Analytic Wavefront Sets of Spherical Distributions on De Sitter Space

TL;DR

This paper characterizes the wavefront set of spherical distributions on de Sitter space, constructing a basis for positive-definite cases and showing their non-vanishing properties using boundary values of kernels.

Contribution

It constructs a basis for spherical distributions on de Sitter space via boundary values of kernels and characterizes their wavefront sets, revealing their non-vanishing nature.

Findings

Constructed a basis for positive-definite spherical distributions.

Characterized the analytic wavefront set for these distributions.

Proved basis elements cannot vanish in any open region.

Abstract

In this work we determine the wavefront set of certain eigendistributions of the Laplace-Beltrami operator on the de Sitter space. Let G = SO_{1,n}(R)_e be the connected component of identity of Lorentz group and let H = SO_{1,n-1}(R)_e, a subset G. The de Sitter space dS^n, is the one-sheeted hyperboloid in R{1,n} isomorphic to G/H. A spherical distribution, is an H-invariant, eigendistribution of the Laplace-Beltrami operator on dS^n. The space of spherical distributions with eigenvalue \lambda, denoted by D'_{\lambda}(dS^n), has dimension 2. In this article we construct a basis for the space of positive-definite spherical distributions as boundary value of sesquiholomorphic kernels on the crown domains, an open G-invariant domain in dS^n_C. It contains dS^n as a G-orbit on the boundary. We characterize the analytic wavefront set for such distributions. Moreover, if a spherical distribution \Theta in D'_{\lambda}(dS^n) has the wavefront set same as one of the basis element, then it must be a constant multiple of that basis element. Using the analytic wavefront sets we show that the basis elements of D'_{\lambda}(dS^n) can not vanish in any open region.