Levi-type Schur–Sergeev duality for general linear super groups

Abstract

In this paper, we investigate a kind of double centralizer property for general linear supergroups. For the super space [Formula: see text] over an algebraically closed field [Formula: see text] whose characteristic is not equal to [Formula: see text], we consider its [Formula: see text]-homogeneous one-dimensional extension [Formula: see text], and the natural action of the supergroup [Formula: see text] on [Formula: see text]. Then we have the tensor product supermodule ([Formula: see text], [Formula: see text]) of [Formula: see text]. We present a kind of generalized Schur–Sergeev duality which is said that the Schur superalgebras [Formula: see text] of [Formula: see text] and the so-called weak degenerate double Hecke algebra [Formula: see text] are double centralizers. The weak degenerate double Hecke algebra is an infinite-dimensional algebra, which has a natural representation on the tensor product space. This notion comes from [B. Shu, Y. Xue and Y. Yao, On enhanced reductive groups (I): Parabolic Schur algebras and the dualities related to degenerate double Hecke algebras, preprint (2013), arXiv:2005.13152 [Math. RT]], with a little modification.