Spherical Distributions on the De Sitter Space and their Spectral Singularities

TL;DR

This paper explores spherical distributions on de Sitter space, their spectral singularities, and connections with group representations, reflection positivity, and wavefront set analysis.

Contribution

It provides a comprehensive survey of the construction of spherical distributions as boundary values of sesquiholomorphic kernels and analyzes their spectral singularities.

Findings

Connections between kernels and reflection positivity

Characterization of spectral singularities via wavefront set

Group representation insights related to spherical distributions

Abstract

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.