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

Karin Baur (1) , Anne Moreau (2)

1 ETH Department of Mathematics
2 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]

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 The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo, Khalgui and Torasso.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.

Mots clés: reductive Lie algebras; quasi-reductive Lie algebras; index; biparabolic Lie algebras; seaweed algebras; regular linear forms

Domaines: Théorie des représentations [math.RT]

Karin Baur, Anne Moreau. Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras.. Annales de l'Institut Fourier, 2011, 61 (2), pp.417-451. ⟨hal-00348974v2⟩