Certain Fourier Operators and their Associated Poisson Summation Formulae on $\mathrm{GL}_1$

In this paper, we explore a possibility to utilize harmonic analysis on $\GL_1$ to understand Langlands automorphic $L$-functions in general, as a vast generalization of the pioneering work of J. Tate. For a split reductive group $G$ over a number field $k$, let $G^\vee(\BC)$ be its complex dual group and $\rho$ be an $n$-dimensional complex representation of $G^\vee(\BC)$. For any irreducible cuspidal automorphic representation $\sig$ of $G(\BA)$, where $\BA$ is the ring of adeles of $k$, we introduce the space $\CS_{\sig,\rho}(\BA^\times)$ of $(\sig,\rho)$-Schwartz functions on $\BA^\times$ and $(\sig,\rho)$-Fourier operator $\CF_{\sig,\rho,\psi}$ that takes $\CS_{\sig,\rho}(\BA^\times)$ to $\CS_{\wt{\sig},\rho}(\BA^\times)$, where $\wt{\sig}$ is the contragredient of $\sig$. By assuming the local Langlands functoriality for the pair $(G,\rho)$, we show that the $(\sig,\rho)$-theta functions $$\Theta_{\sig,\rho}(x,\phi):=\sum_{\alp\in k^\times}\phi(\alp x)$$ converges absolutely for all $\phi\in\CS_{\sig,\rho}(\BA^\times)$, and state conjectures on $(\sigma,\rho)$-Poisson summation formula on $\GL_1$. Then we prove conjectures when $G=\GL_n$ and $\rho$ is the standard representation of $\GL_n(\BC)$ . The proof uses substantially the local theory of Godement-Jacquet for the standard $L$-functions of $\GL_n$ and the Poisson summation formula for the classical Fourier transform on affine spaces. As an application, we provide a spectral interpretation of the critical zeros of the standard $L$-functions $L(s,\pi\times\chi)$ for any irreducible cuspidal automorphic representation $\pi$ of $\GL_n(\BA)$ and idele class character $\chi$ of $k$, which is a reformulation in the adelic framework of the work of A. Connes and is an extension from the Hecke $L$-functions $L(s,\chi)$ to the automorphic $L$-functions $L(s,\pi\times\chi)$.