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Indice du normalisateur du centralisateur d’un élément nilpotent dans une algèbre de Lie semi-simple

Anne Moreau — 2006

Bulletin de la Société Mathématique de France

L’indice d’une algèbre de Lie algébrique complexe est la codimension minimale de ses orbites coadjointes. Si 𝔤 est semi-simple, son indice, ind 𝔤 , est égal à son rang,  rg 𝔤 . Le but de cet article est d’établir une formule générale pour l’indice de 𝔫 ( 𝔤 e ) pour e nilpotent, où 𝔫 ( 𝔤 e ) est le normalisateur dans 𝔤 du centralisateur 𝔤 e de e . Plus précisément, on obtient le résultat suivant, conjecturé par D. Panyushev : ind 𝔫 ( 𝔤 e ) = rg 𝔤 - dim 𝔷 ( 𝔤 e ) , 𝔷 ( 𝔤 e ) est le centre de 𝔤 e . Panyushev obtient l’inégalité ind 𝔫 ( 𝔤 e ) rg 𝔤 - dim 𝔷 ( 𝔤 e ) dans Panyushev 2003...

Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras.

Karin BaurAnne Moreau — 2011

Annales de l’institut Fourier

We say that a finite dimensional Lie algebra is quasi-reductive if it has a linear form whose stabilizer for the coadjoint representation, modulo the center, is a reductive Lie algebra with a center consisting of semisimple elements. Parabolic subalgebras of a semisimple Lie algebra are not always quasi-reductive (except in types A or C by work of Panyushev). The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo,...

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