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

Abstract

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 $^{L}G=\hat G \rtimes \mathrm{Gal}(\bar F/F)$ be the associated $L$-group and $r:{}^L G\to \mathrm{GL}(d,\mathbb{C})$ a continuous homomorphism which is irreducible and does not factor through $\mathrm{Gal}(\bar F/F)$. The families under consideration consist of discrete automorphic representations of $G(\mathbb{A}_F)$ 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 [Progr. Math. 70 (1987), 321--331.] and Serre [J. Amer. Math. Soc. 10 (1997), no. 1, 75--102.]. As an application we study the distribution of the low-lying zeros of the associated family of $L$-functions $L(s,\pi,r)$, 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 the 1-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 $r$ is not isomorphic to its dual $r^\vee$ then the Symmetry type is Unitary. Otherwise there is a bilinear form on $\mathbb{C}^d$ which realizes the isomorphism between $r$ and $r^\vee$. If the bilinear form is symmetric (resp. alternating) then $r$ is real (resp. quaternionic) and the Symmetry type is Symplectic (resp. Orthogonal).