  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.
