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 $\n(\g^ξ)$, for $ξ$ nilpotent, where $\n(\g^ξ)$ is the normaliser in $\g$ of the centraliser $\g^ξ$ of $ξ$. More precisely, we have to show the following result, conjectured by D. Panyushev \cite{Panyushev} : $${\rm ind} \n(\g^ξ) = {\rm rk \g}-\dim \z(\g^ξ),$$ where $\z(\g^ξ)$ is the center of $\g^ξ$. D. Panyushev obtained in \cite{Panyushev} the inequality \hbox{${\rm ind} \n(\g^ξ) \geq {\rm rg \g}-\dim \z(\g^ξ)$} and we show that the maximality of the rank of a certain matrix with entries in the symmetric algebra ${\cal S}(\g^ξ)$ implies the other inequality. The main part of this paper consists of the proof of the maximality of the rank of this matrix.