Journal of Lie Theory Vol. 15, No. 1, pp. 299–320 (2005) 

Analysis on real affine $G$varietiesPablo RamacherPablo RamacherHumboldt–Universität zu Berlin Institut für Reine Mathematik Sitz: Rudower Chaussee 25 D 10099 Berlin, Germany ramacher@mathematik.huberlin.de Abstract: 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 HarishChandra 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. Keywords: Gvarieties, Banach representations, real reductive groups, dense graph theorem, analytic elements, $(\g,K)$modules, reducing series Classification (MSC2000): 57S25, 22E45, 22E46, 22E47, 47D03 Full text of the article: (for faster download, first choose a mirror)
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