On the variety of 1-dimensional representations of finite $W$-algebras in low rank

TL;DR

This paper investigates the structure of 1-dimensional representations of finite W-algebras associated with nilpotent elements in low-rank simple Lie algebras, revealing new examples of non-induced primitive ideals.

Contribution

It provides computational analysis of the variety of 1-dimensional representations and identifies cases of non-induced primitive ideals in low-rank Lie algebras.

Findings

Determined the structure of the variety of 1-dimensional representations for low-rank cases.

Discovered examples of non-induced multiplicity free primitive ideals.

Showed the action of the component group on the representation variety.

Abstract

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e \in \mathfrak g$ be nilpotent. We consider the finite $W$-algebra $U(\mathfrak g,e)$ associated to $e$ and the problem of determining the variety $\mathcal E(\mathfrak g,e)$ of 1-dimensional representations of $U(\mathfrak g,e)$. For $\mathfrak g$ of low rank, we report on computer calculations that have been used to determine the structure of $\mathcal E(\mathfrak g,e)$, and the action of the component group $\Gamma_e$ of the centralizer of $e$ on $\mathcal E(\mathfrak g,e)$. As a consequence, we provide two examples where the nilpotent orbit of $e$ is induced, but there is a 1-dimensional $\Gamma_e$-stable $U(\mathfrak g,e)$-module which is not induced via Losev's parabolic induction functor. In turn this gives examples where there is a "non-induced" multiplicity free primitive ideal of $U(\mathfrak g)$.