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

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 $U(\mathfrak{g})$.

13 pages