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

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.