Asymptotic Expansion for Expanding Spherical Averages in Real Rank One

TL;DR

This paper derives detailed asymptotic formulas for expanding spherical averages on certain geometric spaces, employing harmonic analysis and representation theory to connect orbit averages to differential equations.

Contribution

It introduces a novel approach to analyze non-spherical averages on rank-one Lie groups using harmonic analysis and Casimir operator techniques.

Findings

Asymptotic formulas for expanding averages are established.

Reduction of orbit average analysis to an ODE involving the Casimir operator.

Method applicable to compact quotients of real rank-one Lie groups.

Abstract

This paper develops precise asymptotic formulas for expanding non-spherical averages on compact quotients of real rank-one Lie groups, focusing on $SO(n,1)$ as a model case. Using tools from harmonic analysis and representation theory, the study reduces the analysis of orbit averages to an ordinary (ODE) derived from the action of the Casimir operator.