THE SYMMETRIC INVARIANTS OF CENTRALIZERS AND SLODOWY GRADING
Résumé
Let $\g$ be a finite-dimensional simple Lie algebra of rank $\rg$ over an
algebraically closed field $\k$ of characteristic zero, and let $e$ be a nilpotent
element of $\g$. Denote by $\g^{e}$ the centralizer of $e$ in $\g$
and by $\ai g{e}{}$ the algebra of symmetric invariants of $\g^{e}$.
We say that $e$ is {\em good} if the nullvariety of some $\rg$ homogeneous elements of
$\ai g{e}{}$ in $({\goth g}^{e})^{*}$ has codimension $\rg$. If $e$ is good then
$\ai g{e}{}$ is a polynomial algebra. The main result of this paper stipulates that if
for some homogeneous generators of $\ai g{}{}$, the initial homogeneous components of
their restrictions to $e+\g^{f}$ are algebraically independent, with
$(e,h,f)$ an $\mathfrak{sl}_2$-triple of $\g$, then $e$ is good.
As applications, we pursue the investigations of \cite{PPY}
and we produce (new) examples of nilpotent elements
that verify the above polynomiality condition, in simple Lie algebras of both classical
and exceptional types. We also give a counter-example in type {\bf D}$_{7}$.
Origine : Fichiers produits par l'(les) auteur(s)
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