Spherical Distributions on the De Sitter Space and their Spectral Singularities

A spherical distribution is an eigendistribution of the Laplace-Beltrami operator with certain invariance on the de Sitter space. Let G'=O(1,n;R) be the Lorentz group and H' = O(1,n-1;R) be its subgroup. The authors Olafsson and Sitiraju have constructed the spherical distributions, which are $H'$-invariant, as boundary values of some sesquiholomorphic kernels. In this survey article we will explore the connections of these kernels with reflection positivity and representations of the group G = SO(1,n;R)_e, which is the connected component of the Lorentz group. We will also discuss the singularities of spherical distributions in terms of their wavefront set.