The Index of Centralizers of Elements in Classical Lie Algebras

The index of a finite-dimensional Lie algebra $\mathfrak{g}$ is the minimum of dimensions of the stabilizers $\mathfrak{g}_\alpha$ over all covectors $\alpha\in\mathfrak{g}^*$. Let $\mathfrak{g}$ be a reductive Lie algebra over a field $\mathbb{K}$ of characteristic $\ne2$. Élashvili conjectured that the index of $\mathfrak{g}_\alpha$ is always equal to the index, or, which is the same, the rank of $\mathfrak{g}$. In this article, Élashvili's conjecture is proved for classical Lie algebras. Furthermore, it is shown that if $\mathfrak{g}=\mathfrak{gl}_n$ or $\mathfrak{g}=\mathfrak{sp}_{2n}$ and $e\in\mathfrak{g}$ is a nilpotent element, then the coadjoint action of $\mathfrak{g}_e$ has a generic stabilizer. For $\mathfrak{g}=\mathfrak{so}_n$, we give examples of nilpotent elements $e\in\mathfrak{g}$ such that the coadjoint action of $\mathfrak{g}_e$ does not have a generic stabilizer.