Joint IAS/Princeton University Number Theory Seminar

We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at agiven finite set of primes. For the tame supercuspidals, we prove the limit multiplicity property with error terms. Therebywe obtain a Sato-Tate equidistribution for the Hecke eigenvalues. The main new ingredient is to show that the orbitalintegrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connectedreductive $p$-adic group tend to zero uniformly for every noncentral semisimple element. This is a joint work with Shin andTemplier.