Journal of Lie Theory EMIS ELibM Electronic Journals Journal of Lie Theory
Vol. 13, No. 2, pp. 465--479 (2003)

Previous Article

Next Article

Contents of this Issue

Other Issues

ELibM Journals

ELibM Home



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

M. Firer and O. G. do Rocio

{Marcelo Firer
Instituto de Matemática
Universidade Estadual de Campinas - UNICAMP
Cx. Postal 6065
13.081-970 - Campinas - SP
Osvaldo do Rocio
Centro de Ciências Exatas
Universidade Estadual de Maringá - UEM
Avenida Colombo, 5790
87020-900 - Maringá - PR

Abstract: 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). {\it

Keywords: } semigroups, semi-simple Lie groups, control sets, ideal boundary

Full text of the article:

Electronic version published on: 26 May 2003. This page was last modified: 14 Aug 2003.

© 2003 Heldermann Verlag
© 2003 ELibM and FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition