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تسجيل مستخدم جديدCertain Fourier Operators and their Associated Poisson Summation Formulae on $mathrm{GL}_1$
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In this paper, we explore possibilities to utilize harmonic analysis on $mathrm{GL}_1$ to understand Langlands automorphic $L$-functions in general, as a vast generalization of the pioneering work of J. Tate (cite{Tt50}). For a split reductive group $G$ over a number field $k$, let $G^vee(mathbb{C})$ be its complex dual group and $rho$ be an $n$-dimensional complex representation of $G^vee(mathbb{C})$. For any irreducible cuspidal automorphic representation $sigma$ of $G(mathbb{A})$, where $mathbb{A}$ is the ring of adeles of $k$, we introduce the space $mathcal{S}_{sigma,rho}(mathbb{A}^times)$ of $(sigma,rho)$-Schwartz functions on $mathbb{A}^times$ and $(sigma,rho)$-Fourier operator $mathcal{F}_{sigma,rho,psi}$ that takes $mathcal{S}_{sigma,rho}(mathbb{A}^times)$ to $mathcal{S}_{widetilde{sigma},rho}(mathbb{A}^times)$, where $widetilde{sigma}$ is the contragredient of $sigma$. By assuming the local Langlands functoriality for the pair $(G,rho)$, we show that the $(sigma,rho)$-theta functions [ Theta_{sigma,rho}(x,phi):=sum_{alphain k^times}phi(alpha x) ] converges absolutely for all $phiinmathcal{S}_{sigma,rho}(mathbb{A}^times)$, and state conjectures on $(sigma,rho)$-Poisson summation formula on $mathrm{GL}_1$. One of the main results in this paper is to prove the conjectures when $G=mathrm{GL}_n$ and $rho$ is the standard representation of $mathrm{GL}_n(mathbb{C})$. The proof uses substantially the local theory of Godement-Jacquet (cite{GJ72}) for the standard $L$-functions of $mathrm{GL}_n$ and the Poisson summation formula for the classical Fourier transform on affine spaces.
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For a split reductive group $G$ over a number field $k$, let $rho$ be an $n$-dimensional complex representation of its complex dual group $G^vee(mathbb{C})$. For any irreducible cuspidal automorphic representation $sigma$ of $G(mathbb{A})$, where $ma
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