On normal abelian subgroups in parabolic groups
Annales de l'Institut Fourier, Tome 48 (1998) no. 5, pp. 1455-1482.

Soient G un groupe algébrique réductif, P un sous-groupe parabolique de G avec radical unipotent P u , et A un sous-groupe fermé connexe de P u , normalisé par P. Nous montrons que P opère dans A avec un nombre fini d’orbites, lorsque A est abélien. Ceci généralise un résultat de finitude bien connu, concernant le cas où A est central dans P u . Nous obtenons aussi un résultat analogue pour l’action adjointe de P dans les sous-espaces invariants de l’algèbre de Lie de P u , qui sont des algèbres de Lie abéliennes. Finalement, nous faisons le lien avec un travail de Mal’cev sur les sous-algèbres abéliennes maximales de l’algèbre de Lie de G.

Let G be a reductive algebraic group, P a parabolic subgroup of G with unipotent radical P u , and A a closed connected subgroup of P u which is normalized by P. We show that P acts on A with finitely many orbits provided A is abelian. This generalizes a well-known finiteness result, namely the case when A is central in P u . We also obtain an analogous result for the adjoint action of P on invariant linear subspaces of the Lie algebra of P u which are abelian Lie algebras. Finally, we discuss a connection to some work of Mal’cev on maximal abelian subalgebras of the Lie algebra of G.

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Röhrle, Gerhard. On normal abelian subgroups in parabolic groups. Annales de l'Institut Fourier, Tome 48 (1998) no. 5, pp. 1455-1482. doi : 10.5802/aif.1662. http://www.numdam.org/articles/10.5802/aif.1662/

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