Theorem 4.5, we need to prove that a GL(n − 1)-highest weight vector w in (M ⊗ S^{ℓ}(V^{∗}))^{Y} of weight μ generates an irreducible GL(n − 1)-module if and only if it generates an irreducible GL(n)-module. If the GL(n)-highest weight vector w generates an irreducible GL(n)-module then it certainly generates an irreducible GL(n − 1)-module by [J, II.2.11].

Conversely, let w ∈ M ⊗ S^{ℓ}(V^{∗}) be a GL(n)-highest weight vector of weight μ that generates a reducible GL(n)-module. Then, we can find an operator Y in the negative part of Dist(GL(n)) generated by all F_{i,j}^{(k)} such that Y w ∈ M is a non-zero GL(n)-highest weight vector of weight ν < μ. Observe that ν_{n} ≥ μ_{n} = λ_{n}, while by Lemma 4.1, ν_{n} ≤ λ_{n}. Thus, ν_{n} = λ_{n}, so by weights Y lies in Dist(GL(n−1)). But this implies that the GL(n−1)-module generated by w is also reducible as required.

We remark that Theorem 4.7(i) can also be deduced from [BK1, Corollary 2.10]. Part (ii) is certainly false if we try to weaken the assumption to μ_{n} ≤ λ_{n}. Theorem C follows from Theorem 4.7(ii):

4.8. Corollary. Fix λ, μ ∈ X^{+}(n) with λ_{n} = μ_{n}. Let ℓ = ∑_{i=1}^{n}(λ_{i} − μ_{i}) and μ̄ = (μ_{1}, …, μ_{n−1}). For any submodule M of ∇_{n}(λ),

Hom_{GL(n)}(L_{n}(μ), M ⊗ S^{ℓ}(V^{∗})) ≅ Hom_{GL(n−1)}(L_{n−1}(μ̄), M ↓_{GL(n−1)}).

Proof. This follows immediately from Theorem 4.7(ii) since the jth level M^{j} is the sum of all weight spaces M_{ν} with ν ∈ X^{+}(n) satisfying ν_{1} + ⋯ + ν_{n−1} = λ_{1} + ⋯ + λ_{n−1} − j.

Finally, we deduce Theorem D. For any λ = (λ_{1}, …, λ_{n}) ∈ X^{+}(n), we write λ^{∗} for the dominant weight −w_{0}λ = (−λ_{n}, −λ_{n−1}, …, −λ_{1}) ∈ X^{+}(n).

4.9. Theorem. Fix any λ ∈ X^{+}(n) and μ ∈ X^{+}(n − 1) with λ_{i+1} ≤ μ_{i} ≤ λ_{i} for i = 1, …, n − 1. Let μ̃ = (μ_{1}, …, μ_{n−1}, λ_{n}) ∈ X^{+}(n) and λ̄ = (λ_{1}, …, λ_{n−1}) ∈ X^{+}(n − 1). Then,

Hom_{GL(n−1)}(L_{n−1}(μ), ∇_{n}(λ) ↓_{GL(n−1)}) ≅ Hom_{GL(n−1)}(Δ_{n−1}(\overline{λ^{∗}}), L_{n}(μ̃^{∗}) ↓ GL(n − 1)).

Proof. We note that \overline{λ^{∗}} = (−λ_{n}, …, −λ_{2}) and μ̃^{∗} = (−λ_{n}, −μ_{n−1}, …, −μ_{1}). Let γ = μ̃, γ̄ = μ and ℓ = ∑_{i=1}^{n}(λ_{i} − γ_{i}). Arguing as in Corollary 4.8, it suffices to prove that

Hom_{GL(n−1)}(L_{n−1}(γ̄), ∇_{n}(λ)^{ℓ}) ≅ Hom_{GL(n−1)}(Δ_{n−1}(\overline{λ^{∗}}), L_{n}(γ^{∗})^{ℓ+μ_{1}−λ_{1}}).

Using Theorem 4.7 and the fact that μ_{1} ≤ λ_{1}, we have that

Hom_{GL(n−1)}(L_{n−1}(γ̄), ∇_{n}(λ)^{ℓ}) ≅ Hom_{GL(n)}(L_{n}(γ), ∇_{n}(λ) ⊗ S^{ℓ}(V^{∗}))
≅ Hom_{GL(n)}(Δ_{n}(λ^{∗}), L_{n}(γ^{∗}) ⊗ S^{ℓ}(V^{∗}))
≅ Hom_{GL(n−1)}(Δ_{n−1}(\overline{λ^{∗}}), L_{n}(γ^{∗})^{ℓ+μ_{1}−λ_{1}})

as claimed.

References

[B1] J. Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type A, to appear in Proc. London Math. Soc. (1998).

[B2] J. Brundan, Lowering operators for GL(n) and quantum GL(n), Proc. Symposia in Pure Math. 63 (1998), 95–114.

[BK1] J. Brundan and A. Kleshchev, Modular Littlewood-Richardson coefficients, preprint, University of Oregon, 1998.

14  Jonathan Brundan and Alexander Kleshchev

[BK2] J. Brundan and A. Kleshchev, On translation functors for general linear and symmetric groups, preprint, University of Oregon, 1998.

[BKS] J. Brundan, A. Kleshchev and I. Suprunenko, Semisimple restrictions from GL(n) to GL(n − 1), to appear in J. reine angew. Math. (1998).

[H1] J. E. Humphreys, Modular representations of classical Lie algebras and semisimple groups, J. Algebra 19 (1971), 51–79.

[H2] J. E. Humphreys, Introduction to Lie algebras and representation theory, sixth edition, Springer-Verlag, 1996.

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

[K1] A. 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. 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. Kleshchev, Branching rules for modular representations of symmetric groups I, J. Algebra 178 (1995), 493–511.

[K4] A. Kleshchev, Branching rules for modular representations of symmetric groups II, J. reine angew. Math. 459 (1995), 163–212.

[S] R. Steinberg, Lectures on Chevalley Groups, Yale University, 1967.

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

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

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