Journal of Lie Theory

\def\Cvan{{\rm C_0}} \def\P{{\cal P}} \def\R{{\mathbb R}} \def\C{{\mathbb C}} \def\g{{\frak g}} 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 representation of $G$ on the Banach space $\Cvan(M)$ 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 representation, and by taking into account the algebraic structure of $\P$, we obtain $G$-invariant decompositions and discrete reducing series of $\Cvan(M)$. 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 $\Cvan(M)$, one gets admissible $(\g,K)$-modules as well as $K$-finite Banach representations.