On normal abelian subgroups in parabolic groups

Röhrle, Gerhard

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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