We establish a Schur type duality between a coideal subalgebra of the quantum group of type A and the Hecke algebra of type B with 2 parameters. We identify the ı-canonical basis on the tensor product of the natural representation with Lusztig's canonical basis of the type B Hecke algebra with unequal parameters associated to a weight function.

We generalize a construction in [BW18b] by showing that, for a quantum symmetric pair (U, U ı ) of finite type, the tensor product of a based U ı -module and a based U-module is a based U ı -module. This is then used to formulate a Kazhdan-Lusztig theory for an arbitrary parabolic BGG category O of the ortho-symplectic Lie superalgebras, extending a main result in [BW18a].

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In order to see the behavior of ıcanonical bases at q = ∞, we introduce the notion of ıcrystals associated to an ıquantum group of certain quasi-split type. The theory of ıcrystals clarifies why ıcanonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of ıcrystals whose projective limit can be thought of as the ıcanonical basis of the modified ıquantum group at q = ∞.

ıquantum groups are generalizations of quantum groups which appear as coideal subalgebras of quantum groups in the theory of quantum symmetric pairs. In this paper, we define the notion of classical weight modules over an ıquantum group, and study their properties along the lines of the representation theory of weight modules over a quantum group. In several cases, we classify the finite-dimensional irreducible classical weight modules by a highest weight theory.