Abstract. We reformulate the Kazhdan-Lusztig theory for the BGG category O of Lie algebras of type D via the theory of canonical bases arising from quantum symmetric pairs initiated by Weiqiang Wang and the author. This is further applied to formulate and establish for the first time the KazhdanLusztig theory for the BGG category O of the ortho-symplectic Lie superalgebra osp(2m|2n).

For quantum symmetric pairs (U, U ı ) of Kac-Moody type, we construct ıcanonical bases for the highest weight integrable U-modules and their tensor products regarded as U ı -modules, as well as an ıcanonical basis for the modified form of the ıquantum group U ı . A key new ingredient is a family of explicit elements called ıdivided powers, which are shown to generate the integral form ofU ı . We prove a conjecture of Balagovic-Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi-K-matrix and the constructions of ıcanonical bases, by avoiding a case-by-case rank one analysis and removing the strong constraints on the parameters in a previous work.

Abstract. This is a generalization of the classic work of Beilinson, Lusztig and MacPherson. In this paper (and an Appendix) we show that the quantum algebras obtained via a BLM-type stabilization procedure in the setting of partial flag varieties of type B/C are two (modified) coideal subalgebras of the quantum general linear Lie algebra,U  andU ı . We provide a geometric realization of the Schur-type duality of Bao-Wang between such a coideal algebra and Iwahori-Hecke algebra of type B. The monomial bases and canonical bases of the Schur algebras and the modified coideal algebraU  are constructed. In an Appendix by three authors, a more subtle 2-step stabilization procedure leading tȯ U ı is developed, and then monomial and canonical bases ofU ı are constructed. It is shown thatU ı is a subquotient ofU  with compatible canonical bases. Moreover, a compatibility between canonical bases for modified coideal algebras and Schur algebras is established.

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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.

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For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$ -canonical bases for the highest weight integrable $\textbf U$ -modules and their tensor products regarded as $\textbf {U}^\imath$ -modules, as well as an $\imath$ -canonical basis for the modified form of the $\imath$ -quantum group $\textbf {U}^\imath$ . A key new ingredient is a family of explicit elements called $\imath$ -divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$ . We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi- $K$ -matrix and the constructions of $\imath$ -canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.