Sato–Tate theorem for families and low-lying zeros of automorphic L-functions: With appendices by Robert Kottwitz [A] and by Raf Cluckers, Julia Gordon, and Immanuel Halupczok [B]

© 2015 The Author(s) We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let (Formula presented.) be a reductive group over a number field (Formula presented.) which admits discrete series representations at infinity. Let (Formula presented.) be the associated (Formula presented.)-group and (Formula presented.) a continuous homomorphism which is irreducible and does not factor through (Formula presented.). The families under consideration consist of discrete automorphic representations of (Formula presented.) 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 (Formula presented.)-functions (Formula presented.), assuming from the principle of functoriality that these (Formula presented.)-functions are automorphic. We find that the distribution of the (Formula presented.)-level densities coincides with the distribution of the (Formula presented.)-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 (Formula presented.) is not isomorphic to its dual (Formula presented.) then the symmetry type is unitary. Otherwise there is a bilinear form on (Formula presented.) which realizes the isomorphism between (Formula presented.) and (Formula presented.). If the bilinear form is symmetric (resp. alternating) then (Formula presented.) is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal)