The symmetric invariants of centralizers and Slodowy grading

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 be a nilpotent element of $\mathfrak {g}$. 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 $({\mathfrak g}^{e})^{*}$ has codimension $\ell$. If e is good then $\mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}}$ is a polynomial algebra. The main result of this paper stipulates that if for some homogenous generators of $\mathrm{S}({\mathfrak g}) ^{{\mathfrak g}}$, the initial homogenous components of their restrictions to $e+\mathfrak {g}^{f}$ are algebraically independent, with (e, h, f) an $\mathfrak {sl}_2$-triple of $\mathfrak {g}$, then e is good. As applications, we pursue the investigations of Panyushev et al. (J. Algebra 313:343–391, 2007) and we produce (new) examples of nilpotent elements that satisfy the above polynomiality condition, in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type $\mathbf{D}_{7}$.