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 .This definition is equivalent towhere φ * (x) = φ(-x).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).

Locations

• Pacific Journal of Mathematics - View - PDF