Quantum group of type $A$ and representations of queer Lie superalgebra

Abstract

We establish a maximal parabolic version of the Kazhdan-Lusztig conjecture \cite[Conjecture 5.10]{CKW} for the BGG category ${\mathcal{O}}_{k,\zeta }$ of $\mathfrak{q}(n)$-modules of "$\pm \zeta$-weights", where $k\le n$ and $\zeta \in \mathbb{C}\setminus \frac{1}{2}\mathbb{Z}$. As a consequence, the irreducible characters of these $\mathfrak{q}(n)$-modules in this maximal parabolic category are given by the Kazhdan-Lusztig polynomials of type $A$ Lie algebras. As an application, closed character formulas for a class of $\mathfrak{q}(n)$-modules resembling polynomial and Kostant modules of the general linear Lie superalgebras are obtained.

26 pages