The symmetric invariants of centralizers and Slodowy grading II

TL;DR

This paper proves that for a simple Lie algebra, an element e is good if and only if certain algebraic independence conditions hold for the initial components of generators of symmetric invariants, linking geometric and algebraic properties.

Contribution

It establishes the equivalence between the goodness of e and algebraic independence of initial components of generators, completing a previous partial result.

Findings

e is good if and only if initial components are algebraically independent

The algebra of symmetric invariants is polynomial when e is good

Provides a criterion for goodness based on algebraic independence

Abstract

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $\ell$ over an algebraically closed field $\Bbbk$ 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 ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ the algebra of symmetric invariants of $\mathfrak{g}^{e}$. We say that $e$ is good if the nullvariety of some $\ell$ homogenous elements of ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ in $(\mathfrak{g}^{e})^{*}$ has codimension $\ell$. If $e$ is good then ${\rm S}(\mathfrak{g}^{e})^{\mathfrak{g}^{e}}$ 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 $q_1,\ldots,q_\ell$, the initial homogenous components of their restrictions to $e+\mathfrak{g}^{f}$ are algebraically independent over $\Bbbk$.