The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits

Let $G$ be a simple algebraic group defined over $\mathbb{C}$ and let $e$ be a rigid nilpotent element in $g = \operatorname{Lie} (G)$. In this paper we prove that the finite $W$-algebra $U(\mathfrak{g}, e)$ admits either one or two $1$-dimensional representations. Thanks to the results obtained earlier this boils down to showing that the finite $W$-algebras associated with the rigid nilpotent orbits of dimension 202 in the Lie algebras of type $E_8$ admit exactly two 1‑dimensional representations. As a corollary, we complete the description of the multiplicity-free primitive ideals of $U(\mathfrak{g})$ associated with the rigid nilpotent $G$-orbits of $\mathfrak{g}$. At the end of the paper, we apply our results to enumerate the small irreducible representations of the related reduced enveloping algebras.