Duplex Hecke algebras of type B

Abstract

As a sequel to [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.], in this article we first introduce a so-called duplex Hecke algebras of type [Formula: see text] which is a [Formula: see text]-algebra associated with the Weyl group [Formula: see text] of type [Formula: see text], and symmetric groups [Formula: see text] for [Formula: see text], satisfying some Hecke relations (see Definition 3.1). This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur–Weyl duality of Levi-type (see [B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]), extending the duplex Hecke algebra of type [Formula: see text] arising from the related [Formula: see text]-Schur–Weyl duality of Levi-type (see [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.]). A duplex Hecke algebra of type [Formula: see text] admits natural representations on certain tensor spaces. We then establish a Levi-type [Formula: see text]-Schur–Weyl duality of type [Formula: see text], which reveals the double centralizer property between such duplex Hecke algebras and [Formula: see text]quantum groups studied by Bao and Wang in [H. Bao and W. Wang, A new approach to Kazhdan–Lusztig theory of type B via quantum symmetric pairs, Astérisque 402 (2018)].