Sato–Tate theorem for families and low-lying zeros of automorphic L L -functions

We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands.Let G be a reductive group over a number field F which admits discrete series representations at infinity.Let L G = G Gal( F/F) be the associated L-group and r : L G → GL(d, C) a continuous homomorphism which is irreducible and does not factor through Gal( F/F).The families under consideration consist of discrete automorphic representations of G(A F ) of given weight and level and we let either the weight or the level grow to infinity.We establish a quantitative Plancherel and a quantitative Sato-Tate equidistribution theorem for the Satake parameters of these families.This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321-331, 1987) and Serre (J Am Math Soc 10(1): 1997).As an application we study the distribution of the lowlying zeros of the associated family of L-functions L(s, π, r ), assuming from the principle of functoriality that these L-functions are automorphic.We find that the distribution of the 1-level densities coincides with the distribution of