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 $Gamma_e$ 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 Losevs parabolic induction functor. In turn this gives examples where there is a non-induced multiplicity free primitive ideal of $U(mathfrak g)$.