Non-Euclidean Analogues of Plane Waves

Abstract

The following problem is considered: describe all eigenfunctions of the Laplacian on noncompact symmetric spaces of rank one by means of some “Poisson integral” of hyperfunctions. This problem is solved for functions of special form using analogues of the Beltrami–Klein model for hyperbolic spaces and realizations of the irreducible components of the quasi-regular representations of groups transitive on spheres. Obtained integral representations (so-called Eisenstein integrals) lead to non-Euclidean analogues of plane waves. Such functions exist on all symmetric spaces of noncompact type and appear in the kernel of Helgason’s Fourier transform.