Finite-dimensional half-integer weight modules over queer Lie superalgebras

TL;DR

This paper offers a new perspective on the representation theory of finite-dimensional half-integer weight modules over the queer Lie superalgebra, utilizing Lusztig's canonical basis to compute characters and derive explicit formulas.

Contribution

It introduces a novel interpretation linking half-integer weight modules to integer weight modules via Lusztig's basis, enabling explicit character calculations.

Findings

Computed characters of irreducible modules.

Derived closed-form character formulas for special weight types.

Connected representation theory to Lusztig's canonical basis.

Abstract

We give a new interpretation of representation theory of the finite-dimensional half-integer weight modules over the queer Lie superalgebra $\mathfrak{q}(n)$. It is given in terms of Brundan's work of finite-dimensional integer weight $\mathfrak{q}(n)$-modules by means of Lusztig's canonical basis. Using this viewpoint we compute the characters of the finite-dimensional half-integer weight irreducible modules. For a large class of irreducible modules whose highest weights are of special types (i.e., totally connected or totally disconnected) we derive closed-form character formulas that are reminiscent of Kac-Wakimoto character formula for classical Lie superalgebras.