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Sato–Tate equidistribution for families of Hecke–Maass forms on SL(n, ℝ)∕SO(n)
Algebra & Number Theory  (IF0.938),  Pub Date : 2021-10-16, DOI: 10.2140/ant.2021.15.1343
Jasmin Matz, Nicolas Templier

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.