#### Vol. 15, No. 6, 2021

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Sato–Tate equidistribution for families of Hecke–Maass forms on $\mathrm{SL}(n,\mathbb{R}) / \mathrm{SO}(n)$

### Jasmin Matz and Nicolas Templier

Vol. 15 (2021), No. 6, 1343–1428
##### Abstract

We establish the Sato–Tate equidistribution of Hecke eigenvalues of the family of Hecke–Maass cusp forms on $SL\left(n,ℤ\right)\setminus SL\left(n,ℝ\right)∕SO\left(n\right)$. 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 $n=2$.

##### Keywords
automorphic forms, $L$-functions, Arthur–Selberg trace formula
##### Mathematical Subject Classification
Primary: 11F70
Secondary: 20G30, 33C55, 43A90