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

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 $Γ_e$ 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 $Γ_e$-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 $U(\mathfrak g)$.