Indice du normalisateur du centralisateur d'un element nilpotent dans une algebre de Lie semi-simple
Abstract
Description
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.
54 pages en francais
54 pages en francais