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Pré-Publication, Document De Travail Année : 2014

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}$_{7}$.
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Dates et versions

hal-01082562 , version 1 (13-11-2014)
hal-01082562 , version 2 (18-12-2014)

Identifiants

  • HAL Id : hal-01082562 , version 2

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Jean-Yves Charbonnel, Anne Moreau. THE SYMMETRIC INVARIANTS OF CENTRALIZERS AND SLODOWY GRADING. 2014. ⟨hal-01082562v2⟩
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