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

作者：Wang, Di^*

来源：JOURNAL OF ALGEBRA AND ITS APPLICATIONS, 2023, 22(11): 2350232.

DOI：10.1142/S0219498823502328

• Supergroup
• enhanced vector space
• Schur superalgebra
• Schur duality

In this paper, we investigate a kind of double centralizer property for general linear supergroups. For the super space V = K-m(vertical bar n) over an algebraically closed field K whose characteristic is not equal to 2, we consider its Z(2)-homogeneous one-dimensional extension V = V circle plus Kv, and the natural action of the supergroup (G) over tilde := GL(V) x G(m) on V. Then we have the tensor product supermodule (V-circle times r, rho(r)) of (G) over tilde. We present a kind of generalized Schur-Sergeev duality which is said that the Schur superalgebras S' (m vertical bar n, r) of (G) over tilde and the so-called weak degenerate double Hecke algebra H-r 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.