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

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.

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.

@article{AIF_1998__48_5_1455_0,
     author = {R\"ohrle, Gerhard},
     title = {On normal abelian subgroups in parabolic groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {48},
     number = {5},
     year = {1998},
     pages = {1455-1482},
     doi = {10.5802/aif.1662},
     zbl = {0933.20034},
     mrnumber = {99i:20062},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1998__48_5_1455_0}
}
On normal abelian subgroups in parabolic groups. Annales de l'Institut Fourier, Volume 48 (1998) no. 5, pp. 1455-1482. doi : 10.5802/aif.1662. http://www.numdam.org/item/AIF_1998__48_5_1455_0/

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