Derived subalgebras of centralisers and finite -algebras

Let$\mathfrak{g}=\mbox{Lie}(G)$be the Lie algebra of a simple algebraic group$G$over an algebraically closed field of characteristic$0$. Let$e$be a nilpotent element of$\mathfrak{g}$and let$\mathfrak{g}_e=\mbox{Lie}(G_e)$where$G_e$stands for the stabiliser of$e$in$G$. For$\mathfrak{g}$classical, we give an explicit combinatorial formula for the codimension of$[\mathfrak{g}_e,\mathfrak{g}_e]$in$\mathfrak{g}_e$and use it to determine those$e\in \mathfrak{g}$for which the largest commutative quotient$U(\mathfrak{g},e)^{\mbox{ab}}$of the finite$W$-algebra$U(\mathfrak{g},e)$is isomorphic to a polynomial algebra. It turns out that this happens if and only if$e$lies in a unique sheet of$\mathfrak{g}$. The nilpotent elements with this property are callednon-singularin the paper. Confirming a recent conjecture of Izosimov, we prove that a nilpotent element$e\in \mathfrak{g}$is non-singular if and only if the maximal dimension of the geometric quotients$\mathcal{S}/G$, where$\mathcal{S}$is a sheet of$\mathfrak{g}$containing$e$, coincides with the codimension of$[\mathfrak{g}_e,\mathfrak{g}_e]$in$\mathfrak{g}_e$and describe all non-singular nilpotent elements in terms of partitions. We also show that for any nilpotent element$e$in a classical Lie algebra$\mathfrak{g}$the closed subset of Specm $U(\mathfrak{g},e)^{\mbox{ab}}$consisting of all points fixed by the natural action of the component group of$G_e$is isomorphic to an affine space. Analogues of these results for exceptional Lie algebras are also obtained and applications to the theory of primitive ideals are given.