Sato-Tate equidistribution for families of Hecke-Maass forms on SL(n,R)/SO(n)

TL;DR

This paper proves the Sato-Tate equidistribution of Hecke eigenvalues for families of Hecke-Maass cusp forms on SL(n,R)/SO(n), advancing understanding of their statistical properties and implications for L-functions and Ramanujan conjecture.

Contribution

It establishes Sato-Tate equidistribution for Hecke-Maass forms on SL(n,R)/SO(n) and analyzes associated L-functions and zero distributions.

Findings

Proves Sato-Tate equidistribution on average for these families.

Verifies cuspidality of principal, symmetric square, and exterior square L-functions.

Provides estimates towards Ramanujan conjecture.

Abstract

We establish the Sato-Tate equidistribution of Hecke eigenvalues on average for families of Hecke--Maass cusp forms on SL(n,R)/SO(n). For each of the principal, symmetric square and exterior square L-functions we verify that the families are essentially cuspidal and deduce the level distribution with restricted support of the low-lying zeros. We also deduce average estimates toward Ramanujan.