Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ 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 $$^{L}G=\widehat{G} \rtimes \mathrm{Gal}(\bar{F}/F)$$ be the associated $$L$$ -group and $$r:{}^L G\rightarrow \mathrm{GL}(d,\mathbb {C})$$ 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 (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 $$L$$ -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).