MODULAR REPRESENTATIONS OF THE SUPERGROUP Q(n), I

JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

To Professor Steinberg with admiration

1. Introduction

The representation theory of the algebraic supergroup Q(n) has been studied quite intensively over the complex numbers in recent years, especially by Penkov and Serganova [18, 19, 20] culminating in their solution [21, 22] of the problem of computing the characters of all irreducible finite dimensional representations of Q(n). The characters of one important family of irreducible representations, the so-called polynomial representations, had been determined earlier by Sergeev [24], exploiting an analogue of Schur-Weyl duality connecting polynomial representations of Q(n) to the representation theory of the double covers Ŝ_{n} of the symmetric groups. In [2], we used Sergeev’s ideas to classify for the first time the irreducible representations of Ŝ_{n} over fields of positive characteristic p > 2. In the present article and its sequel, we begin a systematic study of the representation theory of Q(n) in positive characteristic, motivated by its close relationship to Ŝ_{n}.

Let us briefly summarize the main facts proved in this article by purely algebraic techniques. Let G = Q(n) defined over an algebraically closed field k of characteristic p ≠ 2, see §§2-3 for the precise definition. In §4 we construct the superalgebra Dist(G) of distributions on G by reduction modulo p from a Kostant ℤ-form for the enveloping superalgebra of the Lie superalgebra 𝔮(n, ℂ). This provides one of the main tools in the remainder of the paper: there is an explicit equivalence between the category of representations of G and the category of “integrable” Dist(G)- supermodules (see Corollary 5.7).

In §6, we classify the irreducible representations of G by highest weight theory. They turn out to be parametrized by the set

X_{p}^{+}(n) = {(λ_{1}, …, λ_{n}) ∈ ℤ^{n} | λ_{1} ≥ ⋯ ≥ λ_{n} with λ_{i} = λ_{i+1} only if p|λ_{i}}.

For λ ∈ X_{p}^{+}(n), the corresponding irreducible representation is denoted L(λ), and is constructed naturally as the simple socle of an induced representation

H^{0}(λ) := ind_{B}^{G}𝔲(λ)

where B is a Borel subgroup of G and 𝔲(λ) is a certain irreducible representation of B of dimension a power of 2. The main difficulty here is to show that H^{0}(λ) ≠ 0 for λ ∈ X_{p}^{+}(n), which we prove by exploiting the main result of [2] classifying the irreducible polynomial representations of G: so ultimately the proof that H^{0}(λ) ≠ 0 depends on a counting argument involving p-regular conjugacy classes in Ŝ_{n}.

Both authors partially supported by the NSF (grant nos DMS-9801442 and DMS-9900134).

We then turn to considering extensions between irreducible representations. Unlike for reductive algebraic groups, self-extensions are possible, arising from the fact that representations of the “diagonal” subgroup H, which plays the role of maximal torus, are not completely reducible. For example, see Lemma 7.7, there is an odd extension between the trivial module and itself. There is a “linkage principle”, see Theorem 8.10, involving the notion of residue content of a weight λ ∈ X_{p}^{+}(T) which already appeared in work of Leclerc and Thibon [15]. Our proof of the linkage principle is similar to the original Carter-Lusztig proof [3] of the linkage principle for GL(n): there are enough explicitly known central elements in Dist(G) thanks to the work of Sergeev [25].

There is also an analogue of the Steinberg tensor product theorem, see Theorem 9.9. For this, we exploit the Frobenius morphism

F : Q(n) → GL(n)

defined by raising matrix entries to the power p. Given any irreducible representation L of GL(n), its Frobenius twist F^{∗}L gives an irreducible representation of G. We show that any irreducible representation of G can be decomposed as a tensor product of such an F^{∗}L and a restricted irreducible representation. Finally, the restricted irreducible representations of G are precisely the L(λ) which remain irreducible over the Lie superalgebra of G. They are parametrized by the set

\begin{aligned}X_{p}^{+}(n)_{res} = \{(λ_{1}, …, λ_{n}) ∈ X_{p}^{+}(n) | & λ_{i} − λ_{i+1} ≤ p if p ∤ λ_{i}, \\ & λ_{i} − λ_{i+1} < p if p | λ_{i}\}. \end{aligned}

In particular, the determinant representation det of GL(n) gives us a one dimensional representation F^{∗} det of G of highest weight (p, p, …, p). Thus in positive characteristic G has many one dimensional representations, unlike over ℂ when there is only the trivial representation. If M is an arbitrary finite dimensional representation of G, we can tensor M with a sufficiently large power of F^{∗} det to obtain a polynomial representation. So in positive characteristic, the category of polynomial representations is just as hard to understand as the category of all “rational” representations. To further clarify the connection between polynomial and rational representations, we show in §10 that a representation of G is polynomial if and only if it is polynomial over the diagonal subgroup H, following an argument due to Jantzen [12]. For example the representations H^{0}(λ) are polynomial whenever all λ_{i} are non-negative.

Acknowledgements. We would like to thank D. Nakano for pointing out [12] to us.

2. Superschemes and supergroups

THE SUPERGROUP Q(n) 3

Let k be a fixed algebraically closed field of characteristic p ≠ 2. All objects considered here (vector superspaces, superalgebras, superschemes, …) will be defined over k. By a commutative superalgebra, we mean a ℤ_{2}-graded algebra A = A¯0 ⊕ A¯1 such that ab = (−1)^{āb̄}ba for all a, b ∈ A. Here, ā ∈ ℤ_{2} is our notation for the degree of homogeneous a ∈ A, and the preceding formula is to be interpreted for not necessarily homogeneous a, b by extending linearly from the homogeneous case. We will write 𝔰𝔞𝔩𝔤_{k} for the category of all commutative superalgebras and even homomorphisms. For more of the basic notions of superalgebra, we refer to [16, ch.I] and [17, ch.3, §§1–2, ch.4, §1].

A superscheme X = (X, 𝒪_{X}) means a superscheme over k in the sense of [17, ch.4,§1.6]. Let 𝔰𝔰𝔠𝔥_{k} denote the category of all superschemes. In this article, we will always adopt a functorial language for superschemes similar to the language of Demazure and Gabriel [5, ch.I] or Jantzen [11]. Thus, we will identify a superscheme X with its associated functor

Hom_{𝔰𝔰𝔠𝔥_{k}}(Spec ?, X) : 𝔰𝔞𝔩𝔤_{k} → 𝔰𝔢𝔱𝔰.

For example, the affine superscheme 𝔸^{m|n} can be defined as

𝔸^{m|n} := Spec k[x_{1}, …, x_{m}; x′_{1}, …, x′_{n}],

the latter denoting the prime spectrum of the free commutative superalgebra on even generators x_{i} and odd generators x′_{j}. With the functorial language, we are viewing 𝔸^{m|n} instead as the functor defined on a commutative superalgebra A by 𝔸^{m|n}(A) = A⊕m ; ¯0 ⊕ A⊕n ; ¯1 and coordinatewise on morphisms. In general if X is an affine superscheme we write k[X] for its coordinate ring, which we identify with the superalgebra Mor(X, 𝔸^{1|1}) of all natural transformations from the functor X to the functor 𝔸^{1|1}. Then, using functorial language, X = Hom_{𝔰𝔞𝔩𝔤_{k}}(k[X], −).

By a supergroup, we mean a functor G from the category 𝔰𝔞𝔩𝔤_{k} to the category 𝔤𝔯𝔬𝔲𝔭𝔰. We say that G is an algebraic supergroup if it is an affine superscheme (when viewed as a functor to 𝔰𝔢𝔱𝔰) whose coordinate ring k[G] is finitely generated as a ksuperalgebra. In that case, k[G] has a canonical structure of Hopf superalgebra, with comultiplication Δ : k[G] → k[G] ⊗ k[G], antipode S : k[G] → k[G] and counit E : k[G] → k defined as the comorphisms of the multiplication, the inverse and the unit of G. The underlying purely even group G_{ev} is the closed subgroup of G corresponding to the Hopf superideal k[G]k[G]¯1 of the coordinate ring; as a functor, we have that G_{ev}(A) = G(A¯0) for all commutative superalgebras A.

For example, for any vector superspace M, we have the supergroup GL(M) with GL(M, A) (for each commutative superalgebra A) equal to the group of all even Alinear automorphisms of M ⊗ A. For finite dimensional M, GL(M) is an algebraic supergroup, and the underlying even group is the algebraic group GL(M¯0)×GL(M¯1).

A representation of an algebraic supergroup G means a natural transformation ρ : G → GL(M) for some vector superspace M. As usual, there is an equivalent moduletheoretic formulation: a G-supermodule M means a vector superspace equipped with an even structure map η_{M} : M → M ⊗ k[G] making M into a right k[G]-comodule in the usual sense. Given a G-supermodule M, one obtains a representation ρ : G → GL(M) by defining the action of each G(A) on M ⊗ A by g(m ⊗ 1) = ∑_{i} m_{i} ⊗ f_{i}(g), if η_{M}(m) = ∑_{i} m_{i} ⊗ f_{i}. Conversely, given a representation ρ : G → GL(M), one lets η_{M} : M → M ⊗ k[G] be the map m ↦ ρ(id_{k[G]})(m ⊗ 1_{k[G]}), where id_{k[G]} denotes the element of G(k[G]) ≅ Hom_{𝔰𝔞𝔩𝔤_{k}}(k[G], k[G]) corresponding to the identity map.

We write 𝔪𝔬𝔡_{G} for the category of all G-supermodules and arbitrary (not necessarily homogeneous) homomorphisms. So the category 𝔪𝔬𝔡_{G} is not an abelian category. However the underlying even category consisting of the same objects and only even morphisms is abelian, which allows us to make use of all the usual machinery of homological algebra. We also have the parity change functor

Π : 𝔪𝔬𝔡_{G} → 𝔪𝔬𝔡_{G}

defined on objects by letting ΠM equal M as a k[G]-comodule, but with the opposite ℤ_{2}-grading.

In general, we write M ≃ N if G-supermodules M and N are isomorphic in the underlying even category, and M ≅ N if they are isomorphic in 𝔪𝔬𝔡_{G} itself. We say that an irreducible G-supermodule M is of type Q if M ≃ ΠM and of type M otherwise. By the superalgebra analogue of Schur’s lemma,

End_{G}(M) ≃ \{ \begin{matrix}k & if M is of type M, \\ k ⊕ Πk & if M is of type Q. \end{matrix}

We have the right regular representation of G on k[G] defined for each superalgebra A by (gf)(h) = f(hg) for f ∈ k[G] ⊗ A = A[G_{A}], g ∈ G(A) and h ∈ G(B) for all A-superalgebras B. The structure map of the associated G-supermodule is just the comultiplication Δ : k[G] → k[G] ⊗ k[G]. Instead, the left regular representation is defined by the formula (gf)(h) = f(g^{−1}h).

3. The supergroup Q(n)

For the remainder of the article, G will denote the supergroup Q(n). Thus, G is the functor from 𝔰𝔞𝔩𝔤_{k} to the category 𝔤𝔯𝔬𝔲𝔭𝔰 defined on a superalgebra A by letting G(A) be the group of all invertible 2n × 2n matrices (under usual matrix multiplication) of the form

g = ( \begin{matrix}S & S′ \\ −S′ & S \end{matrix} )  (3.1)

where S is an n×n matrix with entries in A¯0 and S′ is an n×n matrix with entries in A¯1. On a morphism f : A → B, G(f) : G(A) → G(B) is the group homomorphism that arises by applying f to each matrix entry. The underlying even group G_{ev} is isomorphic to GL(n), being the functor mapping a superalgebra A to the group of all invertible matrices of the form (3.1) with S′ = 0.

To see that G is an algebraic supergroup, let Mat denote the functor from 𝔰𝔞𝔩𝔤_{k} to 𝔰𝔢𝔱𝔰 defined for a superalgebra A so that Mat(A) is the set of all (not necessarily invertible) matrices of the form (3.1). Then Mat is isomorphic to the affine superscheme 𝔸^{n^{2}|n^{2}}, with coordinate ring k[Mat] being the free commutative superalgebra on even generators s_{i,j} and odd generators s′_{i,j} for 1 ≤ i, j ≤ n, these being the natural transformations Mat → 𝔸^{1|1} picking out the ij-entries of the matrices S and S′ respectively when g ∈ Mat(A) is represented in the form (3.1). By [16, I.7.2], a matrix g ∈ Mat(A) is invertible if and only if det S ∈ A^{×}. Hence, G is the principal open subset of Mat defined by det : g ↦ det S. In particular, the coordinate ring k[G] is the localization of k[Mat] at the function det.

The Hopf superalgebra structure on k[G] is given explicitly in this case by the formulae

Δ(s_{i,j}) = ∑_{k=1}^{n}(s_{i,k} ⊗ s_{k,j} − s′_{i,k} ⊗ s′_{k,j}),

Δ(s′_{i,j}) = ∑_{k=1}^{n}(s_{i,k} ⊗ s′_{k,j} + s′_{i,k} ⊗ s_{k,j}),

E(s_{i,j}) = δ_{i,j}, E(s′_{i,j}) = 0

for 1 ≤ i, j ≤ n. Note k[Mat] is a subbialgebra of k[G], but is not invariant under the antipode.

We have the natural G-supermodule V, namely the vector superspace k^{n|n} with canonical basis v_{1}, …, v_{n}, v′_{1}, …, v′_{n}, where v_{1}, …, v_{n} are even and v′_{1}, …, v′_{n} are odd. Identify elements of V ⊗ A with column vectors so that

∑_{i=1}^{n}(v_{i} ⊗ a_{i} + v′_{i} ⊗ a′_{i}) ⟷ ( a_{1} ; ⋮ ; a_{n} ; a′_{1} ; ⋮ ; a′_{n} ) ∈ A^{n|n}.

Then, the action of G(A) on V ⊗ A is the obvious action on column vectors by left multiplication. In particular, the comodule structure map η satisfies

η(v_{j}) = ∑_{i=1}^{n}(v_{i} ⊗ s_{i,j} − v′_{i} ⊗ s′_{i,j}), η(v′_{j}) = ∑_{i=1}^{n}(v′_{i} ⊗ s_{i,j} + v_{i} ⊗ s′_{i,j})

for each j = 1, …, n. Note V is irreducible of type Q, since it possesses the odd automorphism

J : V → V,  v_{i} ↦ v′_{i}, v′_{i} ↦ −v_{i}.  (3.2)

Let H denote the closed subgroup of G defined on a commutative superalgebra A so that H(A) consists of all matrices of the form (3.1) such that S and S′ are diagonal matrices. Let B (resp. B^{+}) denote the subgroup with each B(A) (resp. B^{+}(A)) consisting of all matrices of the form (3.1) with S and S′ lower (resp. upper) triangular. The supergroup H will play the role of “maximal torus”, while B is the standard Borel subgroup.

There are supergroup epimorphisms

pr : B → H,  pr^{+} : B^{+} → H  (3.3)

defined for each commutative superalgebra A to be the projection of a matrix onto its diagonal part. We set U := ker pr and U^{+} = ker pr^{+}, respectively. Note B (resp. B^{+}) is the semidirect product of U (resp. U^{+}) by H, in the sense of [11, I.2.6].

We call an algebraic supergroup U unipotent if the fixed point space

M^{U} := {m ∈ M | η_{M}(m) = m ⊗ 1_{k[U]}}
= {m ∈ M | u(m ⊗ 1_{A}) = m ⊗ 1_{A} for all

superalgebras A and all u ∈ U(A)}

is non-zero for all non-zero U-supermodules M.

Lemma 3.4. The algebraic supergroups U and U^{+} are unipotent.

Proof. One easily constructs a chain of closed normal subgroups

1 = U_{0} < U_{1} < ⋯ < U_{n(n−1)} = U

and closed subgroups Q_{i} < U such that for each i, U_{i} is the semidirect product of U_{i−1} by Q_{i}, with each Q_{i} isomorphic to one of the additive supergroups 𝔸^{1|0} or 𝔸^{0|1}. Now let M be any non-zero U-supermodule; we need to show that M^{U} ≠ 0. Proceeding by induction on i and using that M^{U_{i}} = (M^{U_{i−1}})^{Q_{i}}, it suffices to prove that the supergroups 𝔸^{1|0} and 𝔸^{0|1} are unipotent. That is well-known for 𝔸^{1|0}, and obvious for 𝔸^{0|1} since its regular representation has precisely two composition factors (namely, k and Πk) both of which are trivial. □

We have the inflation functor

pr^{∗} : 𝔪𝔬𝔡_{H} → 𝔪𝔬𝔡_{B}

coming from the surjection pr of (3.3); we will generally regard H-supermodules as B-supermodules by inflation in this way without further comment. Taking U-fixed points defines a right adjoint to inflation. Using this and Lemma 3.4 shows that every irreducible B-supermodule is the inflation of an irreducible H-supermodule, and similarly for B^{+}.

Let T = H_{ev}. Observe that T is precisely the usual n-dimensional torus of G_{ev} ≅ GL(n) consisting of the diagonal matrices. We denote the character group of T by X(T); it is the free abelian group on generators ε_{1}, …, ε_{n}, where ε_{i} : T → 𝔾_{m} picks out the ith diagonal entry. We have the root system of GL(n), namely, R = R^{+} ∪ −(R^{+}) where

R^{+} = {ε_{i} − ε_{j} | 1 ≤ i < j ≤ n}.

We partially order X(T) by the usual dominance order, so λ ≤ μ if and only if μ − λ is a sum of positive roots. The x^{λ} := x_{1}^{λ_{1}} … x_{n}^{λ_{n}} for λ = ∑_{i=1}^{n} λ_{i}ε_{i} ∈ X(T) form a basis for the algebra ℚ[x_{1}^{±1}, …, x_{n}^{±n}]. Given any finite dimensional T -module M and λ ∈ X(T), we denote by M_{λ} the corresponding weight space, and the character of M is defined as

ch M := ∑_{λ∈X(T)}(dim M_{λ})x^{λ} ∈ ℚ[x_{1}^{±1}, …, x_{n}^{±n}].

Let W denote the symmetric group S_{n}, viewed (for each commutative superalgebra A) as the subgroup of G(A) consisting of matrices of the form (3.1) with S being a permutation matrix and S′ being zero. The longest element of W will be denoted w_{0} as usual. There is a natural left action of W on X(T), hence on ℚ[x_{1}^{±1}, …, x_{n}^{±1}] by permuting the x_{i}. Obviously, if M is a finite dimensional G-supermodule then its character is W -invariant, so lies in the algebra ℚ[x_{1}^{±1}, …, x_{n}^{±n}]^{W} of symmetric functions.

Finally, we describe the “big cell” in this setting. For 1 ≤ m ≤ n, define det_{m} to be the determinant of the m × m matrix (s_{i,j})_{1≤i,j≤m}. We will denote the principal open subset of Mat defined by det_{1} det_{2} … det_{n} by Ω; so Ω is an affine superscheme with coordinate ring k[Ω] being the localization of k[Mat] at det_{1} det_{2} … det_{n}.

Theorem 3.5. Multiplication defines an isomorphism of affine superschemes between B × U^{+} and Ω.

Proof. We need to show that for every commutative superalgebra A, matrix multiplication B(A)×U^{+}(A) → G(A) maps B(A)×U^{+}(A) bijectively onto Ω(A), namely, the set of all g ∈ Mat(A) for which det_{1}(g) det_{2}(g) … det_{n}(g) is a unit in A. This can be proved by induction on n, we leave the details to the reader. □

4. Distributions

Let X be an affine superscheme and x ∈ X(k). Then, there is a general notion of the superspace Dist(X, x) of distributions on X with support at x. The definition is an obvious extension of the purely even case, see [11, I.7]: if I_{x} denotes the kernel of the evaluation map k[X] → k, f ↦ f(x), a superideal of k[X], then

Dist(X, x) := ∑_{n≥0} Dist_{n}(X, x)

where Dist_{n}(X, x) ≅ (k[X]/I_{x}^{n+1})^{∗} is the annihilator in k[X]^{∗} of I_{x}^{n+1}. The tangent space at x,

T_{x}X := (I_{x}/I_{x}^{2})^{∗},

is naturally identified with a subspace of Dist(X, x).

Now suppose that G is an algebraic supergroup and let e ∈ G(k) be the identity element. We make Dist(G) := Dist(G, e) into a cocommutative Hopf superalgebra in a similar way to the purely even case [11, I.7.7], by taking the duals of the Hopf superalgebra structure maps on k[G]. The only significant difference is that now we identify Dist(G) ⊗ Dist(G) ⊆ k[G]^{∗} ⊗ k[G]^{∗} with a subset of (k[G] ⊗ k[G])^{∗} via the map

i : k[G]^{∗} ⊗ k[G]^{∗} ↪ (k[G] ⊗ k[G])^{∗}, (i(u ⊗ v))(a ⊗ b) = (−1)^{v̄ā}u(a)v(b) (4.1)

for u, v ∈ k[G]^{∗}, a, b ∈ k[G]. The supercommutator [., .] on Dist(G) gives T_{e}G the structure of a Lie superalgebra, denoted Lie(G). Also note that if M is a G-supermodule with structure map η : M → M ⊗ k[G], we can view M as a left Dist(G)-supermodule with action um := (id_{M} ¯⊗u)η(m).

We wish to describe Dist(G) explicitly for G = Q(n). The most concrete way to realize this is by reduction modulo p via a ℤ-form of the enveloping superalgebra of the Lie superalgebra 𝔮(n, ℂ). So recall first that 𝔮(n, ℂ) is the Lie superalgebra of all matrices of the form

x = ( \begin{matrix}S & S′ \\ S′ & S \end{matrix} )  (4.2)

under the supercommutator [., .], where S and S′ are n × n matrices over ℂ, and such a matrix is even if S′ = 0 or odd if S = 0. Let 𝒰_{ℂ} be the universal enveloping superalgebra of 𝔮(n, ℂ), see [13, 23]. For 1 ≤ i, j ≤ n, let e_{i,j} (resp. e′_{i,j}) denote the matrix of the form (4.2) where the ij-entry of S (resp. S′) is 1, and all other entries are zero. Thus, the {e_{i,j}, e′_{i,j} | 1 ≤ i, j ≤ n} form a basis of 𝔮(n, ℂ). By the PBW theorem for Lie superalgebras, see [23, §2.3, Cor.1], we obtain a basis for 𝒰_{ℂ} consisting of all monomials

∏_{1≤i,j≤n} e_{i,j}^{a_{i,j}} ∏_{1≤i,j≤n} (e′_{i,j})^{d_{i,j}}

where a_{i,j} are non-negative integers, d_{i,j} ∈ {0, 1}, and the product is taken in any fixed order. We set h_{i} := e_{i,i}, h′_{i} := e′_{i,i} for short.

Define the Kostant ℤ-form 𝒰_{ℤ} to be the ℤ-subalgebra of 𝒰_{ℂ} generated by all elements of the form

e_{i,j}^{(m)}, e′_{i,j}, \binom{h_{r}}{m}, h′_{r}, 1 ≤ i ≠ j ≤ n, 1 ≤ r ≤ n, m ≥ 0,

where e_{i,j}^{(m)} denotes the divided power \frac{e_{i,j}^{m}}{m!} , \binom{h}{m} denotes h(h − 1) … (h − m + 1)/(m!). The standard comultiplication δ : 𝒰_{ℂ} → 𝒰_{ℂ}⊗𝒰_{ℂ} and counit ε : 𝒰_{ℂ} → ℂ are defined as the unique superalgebra homomorphisms with δ(x) = x ⊗ 1 + 1 ⊗ x, ε(x) = 0 for any x ∈ 𝔮(n, ℂ) ⊂ 𝒰_{ℂ}, see e.g. [23, §2.4]. Also the antipode σ : 𝒰_{ℂ} → 𝒰_{ℂ} is defined as the unique superalgebra antiautomorphism with σ(x) = −x for all x ∈ 𝔮(n, ℂ) ⊂ 𝒰_{ℂ}. These maps restrict to give a comultiplication δ : 𝒰_{ℤ} → 𝒰_{ℤ}⊗𝒰_{ℤ}, an antipode σ : 𝒰_{ℤ} → 𝒰_{ℤ} and a counit ε : 𝒰_{ℤ} → ℤ, which make 𝒰_{ℤ} into a Hopf superalgebra over ℤ. Following the proof of [26, Th.2] one verifies the following:

Lemma 4.3. The superalgebra 𝒰_{ℤ} is a ℤ-free ℤ-module with basis given by the set of all monomials of the form

∏_{1≤i≠j≤n} e_{i,j}^{(a_{i,j})} (e′_{i,j})^{d_{i,j}} ∏_{1≤i≤n} \binom{h_{i}}{a_{i,i}} (h′_{i})^{d_{i,i}} ,

for all a_{i,j} ∈ ℤ_{≥0} and d_{i,j} ∈ {0, 1} (the product being taken in some arbitrary but fixed order).

We return to working over our fixed algebraically closed field k. Define 𝒰_{k} := 𝒰_{ℤ} ⊗_{ℤ} k, naturally a Hopf superalgebra over k. We will write h_{i}, \binom{h_{i}}{m}, e_{i,j}^{(m)}, h′_{i}, and

e′_{i,j} for the elements h_{i} ⊗ 1, \binom{h_{i}}{m} ⊗ 1, e_{i,j}^{(m)} ⊗ 1, h′_{i} ⊗ 1, and e′_{i,j} ⊗ 1 of 𝒰_{k}, respectively (in spite of potential ambiguities).

Theorem 4.4. 𝒰_{k} is isomorphic as a Hopf superalgebra to Dist(G).

Proof. In case k = ℂ, the isomorphism i : 𝒰_{k} → Dist(G) is induced uniquely by the Lie superalgebra isomorphism i : 𝔮(n, ℂ) → Lie(G) that maps the basis {e_{i,j}, e′_{i,j} |1 ≤ i, j ≤ n} of 𝔮(n, ℂ) to the unique basis of Lie(G) dual to {s_{i,j}, s′_{i,j} | 1 ≤ i, j ≤ n}. For arbitrary k, the isomorphism is the reduction modulo p of the isomorphism over ℂ, cf. [11, II.1.12]. □

We henceforth identify 𝒰_{k} and Dist(G) in this way. We can also realize each of the superalgebras Dist(H), Dist(B), Dist(B^{+}) etc... as subalgebras of 𝒰_{k}. For instance, Dist(H) is the subalgebra generated by all h′_{i} and \binom{h_{i}}{m}, while Dist(B) is the subalgebra generated by Dist(H) together with all e_{i,j}^{(m)} and e′_{i,j} for i > j.

5. Integrable representations

For λ = ∑_{i=1}^{n} λ_{i}ε_{i} ∈ X(T) and a Dist(G)-supermodule M, define

M_{λ} := \{m ∈ M | \binom{h_{i}}{r}m = \binom{λ_{i}}{r}m for all i = 1, …, n, r ≥ 1\}.

We note by [11, I.7.14] that if M is a Dist(G)-supermodule arising from a Gsupermodule in the standard way, then M_{λ} as defined here coincides with the λweight space M_{λ} taken with respect to the original action of T. Now call a Dist(G)- supermodule M integrable if

(i) M is locally finite, i.e. it is the sum of its finite dimensional Dist(G)- submodules;

(ii) M = ∑_{λ∈X(T)} M_{λ}. It is easy to see that if M is a G-supermodule, then M is integrable when viewed as a Dist(G)-supermodule. Our goal in this section is to prove the converse statement: if M is an integrable Dist(G)-supermodule then the Dist(G)-action lifts uniquely to make M into a G-supermodule.

Introduce the restricted dual Dist(G)^{⋄} of the Hopf superalgebra Dist(G). By definition, this is the set of all f ∈ Dist(G)^{∗} vanishing on some two-sided superideal I (depending on f) such the left module Dist(G)/I is finite dimensional and integrable. If M is any integrable Dist(G)-supermodule with homogeneous basis m_{i}(i ∈ I), we define its coefficient space c(M) to be the subspace of Dist(G)^{∗} spanned by the coefficient functions f_{i,j} defined from

um_{j} = (1 ¯⊗u) (∑_{i∈I} m_{i} ⊗ f_{i,j})  (5.1)

for all u ∈ Dist(G). Note this definition of c(M) is independent of the choice of homogeneous basis. Also, if 0 → N → M → Q → 0 is a short exact sequence of integrable Dist(G)-supermodules, then c(N), c(Q) ⊆ c(M). The following is easy to check (cf. [7, (3.1a)]):

Lemma 5.2. f ∈ Dist(G)^{∗} belongs to Dist(G)^{⋄} if and only if f ∈ c(M) for some integrable Dist(G)-supermodule M.

We wish to give Dist(G)^{⋄} the structure of a Hopf superalgebra. To do this, we identify Dist(G)^{∗} ⊗ Dist(G)^{∗} with a subset of (Dist(G) ⊗ Dist(G))^{∗} as in (4.1), so f ⊗ g corresponds to the function with (f ⊗ g)(u ⊗ v) = (−1)^{ḡū}f(u)g(v). Then, the dual map to the comultiplication δ on Dist(G) gives a multiplication on Dist(G)^{∗}. If M and N are two integrable Dist(G)-supermodules, then M ⊗ N is also integrable, and c(M ⊗ N) = c(M)c(N). It therefore follows from Lemma 5.2 that Dist(G)^{⋄} is a subalgebra of Dist(G)^{∗}. One then checks directly from the definition of Dist(G)^{⋄} that the dual map to the multiplication on Dist(G) embeds Dist(G)^{⋄} into Dist(G)^{⋄}⊗ Dist(G)^{⋄}, so that its restriction gives us a comultiplication on Dist(G)^{⋄}. Moreover, the restriction of σ^{∗} gives an antipode, denoted S : Dist(G)^{⋄} → Dist(G)^{⋄}, evaluation at 1 gives a counit E, and ε gives a unit.

The main point now is to identify the Hopf superalgebras Dist(G)^{⋄} and k[G]. We define a map ∼: k[G] → Dist(G)^{∗} by setting f̃(u) = (−1)f̄ ; ūu(f) for all f ∈ k[G], u ∈ Dist(G). The s_{i,j}, resp. the S(s_{i,j}) map to the coefficient functions of the natural G-supermodule V, resp. it dual. Hence by Lemma 5.2, the image of k[G] under ∼ is contained in Dist(G)^{⋄}. Since the Hopf superalgebra structure on Dist(G) is dual to that on k[G] and the structure on Dist(G)^{⋄} is dual to that on Dist(G), one gets at once that ∼ is a Hopf superalgebra homomorphism. Moreover, if f̃(u) = 0 for all u ∈ Dist(G), then u(f) = 0 for all u ∈ Dist_{n}(G) so that f ∈ I_{e}^{n+1} for each n, i.e. f ∈ ⋂_{n≥0} I_{e}^{n+1}. But this is zero by Krull’s intersection theorem (or a direct calculation), hence f = 0 and ∼ is in fact injective.

Let us henceforward identify k[G] with its image under ∼; then we wish to prove that in fact k[G] = Dist(G)^{⋄}. We need to appeal to the analogous result in the classical case of GL(n). Now, Dist(G_{ev}) is the subalgebra of Dist(G) generated by

\{e_{i,j}^{(m)}, \binom{h_{k}}{m} | 1 ≤ i ≠ j ≤ n, 1 ≤ k ≤ n, m ≥ 0\} .  (5.3)

Just as for Dist(G), we define integrable Dist(G_{ev})-modules, the coefficient space c(M) of an integrable Dist(G_{ev})-module M, the restricted dual Dist(G_{ev})^{⋄} and so on. Note that the restriction of an integrable Dist(G)-supermodule to Dist(G_{ev}) is integrable as a Dist(G_{ev})-module by definition, so restriction of functions gives us a natural Hopf superalgebra homomorphism

ξ : Dist(G)^{⋄} → Dist(G_{ev})^{⋄}.

It is well-known that Dist(G_{ev})^{⋄} is equal to the coordinate ring k[G_{ev}], i.e. it is the localization of the free polynomial algebra k[c_{i,j} | 1 ≤ i, j ≤ n] where c_{i,j} := ξ(s_{i,j}) at determinant. In other words, as goes back at least to Kostant [14] (or see [7, (3.1c)]), we have that:

Lemma 5.4. Dist(G_{ev})^{⋄} is generated as an algebra by {c_{i,j}, S(c_{i,j}) | 1 ≤ i, j ≤ n}.

Fix for the remainder of the section some order for products in the PBW monomials so that every ordered PBW monomial is of the form mu with m being a product of the e′_{i,j} and u ∈ Dist(G_{ev}). Let Υ denote the resulting PBW basis of Dist(G), see Lemma 4.3. Set Γ = {(i, j) | 1 ≤ i, j ≤ n} and for I ⊆ Γ, let m_{I} ∈ Υ denote the ordered PBW monomial ∏_{(i,j)∈I} e′_{i,j}. By Lemma 4.3, we have a direct sum decomposition

Dist(G) = ⨁_{I⊆Γ} m_{I}Dist(G_{ev}),

showing that Dist(G) is a free right Dist(G_{ev})-supermodule with basis {m_{I} | I ⊆ Γ}. For I ⊆ Γ, define η_{I} ∈ Dist(G)^{∗} to be the “indicator function” of the monomial m_{I}, i.e. η_{I}(m_{I}) = 1 and η_{I}(m) = 0 for any other ordered PBW monomial m ∈ Υ different from m_{I}.

Lemma 5.5. For each I ⊆ Γ, η_{I} ∈ k[G].

Proof. Let N = n^{2}, and set M = ⋀^{N}(V ⊗V^{∗}), itself naturally a Dist(G)-supermodule. Since both c(V) and c(V^{∗}) lie in k[G] by definition, we certainly have that c((V ⊗ V^{∗})^{⊗N}) ⊆ k[G]. But M is a quotient of (V ⊗ V^{∗})^{⊗N}, so c(M) ⊆ k[G] too.

Recall that v_{1}, …, v_{n}, v′_{1}, …, v′_{n} denotes the natural basis of V. Set v_{n+i} := v′_{i} for short, and let w_{1}, …, w_{2n} denote the basis for V^{∗} dual to v_{1}, …, v_{2n}. Set z_{i,j} := v_{i} ⊗ w_{j} ∈ V ⊗ V^{∗} for i, j = 1, …, 2n. Fix also some linear order on the set of all pairs {(i, j)|i, j = 1, …, 2n}. Denote by Σ the set of all weakly increasing sequences S = ((i_{1}, j_{1}) ≤ (i_{2}, j_{2}) ≤ ⋯ ≤ (i_{N}, j_{N})) of length N such that (i_{s}, j_{s}) < (i_{s+1}, j_{s+1}) if z_{i_{s},j_{s}} is even. If S ∈ Σ, we denote by z_{S} the canonical image of the element z_{i_{1},j_{1}} ⊗ ⋯ ⊗ z_{i_{N} ,j_{N}} ∈ (V ⊗ V^{∗})^{⊗N} in M. Then ℬ := {z_{S} | S ∈ Σ} is a homogeneous basis for M. Moreover, letting z := z_{S} where S ∈ Σ is the sequence which contains every (i, j) ∈ Γ, z spans the trivial Dist(G_{ev})-submodule ⋀^{N}(V¯0 ⊗ V ∗ ; ¯0) of M.

The crucial step at this point is to observe that the vectors {m_{I}z | I ⊆ Γ} are linearly independent. This follows because they are related to the basis ℬ in a unitriangular way. Let 𝒞 be some homogeneous basis for M extending {m_{I}z|I ⊆ Γ}. For I ⊆ Γ and u ∈ Dist(G), define g_{I}(u) to be the m_{I}z-coefficient of uz when expressed in terms of the basis 𝒞. So, g_{I} ∈ k[G] by the first paragraph. It is clear from the definition that g_{I}(m_{J}) = δ_{I,J} for every I, J ⊆ Γ. Moreover, since z spans a trivial Dist(G_{ev})-module, uz = 0 for all ordered PBW monomials u ∈ Υ not of the form m_{J} for any J ⊆ Γ. Hence g_{I}(u) = 0 for such u, and we have now checked that g_{I} = η_{I}. □

Now we obtain the main result:

Theorem 5.6. Dist(G)^{⋄} = k[G].

Proof. We first claim that (η_{I}f)(m_{I}u) = f(u) and (η_{I}f)(m_{J}u) = 0 for any u ∈ Dist(G_{ev}), J ⊉ I, and f ∈ Dist(G)^{⋄}. Indeed, by definition of the product in Dist(G)^{⋄}, for any J ⊆ Γ we have (η_{I}f)(m_{J}u) = (η_{I} ¯⊗f)(δ(m_{J}u)). But, when expanded in terms of the PBW basis of Dist(G) ⊗ Dist(G), the (m_{I}⊗?)-component of δ(m_{J}u) is equal m_{I} ⊗ u if J = I and 0 if J ⊉ I. This implies the claim.

Now let f ∈ Dist(G)^{⋄} and Δ(f) = ∑_{j} f_{j} ⊗ g_{j}. The restrictions ξ(g_{j}) belong to Dist(G_{ev})^{⋄}, so by Lemma 5.4, there exist degree ¯0 elements h_{j} ∈ k[G] with ξ(g_{j}) = ξ(h_{j}). For any I ⊆ Γ, define f_{I} := ∑_{j} f_{j}(m_{I})η_{I}h_{j}, which is an element of k[G] thanks to 5.5. By the previous paragraph, we have

f_{I}(m_{I}u) = ∑_{j} f_{j}(m_{I})(η_{I}h_{j})(m_{I}u) = ∑_{j} f_{j}(m_{I})h_{j}(u)

= ∑_{j} f_{j}(m_{I})g_{j}(u) = f(m_{I}u).

Similarly f_{I}(m_{J}u) = 0 for any u ∈ Dist(G_{ev}) and any J ⊉ I. Thus, we have proved that given f ∈ Dist(G)^{⋄} one can find a function f_{I} ∈ k[G], with f_{I} = f on m_{I}Dist(G_{ev}) and f_{I} = 0 on ⨁_{J⊉I} m_{J}Dist(G_{ev}).

Now we can prove that f ∈ k[G]. For i = 0, 1, …, n^{2}, define the functions f^{(i)} ∈ k[G] by setting f^{(0)} := f − f_{∅}, and

f^{(i)} = f^{(i−1)} − ∑_{I∈Γ,|I|=i}(f^{(i−1)})_{I}.

By induction on i we prove that f^{(i)} = 0 on ⨁_{J,|J|≤i} m_{J}Dist(G_{ev}). In particular, f^{(n^{2})} is the zero element of Dist(G)^{⋄}. It remains to note that f^{(n^{2})} is obtained from f by subtracting elements of k[G]. □

The theorem has the following important consequence:

Corollary 5.7. The category 𝔪𝔬𝔡_{G} is isomorphic to the category of all integrable Dist(G)-supermodules.

Proof. Suppose M is an integrable Dist(G)-supermodule with homogeneous basis m_{i}(i ∈ I). Define the coefficient functions f_{i,j} ∈ Dist(G)^{⋄} as in (5.1). Let g_{i,j} be the element of k[G] corresponding to f_{i,j} under the identification, i.e. g̃_{i,j} = f_{i,j}. Then, we define a structure map η making M into a G-supermodule by setting

η(m_{j}) = ∑_{i} m_{i} ⊗ g_{i,j}.

Conversely, if M is a G-supermodule, we view M as a Dist(G)-supermodule in the standard way. Now check that the two constructions are inverse to one another and that morphisms of G-supermodules correspond to morphisms of Dist(G)- supermodules. □

Because of this corollary, we will not distinguish between G-supermodules and integrable Dist(G)-supermodules in the sequel.

6. Highest weight theory

Let H be any closed subgroup of an algebraic supergroup G. There are restriction and induction functors

res_{H}^{G} : 𝔪𝔬𝔡_{G} → 𝔪𝔬𝔡_{H}, ind_{H}^{G} : 𝔪𝔬𝔡_{H} → 𝔪𝔬𝔡_{G}.

The induction here is essentially “coalgebra induction” from [6, §3], and is defined as follows. If M is any vector superspace, we can view M ⊗k[G] as a G-supermodule with structure map id_{M} ⊗Δ_{G}; we will denote this by M_{tr} ⊗ k[G] to emphasize that the G-supermodule structure is trivial on M. Let δ : k[G] → k[H] ⊗ k[G] be the comorphism of the multiplication μ̄ : H × G → G. Then ind_{H}^{G}M is defined to be the kernel of the map ∂ = η ⊗ id_{k[G]} − id_{M} ⊗δ in the following exact sequence of G-supermodules:

0 ⟶ ind_{H}^{G}M ⟶ M_{tr} ⊗ k[G] ⟶^{∂} M_{tr} ⊗ k[H]_{tr} ⊗ k[G].  (6.1)

On a morphism f : M → M′ of H-supermodules, ind_{H}^{G}f is simply the restriction of the map f ⊗ id_{k[G]} to the subspace ind_{H}^{G}M ⊆ M_{tr} ⊗ k[G]. We remark that there is an equivalent way of characterizing ind_{H}^{G}M as a subspace of M_{tr} ⊗ k[G]:

ind_{H}^{G}M = (M ⊗ k[G])^{H}  (6.2)

where the H-fixed points are taken with respect to the given action on M and the left regular action on k[G].

Induction and restriction have the familiar properties (see e.g. [6, p.306]): for example, for a G-supermodule M, its structure map η determines a natural even isomorphism M ⟶^{∼} ind_{G}^{G}M. The functor res_{H}^{G} is exact and ind_{H}^{G} is right adjoint to res_{H}^{G}, hence is left exact and sends injectives to injectives. Now one can prove that 𝔪𝔬𝔡_{G} has enough injectives, by following the usual proof in the even case [11, I.3.9]. It therefore makes sense to consider the right derived functors R^{i}ind_{H}^{G} : 𝔪𝔬𝔡_{H} → 𝔪𝔬𝔡_{G}. We have the generalized tensor identity: for any G-supermodule M and H-supermodule N, there is a natural isomorphism

R^{i}ind_{H}^{G}(res_{H}^{G}M ⊗ N) ≃ M ⊗ R^{i}ind_{H}^{G}N  (6.3)

of G-supermodules. (We wrote down a proof based on the arguments of [9, 1.3] and [11, I.4.8].)

Turn now to our case G = Q(n). We wish to use induction from the Borel subgroup B of G to classify the irreducible G-supermodules by their highest weights. We should start with the case Q(1). To understand this case, we analyze its coordinate ring. We know that k[Q(1)] = k[s, s^{−1}, s′] (where s = s_{1,1}, s′ = s′_{1,1}), so it has basis

{s^{m}, s^{m−1}s′ | m ∈ ℤ}.

Let k[Q(1)]_{m} be the subspace spanned by s^{m}, s^{m−1}s′, making k[Q(1)] into a ℤ-graded superalgebra. The comultiplication satisfies

Δ(s^{m}) = s^{m} ⊗ s^{m} − ms^{m−1}s′ ⊗ s^{m−1}s′,
Δ(s^{m−1}s′) = s^{m−1}s′ ⊗ s^{m} + s^{m} ⊗ s^{m−1}s′,

showing that each k[Q(1)]_{m} is a two dimensional subcoalgebra of k[Q(1)]. The dual superalgebra k[Q(1)]_{m}^{∗} is generated by the odd element c_{m} defined from c_{m}(s^{m−1}s′) = 1, c_{m}(s^{m}) = 0, subject only to the relation c_{m}^{2} = m. So it is either a rank one Clifford superalgebra (if m is non-zero modulo p) or a rank one Grassmann superalgebra (if m is zero modulo p). In either case, k[Q(1)]_{m}^{∗} has a unique irreducible supermodule 𝔲(m) up to isomorphism. Now one proceeds in exactly the same way as in [2, §6] to conclude that the corresponding Q(1)-supermodules {𝔲(m)|m ∈ ℤ} form a complete set of pairwise non-isomorphic irreducible Q(1)-supermodules, with 𝔲(m) being of type M with character x_{1}^{m} if p|m and of type Q with character 2x_{1}^{m} if p ∤ m.

Notice that H ≅ Q(1) × ⋯ × Q(1) (n times). So using the previous paragraph and the general theory of representations of direct products of supergroups (which is entirely similar to [2, Lemma 2.9]), we obtain the following parametrization of the irreducible H-supermodules. Given λ = ∑_{i=1}^{n} λ_{i}ε_{i}, we define h_{p′}(λ) to be the number of i = 1, …, n for which p ∤ λ_{i}.

Lemma 6.4. For each λ ∈ X(T), there is a unique irreducible H-supermodule 𝔲(λ) (up to isomorphism) with character 2^{⌊(h_{p′}(λ)+1)/2⌋}x^{λ}. The {𝔲(λ) | λ ∈ X(T)} form a complete set of pairwise non-isomorphic irreducible H-supermodules. Moreover, 𝔲(λ) is of type M if h_{p′}(λ) is even, type Q if h_{p′}(λ) is odd.

As explained in §3, it follows immediately that the (inflations of) {𝔲(λ)|λ ∈ X(T)} give a complete set of pairwise non-isomorphic irreducible B-supermodules. Now we define

H^{0}(λ) := ind_{B}^{G}𝔲(λ),  L(λ) := soc_{G}H^{0}(λ),  (6.5)

for each λ ∈ X(T). Let

X^{+}(T) := {λ = ∑_{i=1}^{n} λ_{i}ε_{i} ∈ X(T) | λ_{1} ≥ ⋯ ≥ λ_{n}},  (6.6)
X_{p}^{+}(T) := {λ ∈ X^{+}(T) | λ_{i} = λ_{i+1} for some 1 ≤ i < n implies p|λ_{i}}.  (6.7)

We refer to weights λ lying in X^{+}(T) as dominant weights, and elements of X_{p}^{+}(T) are p-strict dominant weights. We will show that the {L(λ) | λ ∈ X_{p}^{+}(T)} form a complete set of pairwise non-isomorphic irreducible G-supermodules.

Lemma 6.8. Let ξ : k[G] → k[B] (resp. ξ^{+} : k[G] → k[B^{+}]) denote the comorphism of the inclusion B → G (resp. the inclusion B^{+} → G). Then, for any H-supermodule M, the restriction of the map id_{M} ⊗ξ^{+} : M_{tr} ⊗ k[G] → M_{tr} ⊗ k[B^{+}] to ind_{B}^{G}M ⊂ M ⊗ k[G] defines a monomorphism

ind_{B}^{G}M ↪ ind_{H}^{B^{+}}M

as B^{+}-supermodules.

Proof. To prove injectivity, let η : M → M ⊗k[B] be the structure map of M as a Bsupermodule. Take v ∈ ind_{B}^{G}M with (id_{M} ⊗ξ^{+})(v) = 0. By definition of induction, we have that

η ⊗ id_{k[G]}(v) = id_{M} ⊗([ξ ⊗ id_{k[G]}] ∘ Δ_{G})(v).

So applying id_{M} ⊗ id_{k[B]} ⊗ξ^{+} to both sides, we get that

0 = (η ⊗ id_{k[B^{+}]}) ∘ (id_{M} ⊗ξ^{+})(v) = id_{M} ⊗([ξ ⊗ ξ^{+}] ∘ Δ_{G})(v).

Now Theorem 3.5 implies that the map

(ξ ⊗ ξ^{+}) ∘ Δ_{G} : k[G] → k[B] ⊗ k[B^{+}]

is injective. Hence, v = 0 as required. Finally, the fact that the image lies in ind_{H}^{B^{+}}M ⊆ M ⊗ k[B^{+}] follows from (6.2). □

Lemma 6.9. View k[U^{+}] as a B^{+}-supermodule with structure map being the comorphism ρ^{∗} of the right action ρ : U^{+} × B^{+} → U^{+}, (u, b) ↦ pr^{+}(b)^{−1}ub, where pr^{+} : B^{+} → H is the projection (3.3). Then, for any H-supermodule M,

ind_{H}^{B^{+}}M ≃ M ⊗ k[U^{+}]

as a B^{+}-supermodule.

Proof. By definition, ind_{H}^{B^{+}}M is a B^{+}-submodule of M_{tr}⊗k[B^{+}]. View k[H]⊗k[U^{+}] as a B^{+}-supermodule via the inflation of the right regular H-action on k[H] and as described above on k[U^{+}]. Let μ : H × U^{+} → B^{+} be the superscheme isomorphism induced by multiplication. Then the map

id_{M} ⊗μ^{∗} : M_{tr} ⊗ k[B^{+}] → M_{tr} ⊗ k[H] ⊗ k[U^{+}]

is an isomorphism of B^{+}-supermodules. By definition of induction, it maps ind_{H}^{B^{+}}M to the subspace η(M) ⊗ k[U^{+}] of M ⊗ k[H] ⊗ k[U^{+}], where η : M → M ⊗ k[H] is the structure map of M. Finally, we observe that η(M) ≃ M as B^{+}-supermodules, and the lemma follows. □

Lemma 6.10. Suppose that λ ∈ X(T) such that H^{0}(λ) ≠ 0.

(i) If μ is a weight with H^{0}(λ)_{μ} ≠ 0, then w_{0}λ ≤ μ ≤ λ in the dominance order. (ii) The B^{+}-socle of H^{0}(λ) is precisely its λ-weight space H^{0}(λ)_{λ} ≃ 𝔲(λ). (iii) H^{0}(λ) is finite dimensional.

Proof. By Lemmas 6.8 and 6.9, there is an injective B^{+}-supermodule map

H^{0}(λ) ↪ ind_{H}^{B^{+}}𝔲(λ) ≃ 𝔲(λ) ⊗ k[U^{+}]

which is non-zero by the assumption on λ. By Frobenius reciprocity, ind_{H}^{B^{+}}𝔲(λ) has B^{+}-socle ≃ 𝔲(λ), hence H^{0}(λ) has B^{+}-socle ≃ 𝔲(λ) too. Considering the weights of 𝔲(λ) ⊗ k[U^{+}] gives that dim H^{0}(λ)_{λ} ≤ dim 𝔲(λ) and that H^{0}(λ)_{μ} = 0 unless μ ≤ λ. Parts (i) and (ii) follow easily. Finally, for (iii), each weight space of 𝔲(λ) ⊗ k[U^{+}] is finite dimensional, hence each weight space of H^{0}(λ) is also finite dimensional. But by (i) there are only finitely many non-zero weight spaces, hence H^{0}(λ) itself must be finite dimensional. □

Now we are ready to prove the main result. Note in case p = 0, the following theorem is due to Penkov [18].

Theorem 6.11. For λ ∈ X(T), H^{0}(λ) is non-zero if and only if λ ∈ X_{p}^{+}(T). The modules {L(λ) | λ ∈ X_{p}^{+}(T)} form a complete set of pairwise non-isomorphic irreducible G-supermodules. Moreover L(λ) is of type M if h_{p′}(λ) is even, type Q if h_{p′}(λ) is odd.

Proof. Note it is immediate from Lemma 6.10(ii) that the non-zero L(λ) are irreducible and pairwise non-isomorphic. Moreover, a Frobenius reciprocity argument shows that every irreducible G-supermodule is isomorphic to the socle of some nonzero H^{0}(λ). To check the statement about the type of L(λ) (whenever it is non-zero), note that the B-head of L(λ) is ≅ 𝔲(λ); this follows on conjugating with w_{0} and taking duals from the fact that the B^{+}-socle of L(−w_{0}λ) is 𝔲(−w_{0}λ). So we can calculate using Frobenius reciprocity:

End_{G}(L(λ)) ≅ Hom_{G}(L(λ), H^{0}(λ)) ≅ Hom_{B}(L(λ), 𝔲(λ)) ≅ End_{B}(𝔲(λ)).

Hence, L(λ) has the same type as 𝔲(λ) whenever it is non-zero, see Lemma 6.4. It now just remains to prove that

X_{p}^{+}(T) = {λ ∈ X(T) | H^{0}(λ) ≠ 0}.

Suppose first that λ ∈ X(T) and H^{0}(λ) ≠ 0. Since the set of weights of H^{0}(λ) is invariant under the action of W, Lemma 6.10(i) implies that λ_{1} ≥ ⋯ ≥ λ_{n}. Now suppose that λ_{i} = λ_{i+1} for some i; we need to show that p|λ_{i}. By restricting to the subgroup of G isomorphic to Q(2) corresponding to the ith and (i + 1)th matrix rows and columns, it suffices to do this in the case n = 2, so λ = (λ_{1}, λ_{2}) ∈ X(T) and λ_{1} = λ_{2}. Then, by Lemma 6.10(i), λ is the only non-zero weight of H^{0}(λ). So if we take 0 ≠ v ∈ H^{0}(λ)_{λ} and consider the action of Dist(G), we must have that e′_{2,1}v = 0. Hence,

e′_{1,2}e′_{2,1}v = (h_{1} + h_{2})v = 2λ_{1}v = 0,

so p|λ_{1} as required.

Conversely take λ ∈ X_{p}^{+}(T). We need to show that there exists an irreducible Gsupermodule L with L_{λ} ≠ 0 and L_{μ} = 0 for all μ ≰ λ. Write λ as λ_{−} + λ_{+} where λ_{+} (resp. λ_{−}) is obtained from λ by replacing all negative (resp. positive) parts by zero. By [2, Theorem 10.1], there exist irreducible G-supermodules L(λ_{+}) and L(−w_{0}λ_{−}) with highest weights λ_{+} and −w_{0}λ_{−} respectively. Then, M := L(λ_{+}) ⊗ L(−w_{0}λ_{−})^{∗} is a G-supermodule with M_{λ} ≠ 0 and M_{μ} = 0 for μ ≰ λ. So at least one of the composition factors of M must be an irreducible G-supermodule with highest weight λ. □

Let us finally introduce the G-supermodules

V (λ) := H^{0}(−w_{0}λ)^{∗}  (6.12)

for each λ ∈ X_{p}^{+}(T). Note V (λ) is generated by the B^{+}-stable submodule V (λ)_{λ} ≅ 𝔲(λ). The V (λ) are the “universal” highest weight modules according to the following lemma, proved as in [11, II.2.13]:

Lemma 6.13. Let λ ∈ X_{p}^{+}(T). If M is a G-supermodule which is generated by a B^{+}-stable submodule isomorphic to 𝔲(λ), then M is isomorphic to a quotient of V (λ).

7. Extensions

Next, we consider another basic principle of highest weight theory: it should be possible to compute extensions between L(λ) and other irreducibles by looking at soc_{G}(H^{0}(λ)/L(λ)), cf. [11, II.2.14].

Let 𝔲̃(λ) denote the injective hull of 𝔲(λ) as an H-supermodule. By reducing to the case H = Q(1), one checks:

dim 𝔲̃(λ) = 2^{n−h_{p′}(λ)} dim 𝔲(λ).  (7.1)

For λ ∈ X_{p}^{+}(T), define

H̃^{0}(λ) := ind_{B}^{G}𝔲̃(λ).  (7.2)

Since ind_{B}^{G} is left exact, there is a canonical embedding H^{0}(λ) ↪ H̃^{0}(λ). Moreover, arguing using Lemma 6.8 as we did in the proof of Lemma 6.10,

soc_{B^{+}} H̃^{0}(λ) = soc_{B^{+}}H^{0}(λ) ≃ 𝔲(λ),  (7.3)
soc_{G}H̃^{0}(λ) = soc_{G}H^{0}(λ) ≃ L(λ).  (7.4)

Theorem 7.5. Let λ, μ ∈ X_{p}^{+}(T) with μ ≱ λ.

(i) Ext_{G}^{1}(L(μ), L(λ)) ≃ Hom_{G}(L(μ), H̃^{0}(λ)/L(λ)) ≃ Hom_{G}(L(μ), H^{0}(λ)/L(λ)). (ii) Ext_{G}^{1}(L(λ), L(λ)) ≃ Hom_{G}(L(λ), H̃^{0}(λ)/L(λ)).

Proof. Take arbitrary λ, μ ∈ X_{p}^{+}(T). We have the long exact sequence

0 ⟶ Hom_{G}(L(μ), L(λ)) ⟶^{∼} Hom_{G}(L(μ), H̃^{0}(λ))
⟶ Hom_{G}(L(μ), H̃^{0}(λ)/L(λ)) ⟶ Ext_{G}^{1}(L(μ), L(λ))
⟶ Ext_{G}^{1}(L(μ), H̃^{0}(λ)) ⟶ ….

So it suffices for the first isomorphism in (i) and (ii) to show that Ext_{G}^{1}(L(μ), H̃^{0}(λ)) = 0, providing μ ≯ λ. So suppose that we have an extension

0 ⟶ H̃^{0}(λ) ⟶^{f} M ⟶ L(μ) ⟶ 0.

Let i : H̃^{0}(λ) → 𝔲̃(λ) be the B-supermodule homomorphism induced by the identity map H̃^{0}(λ) → ind_{B}^{G}𝔲̃(λ) under adjointness. By injectivity of 𝔲̃(λ) and the assumption μ ≯ λ, we can find a B-supermodule homomorphism g : M → 𝔲̃(λ) such that g ∘ f = i. This induces a G-supermodule homomorphism ḡ : M → H̃^{0}(λ) which splits f. This proves the claim.

The second isomorphism in (i) is proved similarly but using the fact that

Ext_{H}^{1}(𝔲(λ), 𝔲(μ)) = 0

for λ ≠ μ, instead of injectivity of 𝔲̃(λ). □

As an application of the theorem, we compute Ext_{G}^{1}(k, k). Note 𝔲(0) = k. By (7.1), 𝔲̃(0) has dimension 2^{n}. Set

k̃ := ind_{B}^{G}𝔲̃(0).  (7.6)

Lemma 7.7. There is a non-split short exact sequence

0 ⟶ k ⟶ k̃ ⟶ Πk ⟶ 0

of G-supermodules.

Proof. Let M = Dist(H)⊗_{Dist(T)} (Π^{n}k). This is an integrable Dist(H)-supermodule, hence an H-supermodule. Note M has basis

h_{A} := h′_{a_{1}} … h′_{a_{n}} ⊗ 1

indexed by subsets A = {a_{1} < ⋯ < a_{n}} of {1, …, n}. Let M_{d} denote the subspace of M spanned by all such basis elements with |A| = d, so h′_{i}M_{d} ⊆ M_{d+1} for each i. One easily shows using this that soc_{H}M = M_{n} ≃ k. Hence, M ≃ 𝔲̃(0).

Now ind_{B}^{G}k ≃ k and 𝔲̃(0) has a composition series with all composition factors ≅ k. Hence, all composition factors of ind_{B}^{G}𝔲̃(0) are ≅ k too. This shows that the natural map ind_{B}^{G}𝔲̃(0) ↪ 𝔲̃(0)⊗k[U^{+}] arising from Lemmas 6.8 and 6.9 defines an embedding k̃ = ind_{B}^{G}𝔲̃(0) ↪ 𝔲̃(0). This identifies k̃ with the largest submodule of 𝔲̃(0) for which the action of H can be extended (necessarily uniquely) to an action of G.

So now suppose that N is a submodule of M maximal subject to the condition that the action of Dist(H) extends to an action of Dist(G). Clearly each h_{i} and each e_{i,j}, e′_{i,j} for i ≠ j must act as zero, hence [e_{i,j}, e′_{i,j}] = h′_{i} − h′_{j} acts as zero. This shows that N = {m ∈ M | h′_{1}m = ⋯ = h̄′_{n}m}. Now a straightforward induction gives that N has dimension 2 on basis

h′_{1} … h′_{n} ⊗ 1,  ∑_{i=1}^{n}(−1)^{i}h′_{1} … h′_{i−1}h′_{i+1} … h′_{n} ⊗ 1.

Since N ≃ k̃, this proves the lemma. □

Corollary 7.8. Ext_{G}^{1}(k, k) ≃ Πk.

8. The linkage principle

In [25], Sergeev constructed certain central elements of Dist(G). To describe them explicitly, in the notation of §4, we first define elements x_{i,j}(m), x′_{i,j}(m) ∈ Dist(G) for m ≥ 1, 1 ≤ i, j ≤ n, by setting x_{i,j}(1) = e_{i,j}, x′_{i,j}(1) = e′_{i,j}, and

x_{i,j}(m) = ∑_{s=1}^{n}(e_{i,s}x_{s,j}(m − 1) + (−1)^{m−1}e′_{i,s}x′_{s,j}(m − 1)),  (8.1)

x′_{i,j}(m) = ∑_{s=1}^{n}(e_{i,s}x′_{s,j}(m − 1) + (−1)^{m−1}e′_{i,s}x_{s,j}(m − 1))  (8.2)

for m > 1. The following commutation relations are noted in [25] (they are easily verified using induction on m):

[e_{i,j}, x_{s,t}(m)] = δ_{j,s}x_{i,t}(m) − δ_{i,t}x_{s,j}(m),
[e′_{i,j}, x_{s,t}(m)] = (−1)^{m−1}δ_{j,s}x′_{i,t}(m) − δ_{i,t}x′_{s,j}(m),
[e_{i,j}, x′_{s,t}(m)] = δ_{j,s}x′_{i,t}(m) − δ_{i,t}x′_{s,j}(m),
[e′_{i,j}, x′_{s,t}(m)] = (−1)^{m−1}δ_{j,s}x_{i,t}(m) + δ_{i,t}x_{s,j}(m).

Sergeev’s central elements are the elements

z_{r} := ∑_{i=1}^{n} x_{i,i}(2r − 1),  (8.3)

for r ≥ 1. One checks directly using the above commutator relations that the z_{r} are indeed central. It is even proved in [25] that the elements z_{1}, z_{2}, … generate the center of Dist(G) in case k = ℂ, but we will not need this fact. Instead, for λ = ∑_{i=1}^{n} λ_{i}ε_{i} ∈ X(T), define the integer

z_{r}(λ) := ∑(−2)^{s−1}λ_{i_{1}}λ_{i_{2}} … λ_{i_{s}}(λ_{i_{1}}^{2} − λ_{i_{1}})^{a_{1}}(λ_{i_{2}}^{2} − λ_{i_{2}})^{a_{2}} … (λ_{i_{s}}^{2} − λ_{i_{s}})^{a_{s}},

where the sum is over all 1 ≤ s ≤ r, 1 ≤ i_{1} < i_{2} < ⋯ < i_{s} ≤ n, and a_{1}, a_{2}, …, a_{s} ≥ 0 with a_{1} + ⋯ + a_{s} = r − s.

Lemma 8.4. Let M be a Dist(G)-supermodule and v ∈ M_{λ} be a vector of weight λ annihilated by all e_{i,j}, e′_{i,j} for all 1 ≤ i < j ≤ n. Then, z_{r}v = z_{r}(λ)v.

Proof. Let Ω be the left superideal of Dist(G) generated by all e_{i,j} and e′_{i,j} with i < j. We will write ≡ for congruence modulo Ω throughout the proof. We first show that

x_{i,j}(m) ≡ x′_{i,j}(m) ≡ 0 for each 1 ≤ i < j ≤ n and m ≥ 1.  (8.5)

This follows by induction on m, the induction base being clear. For m > 1, using the inductive assumption and the commutator relations, we get

x_{i,j}(m) = ∑_{s=1}^{n}(e_{i,s}x_{s,j}(m − 1) + (−1)^{m−1}e′_{i,s}x′_{s,j}(m − 1))

≡ ∑_{s=j}^{n}(e_{i,s}x_{s,j}(m − 1) + (−1)^{m−1}e′_{i,s}x′_{s,j}(m − 1))

= ∑_{s=j}^{n} (x_{s,j}(m − 1)e_{i,s} + x_{i,j}(m − 1)
−(−1)^{m−1}x′_{s,j}(m − 1)e′_{i,s} − x_{i,j}(m − 1))
≡ 0.

The proof of (8.5) for x′_{i,j}(m) is similar.

Now let m ≥ 3 be an odd integer. We claim

x_{i,i}(m) ≡ (h_{i}^{2} − h_{i})x_{i,i}(m − 2) − 2 ∑_{s=i+1}^{n} h_{i}x_{s,s}(m − 2).  (8.6)

Indeed, using (8.5) and the commutator relations, we see that

x_{i,i}(m) = ∑_{s=1}^{n}(e_{i,s}x_{s,i}(m − 1) + e′_{i,s}x′_{s,i}(m − 1))
≡ h_{i}x_{i,i}(m − 1) + h′_{i}x′_{i,i}(m − 1)

+ ∑_{s=i+1}^{n}(e_{i,s}x_{s,i}(m − 1) + e′_{i,s}x′_{s,i}(m − 1))

≡ h_{i}x_{i,i}(m − 1) + h′_{i}x′_{i,i}(m − 1).

Similarly, we get

x_{i,i}(m − 1) ≡ h_{i}x_{i,i}(m − 2) − h′_{i}x′_{i,i}(m − 2) − 2 ∑_{s=i+1}^{n} x_{s,s}(m − 2),
x′_{i,i}(m − 1) ≡ h_{i}x′_{i,i}(m − 2) − h′_{i}x_{i,i}(m − 2).

Substituting these formulas into the above expression for x_{i,i}(m) gives (8.6).

Now let r ≥ 1 and y_{i} := h_{i}^{2} − h_{i}. The theorem follows at once from the following formula:

x_{i,i}(2r − 1) ≡ ∑(−2)^{s−1}h_{i_{1}}h_{i_{2}} … h_{i_{s}}y_{i_{1}}^{a_{1}}y_{i_{2}}^{a_{2}} … y_{i_{s}}^{a_{s}},  (8.7)

where the sum is over all 1 ≤ s ≤ r, i = i_{1} < i_{2} < ⋯ < i_{k} ≤ n, and a_{1}, a_{2}, …, a_{s} ≥ 0 with a_{1} + ⋯ + a_{s} = r − s. To prove this, apply induction on r, the induction base being clear. For the induction step, let r > 1. Then by (8.6) and the inductive hypothesis, we have

x_{i,i}(2r − 1) ≡ (h_{i}^{2} − h_{i})x_{i,i}(2r − 3) − 2 ∑_{j=i+1}^{n} h_{i}x_{j,j}(2r − 3)

≡ ∑_{s=1}^{r−1}(−2)^{s−1} ∑ h_{i_{1}}h_{i_{2}} … h_{i_{s}}y_{i_{1}}^{a_{1}+1}y_{i_{2}}^{a_{2}} … y_{i_{s}}^{a_{s}}

+ ∑_{j=i+1}^{n} ∑_{s=1}^{r−1}(−2)^{s} ∑ h_{i}h_{j_{1}}h_{j_{2}} … h_{j_{s}}y_{j_{1}}^{b_{1}}y_{j_{2}}^{b_{2}} … y_{j_{s}}^{b_{s}},

where the first unmarked sum is over all i = i_{1} < i_{2} < ⋯ < i_{s} ≤ n and all a_{1}, a_{2}, …, a_{s} ≥ 0 with a_{1} + ⋯ + a_{s} = r − 1 − s and the second one is over all j = j_{1} < j_{2} < ⋯ < j_{s} ≤ n and all b_{1}, b_{2}, …, b_{s} ≥ 0 with b_{1} + ⋯ + b_{s} = r − 1 − s. The formula (8.7) follows. □

Let ℓ = (p − 1)/2, or ℓ = ∞ in case p = 0. Given j ∈ ℤ, define its residue res(j) to be the unique integer r ∈ {0, 1, …, ℓ} such that j^{2} − j ≡ r^{2} + r (mod p). So:

res(i) = res(j) if and only if (i^{2} − i) ≡ (j^{2} − j) (mod p).  (8.8)

Now let λ = ∑_{i=1}^{n} λ_{i}ε_{i} ∈ X(T). For r ∈ {0, 1, …, ℓ}, define

c_{r} := |{(i, j) | 1 ≤ i ≤ n, 0 < j ≤ λ_{i}, res(j) = r}|
−|{(i, j) | 1 ≤ i ≤ n, λ_{i} < j ≤ 0, res(j) = r}|.

Define the content cont(λ) of λ to be the tuple (c_{0}, c_{1}, …, c_{ℓ}). For example, let n = 6, λ = (9, 7, 4, 0, −5, −8) and p = 5. In the following picture the nodes (i, j) are represented by boxes with the corresponding residues written in them:

2 1 0 0 1 2 1 0 0 1 2 1 0 0 0 0 1 1 1 2 2 2 1 1 1 0 0 0 0 1 1 2 1 −  +

Hence, the content of λ is (2, 4, 1). Also for r, s ∈ {0, 1, …, ℓ}, we set

A_{r}(λ) = |{i | 1 ≤ i ≤ n, res(λ_{i} + 1) = r}|,
B_{r}(λ) = |{i | 1 ≤ i ≤ n, res(λ_{i}) = r}|,
d_{r,s}(λ) = |{i | 1 ≤ i ≤ n, res(λ_{i}) = r, res(λ_{i} + 1) = s}|.

Lemma 8.9. Let λ, μ ∈ X(T) with ∑_{i=1}^{n} λ_{i} = ∑_{i=1}^{n} μ_{i}. Then the following are equivalent:

(i) cont(λ) = cont(μ); (ii) d_{r,r+1}(λ) − d_{r+1,r}(λ) = d_{r,r+1}(μ) − d_{r+1,r}(μ) for all r ∈ {0, 1, …, ℓ − 1}; (iii) A_{r}(λ) − B_{r}(λ) = A_{r}(μ) − B_{r}(μ) for each r ∈ {0, 1, …, ℓ}; (iv) z_{r}(λ) ≡ z_{r}(μ) (mod p) for all r ≥ 1.

Proof. (i)⇔(ii). Let us write λ#μ for the 2n-tuple (λ_{1}, …, λ_{n}, −μ_{1}, …, −μ_{n}). Observe that cont(λ) = cont(μ) if and only if cont(λ#μ) = (0, 0, …, 0), and

d_{r,r+1}(λ) − d_{r+1,r}(λ) = d_{r,r+1}(μ) − d_{r+1,r}(μ)

if and only if

d_{r,r+1}(λ#μ) − d_{r+1,r}(λ#μ) = 0.

In other words, it is sufficient to prove the equivalence of (i) and (ii) in the special case μ = 0.

So suppose μ = 0 and let cont(λ) = (c_{0}, …, c_{ℓ}). We need to show that c_{0} = ⋯ = c_{ℓ} = 0 if and only if d_{r,r+1}(λ) − d_{r+1,r}(λ) = 0 for each r ∈ {0, 1, …, ℓ − 1}, which follows easily from the equations:

c_{0} + c_{1} + ⋯ + c_{ℓ} = 0,
c_{r} − c_{r+1} = d_{r,r+1}(λ) − d_{r+1,r}(λ) for r ∈ {0, 1, …, ℓ − 2},
c_{ℓ−1} − 2c_{ℓ} = d_{ℓ−1,ℓ}(λ) − d_{ℓ,ℓ−1}(λ).

(ii)⇔(iii). This follows from the equations

A_{0} + A_{1} + ⋯ + A_{ℓ} = B_{0} + B_{1} + ⋯ + B_{ℓ} = n,
A_{0} = d_{0,0} + d_{1,0}, B_{0} = d_{0,0} + d_{0,1},
A_{r} = d_{r−1,r} + d_{r+1,r}, B_{r} = d_{r,r−1} + d_{r,r+1} for r ∈ {1, …, ℓ − 1}.

(iii)⇔(iv). Define

G_{λ}(t) := 1 − 2 ∑_{r≥1} z_{r}(λ)t^{r} ∈ k[[t]].

We need to prove that given λ, μ ∈ X(T) with ∑_{i=1}^{n} λ_{i} = ∑_{i=1}^{n} μ_{i}, we have that G_{λ}(t) = G_{μ}(t) if and only if A_{r}(λ) − B_{r}(λ) = A_{r}(μ) − B_{r}(μ) for each r = 0, 1, …, ℓ. We represent G_{λ}(t) as a rational function:

G_{λ}(t) = ∏_{i=1}^{n}(1 − 2 ∑_{r≥1} λ_{i}(λ_{i}^{2} − λ_{i})^{r−1}t^{r})

= ∏_{i=1}^{n}(1 − 2λ_{i}t(1 − (λ_{i}^{2} − λ_{i})t)^{−1})

= ∏_{i=1}^{n} \frac{1 − (λ_{i}^{2} − λ_{i})t − 2λ_{i}t}{1 − (λ_{i}^{2} − λ_{i})t}

= ∏_{i=1}^{n} \frac{1 − (λ_{i}^{2} + λ_{i})t}{1 − (λ_{i}^{2} − λ_{i})t}.

Counting multiplicities of zeros and poles, we see that G_{λ}(t) = G_{μ}(t) if and only if a_{r}(λ) − b_{r}(λ) = a_{r}(μ) − b_{r}(μ) for each r = 0, 1, …, p − 1 where

a_{r}(λ) = |{i = 1, …, n | (λ_{i} + 1)^{2} − (λ_{i} + 1) ≡ r (mod p)}|,
b_{r}(λ) = |{i = 1, …, n | λ_{i}^{2} − λ_{i} ≡ r (mod p)}|.

Finally using (8.8), we have that a_{r}(λ)−b_{r}(λ) = a_{r}(μ)−b_{r}(μ) for each r = 0, …, p−1 if and only if A_{r}(λ) − B_{r}(λ) = A_{r}(μ) − B_{r}(μ) for each r = 0, 1, …, ℓ. □

Now we obtain the main result of the section, being an analogue of the “linkage principle”. The linkage principle in case p = 0 was proved originally by Penkov; see [18] for the shorter statement and proof in that case.

Theorem 8.10. Let λ ∈ X_{p}^{+}(T). All composition factors of H^{0}(λ) are of the form L(μ) for μ ≤ λ with cont(μ) = cont(λ).

Proof. It suffices to show that (z_{r} − z_{r}(λ)) annihilates H^{0}(λ). By Lemma 6.10(ii), the B^{+}-socle of H^{0}(λ) equals H^{0}(λ)_{λ}. So by Lemma 8.4, (z_{r} − z_{r}(λ)) annihilates H^{0}(λ)_{λ}. Therefore (z_{r} −z_{r}(λ))H^{0}(λ) intersects the B^{+}-socle H^{0}(λ)_{λ} trivially, hence is zero. □

Applying Theorem 7.5 (or an obvious direct argument) gives:

Corollary 8.11. For λ, μ ∈ X_{p}^{+}(T) with cont(λ) ≠ cont(μ), we have

Ext_{G}^{1}(L(λ), L(μ)) = 0.

9. Steinberg’s tensor product theorem

Suppose throughout the section that p ≠ 0. We wish next to prove the analogue for G = Q(n) of the Steinberg tensor product theorem [27], following the approach of [4]. We will often now appeal to well-known results about G_{ev} = GL(n). For instance, the coordinate ring k[G_{ev}] is the localization of the free polynomial algebra k[c_{i,j} |1 ≤ i, j ≤ n] in n^{2} even indeterminates at determinant. We have the subgroups T = H_{ev}, B_{ev}, B_{ev}^{+}, U_{ev}, U_{ev}^{+} of G_{ev}, and the algebra of distributions Dist(G_{ev}) was described as a subalgebra of Dist(G) in (5.3). For every λ ∈ X^{+}(T), we have an irreducible G_{ev}-module of highest weight λ which we denote by L_{ev}(λ); it can be defined as the simple socle of the induced module H_{ev}^{0}(λ) = ind_{B_{ev}}^{G_{ev}}k_{λ}, see [11].

For r ≥ 1, we have the Frobenius morphism F^{r} : G → G_{ev} defined for a commutative superalgebra A by letting F^{r} : G(A) → G_{ev}(A) be the group homomorphism obtained by raising matrix entries to the p^{r}-th power (note a^{p^{r}} = 0 for all a ∈ A¯1 so this makes sense). Clearly F^{r} stabilizes the various subgroups H, B, B^{+}, U, U^{+} of G, giving us morphisms also all denoted by F^{r} from each of these supergroups to their even part. We denote the kernel of F^{r} by G_{r} (resp. H_{r}, B_{r}, B_{r}^{+}, U_{r}, U_{r}^{+}). Then, G_{r} is a normal subgroup of G, called the rth Frobenius kernel.

Lemma 9.1. F^{r} : G → G_{ev} is a quotient of G by G_{r} in the category of superschemes, that is, for any morphism of superschemes f : G → X that is constant on G_{r}(A)- cosets in G(A) (for each superalgebra A) there is a unique f̃: G_{ev} → X such that f = f̃∘ F^{r}.

Proof. Let σ : G → G_{ev} be the morphism defined for each A as the projection G(A) → G_{ev}(A) onto the diagonal, see (3.1). Let f : G → X be a morphism of superschemes constant on G_{r}-cosets. For any superalgebra A and any element g ∈ G(A) of the form (3.1), we have that

( \begin{matrix}I & S′S^{−1} \\ −S′S^{−1} & I \end{matrix} )^{−1} ( \begin{matrix}S & S′ \\ −S′ & S \end{matrix} ) = ( \begin{matrix}S & 0 \\ 0 & S \end{matrix} ) ,

i.e. σ(g) = hg for some h ∈ G_{r}(A). Hence, f = (f|_{G_{ev}}) ∘ σ. But F^{r}|_{G_{ev}} : G_{ev} → G_{ev} is the quotient of G_{ev} by G_{r,ev} by the purely even theory [11, I.9.5]. Since f|_{G_{ev}} is constant on G_{r,ev}-cosets in G_{ev}, we get a unique morphism f̃: G_{ev} → X such that f|_{G_{ev}} = f̃∘ (F^{r}|_{G_{ev}}). Hence f = f̃∘ (F^{r}|_{G_{ev}}) ∘ σ = f̃∘ F^{r}. □

Note that k[G_{r}] ≅ k[G]/I_{p^{r}} where

I_{p^{r}} = ⟨s_{i,j}^{p^{r}}, s_{l,l}^{p^{r}} − 1 | 1 ≤ i ≠ j ≤ n, 1 ≤ l ≤ n⟩.

So a basis of k[G_{r}] is given by images of the monomials

∏_{1≤i<j≤n} s_{j,i}^{a_{j,i}} (s′_{j,i})^{d_{j,i}} . ∏_{1≤l≤n} s_{l,l}^{a_{l,l}} (s′_{l,l})^{d_{l,l}} . ∏_{1≤i<j≤n} s_{i,j}^{a_{i,j}} (s′_{i,j})^{d_{i,j}}

for integers a_{i,j} ∈ {0, 1, …, p^{r} − 1}, d_{i,j} ∈ {0, 1}, where the products are taken in any fixed order. In particular, dim k[G_{r}] = (2p^{r})^{n^{2}}, and G_{r} is a finite algebraic supergroup (cf. [11, I.8.1]). Moreover, the p^{r}-th power of each generator of the augmentation ideal I_{e} = ker E of k[G] (i.e. the elements s_{i,j}, s′_{i,j}, s_{r,r} − 1, s′_{r,r} for 1 ≤ i ≠ j ≤ n, 1 ≤ r ≤ n) is an element of I_{p^{r}}, hence the image of I_{e} in k[G_{r}] is a nilpotent ideal. So Dist(G_{r}) can actually be identified with the dual superalgebra k[G_{r}]^{∗}. It follows easily (see [11, I.8.3] and the discussion at the beginning of [2, §5]) that the category of G_{r}-supermodules is isomorphic to the category of Dist(G_{r})- supermodules. As a basis for Dist(G_{r}) ⊂ Dist(G), we can take the ordered PBW monomials

∏_{1≤i<j≤n} e_{j,i}^{(a_{j,i})} (e′_{j,i})^{d_{j,i}} . ∏_{1≤l≤n} \binom{h_{l}}{a_{l,l}} (h′_{l})^{d_{l,l}} . ∏_{1≤i<j≤n} e_{i,j}^{(a_{i,j})} (e′_{i,j})^{d_{i,j}}

for integers a_{i,j} ∈ {0, 1, …, p^{r} − 1}, d_{i,j} ∈ {0, 1}. Similarly, one can describe explicit bases for k[B_{r}], k[U_{r}], Dist(B_{r}), Dist(U_{r}) etc. We obviously have:

Lemma 9.2. Dist(G_{r}) is a free right Dist(B_{r}^{+})-supermodule with basis given by the ordered monomials

{ ∏_{1≤i<j≤n} e_{j,i}^{(a_{j,i})}(e′_{j,i})^{d_{j,i}} | a_{j,i} ∈ {0, 1, …, p^{r} − 1}, d_{j,i} ∈ {0, 1}}.

Identifying G_{r}- (resp. B_{r}^{+}-) supermodules with Dist(G_{r})- (resp. Dist(B_{r}^{+})-) supermodules, we have the coinduction functor

coind_{B_{r}^{+}}^{G_{r}} : 𝔪𝔬𝔡_{B_{r}^{+}} → 𝔪𝔬𝔡_{G_{r}}, coind_{B_{r}^{+}}^{G_{r}}? = Dist(G_{r})⊗_{Dist(B_{r}^{+})}?.

Lemma 9.2 gives that this is an exact functor that is left adjoint to res_{B_{r}^{+}}^{G_{r}}. Indeed, given any B_{r}^{+}-supermodule M, we have that

coind_{B_{r}^{+}}^{G_{r}}M ≃ Dist(U_{r}) ⊗ M  (9.3)

as a vector superspace, so in particular coind_{B_{r}^{+}}^{G_{r}}M has dimension (2p^{r})^{N} dim M for finite dimensional M, where N = n(n − 1)/2.

Now consider the representation theory of H_{r}. For λ ∈ X(T), let 𝔲_{r}(λ) denote the restriction of the H-supermodule 𝔲(λ) to H_{r}. Arguing in the same way as for Lemma 6.4, one shows:

Lemma 9.4. The modules {𝔲_{r}(λ) | λ ∈ X(T)} give a complete set of irreducible H_{r}-supermodules. Moreover, for λ, μ ∈ X(T), we have 𝔲_{r}(λ) ≅ 𝔲_{r}(μ) if and only if λ − μ ∈ p^{r}X(T).

We also write 𝔲_{r}(λ) for its inflation to B_{r} (resp. B_{r}^{+}). By the lemma, the modules {𝔲_{r}(λ) | λ ∈ X(T)} also give a complete set of irreducible B_{r}- (resp. B_{r}^{+}-) supermodules. Given λ ∈ X(T), let

Z_{r}(λ) := coind_{B_{r}^{+}}^{G_{r}}𝔲_{r}(λ).

Note dim Z_{r}(λ) = 2^{⌊(h_{p′}(λ)+1)/2⌋}(2p^{r})^{N}. Let L_{r}(λ) be the G_{r}-head of Z_{r}(λ).

Theorem 9.5. The modules {L_{r}(λ) | λ ∈ X(T)} form a complete set of irreducible G_{r}-supermodules, with L_{r}(λ) ≅ L_{r}(μ) if and only if λ − μ ∈ p^{r}X(T).

Proof. One can easily see that the B_{r}-module 𝔲_{r}(λ) is a quotients of Z_{r}(λ). Moreover, by (9.3), we have dim Hom_{U_{r}}(Z_{r}(λ), k) = dim 𝔲_{r}(λ). Hence, Z_{r}(λ) has irreducible B_{r}-head ≃ 𝔲_{r}(λ). It follows at once that Z_{r}(λ) has irreducible G_{r}-head, so that L_{r}(λ) is an irreducible G_{r}-supermodule. Now, if L is any irreducible G_{r}- supermodule, choose λ ∈ X(T) so that Hom_{B_{r}^{+}}(𝔲_{r}(λ), L) ≠ 0. Then Frobenius reciprocity gives that L is isomorphic to a quotient of Z_{r}(λ), hence L ≅ L_{r}(λ). Finally, since the B_{r}-head of L_{r}(λ) is 𝔲_{r}(λ), we get from Lemma 9.4 that L_{r}(λ) ≅ L_{r}(μ) if and only if λ − μ ∈ p^{r}X(T). □

Now we begin the proof of the Steinberg tensor product theorem. It suffices from now on to consider the special case r = 1, since the tensor product theorem is already proved for G_{ev}. Lemma 9.6. Let L be an irreducible G-supermodule. Then, L is completely reducible as a G_{1}-supermodule.

Proof. Pick L_{1} in the G_{1}-socle of L. Since G_{1} is a normal subgroup of G, we have for each g ∈ G(k) that the translate gL_{1} is an irreducible G_{1}-submodule of L. Hence M := ∑_{g∈G(k)} gL_{1} is a completely reducible G_{1}-submodule of L. To complete the proof, we just need to show that M = L. This will follow from irreducibility of L if we check that M is invariant under the action of Dist(G). By definition, M is invariant under G(k). So M is a G_{ev}-submodule of L, for example by [11, I.2.12(5)] since G(k) = G_{ev}(k) is dense in G_{ev} (cf [11, I.6.16]). So M is invariant under both Dist(G_{ev}) and Dist(G_{1}). But Dist(G) is generated by Dist(G_{ev}) and Dist(G_{1}), so M is invariant under Dist(G). □

Lemma 9.7. Let λ ∈ X_{p}^{+}(T). Then, Dist(G_{1})L(λ)_{λ} is a G_{1}-submodule of L(λ) isomorphic to L_{1}(λ).

Proof. As a B^{+}-supermodule, L(λ)_{λ} ≃ 𝔲(λ), so as a B_{1}^{+}-supermodule it is ≃ 𝔲_{1}(λ). Hence there is an even B_{1}^{+}-supermodule homomorphism 𝔲_{1}(λ) → L(λ) with image L(λ)_{λ}. Applying Frobenius reciprocity, we obtain a G_{1}-supermodule homomorphism Z_{1}(λ) → L(λ) with image Dist(G_{1})L(λ)_{λ}. By Lemma 9.6, Dist(G_{1})L(λ)_{λ} is completely reducible, while Z_{1}(λ) has irreducible G_{1}-head. So in fact Dist(G_{1})L(λ)_{λ} is an irreducible G_{1}-supermodule isomorphic to the head L_{1}(λ) of Z_{1}(λ). □

Now we introduce the restricted weights (cf. [2, §9]): call λ = ∑_{i=1}^{n} λ_{i}ε_{i} ∈ X_{p}^{+}(T) restricted if λ_{i} − λ_{i+1} ≤ p when p ∤ λ_{i}, and λ_{i} − λ_{i+1} < p when p | λ_{i}, for each i = 1, …, n−1. Let X_{p}^{+}(T)_{res} denote the set of all restricted λ ∈ X_{p}^{+}(T). The proof of the next lemma is based on the argument in [1, 6.4].

Lemma 9.8. For λ ∈ X_{p}^{+}(T)_{res}, the irreducible G-supermodule L(λ) is also irreducible as a G_{1}-supermodule, and res_{G_{1}}^{G}L(λ) ≃ L_{1}(λ).

Proof. Let M = Dist(G_{1})L(λ)_{λ}. By Lemma 9.7, M ≃ L_{1}(λ). So we just need to show that M = L(λ), which will follow if we can show that M is invariant under the action of Dist(G). But Dist(G) is generated by Dist(G_{ev}) and Dist(G_{1}), so we just need to check that M is invariant under the action of Dist(G_{ev}). Since B_{ev}^{+} normalizes G_{1} and L(λ)_{λ} is a B_{ev}^{+}-submodule of L(λ), M is invariant under the action of Dist(B_{ev}^{+}); in particular, M is the sum of its weight spaces. Therefore we just need to prove that:

(∗) e_{i+1,i}^{(r)}v ∈ M for all v ∈ M_{μ}, i = 1, …, n − 1, r ≥ 1 and all μ ∈ X(T). We prove (∗) by downward induction on the weight μ.

To start the induction, we need to consider the case μ = λ. Here we need to show that e_{i+1,i}^{(r)}v ∈ M for all r ≥ 1 and v ∈ L(λ)_{λ}. But e_{i+1,i}^{(r)}v = 0 for r > λ_{i} − λ_{i+1} by

SL(2) theory, while clearly e_{i+1,i}^{(r)}v ∈ M for r < p. This just leaves the case when

λ_{i} − λ_{i+1} = p, when p ∤ λ_{i} by assumption; we need to check that e_{i+1,i}^{(p)}v ∈ M for all v ∈ L(λ)_{λ}. In fact we will show more, namely, that

e_{i+1,i}^{(p)}v = e_{i+1,i}^{(p−1)}e′_{i+1,i}hv, where h = \frac{h′_{i} − h′_{i+1}}{λ_{i} + λ_{i+1}} .

Well, consider m = e_{i+1,i}^{(p)}v − e_{i+1,i}^{(p−1)}e′_{i+1,i}hv. If m ≠ 0, then (since L(λ)^{U^{+}} = L(λ)_{λ}) we can find a PBW monomial x in Dist(U^{+}) such that xm is a non-zero vector in L(λ)_{λ}. But by weights, the only possibilities for the monomial x are e_{i,i+1}^{(p)} and

e′_{i,i+1}e_{i,i+1}^{(p−1)}. We note that (h′_{i} − h′_{i+1})hv = v. So using the commutator relations (cf. [26, Lemma 5]), we get that:

e_{i,i+1}^{(p)}m = \binom{h_{i} − h_{i+1}}{p}v − \binom{h_{i} − h_{i+1} − 1}{p − 1}e_{i,i+1}e′_{i+1,i}hv
= v − (h′_{i} − h′_{i+1})hv = 0,
e′_{i,i+1}e_{i,i+1}^{(p−1)}m = e′_{i,i+1}e_{i+1,i}\binom{h_{i} − h_{i+1} − 1}{p − 1}v

−e′_{i,i+1}\binom{h_{i} − h_{i+1}}{p − 1}e′_{i+1,i}hv

−e′_{i,i+1}e_{i+1,i}\binom{h_{i} − h_{i+1} − 2}{p − 2}e_{i,i+1}e′_{i+1,i}hv

= (h′_{i} − h′_{i+1})v − 0 − (h′_{i} − h′_{i+1})^{2}hv = 0.

This shows that m = 0, as required.

Now take μ < λ and assume (∗) has been proved for all greater μ. Any v ∈ M_{μ} can be written as e_{s,s+1}w or e′_{s,s+1}w for some s and w ∈ M_{μ+ε_{s}−ε_{s+1}}. Then,

e_{i,i+1}^{(r)}v = e_{i,i+1}^{(r)}e_{s,s+1}w or e_{i,i+1}^{(r)}e′_{s,s+1}w.

Now one uses the commutator relations once more together with the inductive hypothesis to see that in either case the expression on the right hand side lies in M. □

Given any G_{ev}-module M (viewed as a supermodule concentrated in degree ¯0) we can inflate through F = F^{1} : G → G_{ev} to obtain a G-supermodule F^{∗}M, the Frobenius twist of M. Thus, we have a functor

F^{∗} : 𝔪𝔬𝔡_{G_{ev}} → 𝔪𝔬𝔡_{G}.

Conversely, given a G-supermodule N, there is an induced G_{ev}-module structure on the fixed point space N^{G_{1}}: the representation G → GL(N^{G_{1}}) is constant on G_{1}-cosets so factors to G_{ev} → GL(N^{G_{1}}) by Lemma 9.1. (In case N is infinite dimensional, equal to the direct limit of its finite dimensional submodules, one needs here to use the fact that taking fixed points commutes with direct limits.) Thus we have a functor

?^{G_{1}} : 𝔪𝔬𝔡_{G} → 𝔪𝔬𝔡_{G_{ev}}

which is right adjoint to F^{∗}. Now we can prove the main result of the section:

Theorem 9.9. For λ ∈ X_{p}^{+}(T)_{res}, μ ∈ X^{+}(T),

L(λ + pμ) ≅ L(λ) ⊗ F^{∗}L_{ev}(μ)

as a G-supermodule.

Proof. For λ ∈ X_{p}^{+}(T)_{res}, L(λ) is irreducible as a G_{1}-supermodule by Lemma 9.8. By Lemma 9.7,

H := Hom_{G_{1}}(L(λ), L(λ + pμ))¯0

is non-zero (replacing L(λ + pμ) by ΠL(λ + pμ) if necessary). In what follows, we view H as a G-supermodule by conjugation, so the action of u ∈ Dist(G) is given by (uf)(l) = ∑_{i} u_{i}f(σ(u′_{i})l) for f ∈ H, l ∈ L(λ) if δ(u) = ∑_{i} u_{i} ⊗ u′_{i} (where σ is the antipode on Dist(G)). One checks directly from this that the map

θ : H ⊗ L(λ) → L(λ + pμ), f ⊗ l ↦ f(l)

is an even G-supermodule homomorphism. Since it is non-zero and L(λ + pμ) is irreducible, θ is surjective. On the other hand, by Schur’s lemma,

dim Hom_{G_{1}}(L(λ), L(λ + pμ))¯0 ⊗ L(λ)
≤ (dim L(λ + pμ)/ dim L(λ)) dim L(λ)
= dim L(λ + pμ),

hence θ is in fact an isomorphism. Finally, since the action of G_{1} on H is trivial, we must have that H ≃ F^{∗}M for some G_{ev}-module M. Moreover, M must be irreducible since L(λ + pμ) is irreducible. But the highest weight of H is pμ, hence M ≅ L_{ev}(μ). □

10. Polynomial representations

Recall that k[Mat] = k[s_{i,j}, s′_{i,j} | 1 ≤ i, j ≤ n], and k[G] is the localization of k[Mat] at det. We call a representation M of G = Q(n) polynomial if the image of the structure map η : M → M ⊗ k[G] lies in M ⊗ k[Mat].

More generally, given any standard Levi subgroup G_{γ} of G, we can talk about polynomial representations of G_{γ}, defined in an entirely analogous way. In particular, we can talk about polynomial representations of H: an H-supermodule M with structure map η : M → M ⊗ k[H] is polynomial if η(M) ⊆ M ⊗ k[x_{i}, x′_{i} | 1 ≤ i ≤ n] where x_{i}, x′_{i} are the restrictions of the coordinate functions s_{i,i}, s′_{i,i} to H respectively.

The natural G-supermodule V is a polynomial representation, as is any subquotient of V^{⊗d} for any d ≥ 0, but the dual G-supermodule V^{∗} is not polynomial. Thus, the natural duality ∗ on finite dimensional G-supermodules does not respect polynomial representations. However, there is another duality denoted τ which does take polynomial representations to polynomial representations. To define this, let τ : k[G] → k[G] be the unique linear map which maps 1 ↦ 1, s_{i,j} ↦ s_{j,i}, s′_{i,j} ↦ s′_{j,i} and satisfies τ(fg) = τ(g)τ(f) for all f, g ∈ k[G]. By checking on the generators s_{i,j}, s′_{i,j}, det^{−1}, one verifies:

Lemma 10.1. Δ ∘ τ = T ∘ (τ ⊗ τ) ∘ Δ where T : k[G] ⊗ k[G] → k[G] ⊗ k[G] is the unsigned twist f_{1} ⊗ f_{2} ↦ f_{2} ⊗ f_{1}.

Now let M be a finite dimensional G-supermodule with structure map η : M → M ⊗ k[G]. Pick a basis m_{1}, …, m_{r} for M and write η(m_{j}) = ∑_{i=1}^{r} m_{i} ⊗ c_{i,j} for c_{i,j} ∈ k[G]. Let f_{1}, …, f_{r} be the basis for M^{∗} dual to m_{1}, …, m_{r}. Define M^{τ} to be the G-supermodule equal to M^{∗} as a vector superspace with structure map

η^{τ} : M^{τ} → M^{τ} ⊗ k[G], f_{j} ↦ ∑_{i=1}^{r} f_{i} ⊗ τ(c_{j,i}).

Note the definition of η^{τ} is independent of the choice of basis, and it is easily checked to be a comodule structure map using Lemma 10.1. Obviously, (M^{τ})^{τ} ≃ M and, since τ leaves k[Mat] invariant, M^{τ} is polynomial if and only if M is polynomial. Moreover, M and M^{τ} have the same character since τ fixes each s_{i,i}.

Let

Λ(T) = {λ = ∑_{i=1}^{n}λ_{i}ε_{i} ∈ X(T) | λ_{i} ≥ 0 for each i = 1, …, n}, (10.2)
Λ_{p}^{+}(T) = Λ(T) ∩ X_{p}^{+}(T).  (10.3)

If M is a polynomial representation of G, it is polynomial over H so in particular all its weights lie in Λ(T). Hence all its composition factors are of the form L(λ) for λ ∈ Λ_{p}^{+}(T). We can state [2, Theorem 10.1] (see also [24] over ℂ) as follows:

Theorem 10.4. The modules {L(λ) | λ ∈ Λ_{p}^{+}(T)} give a complete set of pairwise non-isomorphic irreducible polynomial representations of G.

Note however that there exist non-polynomial extensions of polynomial L(λ)’s, unlike the situation for GL(n). For example, the G-supermodule k̃ from (7.6) is an extension of two copies of the trivial G-supermodule that is not even polynomial over H. The goal in the remainder of the section is to prove that a G-supermodule is polynomial if and only if it is polynomial on restriction to H. There are several proofs of the analogous statement for GL(n) in the literature, see [7], [8], [10, Prop.3.4] and [12, Theorem 5.3]. Of these, Jantzen’s argument from [12] adapts easily to our situation. First we record the following lemma which will easily settle the case p = 0:

Lemma 10.5. If p = 0, every representation of G that is polynomial on restriction to H is completely reducible.

Proof. This is trivial to check directly if G = Q(1). Hence, since H is a direct product of copies of Q(1), every polynomial representation of H is completely reducible.

As observed above, if M is a polynomial representation of G that is polynomial over H then all its composition factors are of the form L(λ) for λ ∈ Λ_{p}^{+}(T). So take λ, μ ∈ Λ_{p}^{+}(T) with μ ≯ λ and suppose we have an extension

0 ⟶ L(λ) ⟶ M ⟶ L(μ) ⟶ 0

where M is a G-supermodule that is polynomial over H. To prove the lemma, it suffices to show that the extension splits. By the linkage principle in characteristic 0, H^{0}(λ) = L(λ) for λ ∈ Λ_{p}^{+}(T). Thus, L(λ) is the induced module ind_{B}^{G}𝔲(λ). Using this, one constructs a splitting M → L(λ) of the above short exact sequence in exactly the same way as in the proof that Ext_{G}^{1}(L(μ), H̃^{0}(λ)) = 0 in Theorem 7.5, replacing the injectivity of 𝔲̃(λ) there with the complete reducibility of M as an H-supermodule. □

Now suppose for the next lemma that p > 0. Then we can consider the “thickened” Frobenius kernel G_{r}T defined to be the closed subgroup (F^{r})^{−1}T of G, so (G_{r}T)(A) = G_{r}(A)T (A) for each superalgebra A. The coordinate ring k[G_{r}T] is the quotient of k[G] by the ideal generated by {s_{i,j}^{p^{r}} | i ≠ j}. Note also that H is a subgroup of G_{r}T. On G_{r}T, we have that det^{p^{r}} = (s_{1,1} … s_{n,n})^{p^{r}}, hence det^{−1} = det^{p^{r}−1}(s_{1,1} … s_{n,n})^{−p^{r}}. So

k[G_{r}T] = k[Mat][s_{1,1}^{−1}, …, s_{n,n}^{−1}]/(s_{i,j}^{p^{r}} | i ≠ j).  (10.6)

We also define the following subsets of k[G_{r}T]:

R := k[Mat]/(s_{i,j}^{p^{r}} | i ≠ j),  (10.7)
R_{l} := k[Mat][s_{1,1}^{−1}, …, s_{l−1,l−1}^{−1}, s_{l+1,l+1}^{−1}, …, s_{n,n}^{−1}]/(s_{i,j}^{p^{r}} | i ≠ j), (10.8)

for each l = 1, …, n. We call a G_{r}T -supermodule M with structure map η : M → M ⊗ k[G_{r}T] polynomial if η(M) ⊆ M ⊗ R.

Lemma 10.9. If p > 0, every G_{r}T -supermodule that is polynomial on restriction to H is polynomial over G_{r}T.

Proof. Let u_{i,j}, u′_{i,j} (i > j), v_{i,j}, v′_{i,j} (i < j) and x_{i}, x′_{i} (1 ≤ i ≤ n) be the standard coordinate functions (restrictions of various s_{i,j}, s′_{i,j}) on U_{r}, U_{r}^{+} and H respectively. Consider the morphism

μ : U_{r} × H × U_{r}^{+} → G_{r}T

induced by multiplication. We claim that μ is an isomorphism of superschemes, and moreover the functions u_{i,j}, u′_{i,j}, v_{i,j}, v′_{i,j} and x_{i}, x′_{i} belong to R_{n} when viewed as elements of k[G_{r}T] via the isomorphism μ. This is proved by induction on n in a similar way to [12, Lemma 5.1]. For example, for n = 2, the matrix

g = [ \begin{matrix}a & b & a′ & b′ \\ c & d & c′ & d′ \\ −a′ & −b′ & a & b \\ −c′ & −d′ & c & d \end{matrix} ] ∈ (G_{r}T)(A)

can be factorized as

g = [ \begin{matrix}1 & 0 & 0 & 0 \\ \frac{ca+c′a′}{a^{2}} & 1 & \frac{c′a−ca′}{a^{2}} & 0 \\ 0 & 0 & 1 & 0 \\ \frac{ca′−c′a}{a^{2}} & 0 & \frac{ca+c′a′}{a^{2}} & 1 \end{matrix} ] × [ \begin{matrix}a & 0 & a′ & 0 \\ 0 & q & 0 & q′ \\ −a′ & 0 & a & 0 \\ 0 & −q′ & 0 & q \end{matrix} ]

× [ \begin{matrix}1 & \frac{ab+a′b′}{a^{2}} & 0 & \frac{ab′−a′b}{a^{2}} \\ 0 & 1 & 0 & 0 \\ 0 & \frac{a′b−ab′}{a^{2}} & 1 & \frac{ab+a′b′}{a^{2}} \\ 0 & 0 & 0 & 1 \end{matrix} ] ,

where

q = d − \frac{abc + ab′c′ + a′b′c − a′bc′}{a^{2}} ,
q′ = d′ − \frac{ab′c + a′b′c′ + abc′ − a′bc}{a^{2}}.

The point is that d^{−1} does not appear in these expressions.

Now let M be a G_{r}T -supermodule that is polynomial over H, with structure map η : M → M ⊗ k[G_{r}T]. Let m_{i} (i ∈ I) be a homogeneous basis for M and write

η(m_{j}) = ∑_{i∈I} m_{i} ⊗ c_{i,j}  (10.10)

for some c_{i,j} ∈ k[G_{r}T]. Given g ∈ (G_{r}T)(A) for some superalgebra A, write g = uhv for u ∈ U_{r}, h ∈ H, v ∈ U_{r}^{+}. Then,

c_{i,j}(g) = ∑_{l,m} c_{i,l}(u)c_{l,m}(h)c_{m,j}(v).

Now, c_{i,l}(u) and c_{m,j}(v) are polynomial functions in the coordinates u_{i,j}, u′_{i,j} on U_{r} and v_{i,j}, v′_{i,j} of U_{r}^{+} respectively. Moreover, by the assumption that M is polynomial over H, c_{l,m}(h) is a polynomial in the coordinates x_{i}, x′_{i} on H. This shows each c_{i,j} ∈ R_{n} thanks to the previous paragraph.

The symmetric group W acts on G_{r}T by conjugation, hence for each w ∈ W we obtain a new G_{r}T -supermodule ^{w}M by twisting the action by w. The matrix coefficients of^{w}M are the functions w·c_{i,j} where the c_{i,j}’s are as defined in (10.10) and (w · c_{i,j})(g) = c_{i,j}(w^{−1}gw). Clearly each ^{w}M is polynomial over H too, so applying the previous paragraph to ^{w}M instead shows that w · c_{i,j} ∈ R_{n}, hence c_{i,j} ∈ R_{w^{−1}n}, for all w ∈ W. But R = ⋂_{l=1}^{n} R_{l}. Thus we have shown that each c_{i,j} ∈ R, hence M is polynomial over G_{r}T. □

Theorem 10.11. A G-supermodule M is polynomial if and only if it is polynomial on restriction to H.

Proof. Obviously, any polynomial representation of G is polynomial over H. So suppose instead that M is a G-supermodule that is polynomial over H. In case p = 0, Lemma 10.5 shows that M is completely reducible. Hence M is a direct sum of L(λ)’s for λ ∈ Λ_{p}^{+}(T). But all such L(λ) are polynomial by Theorem 10.4. Now assume p > 0. By Lemma 10.9, M is polynomial over G_{r}T for all r ≥ 1. This implies that M is polynomial over G by the argument of [12, Corollary 5.4]. □

Corollary 10.12. If M is a B-supermodule that is polynomial over H, then ind_{B}^{G}M is polynomial over G. In particular, each H^{0}(λ) for λ ∈ Λ_{p}^{+}(T) is polynomial.

Proof. It suffices to prove this for finite dimensional M, in which case ind_{B}^{G}M is finite dimensional too by Lemma 6.10(iii). Set

P := M^{τ} , Q := (ind_{B}^{G}M)^{τ} .

By the theorem, we just need to check that Q is polynomial over H. Since we can conjugate by W, it suffices in turn to check that Q is polynomial over the subgroup H(1) ≅ Q(1) embedded into G in the bottom right hand corner. Let P_{0} (resp. Q_{0}) denote the sum of all weight spaces of P (resp. Q) with ε_{n}-component equal to zero, and P_{>0} (resp. Q_{>0}) denote the sum of all weight spaces with ε_{n}-component greater than zero. All weights of M lie in Λ(T), so the same is true for Q by Lemma 6.10(i). Hence,

P = P_{0} ⊕ P_{>0}, Q = Q_{0} ⊕ Q_{>0}.

All H(1)-supermodules of weight > 0 are polynomial. Hence, Q_{>0} is certainly polynomial over H(1). Moreover, P_{0} is polynomial over H(1) by assumption.

By Lemmas 6.8 and 6.9, there is a B^{+}-homomorphism ind_{B}^{G}M ↪ M ⊗ k[U^{+}]. Considering the U^{+}-fixed points, we get an H-homomorphism ind_{B}^{G}M → M that is injective on the B^{+}-socle of ind_{B}^{G}M. Applying τ, we get an H-homomorphism ρ : P → Q whose image generates Q as a B-supermodule.

Let B(n − 1) denote the Borel subgroup of Q(n − 1) embedded into G in the top left hand corner. Then, by the previous paragraph, Q_{0} is generated as a B(n − 1)- supermodule by ρ(P_{0}). Thus,

Q_{0} = ∑_{1≤j<i<n, r≥0} (e_{i,j}^{(r)}ρ(P_{0}) + e_{i,j}^{(r)}e′_{i,j}ρ(P_{0})) .

Since each such e_{i,j}^{(r)} and e′_{i,j} centralizes H(1), this shows that Q_{0} is a sum of H(1)- homomorphic images of P_{0}. Since P_{0} is polynomial over H(1), this implies that Q_{0} is polynomial over H(1) too, completing the proof. □

Remark 10.13. In particular, the corollary shows that the supermodules

V (λ) ≅ H^{0}(λ)^{τ}  (10.14)

defined in (6.12) are polynomial for λ ∈ Λ_{p}^{+}(T). Comparing Lemma 6.13 and [2, Lemma 8.3] now shows that the V (λ) as defined here coincide with the V (λ) introduced in [2] for λ ∈ Λ_{p}^{+}(T).

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Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

E-mail address: brundan@darkwing.uoregon.edu, klesh@math.uoregon.edu