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

 Joint IAS/Princeton University Number Theory Seminar Topic: Asymptotic behavior of supercuspidal representations and Sato-Tate equidistribution for families Speaker: Ju-Lee Kim Affiliation: Massachusetts Institute of Technology; Visiting Professor, School of Mathematics Date: Thursday, September 29 Time/Room: 4:30pm - 5:30pm/Fine 214, Princeton University

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, we prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues. 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. This is a joint work with Shin and Templier.