On finite dimensional representations of finite W-superalgebras

TL;DR

This paper investigates finite dimensional representations of finite W-superalgebras associated with basic Lie superalgebras, establishing a conjecture link to finite W-algebras and providing classification tools for simple modules.

Contribution

It formulates and proves a version of Premet's conjecture for finite W-superalgebras, connecting their simple modules to those of W-algebras and offering an algorithm for character computation.

Findings

Proves Premet's conjecture for finite W-superalgebras.

Establishes a bijection between simple W-supermodules and W-modules for basic Lie superalgebras.

Provides an algorithm to compute characters of finite-dimensional simple W-supermodules.

Abstract

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}+\mathfrak{g}_{\bar{1}}$ be a basic Lie superalgebra, $\mathcal{W}_0$ (resp.$\mathcal{W}$) be the finite W-(resp.super-) algebras constructed from a fixed nilpotent element in $\mathfrak{g}_{\bar{0}}$. Based on a relation between finite W-algebra $\mathcal{W}_0$ and W-superalgebra $\mathcal{W}$ found recently by the author and Shu, we study the finite dimensional representations of finite W-superalgebras in this paper. We first formulate and prove a version of Premet's conjecture for the finite W-superalgebras from basic simple Lie superalgebras. As in the W-algebra case, the Premet's conjecture is very close to give a classification to the finite dimensional simple $\mathcal{W}$-modules. In the case of $\ggg$ is Lie superalgebras of basic type \Rmnum{1}, we prove the set of simple $\mathcal{W}$-supermodules is bijective with that of simple $\mathcal{W}_0$-modules; presenting a triangular decomposition to the tensor product of $\mathcal{W}$ with a Clifford algebra, we also give an algorithm to compute the character of the finite dimensional simple $\mathcal{W}$-supermodules with integral central character.