Integral Representation of Positive Eigenfunctions of Convolution Operators

Abstract

For the integral representation of λ-eigenfunctions of the Laplacian, it is important to have an explicit description of ∂_eX(λ), the set of minimal eigenfunctions. When X is a general symmetric space of non-compact type, these eigenfunctions were first determined by Karpelevič [K3]. In this chapter they are determined by using convolution equations (see Theorems 13.1, 13.23, and 13.28), a method first used by Furstenberg for semisimple Lie groups. This method is to used prove analogous results for convolution equations on a general class of groups that includes local field analogues of G as well as reductive Lie groups.