Modular representations of the supergroup Q(n), I

Abstract

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 ofQ(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 Ŝn of the symmetric groups. In [2], we used Sergeev’s ideas to classify for the first time the irreducible representations of Ŝn 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 Ŝn. 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 6= 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) = {(λ1, . . . , λn) ∈ Z | λ1 ≥ · · · ≥ λn with λi = λi+1 only if p|λi}. 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(λ) := indBu(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 H0(λ) 6= 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 H0(λ) 6= 0 depends on a counting argument involving p-regular conjugacy classes in Ŝn.