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

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 $g\in G$, $\lambda\in\b R^n$, $$|\phi_{i\rho+\lambda}(g)- e^{i\lambda\cdot m_{\bf 1}(g)}|\le C\|\lambda\|^2.$$