Uniform oscillatory behavior of spherical functions of $GL_n/U_n$ at the identity and a central limit theorem

TL;DR

This paper establishes a central limit theorem for the singular spectrum of a random walk on $GL_n( F)$, using oscillatory properties of spherical functions to derive explicit formulas for the distribution's drift and covariance.

Contribution

It introduces a novel oscillatory analysis of spherical functions of $(GL_n( F),U_n( F))$ and applies it to prove a CLT for the singular spectrum of associated random walks.

Findings

Central limit theorem for the singular spectrum with explicit formulas

Oscillatory behavior of spherical functions near the identity

Explicit analytic expressions for drift and covariance

Abstract

Let $\mathbb F=\mathbb R$ or $\mathbb C$ and $n\in\b N$. Let $(S_k)_{k\ge0}$ be a time-homogeneous random walk on $GL_n(\b F)$ associated with an $U_n(\b F)$-biinvariant measure $\nu\in M^1(GL_n(\b F))$. We derive a central limit theorem for the ordered singular spectrum $\sigma_{sing}(S_k)$ with a normal distribution as limit with explicit analytic formulas for the drift vector and the covariance matrix. The main ingredient for the proof will be a oscillatory result for the spherical functions $\phi_{i\rho+\lambda}$ of $(GL_n(\b F),U_n(\b F))$. More precisely, we present a necessarily unique mapping $m_{\bf 1}:G\to\b R^n$ such that for some constant $C$ and all $g\in G$, $\lambda\in\b R^n$, $$|\phi_{i\rho+\lambda}(g)- e^{i\lambda\cdot m_{\bf 1}(g)}|\le C\|\lambda\|^2.$$