On invariant $1$-dimensional representations of a finite $W$-algebra

Abstract: Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$ and $G$ be the corresponding simply connected algebraic group. Consider a nilpotent element $e\in \mathfrak{g}$, the corresponding element $\chi=(e, \bullet)$ in $\mathfrak{g}^*$, and the coadjoint orbit $\mathbb{O}=G\chi$. We are interested in the set $\mathfrak{J}\mathfrak{d}^1(\mathcal{W})$ of codimension $1$ ideals $J\subset \mathcal{W}$ in a finite $W$-algebra $\mathcal{W}=U(\mathfrak{g}, e)$. We have a natural action of the component group $\Gamma=Z_G(\chi)/Z_G^\circ(\chi)$ on $\mathfrak{J}\mathfrak{d}^1(\mathcal{W})$. Denote the set of $\Gamma$-stable points of $\mathfrak{J}\mathfrak{d}^1(\mathcal{W})$ by $\mathfrak{J}\mathfrak{d}^{1}(\mathcal{W})^{\Gamma}$. For a classical $\mathfrak{g}$ Premet and Topley proved that $\mathfrak{J}\mathfrak{d}^{1}(\mathcal{W})^{\Gamma}$ is isomorphic to an affine space. In this paper we will give an easier and shorter proof of this fact.