Orbital integrals on Lorentzian symmetric spaces

TL;DR

This paper investigates how to recover functions from their orbital integrals on Lorentzian symmetric spaces, extending previous results to include solvable transvection groups and odd-dimensional cases.

Contribution

It extends the inversion formulas for orbital integrals to indecomposable Lorentzian symmetric spaces with solvable transvection groups, including odd-dimensional cases.

Findings

Derived inversion formulas for solvable Lorentzian symmetric spaces.

Extended Helgason's limit formula to odd-dimensional isotropic cases.

Provided explicit descriptions of orbital integrals on solvable spaces.

Abstract

In this paper, we address the problem of determining a function in terms of its orbital integrals on Lorentzian symmetric spaces. It has been solved by S. Helgason for even-dimensional isotropic Lorentzian symmetric spaces via a limit formula involving the Laplace-Beltrami operator. The result has been extended by J. Orloff for rank-one semisimple pseudo-Riemannian symmetric spaces giving the keys to treat the odd-dimensional isotropic Lorentzian symmetric spaces. Indecomposable Lorentzian symmetric spaces are either isotropic or have solvable transvection group. We study orbital integrals including an inversion formula on the solvable ones which have been explicitly described by M. Cahen and N. Wallach.