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Journal of Lie Theory 13 (2003), No. 2, 465--479
Copyright Heldermann Verlag 2003

Invariant Control Sets on Flag Manifolds and Ideal Boundaries of Symmetric Spaces

Marcelo Firer
Instituto de Matemática, Universidade Estadual de Campinas, Cx. P. 6065, 13.081-970 Campinas SP, Brasil

Osvaldo G. do Rocio
Centro de Ciencias Exatas, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá PR, Brasil


Let $G$ be a semisimple real Lie group of non-compact type, $K$ a maximal compact subgroup and $S\subseteq G$ a semigroup with nonempty interior. We consider the ideal boundary $\partial_{\infty}(G/K)$ of the associated symmetric space and the flag manifolds $G/P_{\Theta}$. We prove that the asymptotic image $\partial_{\infty} (Sx_{0})\subseteq \partial_{\infty}(G/K)$, where $x_{0}\in G/K$ is any given point, is the maximal invariant control set of $S$ in $\partial_{\infty}(G/K)$. Moreover there is a surjective projection $$\pi\colon\partial_{\infty}(Sx_{0}) \rightarrow \bigcup\limits_{\Theta\subseteq\Sigma}C_{\Theta},$$ where $C_{\Theta}$ is the maximal invariant control set for the action of $S$ in the flag manifold $G/P_{\Theta}$, with $P_{\Theta}$ a parabolic subgroup. The points that project over $C_{\Theta}$ are exactly the points of type $\Theta$ in $\partial_{\infty}(Sx_{0})$ (in the sense of the type of a cell in a Tits Building).

Keywords: Semigroups, semi-simple Lie groups, control sets, ideal boundary.

MSC: 20M20; 93B29, 22E46

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