Kazhdan-Lusztig theory of super type D and quantum symmetric pairs

Abstract

Abstract. 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 KazhdanLusztig theory for the BGG category O of the ortho-symplectic Lie superalgebra osp(2m|2n).

Positivity VS Negativity of Canonical Bases

“…Via a stabilization procedure these algebras give rise to a limit algebra which was shown to be isomorphic to the modified quantum coideal algebra iU (gl n ) of gl n , and which also admits a stably canonical basis. The appearance of the quantum coideal algebra was inspired by [3] where a new approach to Kazhdan-Lusztig theory of type B/C via a new theory of canonical bases arising from quantum coideal algebras was developed. Even though the constructions for n odd and even are quite different with the case of even n being more challenging [3], one can carry out the construction in the even n case by relating to the odd n case via a more subtle two-step stabilization [2].…”

Branching Rules for Finite-Dimensional U q ( 𝖘 𝖚 ( 3 ) ) $\mathcal {U}_{q}(\mathfrak {su}(3))$ -Representations with Respect to a Right Coideal Subalgebra

“…The motivating example for the development for this theory was given by Koornwinder [15], who studied scalar-valued spherical functions on the quantum analogue of (SU (2), U(1)) considering twisted primitive elements in the quantised universal enveloping algebra of U q (sl (2)). Koornwinder identified all scalar-valued spherical functions with Askey-Wilson polynomials in two free parameters.…”

“…One recent extension of this situation [1], where higher-dimensional representations of the coideal subalgebra B are involved, arises with the study of matrix-valued spherical functions of the quantum analogue of (SU(2) × SU(2), SU (2)) where the subgroup is diagonally embedded. The quantum symmetric pair is given by the quantised universal enveloping algebra of U q (g), where g = su(2) ⊕ su (2), and a right coideal subalgebra B that can be identified with U q (su (2)).…”

“…The quantum symmetric pair is given by the quantised universal enveloping algebra of U q (g), where g = su(2) ⊕ su (2), and a right coideal subalgebra B that can be identified with U q (su (2)). As in the Lie group setting [8,11,12,27], the explicit knowledge of the branching rules plays a fundamental role in the explicit determination of the matrix-valued spherical functions.…”

“…[6] and Bao, see e.g. [2], it is indicated how modern representation theory for quantum groups, such as Schur-Jimbo duality, canonical bases, Kazhdan-Lusztig theory, categorification, etc., can be extended to quantum symmetric pairs. Moreover, the role of quantum symmetric pairs in integrable systems in mathematical physics is explained in [3] in conjuction with other types of boundary conditions.…”