Indice du normalisateur du centralisateur d'un element nilpotent dans une algebre de Lie semi-simple

Abstract

The index of a complex Lie algebra is the minimal codimension of its coadjoint orbits. Let us suppose $\g$ semisimple, then its index, ${\rm ind} \g$, is equal to its rank, ${\rm rk \g}$. The goal of this paper is to establish a simple general formula for the index of $\n(\g^{\xi})$, for $\xi$ nilpotent, where $\n(\g^{\xi})$ is the normaliser in $\g$ of the centraliser $\g^{\xi}$ of $\xi$. More precisely, we have to show the following result, conjectured by D. Panyushev \cite{Panyushev} : $${\rm ind} \n(\g^{\xi}) = {\rm rk \g}-\dim \z(\g^{\xi}),$$ where $\z(\g^{\xi})$ is the center of $\g^{\xi}$. D. Panyushev obtained in \cite{Panyushev} the inequality \hbox{${\rm ind} \n(\g^{\xi}) \geq {\rm rg \g}-\dim \z(\g^{\xi})$} and we show that the maximality of the rank of a certain matrix with entries in the symmetric algebra ${\cal S}(\g^{\xi})$ implies the other inequality. The main part of this paper consists of the proof of the maximality of the rank of this matrix.