Asymptotic Expansion for Expanding Spherical Averages in Real Rank One

This paper establishes an asymptotic expansion for expanding sector average on compact quotients of real rank-one Lie groups, with $SO(n,1)$ as a model case. Using tools from harmonic analysis and representation theory, the problem is reduced, via the action of the Casimir operator, to an ordinary differential equation(ODE) governing the evolution along the expanding $A$-direction. The asymptotic expansion is then derived from the analysis of this ODE.