The low lying zeros of a GL(4) and a GL(6) family of $L$-functions

We investigate the large weight ($k\to\infty$) limiting statistics for the low lying zeros of a GL(4) and a GL(6) family of $L$-functions, $\{L(s,\phi \times f): f \in H_k\}$ and $\{L(s,\phi \times {\rm sym}^2 f): f \in H_k\}$; here $\phi$ is a fixed even Hecke–Maass cusp form and $H_k$ is a Hecke eigenbasis for the space $H_k$ of holomorphic cusp forms of weight $k$ for the full modular group. Katz and Sarnak conjecture that the behavior of zeros near the central point should be well modeled by the behavior of eigenvalues near 1 of a classical compact group. By studying the 1- and 2-level densities, we find evidence of underlying symplectic and SO(even) symmetry, respectively. This should be contrasted with previous results of Iwaniec–Luo–Sarnak for the families $\{L(s,f): f\in H_k\}$ and $\{L(s,{\rm sym}^2f): f\in H_k\}$, where they find evidence of orthogonal and symplectic symmetry, respectively. The present examples suggest a relation between the symmetry type of a family and that of its twistings, which will be further studied in a subsequent paper. Both the GL(4) and the GL(6) families above have all even functional equations, and neither is naturally split from an orthogonal family. A folklore conjecture states that such families must be symplectic, which is true for the first family but false for the second. Thus, the theory of low lying zeros is more than just a theory of signs of functional equations. An analysis of these families suggest that it is the second moment of the Satake parameters that determines the symmetry group.