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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Morikuni Goto
Journal: Trans. Amer. Math. Soc. 247 (1979), 301-315
MSC: Primary 57S20; Secondary 22D05
MathSciNet review: 517696
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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

$\displaystyle M = \,\bigcup\limits_{\lambda \in \wedge } \,{Q_\lambda },\,\,\,\... ...,}}\, \cap \,\,{Q_\mu } = \,\emptyset \,\,\,\,\left( {\lambda \ne \mu } \right)$

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.

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Article copyright: © Copyright 1979 American Mathematical Society

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