On the one-dimensional representations of finite $W$-superalgebras for $\mathfrak{gl}_{M|N}$

TL;DR

This paper classifies all one-dimensional representations of finite W-superalgebras associated with the general linear Lie superalgebra, using shifted super Yangians to analyze their structure.

Contribution

It provides a complete classification of one-dimensional representations for finite W-superalgebras of gl_{M|N}, a novel result in the representation theory of Lie superalgebras.

Findings

Complete classification of one-dimensional representations.

Use of shifted super Yangians to analyze algebra structure.

Identification of commutative quotients of W-superalgebras.

Abstract

Let $\mathfrak{g}=\mathfrak{gl}_{M|N}(\mathbb{k})$ be the general linear Lie superalgebra over an algebraically closed field $\mathbb{k}$ of characteristic zero. Fix an arbitrary even nilpotent element $e$ in $\mathfrak{g}$ and let $U(\mathfrak{g},e)$ be the finite $W$-superalgebra associated to the pair $(\mathfrak{g},e)$. In this paper we will give a complete classification of one-dimensional representations for $U(\mathfrak{g},e)$. To achieve this, we use the tool of shifted super Yangians to determine the commutative quotients of the finite $W$-superalgebras.