  We show that Hecke algebra of type B and a coideal subalgebra of the type A
quantum group satisfy a double centralizer property, generalizing the
Schur-Jimbo duality in type A. The quantum group of type A and its coideal
subalgebra form a quantum symmetric pair. A new theory of canonical bases
arising from quantum symmetric pairs is initiated. It is then applied to
formulate and establish for the first time a Kazhdan-Lusztig theory for the BGG
category O of the ortho-symplectic Lie superalgebras $\mathfrak{osp}(2m+1|2n)$.
In particular, our approach provides a new formulation of the Kazhdan-Lusztig
theory for Lie algebras of type B/C.
