Super duality and Crystal bases for quantum orthosymplectic superalgebras II

Let $\mathcal{O}^{int}_q(m|n)$ be a semisimple tensor category of modules over a quantum ortho-symplectic superalgebra of type $B, C, D$ introduced in the author's previous work. It is a natural counterpart of the category of finitely dominated integrable modules over a quantum group of type $B, C, D$ from a viewpoint of super duality. Continuing the previous work on type $B$ and $C$, we classify the irreducible modules in $\mathcal{O}^{int}_q(m|n)$, and prove the existence and uniqueness of their crystal bases in case of type $D$. A new combinatorial model of classical crystals of type $D$ is introduced, whose super analogue gives a realization of crystals for the highest weight modules in $\mathcal{O}^{int}_q(m|n)$.