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 \(\mathfrak {J}\mathfrak {d}^{1}(\mathcal {W})^{\Gamma }\) 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 \(\Gamma \) -invariant 1-dimensional representations of \(\mathcal {W}\) are obtained by a parabolic induction introduced by Losev.
