Analysis on real affine G-varieties

摘要

We consider the action of a real linear algebraic group G on a smooth, real affine algebraic variety M C R, and study the corresponding left regular representation of G 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](.)e(-r2), where C[M] denotes the algebra of regular functions of M and r the distance function in V. 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 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 CO(M), one gets admissible (g, K)-modules as well as K-finite Banach representations.