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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An abstract Borel density theorem

Author: Martin Moskowitz
Journal: Proc. Amer. Math. Soc. 78 (1980), 19-22
MSC: Primary 22E40
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 $ \rho $ is strongly admissible if each $ {\Lambda ^r}\rho $ is admissible. Although this notion is somewhat technical it is satisfied for certain pairs $ (G,\rho )$; e.g., if G is minimally almost periodic and $ \rho $ arbitrary, if G is complex analytic and $ \rho $ holomorphic. If G is real analytic with radical R, $ G/R$ has no compact factors and R acts under $ \rho $ with real eigenvalues, then $ \rho $ 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 $ \rho $ is a strongly admissible representation of G on V, then each G-invariant measure $ \mu $ on $ \mathcal{G}(V)$, the Grassmann space of V, has support contained in the G-fixed point set.

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

If G is an algebraic subgroup of $ {\text{Gl}}(V)$ and each rational representation is admissible, then H is Zariski dense in G.

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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

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