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

TL;DR

This paper proves a maximal parabolic Kazhdan-Lusztig conjecture for queer Lie superalgebra modules, providing explicit character formulas and linking their representation theory to type A Lie algebra polynomials.

Contribution

It establishes a maximal parabolic Kazhdan-Lusztig conjecture for $rak{q}(n)$-modules, connecting their characters to type A Kazhdan-Lusztig polynomials and deriving explicit character formulas.

Findings

Irreducible characters are given by type A Kazhdan-Lusztig polynomials.

Provides closed-form character formulas for certain $rak{q}(n)$-modules.

Validates the maximal parabolic Kazhdan-Lusztig conjecture for these modules.

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 $k\leq n$ 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.