An abstract Borel density theorem

Author:
Martin Moskowitz

Journal:
Proc. Amer. Math. Soc. **78** (1980), 19-22

MSC:
Primary 22E40

DOI:
https://doi.org/10.1090/S0002-9939-1980-0548076-2

MathSciNet review:
548076

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Abstract: In this paper an abstract form of the Borel density theorem and related results is given centering around the notion of the author's of a (finite dimensional) ``admissible'' representation. A representation is strongly admissible if each is admissible. Although this notion is somewhat technical it is satisfied for certain pairs ; e.g., if *G* is minimally almost periodic and arbitrary, if *G* is complex analytic and holomorphic. If *G* is real analytic with radical *R*, has no compact factors and *R* acts under with real eigenvalues, then is strongly admissible. If in addition *G* is algebraic/**R**, then each **R**-rational representation is admissible. The results are proven in three stages where *V* is defined either over **R** or **C**.

If is a strongly admissible representation of *G* on *V*, then each *G*-invariant measure on , the Grassmann space of *V*, has support contained in the *G*-fixed point set.

If is a strongly admissible representation of *G* on *V* and has finite volume, then each *H*-invariant subspace of *V* is *G*-invariant.

If *G* is an algebraic subgroup of and each rational representation is admissible, then *H* is Zariski dense in *G*.

**[1]**A. Borel,*Density properties for certain subgroups of semisimple groups without compact components*, Ann. of Math.**72**(1960), 179-188. MR**0123639 (23:A964)****[2]**-,*Linear algebraic groups*, Benjamin, New York, 1969. MR**0251042 (40:4273)****[3]**H. Furstenberg,*A note on Borel's density theorem*, Proc. Amer. Math. Soc.**55**(1976), 209-212. MR**0422497 (54:10484)****[4]**M. Moskowitz,*On the density theorems of Borel and Furstenberg*, Ark. Mat. (1)**16**(1978), 11-27. MR**0507233 (58:22393)****[5]**S. P. Wang,*Homogeneous spaces with finite invariant volume*, Amer. J. Math.**98**(1976), 311-324. MR**0447477 (56:5789)****[6]**-,*On density properties of S-subgroups of locally compact groups*, Ann. of Math.**94**(1971), 325-329. MR**0291351 (45:444)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0548076-2

Keywords:
Algebraic linear group,
complex analytic group,
radical,
Levi factor,
homogeneous space of finite volume,
Zariski density,
support of a finite *G*-invariant measure,
*G*-fixed point set,
Grassmann manifold

Article copyright:
© Copyright 1980
American Mathematical Society