A duality of a twisted group algebra of the hyperoctahedral group and the queer Lie superalgebra

Abstract

We establish a duality relation between one of the twisted group algebras of the hyperoctahedral groupf H_k and a Lie superalgebra q(n_0) \oplus q(n_1) for any integers k and n_0, n_1, where q(n_0) and q(n_1) denote the ``queer'' Liesuperalgebras. Note that this twisted group algebra \B'_k belongs to a different cocycle from the one \B_k used by A. N. Sergeev in [8] and by the present author in [11]. We will use the supertensor product \C_k \otimes \B'_k of the 2^k-dimensional Clifford algebra \C_k and \B'_k, as an intermediary for establishing our duality. We show that the algebra \C_k \otimes B'_k and q(n_0) \oplus q(n_1) act on the k-fold tensor product W=V^{\otimes k} of the natural representation V of q(n_0+n_1) ``as mutual centralizers of each other'' (Theorem 4.1). Moreover, we show that \B'_k and q(n_0) \oplus q(n_1) act on a subspace W' of W ``as mutual centralizers of each other'' (Theorem 4.2). This duality relation gives a formula for character values of simple B'_k-modules. This formula is di fferent from a formula (Theorem D) obtained by J. R. Stembridge (cf. [10, Lem 7.5]).