Semi-algebraic groups and the local closure of an orbit in a homogeneous space

Abstract:

Let L be a topological group acting on a locally compact Hausdorff space M as a transformation group. Let m be in M. A subset Q of M is called the local closure of the orbit Lm if Q is the smallest locally compact invariant subset of M with $m \in Q$. A partition \[ M = \bigcup \limits _{\lambda \in \wedge } {Q_\lambda }, {Q_{\lambda }} \cap {Q_\mu } = \emptyset \left ( {\lambda \ne \mu } \right )\] is called an LC-partition of M with respect to the L action if each ${Q_\lambda }$ is the local closure of Lm for any m in ${Q_\lambda }$. Theorem. Let G be a connected Lie group, and let A and B be subgroups of G with only finitely many connected components. Suppose that B is closed. Then the factor space $G/B$ has an LC-partition with respect to the A action.