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 $e \in \mathfrak g$benilpotent\mathrm{.}Weconsiderthefinite$W$-algebra$U(\mathfrak g,e)$associatedto$e$andtheproblemofdeterminingthevariety$\mathcal E(\mathfrak g,e)$of1-dimensionalrepresentationsof$U(\mathfrak g,e)$\mathrm{.}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 ${\mathrm{\Gamma }}_e$ 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 ${\mathrm{\Gamma }}_e$-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 $U(\mathfrak{g})$.