Abstract: Inspired by the work of Gukov-Witten, we investigate stringy E-polynomials of nilpotent orbit closures of type $B_n$ and $C_n$. Classically, there is a famous Springer duality between special orbits. Therefore, it is natural to speculate that the mirror symmetry we seek may coincide with Springer duality in the context of special orbits. Unfortunately, such a naive statement fails. To remedy the situation, we propose a conjecture which asserts the mirror symmetry for certain parabolic/induced covers of special orbits. Then, we prove the conjecture for Richardson orbits and obtain certain partial results in general.  In the mirror symmetry, we find an interesting seesaw phenonem where Lusztig's canonical quotient group plays an important role. This talk is based on the joint work with Baohua Fu and Yongbin Ruan, https://arxiv.org/abs/2207.10533.