  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. Instead of relying on the trace formula, which
was the primary tool in preceding studies on the subject, we use Fourier
integral operator methods. This allows us to treat not only spherical, but also
non-spherical Hecke-Maass forms with corresponding remainder estimates. Our
asymptotic formulas are available for arbitrary simple and connected algebraic
groups over number fields with cocompact arithmetic subgroups.
