The symmetric invariants of centralizers and Slodowy grading II

Abstract

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $\ell$ over an algebraically closed field $\mathbb{k}$ of characteristic zero, and let $(e,h,f)$ be an ${\mathfrak{sl}}_2$-triple of g. Denote by ${\mathfrak{g}}^e$ the centralizer of $e$ in $\mathfrak{g}$ and by $\mathrm{S}({\mathfrak{g}}^e{)}^{{\mathfrak{g}}^e}$ the algebra of symmetric invariants of ${\mathfrak{g}}^e$. We say that $e$ is good if the nullvariety of some $\ell$ homogenous elements of $\mathrm{S}({\mathfrak{g}}^e{)}^{{\mathfrak{g}}^e}$ in $({\mathfrak{g}}^e{)}^{\ast }$ has codimension $\ell$. If $e$ is good then $\mathrm{S}({\mathfrak{g}}^e{)}^{{\mathfrak{g}}^e}$ is a polynomial algebra. In this paper, we prove that the converse of the main result of arXiv:1309.6993 is true. Namely, we prove that $e$ is good if and only if for some homogenous generating sequence $q_1,\dots ,q_{\ell }$, the initial homogenous components of their restrictions to $e+{\mathfrak{g}}^f$ are algebraically independent over $\mathbb{k}$.

arXiv comment26 pages in English. Minor corrections, improved exposition