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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 $π$ 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 $π$-Schwartz space $\mathcal{S}_π(\mathbb{A}^\times)$ and $π$-Fourier transform $\mathcal{F}_{π,ψ}$ with a non-trivial additive character $ψ$ 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 $L(s,π)$, 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 $H_{π,s}$ and their dual kernels $K_{π,s}$ for any irreducible cuspidal automorphic representation $π$ of $\mathrm{GL}_n(\mathbb{A})$ and show that $H_{π,s}$ and $K_{π,1-s}$ are related by the nonlinear $π_\infty$-Fourier transform if and only if $s\in\mathbb{C}$ is a zero of $L_f(s,π_f)=0$, the finite part of the standard automorphic $L$-function $L(s,π)$, which are the $(\mathrm{GL}_n,π)$-versions of a Clozel's Theorem, where the Tate kernel with $n=1$ and $π$ the trivial character are considered.