Asymptotic behavior of supercuspidal representations and Sato-Tate equidistribution for families

TL;DR

This paper investigates the asymptotic properties of families of automorphic representations, demonstrating Sato-Tate equidistribution for Hecke eigenvalues by analyzing orbital integrals of tame supercuspidal representations.

Contribution

It proves the limit multiplicity property with error terms for tame supercuspidals and establishes Sato-Tate equidistribution in this context.

Findings

Limit multiplicity property with error terms established

Sato-Tate equidistribution for Hecke eigenvalues proven

Orbital integrals tend to zero for noncentral semisimple elements

Abstract

We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes. For the tame supercuspidals constructed by J.-K. Yu we prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues of these families. The main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive $p$-adic group tend to zero uniformly for every noncentral semisimple element.