Vol. 15, No. 6, 2021

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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(n, )SL(n, )SO(n). 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
Milestones
Received: 1 September 2018
Revised: 24 November 2020
Accepted: 20 December 2020
Published: 16 October 2021
Authors
Jasmin Matz
Department of Mathematical Science
Universitetsparken 5
Copenhagen
Denmark
Nicolas Templier
Department of Mathematics
Cornell University
Ithaca, NY
United States