A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs

TL;DR

This paper introduces a novel approach to Kazhdan-Lusztig theory of type B using quantum symmetric pairs, establishing new dualities and canonical bases, and extending the theory to Lie superalgebras and types B/C.

Contribution

It develops a new framework linking Hecke algebras, quantum groups, and symmetric pairs, and applies it to Kazhdan-Lusztig theory for Lie superalgebras and classical types.

Findings

Establishes a double centralizer property for type B Hecke algebra and quantum groups.

Introduces a new theory of canonical bases from quantum symmetric pairs.

Formulates and proves Kazhdan-Lusztig theory for the category O of ortho-symplectic Lie superalgebras.

Abstract

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.