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

The index of a complex Lie algebra is the minimal codimension of its coadjoint orbits. Let us suppose $\text{g}$ semisimple, then its index, ${\rm ind} \g$,isequaltoitsrank,${\rm rk \g}$. The goal of this paper is to establish a simple general formula for the index of $\text{n}({\text{g}}^{\xi })$, for $\xi$ nilpotent, where $\text{n}({\text{g}}^{\xi })$ is the normaliser in $\text{g}$ of the centraliser ${\text{g}}^{\xi }$ of $\xi$. More precisely, we have to show the following result, conjectured by D. Panyushev \cite{Panyushev} : $$\mathrm{i}\mathrm{n}\mathrm{d}\text{n}({\text{g}}^{\xi })=\mathrm{r}\mathrm{k}\text{g}-\dim \text{z}({\text{g}}^{\xi }),$$ where $\text{z}({\text{g}}^{\xi })$ is the center of ${\text{g}}^{\xi }$. 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 $\mathcal{S}({\text{g}}^{\xi })$ implies the other inequality. The main part of this paper consists of the proof of the maximality of the rank of this matrix.