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

M. Firer; O. G. do Rocio

doi:10.5802/jolt.315

M. Firer; O. G. do Rocio

Journal of Lie Theory, Volume 13 (2003) no. 2, pp. 465-479

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).

EuDML Zbl

DOI: 10.5802/jolt.315

Classification: 20M20, 93B29, 22E46
Keywords: Semigroups, semi-simple Lie groups, control sets, ideal boundary

@article{JOLT_2003_13_2_a9,
     author = {M. Firer and O. G. do Rocio},
     title = {Invariant {Control} {Sets} on {Flag} {Manifolds} and {Ideal} {Boundaries} of {Symmetric} {Spaces}},
     journal = {Journal of Lie Theory},
     pages = {465--479},
     year = {2003},
     volume = {13},
     number = {2},
     doi = {10.5802/jolt.315},
     zbl = {1030.22005},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.315/}
}

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M. Firer; O. G. do Rocio. Invariant Control Sets on Flag Manifolds and Ideal Boundaries of Symmetric Spaces. Journal of Lie Theory, Volume 13 (2003) no. 2, pp. 465-479. doi: 10.5802/jolt.315

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