Character formulae for queer Lie superalgebras and canonical bases of types $A/C$

TL;DR

This paper formulates Kazhdan-Lusztig conjectures for queer Lie superalgebras using canonical bases of quantum groups of types A and C, and provides character formulas for finite-dimensional modules.

Contribution

It introduces new Kazhdan-Lusztig conjectures for queer Lie superalgebras based on canonical bases of quantum groups, linking representation theory with quantum algebra structures.

Findings

Formulated Kazhdan-Lusztig conjectures for $rak{q}(n)$-modules

Established character formulas for finite-dimensional irreducible modules

Connected module characters with canonical bases of quantum groups

Abstract

For the BGG category of $\mathfrak{q}(n)$-modules of half-integer weights, a Kazhdan-Lusztig conjecture \`a la Brundan is formulated in terms of categorical canonical basis of the $n$th tensor power of the natural representation of the quantum group of type $C$. For the BGG category of $\mathfrak{q}(n)$-modules of congruent non-integral weights, a Kazhdan-Lusztig conjecture is formulated in terms of canonical basis of a mixed tensor of the natural representation and its dual of the quantum group of type $A$. We also establish a character formula for the finite-dimensional irreducible $\mathfrak{q}(n)$-modules of half-integer weights in terms of type $C$ canonical basis of the corresponding $q$-wedge space.