Primitive ideals, non-restricted representations and finite $W$-algebras

Let $G$ be a simple algebraic group over $\mathbb C$ and $\mathfrak g={\rm Lie}G$. Let $(e,h,f)$ be an $\mathfrak{sl}_2$-triple in $\mathfrak g$ and $(\cdot,\cdot)$ the $G$-invariant bilinear form on $\mathfrak g$ such that $(e,f)=1$. Let $\chi\in\mathfrak g^*$ be such that $\chi(x)=(e,x)$ for all $x\in\mathfrak g$ and let $H_{\chi}$ denote the enveloping algebra of the Slodowy slice $e+{\rm Ker\,ad} f$. Let $\mathcal I$ be a primitive ideal of the universal enveloping algebra $U(\mathfrak g)$ whose associated variety is the closure of the coadjoint orbit $\mathcal O_{\chi}$. We prove in this note that if $\mathcal I$ has rational infinitesimal character, then there is a finite dimensional irreducible $H_{\chi}$-module V such that $\mathcal I={\rm Ann}_{U(\mathfrak g)}(Q_{\chi}\otimes_{H_{\chi}}V)$, where $Q_{\chi}$ is the generalised Gelfand–Graev $\mathfrak g$-module associated with the triple $(e,h,f)$. In conjunction with well-known results of Barbasch and Vogan this implies that all finite $W$-algebras possess finite dimensional irreducible representations.