Kazhdan-Lusztig theory of type D. On the other hand, the proof of the Kazhdan-Lusztig theory for the category $\mc{O}^{\bf b}$ of the Lie superalgebra $\mathfrak{osp}(2m|2n)$ with respect to ${\bf b}$ of the second type follows by induction on $m$, where the base case $m=0$ follows from the classical Kazhdan-Lusztig theory of type C. The induction processes of the two types are actually similar, where we compare category $\mc{O}$’s with respect to adjacent Borel subalgebras (switching adjacent $0$ and $1$ in the sequence ${\bf b}$), as well as compare the parabolic category $\mc{O}$ with the full category $\mc{O}$. We also study certain infinite rank limits of the parabolic category $\mc{O}$.

[**Acknowledgement.**]{} The author would like to thank Weiqiang Wang for helpful discussion and suggestion on this paper. This research is partially supported by the AMS-Simons Travel Grant. A part of this paper was written when the author was visiting the Max-Planck-Institute in Bonn during summer 2015. He would like to thank the institute for its excellent working environment and support.

Preliminaries on quantum groups {#sec:prelim}
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In this preliminary section, we review some basic definitions and constructions on quantum groups from Lusztig’s book [@Lu94]. We also introduce the involution $\inv$ and 