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 . A partition

*partition*of

*M*with respect to the

*L*action if each is the local closure of

*Lm*for any

*m*in .

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 has an LC-partition with respect to the A action.*

**[1]**Morikuni Goto,*Orbits of one-parameter groups. III. Lie group case*, J. Math. Soc. Japan**23**(1971), 95–102. MR**0279238****[2]**Morikuni Goto,*Products of two semi-algebraic groups*, J. Math. Soc. Japan**25**(1973), 71–74. MR**0315050****[3]**Morikuni Goto and Hsien-chung Wang,*Non-discrete uniform subgroups of semisimple Lie groups*, Math. Ann.**198**(1972), 259–286. MR**0354934****[4]**Sigurđur Helgason,*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455****[5]**Kenkichi Iwasawa,*On some types of topological groups*, Ann. of Math. (2)**50**(1949), 507–558. MR**0029911****[6]**L. Pukanszky,*Unitary representations of solvable Lie groups*, Ann. Sci. École Norm. Sup. (4)**4**(1971), 457–608. MR**0439985**

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DOI:
https://doi.org/10.1090/S0002-9947-1979-0517696-X

Article copyright:
© Copyright 1979
American Mathematical Society