Geometric Schur duality of classical type

This is a generalization of the classic work of Beilinson, Lusztig and MacPherson. In this paper (and an Appendix) we show that the quantum algebras obtained via a BLM-type stabilization procedure in the setting of partial flag varieties of type $B/C$ are two (modified) coideal subalgebras of the quantum general linear Lie algebra, $\dot{\mathbf U}^{\jmath}$ and $\dot{\mathbf U}^{\imath}$. We provide a geometric realization of the Schur-type duality of Bao-Wang between such a coideal algebra and Iwahori-Hecke algebra of type $B$. The monomial bases and canonical bases of the Schur algebras and the modified coideal algebra $\dot{\mathbf U}^{\jmath}$ are constructed. In an Appendix by three authors, a more subtle $2$-step stabilization procedure leading to $\dot{\mathbf U}^{\imath}$ is developed, and then monomial and canonical bases of $\dot{\mathbf U}^{\imath}$ are constructed. It is shown that $\dot{\mathbf U}^{\imath}$ is a subquotient of $\dot{\mathbf U}^{\jmath}$ with compatible canonical bases. Moreover, a compatibility between canonical bases for modified coideal algebras and Schur algebras is established.