Analysis on real affine G-varieties

We consider the action of a real linear algebraic group $G$ on a smooth, real affine algebraic variety $M\subset \R^n$, and study the corresponding left regular $G$-representation on the Banach space $C_0(M)$ of continuous, complex valued functions on $M$ vanishing at infinity. We show that the differential structure of this representation is already completely characterized by the action of the Lie algebra $\g$ of $G$ on the dense subspace $\P=\C[M] \cdot e^{-r^2}$, where $\C[M]$ denotes the algebra of regular functions of $M$ and $r$ the distance function in $\R^n$. We prove that the elements of this subspace constitute analytic vectors of the considered $G$-representation, and, using this fact, we construct discrete reducing series in $C_0(M)$. In case that $G$ is reductive, $K$ a maximal compact subgroup, $\P$ turns out to be a $(\g,K)$-module in the sense of Harish-Chandra and Lepowsky, and by taking suitable subquotients of $\P$, respectively $C_0(M)$, one gets admissible $(\g,K)$-modules as well as $K$-finite Banach representations.