$\imath$Schur duality and Kazhdan-Lusztig basis expanded

Expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. We formulate an $\imath$Schur duality between an $\imath$quantum group of type AIII (allowing black nodes in its Satake diagram) and a Hecke algebra of type B acting on a tensor space, providing a common generalization of Jimbo-Schur duality and Bao-Wang's quasi-split $\imath$Schur duality. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the $\imath$canonical basis on the tensor space. An inversion formula for quasi-parabolic KL polynomials is established via the $\imath$Schur duality.