An abstract Borel density theorem
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.
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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