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

We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let be a reductive group over a number field which admits discrete series representations at infinity. Let be the associated -group and a continuous homomorphism which is irreducible and does not factor through . The families under consideration consist of discrete automorphic representations of 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):75-102, 1997). As an application we study the distribution of the low-lying zeros of the associated family of -functions , assuming from the principle of functoriality that these -functions are automorphic. We find that the distribution of the -level densities coincides with the distribution of the -level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz-Sarnak heuristics. We provide a criterion based on the Frobenius-Schur indicator to determine this symmetry type. If is not isomorphic to its dual then the symmetry type is unitary. Otherwise there is a bilinear form on which realizes the isomorphism between and . If the bilinear form is symmetric (resp. alternating) then is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).