Canonical bases for tensor products and super Kazhdan-Lusztig theory

We generalize a construction in [BW18] (arXiv:1610.09271) by showing that the tensor product of a based ${\mathbf{\text{U}}}^{\mathrm{ı}}$-module and a based $\mathbf{\text{U}}$-module is a based ${\mathbf{\text{U}}}^{\mathrm{ı}}$-module. This is then used to formulate a Kazhdan-Lusztig theory for an arbitrary parabolic BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebras, extending a main result in [BW13] (arXiv:1310.0103).