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 $$\mathbf{D}_{7}$$ .