Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

Abstract

In this paper, we describe the asymptotic distribution of Hecke eigenvalues in the Laplace eigenvalue aspect for certain families of Hecke-Maass forms on compact arithmetic quotients. In particular, we treat not only spherical, but also non-spherical Hecke-Maass forms, including remainder estimates. Our asymptotic formulas are available for arbitrary simple connected algebraic groups over number fields with cocompact arithmetic subgroups. In preceding studies on distributions of Hecke eigenvalues the trace formula played an important role. Since the latter is currently not available for our families, we rely on Fourier integral operator methods instead. Contents 1. Introduction 1 2. Asymptotics for Hecke eigenvalues and low-lying zeros for PGL(n, R) 2 3. Spectral asymptotics for kernels of Hecke operators 6 4. Non-equivariant asymptotics for Hecke eigenvalues and Sato-Tate equidistribution 13 5. Equivariant asymptotics for Hecke eigenvalues and Sato-Tate equidistribution 19 6. Examples 27 References 28