Asymptotic Schur orthogonality for lattices
Let $π:G\to \mathcal{U}(L^{2}(G/P,ν))$ be the boundary representation of a non-compact connected semisimple Lie group $G$ with finite center on its Furstenberg-Poisson boundary $(G/P,ν)$. Let $Γ\subset G$ be a lattice in $G$ (uniform or not). We show that for any \emph{continuous} functions $φ,ψ,φ',ψ'$ on $G/P$, $$ \lim_{n\to\infty}\frac{1}{|Γ_{n}|}\sum_{γ\in Γ_{n}}\frac{\langleπ(γ)φ,ψ\rangle\overline{\langleπ(γ)φ',ψ'\rangle}}{Ξ^{2}(γ)}=\langle φ,φ'\rangle\overline{\langleψ,ψ'\rangle}, $$ where $Γ_{n}$ is a ball in $Γ$ with radius $n$ relative to a natural length function and with center the identity element of $G$, and $Ξ$ is the restriction to $Γ$ 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 $φ,ψ,φ',ψ'$ in $L^{2}(G/P,ν)$ if and only if the lattice $Γ$ is uniform.