The Voronoi Summation Formula for ${\mathrm{GL}}_n$ and the Godement-Jacquet Kernels

Abstract

Let $\mathbb{A}$ be the ring of adeles of a number field $k$ and $\pi$ be an irreducible cuspidal automorphic representation of ${\mathrm{GL}}_n(\mathbb{A})$. In the previous work of the first author with Zhilin Luo, they introduced $\pi$-Schwartz space ${\mathcal{S}}_{\pi }({\mathbb{A}}^{\times })$ and $\pi$-Fourier transform ${\mathcal{F}}_{\pi ,\psi }$ with a non-trivial additive character $\psi$ of $k\backslash \mathbb{A}$, proved the associated Poisson summation formula over ${\mathbb{A}}^{\times }$, based on the Godement-Jacquet theory for the standard $L$-functions $L(s,\pi )$, and provided interesting applications. In this paper, in addition to the further development of the local theory, we found two global applications. First, we find a Poisson summation formula proof of the Voronoi summation formula for ${\mathrm{GL}}_n$ over a number field, which was first proved by A. Ichino and N. Templier. Then we introduce the notion of the Godement-Jacquet kernels $H_{\pi ,s}$ and their dual kernels $K_{\pi ,s}$ for any irreducible cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_n(\mathbb{A})$ and show that $H_{\pi ,s}$ and $K_{\pi ,1-s}$ are related by the nonlinear ${\pi }_{\infty }$-Fourier transform if and only if $s\in \mathbb{C}$ is a zero of $L_f(s,{\pi }_f)=0$, the finite part of the standard automorphic $L$-function $L(s,\pi )$, which are the $({\mathrm{GL}}_n,\pi )$-versions of a Clozel's Theorem, where the Tate kernel with $n=1$ and $\pi$ the trivial character are considered.