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

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbf{C}$ and $G$ be the corresponding simply connected algebraic group. Consider a nilpotent element $e\in \mathfrak{g}$, the corresponding element $\chi =(e,\bullet )$ in ${\mathfrak{g}}^{\ast }$, and the coadjoint orbit $\mathbf{O}=G\chi$. 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 $\mathrm{\Gamma }=Z_G(\chi )\mathrm{/}{Z}_G^{\circ }(\chi )$ on $\mathfrak{J}{\mathfrak{d}}^1(\mathcal{W})$. Denote the set of $\mathrm{\Gamma }$-stable points of $\mathfrak{J}{\mathfrak{d}}^1(\mathcal{W})$ by $\mathfrak{J}{\mathfrak{d}}^1(\mathcal{W}{)}^{\mathrm{\Gamma }}$. For a classical $\mathfrak{g}$ Premet and Topley proved that $\mathfrak{J}{\mathfrak{d}}^1(\mathcal{W}{)}^{\mathrm{\Gamma }}$ are parameterized by the points of an affine space. In this paper we will give an alternative proof of this fact using completely different approach. As a corollary of our construction, we imply that for classical $\mathfrak{g}$ all $\mathrm{\Gamma }$-invariant $1$-dimensional representations of $\mathcal{W}$ are obtained by a parabolic induction introduced by Losev.