Asymptotic Schur orthogonality for lattices

Abstract

Let $\pi:G\to \mathcal{U}(L^{2}(G/P,\nu))$ be the boundary representation of a non-compact connected semisimple Lie group $G$ with finite center on its Furstenberg-Poisson boundary $(G/P,\nu)$. Let $\Gamma\subset G$ be a lattice in $G$ (uniform or not). We show that for any \emph{continuous} functions $\varphi,\psi,\varphi',\psi'$ on $G/P$, $$ \lim_{n\to\infty}\frac{1}{|\Gamma_{n}|}\sum_{\gamma\in \Gamma_{n}}\frac{\langle\pi(\gamma)\varphi,\psi\rangle\overline{\langle\pi(\gamma)\varphi',\psi'\rangle}}{\Xi^{2}(\gamma)}=\langle \varphi,\varphi'\rangle\overline{\langle\psi,\psi'\rangle}, $$ where $\Gamma_{n}$ is a ball in $\Gamma$ with radius $n$ relative to a natural length function and with center the identity element of $G$, and $\Xi$ is the restriction to $\Gamma$ of the Harish-Chandra function of $G$. As a corollary, we deduce that when the real rank of $G$ is one, then the analogous convergence holds for any $\varphi,\psi,\varphi',\psi'$ in $L^{2}(G/P,\nu)$ if and only if the lattice $\Gamma$ is uniform.