Improved sup-norm bounds for locally symmetric spaces

Abstract

Let $X=G/K$ be a symmetric space of noncompact type, of dimension $n$ and rank $r$, and let $Y=Γ\backslash X$. Sarnak&#39;s local bound for an $L^2$-normalized spherical joint eigenfunction with regular tempered parameter of size $T$ is $\|ϕ\|_\infty\ll T^{(n-r)/2}$. We prove $o(T^{(n-r)/2})$ locally uniformly on every quotient. On finite-volume real hyperbolic manifolds this is uniform in the expanding cusp range $y\leq T^β$, $β<1/2$. If the injectivity radius is bounded below, we prove the global estimate $T^{(n-r)/2}(\log T)^{-r/2}$. The proof makes use of a novel kernel argument and a uniform bound on the spherical function.