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

Abstract

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 $ρ$ 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 $\CS_{\sig,ρ}(\BA^\times)$ of $(\sig,ρ)$-Schwartz functions on $\BA^\times$ and $(\sig,ρ)$-Fourier operator $\CF_{\sig,ρ,ψ}$ that takes $\CS_{\sig,ρ}(\BA^\times)$ to $\CS_{\wt{\sig},ρ}(\BA^\times)$, where $\wt{\sig}$ is the contragredient of $\sig$. By assuming the local Langlands functoriality for the pair $(G,ρ)$, we show that the $(\sig,ρ)$-theta functions \[ Θ_{\sig,ρ}(x,ϕ):=\sum_{\alp\in k^\times}ϕ(\alp x) \] converges absolutely for all $ϕ\in\CS_{\sig,ρ}(\BA^\times)$, and state conjectures on $(σ,ρ)$-Poisson summation formula on $\GL_1$. Then we prove conjectures when $G=\GL_n$ and $ρ$ 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 $L(s,π\timesχ)$ for any irreducible cuspidal automorphic representation $π$ of $\GL_n(\BA)$ and idele class character $χ$ 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 $L(s,χ)$ to the automorphic $L$-functions $L(s,π\timesχ)$.