difficulty, that is, simple $\U_q(\mathfrak{sl}_{k})$-modules and their tensor products admit $\imath$-canonical bases. In the ongoing work [@BW16], we generalize the construction of $\imath$-canonical bases to more general quantum symmetric pairs (see also Remark \[rem:BW16\]).

Thanks to the (weak) Schur type dualities for the type D (and type C), the classical Kazhdan-Lusztig theory of type D (and type C, respectively) can be reformulated in terms of $\imath$-canonical bases on $\VV^{\otimes m}$ (on $(\VV^*)^{\otimes n}$, respectively). More precisely speaking, the entries of the transition matrix between the $\imath$-canonical basis and the standard basis (i.e., the monomial basis) are exactly the Kazhdan-Lusztig polynomials of type D (and type C, respectively.)

We apply the theory of $\imath$-canonical bases to the BGG category $\mathcal{O}$ of the Lie superalgebra $\mathfrak{osp}(2m|2n)$ in Section \[sec:rep\]. In this section we consider the infinite rank limit of the quantum symmetric pair $(\U_q(\mathfrak{sl}_{\infty}), {\U^{\imath}}_q(\mathfrak{sl}_{\infty}))$. The theory of the super duality developed in [@CLW11] plays the essential role.

For a $0^m1^n$-sequence ${\bf b}$ (which consists of $m$ zeros and $n$ ones), we define a tensor space $\mathbb{T}^{\bf b}$ using $m$ copies of $\VV$ and $n$ copies of $\VV^*$ with the tensor order prescribed by ${\bf b}$ (with 0 corresponds to V). In 