On spherical nilpotent orbits and beyond

Panyushev, Dmitri I.

Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1453-1476.

Nous continuons nos investigations concernant la complexité des orbites nilpotentes dans une algèbre de Lie semi-simple. Nous donnons une caractérisation des orbites nilpotentes sphériques au moyen d’une sous-algèbre de Levi minimale qui les rencontre. Ceci fournit une sorte de forme canonique pour ces orbites. Nous obtenons une description des orbites minimales sphériques pour toutes les algèbres de Lie simples. La théorie obtenue pour la représentation adjointe s’étend aux $\theta$-groupes de Vinberg. Nous en déduisons une description des orbites nilpotentes sphériques pour la représentation associée à un espace symétrique.

We continue investigations that are concerned with the complexity of nilpotent orbits in semisimple Lie algebras. We give a characterization of the spherical nilpotent orbits in terms of minimal Levi subalgebras intersecting them. This provides a kind of canonical form for such orbits. A description minimal non-spherical orbits in all simple Lie algebras is obtained. The theory developed for the adjoint representation is then extended to Vinberg’s $\theta$-groups. This yields a description of spherical nilpotent orbits for the isotropy representation of a symmetric variety.

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