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 $\rho$ 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.