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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-Wangs 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.
We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was initiated by Brund
Let $(bf U, bf U^imath)$ be a quantum symmetric pair of Kac-Moody type. The $imath$quantum groups $bf U^imath$ and the universal $imath$quantum groups $widetilde{bf U}^imath$ can be viewed as a generalization of quantum groups and Drinfeld doubles $w
We use the Kazhdan-Lusztig tensoring to define affine translation functors, describe annihilating ideals of highest weight modules over an affine Lie algebra in terms of the corresponding VOA, and to sketch a functorial approach to ``affine Harish-Chandra bimodules.
We establish a three-parameter Schur duality between the affine Hecke algebra of type C and a coideal subalgebra of quantum affine $mathfrak{sl}_n$. At the equal parameter specializations, we obtain Schur dualities of types BCD.
We introduce the notion of minimal reduction type of an affine Springer fiber, and use it to define a map from the set of conjugacy classes in the Weyl group to the set of nilpotent orbits. We show that this map is the same as the one defined by Lusz