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

Let $k$ be a number field and ${\mathbb{A}}={\mathbb{A}}_{k}$ the associated ring of adeles. For any irreducible cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_{n}({\mathbb{A}})$ for any integer $n\geq 1$, the Godement-Jacquet theory ([GJ72]) establishes the analytic continuation and functional equation for the standard $L$-functions $L(s,\pi)$. The key input from the harmonic analysis to the Godement-Jacquet theory is the classical Fourier analysis on the affine space ${\mathrm{M}}_{n}$, the space of $n\times n$-matrices, and the associated Poisson summation formula. From the point of view in the Braverman-Kazhdan-Ngô program ([BK00] and [Ngo20]), this classical theory of Fourier analysis should be reformulated on the group ${\mathrm{GL}}_{n}({\mathbb{A}})$. In such a reformulation, the classical (additive) Fourier transform is converted to a convolution integral with a kernel function and the Poisson summation formula is converted to a theta inversion formula, which is a generalization of the classical theta inversion formula. We refer to [JL23, Section 2] for details.

In [JL22, JL23], the Godement-Jacquet theory has been reformulated as harmonic analysis on ${\mathrm{GL}}_{1}$ for $L(s,\pi)$ as a vast generalization of the 1950 thesis of J. Tate ([Tat67]). More precisely, for any irreducible cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_{n}({\mathbb{A}})$, the space of $\pi$-Schwartz functions on ${\mathrm{GL}}_{1}({\mathbb{A}})={\mathbb{A}}^{\times}$ is defined, which is denoted by ${\mathcal{S}}_{\pi}({\mathbb{A}}^{\times})$, and the $\pi$-Foruier transform (or operator) ${\mathcal{F}}_{\pi,\psi}$ is defined for any given non-trivial additive character $\psi$ of $k\backslash{\mathbb{A}}$, which takes the $\pi$-Schwartz space ${\mathcal{S}}_{\pi}({\mathbb{A}}^{\times})$ to the $\widetilde{\pi}$-Schwartz space ${\mathcal{S}}_{\widetilde{\pi}}({\mathbb{A}}^{\times})$, where $\widetilde{\pi}$ is the contragredient of $\pi$. We refer to Section 2 for details. The $\pi$-Poisson summation formula as proved in [JL23, Theorem 4.7] (or recalled in Theorem 2.1) takes the following form.

This ${\mathrm{GL}}_{1}$-reformulation in [JL22, JL23] of the Godement-Jacquet theory has found following nice applications, among others:

1. (1)

   The local theory of such a ${\mathrm{GL}}_{1}$-reformulation as developed in [JL22, JL23] proves that the (nonlinear) Fourier transform that is responsible for the local functional equation is given by a convolution operator with an explicitly defined kernel function $k_{\pi_{\nu},\psi_{\nu}}$ on ${\mathrm{GL}}_{1}$, which will be related to a Bessel function in Section 4 of this paper. As consequences, all the Langlands local gamma factors take the form of those in the theory of I. Gelfand, M. Graev, and I. Piatetski-Shapiro in [GGPS69] and of A. Weil in [Wei95], where the local gamma factors associated with quasi-characters of ${\mathrm{GL}}_{1}$ were considered.

2. (2)

   The global theory of such a ${\mathrm{GL}}_{1}$-reformation as developed in [JL23] gives the adelic formulation of A. Connes’ theorem ([Con99, Theorem III.1]) for $L(s,\pi)$, and the complete version of C. Soulé’s theorem ([Sou01, Theorem 2]) that provides a spectral interpretation of the zeros of $L(s,\pi)$. We refer to [JL23, Theorem 8.1] for details.

3. (3)

   In Section 5 of this paper, the local and global theory of such a ${\mathrm{GL}}_{1}$-reformulation in [JL22, JL23] provides a Poisson summation formula proof of the Voronoi formula for any irreducible cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_{n}$, which was previously proved by S. Miller and W. Schmid in [MS11] and by A. Ichino and N. Templier in [IT13] by using the Rankin-Selberg convolution of H. Jacquet, I. Piatetski-Shapiro and J. Shalika in [JPSS83, CPS04].

4. (4)

   In Section 6 of this paper, the local and global theory of such a ${\mathrm{GL}}_{1}$-reformulation in [JL22, JL23] defines the Godement-Jacquet kernels for $L(s,\pi)$ and proves Theorems 6.9 and 6.13, which are the $({\mathrm{GL}}_{n},\pi)$-versions of Clozel’s theorem ([Clo22, Theorem 1.1]) for any irreducible cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_{n}$. In [Clo22], Clozel formulates and proves such a theorem for the Tate kernel associated with the Dedekind zeta function $\zeta_{k}(s)$ for any number field $k$.

The classical Voronoi summation formula and its recent extension to the ${\mathrm{GL}}_{n}$-version have been one of the most powerful tools in Number Theory and relevant areas in Analysis. We refer to an enlightening survey paper by S. Miller and W. Schmid ([MS04b]) for a detailed account of the current state of the art of the Voronoi summation formula and its applications to important problems in Number Theory.

The Voronoi summation formula for ${\mathrm{GL}}_{n}$ was first studied by S. Miller and W. Schmid in [MS06] for $n=3$ and in [MS11] for general $n$. They use two approaches. One is based on classical harmonic analysis that has been developed in their earlier paper ([MS04a]), and the other is based on the adelic version of the Rankin-Selberg convolutions for ${\mathrm{GL}}_{n}\times{\mathrm{GL}}_{1}$, which was developed by H. Jacquet, I. Piatetski-Shapiro and J. Shalika in [JPSS83] and by J. Cogdell and Piatetski-Shapiro in [CPS04] (and also by Jacquet in [Jac09] for the Archimedean local theory). The classical approach to the Voronoi formula for ${\mathrm{GL}}_{n}$ has also been discussed in [GL06] and [GL08]. A complete treatment of the adelic approach to the Voronoi formula for ${\mathrm{GL}}_{n}$ over a general number field was given by A. Ichino and N. Templier in [IT13]. We recall their general Voromoi formula for ${\mathrm{GL}}_{n}$ below.

For each irreducible cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_{n}({\mathbb{A}})$, the Voronoi summation formula is an identity of two summations. One side of the identity is given by certain data associated with $\pi$ and the other side is given by certain corresponding data associated with $\widetilde{\pi}$, the contragredient of $\pi$. Let $\psi=\otimes_{\nu}\psi_{\nu}$ be a non-trivial additive character on $k\backslash{\mathbb{A}}$. At each local place $\nu$ of $k$, to a smooth compactly supported function $w_{\nu}(x)\in{\mathcal{C}}_{c}^{\infty}(k_{\nu}^{\times})$ is associated a dual function $\widetilde{w}_{\nu}(x)$ such that the following functional equation

holds for all $s\in{\mathbb{C}}$ and all unitary characters $\chi_{\nu}$ of $k_{\nu}^{\times}$. Since any irreducible cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_{n}({\mathbb{A}})$ is generic, i.e. it has a non-zero Whittaker-Fourier coefficient. If we write $\pi=\otimes_{\nu\in|k|}\pi_{\nu}$, where $|k|$ denotes the set of all local places of $k$, then at any local place $\nu$, the local component $\pi_{\nu}$ is an irreducible admissible and generic representation of ${\mathrm{GL}}_{n}(k_{\nu})$. Let ${\mathcal{W}}(\pi_{\nu},\psi_{\nu})$ be the local Whittaker model of $\pi_{\nu}$, and $W_{\nu}(g)$ be any Whittaker function on ${\mathrm{GL}}_{n}(k_{\nu})$ that belongs to ${\mathcal{W}}(\pi_{\nu},\psi_{\nu})$ (see Section 3 for the details).

Let $S$ be a finite set of $|k|$ including all Archimedean places and the local places $\nu$ where $\pi_{\nu}$ or $\psi_{\nu}$ is ramified. As usual, we write ${\mathbb{A}}={\mathbb{A}}_{S}\times{\mathbb{A}}^{S}$, where ${\mathbb{A}}_{S}=\prod_{\nu\in S}k_{\nu}$, which is naturally embedded as a subring of ${\mathbb{A}}$, and ${\mathbb{A}}^{S}$ the subring of adeles ${\mathbb{A}}$ with trivial component above $S$. At $\nu\notin S$, we take the unramified Whittaker vector ${}^{\circ}W_{\nu}$ of $\pi_{\nu}$, which is so normalized that ${}^{\circ}W_{\nu}({\mathrm{I}}_{n})=1$. Denote by ${}^{\circ}W^{S}:=\prod_{\nu\notin S}{{}^{\circ}W_{\nu}}$, which is the normalized unramified Whittaker function of $\pi^{S}=\otimes_{\nu\notin S}\pi_{\nu}$. Similarly, we define ${}^{\circ}\widetilde{W}^{S}=\prod_{\nu\notin S}{{}^{\circ}\widetilde{W}}_{\nu}$ to be the (normalized) unramified Whittaker function of $\widetilde{\pi}^{S}=\otimes_{\nu\notin S}\widetilde{\pi}_{\nu}$. We recall that the functions ${{}^{\circ}W^{S}}$ and ${{}^{\circ}\widetilde{W}^{S}}$ are related by the following

for all $g\in{\mathrm{G}}_{n}({\mathbb{A}}^{S})$, where $w_{n}$ is the longest Weyl element of ${\mathrm{G}}_{n}={\mathrm{GL}}_{n}$ as defined in (3.3). The following is the Voronoi formula proved in [IT13, Theorem 1]. The unexplained notation will be defined in Sections 3 and 5.

The proof of Theorem 1.2 in [IT13] is based on the local and global theory of the Rankin-Selberg convolution for ${\mathrm{GL}}_{n}\times{\mathrm{GL}}_{1}$ ([JPSS83, CPS04, Jac09]). It is important to mention that Theorem 1.2 and its proof has be extended by A. Corbett to cover an even more general situation with more applications in Number Theory ([Cor21, Theorem 3.4]).

From the historical development of the Voronoi summation formula, one expects that there should be a proof of the Voronoi formula via a certain kind of Poisson summation formula. In other words, the two sides of the Voronoi formula should be related by a certain kind of Fourier transform and the identity should be deduced from the corresponding Poisson summation formula. In the current proof of Theorem 1.2, such important ingredients from the harmonic analysis were missing, although there were discussions in [IT13] and [Cor21] on the local Bessel transform with the kernels deduced from the local functional equation in the local theory of the Rankin-Selberg convolution in [JPSS83] and [Jac09], and the identity was deduced from explicit computations from the global zeta integrals of the Rankin-Selberg convolution ([JPSS83] and [CPS04]). Over the Archimedean local fields, Z. Qi has developed in [Qi20] a theory of fundamental Bessel functions of high rank and formulated those Bessel transforms in the framework of general Hankel transforms that are integral transforms with Bessel functions as the kernel functions.

The first global result of this paper is to show that Theorem 1.2 is a special case of Theorem 1.1. From our proof, it will be clear that any variant of Theorem 1.2 (see [Cor21, Theorem 3.4] for instance) is also a special case of Theorem 1.1. Note that the $\pi$-Poisson summation formula on ${\mathrm{GL}}_{1}$ in Theorem 1.1) relies heavily on the work of R. Godement and H. Jacquet ([GJ72]). Hence our proof of Theorem 1.2 is in principle based on the local and global theory of the Godement-Jacquet integrals for the standard $L$-functions of ${\mathrm{GL}}_{n}\times{\mathrm{GL}}_{1}$.

In order to carry out such a proof, we have to understand the nature of the functions occurring on the both side of the Voromoi formula in Theorem 1.2, locally and globally, and show that they are $\pi$-Schwartz function on ${\mathbb{A}}^{\times}$ and are related by the $\pi$-Fourier transform ${\mathcal{F}}_{\pi,\psi}$ in the sense of [JL22, JL23]. More precisely, for any irreducible smooth representation $\pi_{\nu}$ of ${\mathrm{GL}}_{n}(k_{\nu})$, which is of Casselman-Wallach type if $\nu$ is an infinite local place of $k$, we define the $\pi_{\nu}$-Bessel function ${\mathfrak{b}}_{\pi_{\nu},\psi_{\nu}}(x)$ on $k_{\nu}^{\times}$ (Definitions 4.2, 4.9 and 4.12) and obtain a series of results on the relations between the $\pi_{\nu}$-Bessel functions ${\mathfrak{b}}_{\pi_{\nu},\psi_{\nu}}(x)$, the $\pi_{\nu}$-Fourier transforms ${\mathcal{F}}_{\pi_{\nu},\psi_{\nu}}$ and the $\pi_{\nu}$-kernel functions $k_{\pi_{\nu},\psi_{\nu}}(x)$ as introduced and studied in [JL22, JL23] (see (2.18) and (2.20) for details), and on new formulas for the dual functions $\widetilde{w_{\nu}}(x)$ of $w_{\nu}(x)\in{\mathcal{C}}^{\infty}_{c}(k_{\nu}^{\times})$. After all the local preparation, we deduce the Voromoi formula in Theorem 1.2 from the $\pi$-Poisson summation formula in Theorem 1.1. We summarize those local results as the following theorem.

The proof of Theorem 1.3 is given in Sections 3 and 4. More precisely, Part (1) of Theorem 1.3 is Proposition 3.5. Part (2) is Corollary 3.6. Part (3) is a combination of Propositions 4.3, 4.8, and 4.13. Part (4) is an easy consequence of Parts (2) and (3), which is Corollary 4.15.

It is important to point out that the $\pi_{\nu}$-Bessel functions ${\mathfrak{b}}_{\pi_{\nu},\psi_{\nu}}(x)$ in the $p$-adic case is defined by means of the Whittaker model of $\pi_{\nu}$ following the general framework of E. Baruch in [Bar05]. Hence we have to assume in the $p$-adic case that $\pi_{\nu}$ is generic in the definition of the $\pi_{\nu}$-Bessel functions ${\mathfrak{b}}_{\pi_{\nu},\psi_{\nu}}(x)$. However, in the real or complex case, we follow the general theory of Z. Qi in [Qi20] on Bessel functions of high rank, which works for general irreducible smooth representations of ${\mathrm{GL}}_{n}$ of Casselman-Wallach type. Hence in the real or complex case, the definition of the $\pi_{\nu}$-Bessel functions ${\mathfrak{b}}_{\pi_{\nu},\psi_{\nu}}(x)$ does not require that $\pi_{\nu}$ is generic. Since the $\pi_{\nu}$-kernel function $k_{\pi_{\nu},\psi_{\nu}}$ as in (2.20) is defined based on the Godement-Jacquet theory, which does not require that $\pi_{\nu}$ is generic, the uniform result in Part (3) of Theorem 1.3 suggests that one may define the $\pi_{\nu}$-kernel function $k_{\pi_{\nu},\psi_{\nu}}$ to be the $\pi_{\nu}$-Bessel functions ${\mathfrak{b}}_{\pi_{\nu},\psi_{\nu}}(x)$ in the $p$-adic case when $\pi_{\nu}$ is not generic. Finally, let us mention that the $\pi_{\nu}$-Bessel functions ${\mathfrak{b}}_{\pi_{\nu},\psi_{\nu}}(x)$ in the real case as given in Definition 4.12 is more general than the one defined in [Qi20], and refer to Remark 4.14 for details.

Write $|k|=|k|_{\infty}\cup|k|_{f}$, where $|k|_{\infty}$ denotes the subset of $|k|$ consisting of all Archimedean local places of $k$, and $|k|_{f}$ denotes the subset of $|k|$ consisting of all finite local places of $k$. Write ${\mathbb{A}}_{\infty}=\prod_{\nu\in|k|_{\infty}}k_{\nu}$. For $x=(x_{\nu})\in{\mathbb{A}}_{\infty}$, set $|x|_{\infty}:=\prod_{\nu\in|k|_{\infty}}|x_{\nu}|_{\nu}$. Let ${\mathcal{O}}={\mathcal{O}}_{k}$ be the ring of algebraic integers of $k$. L. Clozel defines in [Clo22] the Tate kernel:

for $x\in{\mathbb{A}}_{\infty}^{\times}$ with $X=|x|_{\infty}$, where ${\mathfrak{a}}$ runs over nonzero integral ideals ${\mathfrak{a}}\subset{\mathcal{O}}$, and $\kappa={\mathrm{Res}}_{s=1}\zeta_{k}(s)$. Here $\zeta_{k}(s)=\sum_{{\mathfrak{a}}\subset{\mathcal{O}}}{\mathfrak{N}}({\mathfrak{a}})^{-s}$ is the Dedekind zeta function of $k$ with ${\mathfrak{N}}({\mathfrak{a}}):=|N_{k/{\mathbb{Q}}}{\mathfrak{a}}|$, the absolute norm of ${\mathfrak{a}}$; and the dual kernel

where $\mathcal{D}=\mathcal{D}_{k}$ is the difference of ${\mathcal{O}}$ and $\mathcal{D}^{-1}$ is the inverse difference; and $D={\mathfrak{N}}(\mathcal{D})$ is the absolute value of the discriminant. Theorem 1.1 of [Clo22] expresses the relation between those tempered distributions $H_{s}(x)$ and $K_{s}(x)$ on ${\mathbb{A}}_{\infty}^{\times}$ in terms of the condition: $\zeta_{k}(s)=0$ with $\sigma={\mathrm{Re}}(s)\in(0,1)$, which is more precisely stated as follows.

The second global result of this paper is to define the kernel functions for any irreducible cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_{n}({\mathbb{A}})$, which will be called the Godement-Jacquet kernels, and prove the analogy of Theorem 1.4 for the Godement-Jacquet kernels and the standard $L$-functions $L(s,\pi)$ (see Theorems 6.9 and 6.13 for the exact statements).

Let $\pi$ be an irreducible cuspidal automorphic representation of ${\mathrm{GL}}_{n}({\mathbb{A}})$ and write $\pi=\pi_{\infty}\otimes\pi_{f}$, where $\pi_{f}:=\otimes_{p<\infty}\pi_{p}$, and write the standard $L$-function of $\pi$ as

for ${\mathrm{Re}}(s)$ sufficiently positive. As usual, $L(s,\pi)$ is called the complete $L$-function associated with $\pi$, and $L_{f}(s,\pi_{f})$ is called the finite part of the $L$-function associated with $\pi$. The local and global theory of R. Godement and H. Jacquet in [GJ72] introduces the global zeta integrals for $L(s,\pi)$ and proves that $L(s,\pi)$ has analytic continuation to an entire function in $s\in{\mathbb{C}}$ and satisfies the functional equation

Following the reformulation as developed in [JL23], for an irreducible cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_{n}({\mathbb{A}})$, there exists a $\pi$-Schwartz space ${\mathcal{S}}_{\pi}({\mathbb{A}}^{\times})$ as defined in (2.23), which defines the ${\mathrm{GL}}_{1}$-zeta integral

for any $\phi\in{\mathcal{S}}_{\pi}({\mathbb{A}}^{\times})$. By [JL23, Theorem 4.6] the zeta integral ${\mathcal{Z}}(s,\phi)$ converges absolutely for ${\mathrm{Re}}(s)>\frac{n+1}{2}$, admits analytic continuation to an entire function in $s\in{\mathbb{C}}$, and satisfies the functional equation

where ${\mathcal{F}}_{\pi,\psi}$ is the $\pi$-Fourier transform as defined in (2.25). From the global functional equation in (1.7), we introduce the notion of the Godement-Jacquet kernels for $L(s,\pi)$ in Definition 6.5, which can be briefly explained as follows.

Write $x\in{\mathbb{A}}^{\times}$ as $x=x_{\infty}\cdot x_{f}$ with $x_{\infty}\in{\mathbb{A}}_{\infty}^{\times}$ and $x_{f}\in{\mathbb{A}}_{f}^{\times}$. Set ${\mathbb{A}}^{>1}:=\{x\in{\mathbb{A}}^{\times}\ \colon\ |x|_{\mathbb{A}}>1\}$. For $x\in{\mathbb{A}}^{>1}$, we have that $|x|=|x_{\infty}|_{{\mathbb{A}}}\cdot|x_{f}|_{{\mathbb{A}}}>1$ and $|x_{f}|_{{\mathbb{A}}}>|x_{\infty}|_{{\mathbb{A}}}^{-1}$. Write

For $\phi=\phi_{\infty}\otimes\phi_{f}\in{\mathcal{S}}_{\pi}({\mathbb{A}}^{\times})$ with $\phi_{\infty}\in{\mathcal{S}}_{\pi_{\infty}}({\mathbb{A}}_{\infty}^{\times})$ and $\phi_{f}\in{\mathcal{S}}_{\pi_{f}}({\mathbb{A}}_{f}^{\times})$, we write

for any $s\in{\mathbb{C}}$, where the inner integral is taken over the domain $\{x_{f}\in{\mathbb{A}}_{f}^{\times}\ \colon\ |x_{f}|_{{\mathbb{A}}}>|x_{\infty}|_{{\mathbb{A}}}^{-1}\}$. Proposition 6.1 shows that the integral converges absolutely for any $s\in{\mathbb{C}}$ and is holomorphic in $s\in{\mathbb{C}}$. By the Fubini theorem and the support in ${\mathbb{A}}_{f}^{\times}$ of $\phi_{f}$, which is a fractional ideal of $k$ (Proposition 6.6) , the inner integral $\int_{{\mathbb{A}}_{f}^{\times}}^{>|x_{\infty}|_{{\mathbb{A}}}^{-1}}\phi_{f}(x_{f})|x_{f}|_{{\mathbb{A}}}^{s-\frac{1}{2}}\,\mathrm{d}^{\times}x_{f}$ converges absolutely for any $s\in{\mathbb{C}}$ and any $x_{\infty}\in{\mathbb{A}}_{\infty}^{\times}$. The Godement-Jacquet kernel for $L(s,\pi)$ is defined by

for $x_{\infty}\in{\mathbb{A}}_{\infty}^{\times}$ and for all $s\in{\mathbb{C}}$. Clozel defines in [Clo22] the dual kernel for the case of ${\mathrm{GL}}_{1}$ with $\pi$ the trivial character. We define here the dual kernel of the Godement-Jacquet kernel $H_{\pi,s}(x_{\infty},\phi_{f})$ for $L(s,\pi)$ to be

for $x_{\infty}\in{\mathbb{A}}_{\infty}^{\times}$ and for all $s\in{\mathbb{C}}$. Proposition 6.6 shows that both kernel functions $H_{\pi,s}(x_{\infty},\phi_{f})$ and $K_{\pi,s}(x_{\infty},\phi_{f})$ on ${\mathbb{A}}_{\infty}^{\times}$ can be extended uniquely to tempered distributions on ${\mathbb{A}}_{\infty}$ for any $\phi_{f}\in{\mathcal{S}}_{\pi_{f}}({\mathbb{A}}_{f}^{\times})$ and for any $s\in{\mathbb{C}}$, by using the work of S. Miller and W. Schmid in [MS04a]. We are able to match the kernels $H_{\pi,s}(x_{\infty},\phi_{f})$ and $K_{\pi,s}(x_{\infty},\phi_{f})$ with the Euler product expression or Dirichlet series expression of the finite part $L$-function $L_{f}(s,\pi_{f})$ by specifically choosing the $\pi_{f}$-Schwartz functions $\phi_{f}\in{\mathcal{S}}_{\pi_{f}}({\mathbb{A}}_{f}^{\times})$, and prove in Section 6 the $\pi$-versions of the Clozel theorem (Theorem 1.4). Here is an overly simplified version of Theorem 6.9, to which we refer the details.

Theorem 6.13 is a more precise version of Theorem 1.5 (and Theorem 6.9), which is the exact $\pi$-analogy of Theorem 1.4.

We recall the $\pi$-Poisson summation formula on ${\mathrm{GL}}_{1}$ developed by Z. Luo and the first named author of this paper in [JL23] in Section 2. Sections 2.1 and 2.2 are to review briefly the local $\pi$-Schwartz spaces and local $\pi$-Fourier operators developed in [JL22] and [JL23]. Based on their work as well as the work of [GJ72], we recall the formulation of the $\pi$-Poission summation formula on ${\mathrm{GL}}_{1}$ in [JL23] in Section 2.3.

Sections 3 and 4 are devoted to understand the duality between the function $w_{\nu}(x)$ and the function $\widetilde{w_{\nu}}(x)$ by means of the harmonic analysis on ${\mathrm{GL}}_{1}$ as developed in [JL22] and [JL23], and to prove our main local results (Theorem 1.3). By comparing the Godement-Jacquet theory with the ${\mathrm{GL}}_{n}\times{\mathrm{GL}}_{1}$ Rankin-Selberg convolution, we are able to express the dual function $\widetilde{w_{\nu}}(x)$ of $w_{\nu}(x)$ in terms of the $\pi_{\nu}$-Fourier transform ${\mathcal{F}}_{\pi_{\nu},\psi_{\nu}}$ up to certain normalization (Proposition 3.5), based on Proposition 3.1 that identifies the $\pi_{\nu}$-Schwartz space ${\mathcal{S}}_{\pi_{\nu}}(k_{\nu}^{\times})$, as introduced in [JL23] and recalled in (2.5), with the $\pi_{\nu}$-Whittaker-Schwartz space ${\mathcal{W}}_{\pi_{\nu}}(k_{\nu}^{\times})$ as defined in (3.1). As a consequence, we obtain a formula that express the dual function $\widetilde{w_{\nu}}(x)$ of $w_{\nu}(x)$ as a convolution of the $\pi_{\nu}$-kernel function $k_{\pi_{\nu},\psi_{\nu}}(x)$ with $w_{\nu}(x)$, up to certain normalization (Corollary 3.6). In Section 4, we introduce the notion of $\pi_{\nu}$-Bessel functions ${\mathfrak{b}}_{\pi_{\nu},\psi_{\nu}}(x)$ (Definitions 4.2, 4.9 and 4.12) and prove the precise relation between the $\pi_{\nu}$-Bessel functions and $\pi_{\nu}$-kernel functions as defined in (2.20) (Propositions 4.3, 4.8, and 4.13). In the $p$-adic case, the $\pi_{\nu}$-Bessel function ${\mathfrak{b}}_{\pi_{\nu},\psi_{\nu}}(x)$ on $k_{\nu}^{\times}$ is introduced following the work of E. Baruch in [Bar05], which is recalled in Section 4.1. In the real or complex case, we introduce the $\pi_{\nu}$-Bessel function ${\mathfrak{b}}_{\pi_{\nu},\psi_{\nu}}(x)$ on $k_{\nu}^{\times}$ by following the general theory of Bessel functions of high rank by Z. Qi in [Qi20]. It should be mentioned that the $\pi_{\nu}$-Bessel function ${\mathfrak{b}}_{\pi_{\nu},\psi_{\nu}}(x)$ on ${\mathbb{R}}^{\times}$ in the real case is more general than the one considered in [Qi20] (Remark 4.14). As expected, when $n=2$ our results recover the previous known results as discussed by J. Cogdell in [Cog14] and by D. Soudry in [Sou84].

With all these ingredients, we are able to give a new proof of the Voronoi formula for ${\mathrm{GL}}_{n}$ (Theorem 1.2) as proved in [IT13] in Section 5. In fact, the proof of the Voronoi formula in [IT13] is based on the Rankin-Selberg convolution for ${\mathrm{GL}}_{n}\times{\mathrm{GL}}_{1}$ in [JPSS83], [CPS04] and [Jac09]. And our proof is based on the Godement-Jacquet theory in [GJ72] and its reformulation in [JL22, JL23]. The main idea is that the Voronoi summation formula for ${\mathrm{GL}}_{n}$ as in Theorem 1.2 is a special case of the $\pi$-Poission summation formula on ${\mathrm{GL}}_{1}$ as in Theorem 2.1 ([JL23, Theorem 4.7]), after the long computations carried out in Section 3 of this paper and in [JL22, JL23]. Those computations enable us to express the summands on the dual side (the right-hand side) of the Voronoi formula in Theorem 1.2 as the global $\pi$-Fourier transform of the summands on the given side (the left-hand side), which is Proposition 5.4.

In Section 6.1, in order to define the Godement-Jacquet kernels $H_{\pi,s}(x,\phi_{f})$ and their dual kernels $K_{\pi,s}(x,\phi_{f})$ (Definition 6.5) for any irreducible cuspidal automorphic representation $\pi$ of ${\mathrm{GL}}_{n}({\mathbb{A}})$, we develop further properties (Propositions 6.1 and 6.3, and Corollary 6.4) of the global zeta integrals ${\mathcal{Z}}(s,\phi)$, as defined in (1.6), by using the $\pi$-Fourier transform ${\mathcal{F}}_{\pi,\psi}$ and the associated $\pi$-Poisson summation formula as developed in [JL23]. In Proposition 6.6, we show that both kernel functions $H_{\pi,s}(x_{\infty},\phi_{f})$ and $K_{\pi,s}(x_{\infty},\phi_{f})$ on ${\mathbb{A}}_{\infty}^{\times}$ can be extended uniquely to tempered distributions on ${\mathbb{A}}_{\infty}$ for any $\phi_{f}\in{\mathcal{S}}_{\pi_{f}}({\mathbb{A}}_{f}^{\times})$ and for any $s\in{\mathbb{C}}$. In Section 6.2, guided by Theorem 1.4, we prove in Proposition 6.7 that if $s\in{\mathbb{C}}$ is a zero of $L_{f}(s,\pi_{f})$, then the kernel $H_{\pi,s}(x_{\infty},\phi_{f})$ is equal to the negative of $\pi_{\infty}$-Fourier transform of the dual kernel $K_{\pi,1-s}(x_{\infty},\phi_{f})$. For any $\phi_{\infty}\in{\mathcal{S}}_{\pi_{\infty}}({\mathbb{A}}_{\infty}^{\times})$, take $\phi^{\star}=\phi_{\infty}\otimes\phi_{f}^{\star}$, where $\phi_{f}^{\star}:=\otimes_{\nu}\phi_{\nu}$ with $\phi_{\nu}$ as given in Proposition 6.8. Theorem 6.9 proves the $\pi$-version of Theorem 1.4 for the Euler product expression of $L_{f}(s,\pi_{f})$. With the help of Lemma 6.10, we obtain the Dirichlet series expression of the kernels in Propositions 6.11 and 6.12. Finally, Theorem 6.13 establishes the $\pi$-version of Theorem 1.4 for the Dirichlet series expression of $L_{f}(s,\pi_{f})$.

Let $|k|$ be the set of all local places of $k$. For any local place $\nu$, we denote by $F=k_{\nu}$, the local field of $k$ at $\nu$. If $F$ is non-Archimedean, we denote by ${\mathfrak{o}}={\mathfrak{o}}_{F}$ the ring of integers and by ${\mathfrak{p}}={\mathfrak{p}}_{F}$ the maximal ideal of ${\mathfrak{o}}$. Let ${\mathrm{G}}_{n}={\mathrm{GL}}_{n}$ be the general linear group defined over $F$. Fix the maximal compact subgroups $K$ of ${\mathrm{G}}_{n}(F)$, where $K={\mathrm{GL}}_{n}({\mathfrak{o}}_{F})$ if $F$ is non-Archimedean, $K={\mathrm{O}}_{n}$ if $F={\mathbb{R}}$, and $K={\mathrm{U}}_{n}$ if $F={\mathbb{C}}$.

Let ${\mathrm{M}}_{n}(F)$ be the space of all $n\times n$ matrices over $F$ and ${\mathcal{S}}({\mathrm{M}}_{n}(F))$ be the space of Schwartz functions on ${\mathrm{M}}_{n}(F)$. When $F$ is Archimedean, it is the space of usual Schwartz functions on the affine space ${\mathrm{M}}_{n}(F)$, and when $F$ is $p$-adic, it consists of all locally constant, compactly supported functions on ${\mathrm{M}}_{n}(F)$. Let $|\cdot|_{F}$ be the normalized absolute value on the local field $F$, which is the modular function of the multiplication of $F^{\times}$ on $F$ with respect to the self-dual additive Haar measure $\,\mathrm{d}^{+}x$ on $F$. As a reformulation of the local Godement-Jacquet theory in [JL23, Section 2.2], the (standard) Schwartz space on ${\mathrm{G}}_{n}(F)$ is defined to be

where ${\mathcal{C}}^{\infty}({\mathrm{G}}_{n}(F))$ denotes the space of all smooth functions on ${\mathrm{G}}_{n}(F)$. By [JL23, Prposition 2.5], the Schwartz space ${\mathcal{S}}_{\rm std}({\mathrm{G}}_{n}(F))$ is a subspace of $L^{2}({\mathrm{G}}_{n}(F),\,\mathrm{d}g)$, which is the space of square-integrable functions on ${\mathrm{G}}_{n}(F)$.

Consider the determinant map $\det={\det}_{F}\colon{\mathrm{M}}_{n}(F)={\mathrm{GL}}_{n}(F)\to F.$ When restricted to $F^{\times}$, we obtain that

and the fibers of the determinant map $\det$ are of the form:

When $x=1$, the fiber is the kernel of the map, i.e. $\ker(\det)={\mathrm{SL}}_{n}(F)$. In general, each fiber ${\mathrm{G}}_{n}(F)_{x}$ is an ${\mathrm{SL}}_{n}(F)$-torsor. Let $\,\mathrm{d}^{+}g$ be the self-dual Haar measure on ${\mathrm{M}}_{n}(F)$ with respect to the standard Fourier transform defined by (2.7) below. On ${\mathrm{G}}_{n}(F)$, we fix the Haar measure $\,\mathrm{d}g=|\det g|_{F}^{-n}\cdot\,\mathrm{d}^{+}g$. Let $\,\mathrm{d}_{1}g$ be the induced Haar measure $\,\mathrm{d}g$ from ${\mathrm{G}}_{n}(F)$ to ${\mathrm{SL}}_{n}(F)$. It follows that the Haar measure $\,\mathrm{d}_{1}g$ induces an ${\mathrm{SL}}_{n}(F)$-invariant measure $\,\mathrm{d}_{x}g$ on each fiber ${\mathrm{G}}_{n}(F)_{x}$.

Let $\Pi_{F}({\mathrm{G}}_{n})$ be the set of equivalence classes of irreducible smooth representations of ${\mathrm{G}}_{n}(F)$ when $F$ is non-Archimedean; and of irreducible Casselman-Wallach representations of ${\mathrm{G}}_{n}(F)$ when $F$ is Archimedean. For $\pi\in\Pi_{F}({\mathrm{G}}_{n})$, we denote by ${\mathcal{C}}(\pi)$ the space of all matrix coefficients of $\pi$. Write $\xi=|\det g|_{F}^{\frac{n}{2}}\cdot f(g)\in{\mathcal{S}}_{\rm std}({\mathrm{G}}_{n}(F))$ with some $f\in{\mathcal{S}}({\mathrm{M}}_{n}(F))$ as in (2.1). For $\varphi_{\pi}\in{\mathcal{C}}(\pi)$, as in [JL23, Section 3.1], we define