Duplex Hecke algebras of type B

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 $B$ which is a $\mathbb{Q}(q)$-algebra associated with the Weyl group $\mathcal{𝒲}(B)$ of type $B$, and symmetric groups ${𝔖}_l$ for $l=0,1,\dots ,m$, 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 $A$ arising from the related $q$-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 $B$ admits natural representations on certain tensor spaces. We then establish a Levi-type $q$-Schur–Weyl duality of type $B$, which reveals the double centralizer property between such duplex Hecke algebras and $ı$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, Astérisque402 (2018)].