Let \({\mathbb {A}}\) be the ring of adeles of a number field k and \(\pi \) be an irreducible cuspidal automorphic representation of \({\textrm{GL}}_n({\mathbb {A}})\) . In Jiang and Luo (Pac J Math 318:339–374. https://doi.org/10.2140/pjm.2022.318.339, 2022, Pac J Math 326: 301–372. https://doi.org/10.2140/pjm.2023.326.301, 2023), the authors 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 \({\textrm{GL}}_n\) over a number field, which was first proved by Ichino and Templier (Am J Math 135:65–101. https://doi.org/10.1353/ajm.2013.0005, 2013, Theorem 1). 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 \({\textrm{GL}}_n({\mathbb {A}})\) and show in Theorems 6.10 and 6.15 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 \(({\textrm{GL}}_n,\pi )\) -versions of Clozel (J Number Theory 261: 252–298 https://doi.org/10.1016/j.jnt.2024.02.018, 2024, Theorem 1.1), where the Tate kernel with \(n=1\) and \(\pi \) the trivial character are considered.
