Wang and the author initiated a theory of canonical bases arising from quantum symmetric pairs. We showed that a coideal subalgebra of $\U_q(\mathfrak{sl}_{k})$ centralizes the Hecke algebra of type B (of equal parameters) when acting on $\VV^{\otimes m}$, the tensor product of the natural representation $\VV$ of $\U_q(\mathfrak{sl}_{k})$. We constructed a (new) $\imath$-canonical basis on $\VV^{\otimes m}$, which allows a reformulation of the Kazhdan-Lusztig theory of type B independent of the Hecke algebra. The theory was further applied to formulate and establish for the first time the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m+1|2n)$. The geometric realization of the coideal subalgebras considered there and the canonical bases on the modified coideal subalgebras have been given in [@BKLW] and [@LW15] using partial flag varieties of type B/C.

On the other hand, the problem of determining the irreducible characters in the BGG category $\mc{O}$ of the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m|2n)$ is still open since 1970’s.

In this paper, we provide a complete solution to the irreducible character problem in the BGG category $\mc{O}$ of modules of integral and half-integral weights of the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m|2n)$. We adapt the theory of canonical bases from [@BW13] to 