Homogeneous spaces admitting transitive semigroups

Abstract: Let $G$ be a semi-simple Lie group with finite center and $S\subset G$ a semigroup with $% %TCIMACRO{\TeXButton{int}{\inte}} %BeginExpansion \inte% %EndExpansion {}S\neq \emptyset $. A closed subgroup $L\subset G$ is said to be $S$% -admissible if $S$ is transitive in $G/L$. In \cite{smtt} it was proved that a necessary condition for $L$ to be $S$-admissible is that its action in $% B\left( S\right) $ is minimal and contractive where $B\left( S\right) $ is the flag manifold associated with $S$, as in \cite{smt}. It is proved here, under an additional assumption, that this condition is also sufficient provided $S$ is a compression semigroup. A subgroup with a finite number of connected components is admissible if and only if its component of the identity is admissible, and if $L$ is a connected admissible group then $L$ is reductive and its semi-simple component $E$ is also admissible. Moreover, $E$ is transitive in $B\left( S\right) $ which turns out to be a flag manifold of $E$.

Keywords: semigroups, semi-simple groups, flag manifolds, transitive groups, minimal actions

Classification (MSC91): 20M20, 54H15, 93B

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