We reformulate the Kazhdan-Lusztig theory for the BGG category O of Lie algebras of type D via the theory of canonical bases arising from quantum symmetric pairs initiated by Weiqiang Wang and the author. This is further applied to formulate and establish for the first time the Kazhdan-Lusztig theory for the BGG category O of the ortho-symplectic Lie superalgebra osp(2m|2n). 247 248 HUANCHEN BAO group U q (sl k ) of type A. Brundan's conjecture was proved first by Cheng, Lam and Wang [CLW15]

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... later by Brundan, Losev and Webster [BLW]. Recently in [BW13], Weiqiang Wang and the author initiated a theory of canonical bases arising from quantum symmetric pairs. We showed that a coideal subalgebra of U q (sl k ) centralizes the Hecke algebra of type B (of equal parameters) when acting on V ⊗m , the tensor product of the natural representation V of U q (sl k ). We constructed a (new) ı-canonical basis on V ⊗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 O of the ortho-symplectic Lie superalgebra 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 [LW] using partial flag varieties of type B/C. On the other hand, the problem of determining the irreducible characters in the BGG category O of the ortho-symplectic Lie superalgebra osp (2m|2n) is still open since the 1970s.