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 $\phi ,\psi ,{\phi }^{\prime },{\psi }^{\prime }$ on $G/P$, $$\underset{n\to \infty }{\lim }\frac{1}{|{\Gamma }_n|}\sum _{\gamma \in {\Gamma }_n}\frac{⟨\pi (\gamma )\phi ,\psi ⟩\overline{⟨\pi (\gamma ){\phi }^{\prime },{\psi }^{\prime }⟩}}{{\Xi }^2(\gamma )}=⟨\phi ,{\phi }^{\prime }⟩\overline{⟨\psi ,{\psi }^{\prime }⟩},$$ 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 $\phi ,\psi ,{\phi }^{\prime },{\psi }^{\prime }$ in $L^2(G/P,\nu )$ if and only if the lattice $\Gamma$ is uniform.