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}.
Categories: math.RT