Supercharacters of queer Lie superalgebras

TL;DR

This paper proves that irreducible finite-dimensional modules over the queer Lie superalgebra have trivial supercharacters, confirming a conjecture and revealing new structural properties of these modules.

Contribution

It establishes the triviality of supercharacters for finite-dimensional irreducible modules over the queer Lie superalgebra, confirming a conjecture by Gorelik, Grantcharov, and Mazorchuk.

Findings

The space of invariants under the even part is trivial for these modules.

Supercharacters of finite-dimensional irreducible modules are trivial.

Confirmed a conjecture about supercharacter triviality in queer Lie superalgebras.

Abstract

Let $\mathfrak g=\mathfrak g_{\bar 0}\oplus\mathfrak g_{\bar 1}$ be the queer Lie superalgebra and let $L$ be a finite-dimensional non-trivial irreducible $\mathfrak g$-module. Restricting the $\mathfrak g$-action on $L$ to $\mathfrak g_{\bar 0}$, we show that the space of $\mathfrak g_{\bar 0}$-invariants $L^{\mathfrak g_{\bar 0}}$ is trivial. As a consequence we establish a conjecture first formulated by Gorelik, Grantcharov and Mazorchuk on the triviality of the supercharacter of irreducible $\mathfrak g$-modules in the case when the modules are finite dimensional.