terms of the canonical bases (i.e., Kazhdan-Lusztig bases) of Hecke algebras. On the other hand, the classification of finite-dimensional simple Lie superalgebras over complex numbers has been obtained by Kac ([@Kac77]) in 1970’s, while the representation theory of Lie superalgebras turns out to be very difficult. One of the main reasons is that the corresponding Weyl group of a Lie superalgebra is not enough to control the linkage principle in the BGG category $\mc{O}$. Thus the relevant Hecke algebras do not play significant roles in the representation theory of Lie superalgebras as in the representation theory of semisimple Lie algebras.

The Lie superalgebras $\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(m|2n)$, which generalize the classical Lie algebras, are arguably the most important classes of Lie superalgebras. In 2003, Brundan in [@Br03] formulated a Kazhdan-Lusztig type conjecture for the full category $\mc{O}$ of general linear Lie superalgebras. The Jimbo-Schur ([@Jim]) duality plays a crucial role in Brundan’s conjecture, which allows a reformulation of the Kazhdan-Lusztig theory in type A in terms of the canonical bases of the quantum group $\U_q(\mathfrak{sl}_{k})$ of type A. Brundan’s conjecture was proved first by Cheng, Lam and Wang [@CLW15] and later by Brundan, Losev and Webster [@BLW].

Recently in [@BW13], Weiqiang 