Analysis on real affine G-varieties

الأصل · EN

We consider the action of a real linear algebraic group G on a smooth, real affine algebraic variety M⊂ ⁿ, and study the corresponding left regular G-representation on the Banach space C₀(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 of G on the dense subspace ¶=[M] · e⁻ʳ², where [M] denotes the algebra of regular functions of M and r the distance function in 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 C₀(M). In case that G is reductive, K a maximal compact subgroup, ¶ turns out to be a (,K)-module in the sense of Harish-Chandra and Lepowsky, and by taking suitable subquotients of ¶, respectively C₀(M), one gets admissible (,K)-modules as well as K-finite Banach representations.