Asymptotic Schur orthogonality for lattices

Let $\pi :G\to \mathcal{U}(L^2(G\mathrm{/}P,\nu ))$ be the boundary representation of a non-compact connected semisimple Lie group $G$ with finite center on its Furstenberg-Poisson boundary $(G\mathrm{/}P,\nu )$. Let $\mathrm{\Gamma }\subset G$ be a lattice in $G$ (uniform or not). We show that for any \emph{continuous} functions $\phi ,\psi ,{\phi }^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }}$ on $G\mathrm{/}P$, $$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{\mathrm{\mid }{\mathrm{\Gamma }}_n\mathrm{\mid }}\sum _{\gamma \in {\mathrm{\Gamma }}_n}\frac{⟨\pi (\gamma )\phi ,\psi ⟩\overline{⟨\pi (\gamma ){\phi }^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }}⟩}}{{\mathrm{\Xi }}^2(\gamma )}=⟨\phi ,{\phi }^{\mathrm{\prime }}⟩\overline{⟨\psi ,{\psi }^{\mathrm{\prime }}⟩},$$ where ${\mathrm{\Gamma }}_n$ is a ball in $\mathrm{\Gamma }$ with radius $n$ relative to a natural length function and with center the identity element of $G$, and $\mathrm{\Xi }$ is the restriction to $\mathrm{\Gamma }$ 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 $\phi ,\psi ,{\phi }^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }}$ in $L^2(G\mathrm{/}P,\nu )$ if and only if the lattice $\mathrm{\Gamma }$ is uniform.