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Asymptotic Expansion for Expanding Spherical Averages in Real Rank One

Zhiyuan Deng, Yutian Sun

TL;DR

This work analyzes sharp asymptotics for expanding non-spherical averages on compact quotients of real rank-one groups, modeling with $G=\mathrm{SO}(n,1)^\circ$ and $X=\Gamma\backslash G$. The authors reduce the problem to a one-dimensional ODE by exploiting joint eigenvectors of the Casimir operators $\Omega_G$ and $\Omega_M$ and develop an explicit ODE-driven framework that yields a full asymptotic expansion in terms of exponential terms $e^{(\lambda_j - m)t}$, with precise remainder bounds. The expansion is carried out across regimes determined by the discriminant $\mathcal{D}$ (including imaginary, real nonzero, and zero cases), with iterative and refined procedures to extract main terms, absorbing tails into leading coefficients. Their approach leverages a detailed blend of representation theory (induced representations, branching laws, Weyl laws) and analytic ODE methods to obtain fully explicit asymptotics for a broad class of base vectors and test functions, culminating in a general, vector-valued expansion for arbitrary $v$ on $X$. This provides sharp, computable rates for equidistribution of expanding orbits in hyperbolic spaces and strengthens the spectral-analytic toolkit for real rank-one dynamics and number-theoretic applications.

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.

Asymptotic Expansion for Expanding Spherical Averages in Real Rank One

TL;DR

This work analyzes sharp asymptotics for expanding non-spherical averages on compact quotients of real rank-one groups, modeling with and . The authors reduce the problem to a one-dimensional ODE by exploiting joint eigenvectors of the Casimir operators and and develop an explicit ODE-driven framework that yields a full asymptotic expansion in terms of exponential terms , with precise remainder bounds. The expansion is carried out across regimes determined by the discriminant (including imaginary, real nonzero, and zero cases), with iterative and refined procedures to extract main terms, absorbing tails into leading coefficients. Their approach leverages a detailed blend of representation theory (induced representations, branching laws, Weyl laws) and analytic ODE methods to obtain fully explicit asymptotics for a broad class of base vectors and test functions, culminating in a general, vector-valued expansion for arbitrary on . This provides sharp, computable rates for equidistribution of expanding orbits in hyperbolic spaces and strengthens the spectral-analytic toolkit for real rank-one dynamics and number-theoretic applications.

Abstract

This paper develops precise asymptotic formulas for expanding non-spherical averages on compact quotients of real rank-one Lie groups, focusing on 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.
Paper Structure (27 sections, 48 theorems, 486 equations)

This paper contains 27 sections, 48 theorems, 486 equations.

Table of Contents

  1. Introduction
  2. Expression of the Casimir operator L_Omega for Semisimple Connected Lie Group
  3. The Casimir Operator
  4. Explicit Formula of L_Omega
  5. Preliminaries on Harmonic analysis on SO(n,1)
  6. Lie-Algebraic Framework for SO(n,1)
  7. Representation Theory of Orthogonal Groups
  8. Representation of K
  9. Representations of SO(n,1) (connected)
  10. Complete List of Irreducible Unitary Representations of SO(n,1)
  11. Restriction of Induced Representations and Branching Law between Compact Subgroups
  12. Weyl Law
  13. Sobolev Norms and Embedding Inequality
  14. Construction of an ODE and Its Solutions
  15. Reduction to an ODE
  16. ...and 12 more sections

Key Result

Theorem 1.1

There exists a countable family of complex numbers $\{\lambda_j\}_{j\ge1}$ satisfying $\operatorname{Re}(\lambda_j)\in\bigl(-\infty,\tfrac{1-n}{2}+\nu(\Gamma)\bigr]$ for all $j\ge1$, where $\nu(\Gamma)>0$ is the spectral gap parameter associated with the right regular representation of $G$ on $L^{2} admit the following pointwise expansion for $t>0$: where the double series over $j$ and $m$ conver

Theorems & Definitions (82)

  • Theorem 1.1: Pointwise asymptotic expansion
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Theorem 2.5: Derivative formula for the Casimir operator
  • Corollary 3.1
  • Theorem 3.2: Matrix coefficient decay for $SO^{0}(n,1)$, following Trombi TROMBI197883
  • Theorem 3.3
  • ...and 72 more