Group-theoretic compactification of Bruhat–Tits buildings

Yves Guivarc'h; Bertrand Rémy

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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...

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