Calculs explicites dans une algèbre de Lie semi-simple effectués avec GAP4

Anne Moreau (1)

1 IMJ - Institut de Mathématiques de Jussieu

In \cite{indice}, we show the following result, conjectured by D. Panyushev \cite{Panyushev}, for $\g$ a semisimple Lie algebra: {\rm ind}~\n(\g^{e}) = {\rm rk}~\g-\dim \z(\g^{e}, where $\n(\g^{e})$ and $\z(\g^{e})$ are, respectively, the normaliser and the centre of the centraliser $\g^{e}$ of a nilpotent element $e$. This result is proved in \cite{indice} when $\g$ is a classical simple Lie algebra and when $e$ satisfies a certain property $(P)$. We present in this paper the computations, made using GAP4, which prove that distinguished, non-regular, nilpotent orbits in $E_6$, $E_7$, $E_8$ and $F_4$ satisfy the property $(P)$. This work completes the proof, presented in \cite{indice}, of the equality (\ref{princ}). The complete proof of this result was already presented in \cite{indice_arxiv}.

Mots clés: indice; GAP; algèbres de Lie exceptionnelles; éléments nilpotents; sl2-triplets

Domaines: Théorie des représentations [math.RT]

Anne Moreau. Calculs explicites dans une algèbre de Lie semi-simple effectués avec GAP4. 2005. ⟨hal-00004346⟩