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 ρ \rho is strongly admissible if each Λ r ρ {\Lambda ^r}\rho is admissible. Although this notion is somewhat technical it is satisfied for certain pairs ( G , ρ ) (G,\rho ) ; e.g., if G is minimally almost periodic and ρ \rho arbitrary, if G is complex analytic and ρ \rho holomorphic. If G is real analytic with radical R, G / R G/R has no compact factors and R acts under ρ \rho with real eigenvalues, then ρ \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 ρ \rho is a strongly admissible representation of G on V, then each G-invariant measure μ \mu on G ( V ) \mathcal {G}(V) , the Grassmann space of V, has support contained in the G-fixed point set. If ρ \rho is a strongly admissible representation of G on V and G / H G/H has finite volume, then each H-invariant subspace of V is G-invariant. If G is an algebraic subgroup of Gl ( V ) {\text {Gl}}(V) and each rational representation is admissible, then H is Zariski dense in G.