The Voronoi summation formula for GLn and the Godement–Jacquet kernels

Let A be the ring of adeles of a number field k and π be an irreducible cuspidal automorphic representation of GLn(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 π-Schwartz space Sπ(A×) and π-Fourier transform Fπ,ψ with a non-trivial additive character ψ of k\A, proved the associated Poisson summation formula over A×, 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 GLn 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 Hπ,s and their dual kernels Kπ,s for any irreducible cuspidal automorphic representation π of GLn(A) and show in Theorems 6.10 and 6.15 that Hπ,s and Kπ,1-s are related by the nonlinear π∞-Fourier transform if and only if s∈C is a zero of Lf(s,πf)=0, the finite part of the standard automorphic L-function L(s,π), which are the (GLn,π)-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 π the trivial character are considered.