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

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 $ν\in M^1(GL_n(\b F))$. We derive a central limit theorem for the ordered singular spectrum $σ_{sing}(S_k)$ 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 $ϕ_{iρ+λ}$ 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 $g\in G$, $λ\in\b R^n$, $$|ϕ_{iρ+λ}(g)- e^{iλ\cdot m_{\bf 1}(g)}|\le C\|λ\|^2.$$