Improved sup-norm bounds for locally symmetric spaces

Let $X=G\mathrm{/}K$ be a symmetric space of noncompact type, of dimension $n$ and rank $r$, and let $Y=\mathrm{\Gamma }\mathrm{\backslash }X$. Sarnak’s local bound for an $L^2$-normalized spherical joint eigenfunction with regular tempered parameter of size $T$ is $\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}\ll T^{(n-r)\mathrm{/}2}$. We prove $o(T^{(n-r)\mathrm{/}2})$ locally uniformly on every quotient. On finite-volume real hyperbolic manifolds this is uniform in the expanding cusp range $y\le T^{\beta }$, $\beta <1\mathrm{/}2$. If the injectivity radius is bounded below, we prove the global estimate $T^{(n-r)\mathrm{/}2}(\log T{)}^{-r\mathrm{/}2}$.