Jasmin Matz, Nicolas Templier

We establish the Sato–Tate equidistribution of Hecke eigenvalues of the family of Hecke–Maass cusp forms on $SL\left(n,\mathbb{Z}\right)\setminus SL\left(n,\mathbb{R}\right)\u2215SO\left(n\right)$. As part of the proof, we establish a uniform upper-bound for spherical functions on semisimple Lie groups which is of independent interest. For each of the principal, symmetric square and exterior square $L$-functions, we deduce the level distribution with restricted support of the low-lying zeros. We also deduce average estimates toward Ramanujan, including an improvement on the previous literature in the case $n=2$.