Canonical Bases Arising from Quantum Symmetric Pairs and Kazhdan-Lusztig Theory

A breakthrough in representation theory is the discovery of canonical bases of quantum groups by Lusztig. In type A, the canonical bases can be used to reformulate the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of general linear Lie algebras, which enables further generalization to Brundan's Kazhdan-Lusztig conjecture for general linear Lie superalgebras.

In this dissertation, we first show a coideal subalgebra of the quantum group of type A and the Hecke algebra of type B satisfy a double centralizer property, generalizing the Schur-Jimbo duality. The quantum group of type A and its coideal subalgebra form a quantum symmetric pair.
Then we initiate a theory of canonical bases arising from quantum symmetric pairs.
We show simple integrable modules of the quantum group of type A and their tensor products admit new canonical bases different from Lusztig's canonical bases.
Finally we use such new canonical bases to formulate and establish the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m+1|2n)$ for the first time. The non-super specialization of our theory amounts to a new formulation of the classical Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of the Lie algebras of type B/C.