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

Baur, Karin; Moreau, Anne

Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 417-451.

Une algèbre de Lie de dimension finie est dite quasi-réductive si elle possède une forme linéaire dont le stablisateur pour la représentation coadjointe, modulo le centre, est une algèbre de Lie réductive avec un centre formé d’éléments semi-simples. Les sous-algèbres paraboliques d’une algèbre de Lie semi-simple ne sont pas toujours quasi-réductives (sauf en types A ou C d’après un résultat de Panyushev). Récemment, Duflo, Khalgui and Torasso ont terminé la classification des sous-algèbres paraboliques quasi-réductives dans le cas classique. Dans cet article nous étudions la quasi-réductivité des sous-algèbres biparaboliques des algèbres de Lie réductives. Les sous-algèbres biparaboliques sont les intersections de deux sous-algèbres paraboliques dont la somme est l’algèbre de Lie ambiante. Notre principal résultat est la complétion de la classification des sous-algèbres paraboliques quasi-réductives des algèbres de Lie réductives.

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 The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo, Khalgui and Torasso. In this paper, we investigate the quasi-reductivity of biparabolic subalgebras of reductive Lie algebras. Biparabolic (or seaweed) subalgebras are the intersection of two parabolic subalgebras whose sum is the total Lie algebra. As a main result, we complete the classification of quasi-reductive parabolic subalgebras of reductive Lie algebras by considering the exceptional cases.

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