Asymptotic Expansion for Expanding Spherical Averages in Real Rank One

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.

💡 Research Summary

The paper studies the long‑time behavior of non‑spherical averages on compact quotients of real‑rank‑one Lie groups, with the model case (G=SO(n,1)^\circ). For a unitary representation ((\pi,\mathcal H)) of (G), a smooth density (\varphi\in C^\infty(K)), and a vector (v\in\mathcal H), the authors consider the average
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