An analogue of Krein’s theorem for semisimple Lie groups

Abstract

We give an integral representation of K -positive definite functions on a real rank n connected, noncompact, semisimple Lie group with finite centre. Moreover, we characterize the λ's for which the τ -spherical function φ τ σ,λ is positive definite for the group G = Spin e (n, 1) and the complex spin representation τ . IntroductionA continuous function f on ‫ޒ‬ is said to be positive definite if for any real numbers x 1 , . . . , x m and complex numbers ξ 1 , . . . , ξ m the following holds:This definition is equivalent to. Also, an even continuous function f on ‫ޒ‬ is said to be evenly positive definite ifwhere C ∞ c ‫)ޒ(‬ e denotes the set of infinitely differentiable compactly supported even functions on ‫.ޒ‬ Then it is clear that the set of even positive definite functions is a subset of the set of evenly positive definite functions. Bochner's theorem and M. G. Krein's theorem respectively give integral representations of positive definite functions and evenly positive definite functions. Precisely, for a positive definite function f on ‫,ޒ‬ there exists a finite positive measure µ on ‫ޒ‬ such that f (x) = ‫ޒ‬ e iλx dµ(λ).MSC2010: primary 43A85; secondary 22E30.

An Analogue of Bochner’s Theorem for Damek-Ricci Spaces

“…We guess the image space for finite, positive, radial measures under the spherical transform and write it as a conjecture. We refer [2,3,7,8] for further study in this literature.…”