The spherical Bochner theorem on semisimple Lie groups

Abstract

Let G be a connected semisimple Lie group with finite center and K a maximal compact subgroup. Denote (i) Harish-Chandra's Schwartz spaces by C p( G)(0< p⩽2), (ii) the K-biinvariant elements in C p(G) by I p(G), (iii) the positive definite (zonal) spherical functions by P , and (iv) the spherical transform on C p( G) by ϑ → \\ ̂ gj. For T a positive definite distribution on G it is established that (i) T extends uniquely onto C l( G), (ii) there exists a unique measure μ of polynomial growth on P such that T[ ψ]=∫ pψdμ for all ψϵI 1( G) (iii) all measures μ of polynomial growth on P are obtained in this way, and (iv) T may be extended to a particular I p(G) space (1 ⩽ p ⩽ 2) if and only if the support of μ lies in a certain easily defined subset of P . These results generalize a well-known theorem of Godement, and the proofs rely heavily on the recent harmonic analysis results of Trombi and Varadarajan.