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Register a new userCertain 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 $mathbb{A}$ is the ring of adeles of $k$, in cite{JL21}, the authors introduce the $(sigma,rho)$-Schwartz space $mathcal{S}_{sigma,rho}(mathbb{A}^times)$ and $(sigma,rho)$-Fourier operator $mathcal{F}_{sigma,rho}$, and study the $(sigma,rho,psi)$-Poisson summation formula on $mathrm{GL}_1$, under the assumption that the local Langlands functoriality holds for the pair $(G,rho)$ at all local places of $k$, where $psi$ is a non-trivial additive character of $kbackslashmathbb{A}$. Such general formulae on $mathrm{GL}_1$, as a vast generalization of the classical Poisson summation formula, are expected to be responsible for the Langlands conjecture (cite{L70}) on global functional equation for the automorphic $L$-functions $L(s,sigma,rho)$. In order to understand such Poisson summation formulae, we continue with cite{JL21} and develop a further local theory related to the $(sigma,rho)$-Schwartz space $mathcal{S}_{sigma,rho}(mathbb{A}^times)$ and $(sigma,rho)$-Fourier operator $mathcal{F}_{sigma,rho}$. More precisely, over any local field $k_ u$ of $k$, we define distribution kernel functions $k_{sigma_ u,rho,psi_ u }(x)$ on $mathrm{GL}_1$ that represent the $(sigma_ u,rho)$-Fourier operators $mathcal{F}_{sigma_ u,rho,psi_ u}$ as convolution integral operators, i.e. generalized Hankel transforms, and the local Langlands $gamma$-functions $gamma(s,sigma_ u,rho,psi_ u)$ as Mellin transform of the kernel function. As consequence, we show that any local Langlands $gamma$-functions are the gamma functions in the sense of Gelfand, Graev, and Piatetski-Shapiro in cite{GGPS}.
Following the paradigm of cite{MR3117742}, we are going to explore the stable transfer factors for $mathrm{Sym}^{n}$ lifting from $mathrm{GL}_{2}$ to $mathrm{GL}_{n+1}$ over any local fields $F$ of characteristic zero with residue characteristic not equal to $2$. When $F=mathbb{C}$ we construct an explicit stable transfer factor for any $n$. When $n$ is odd, we provide a reduction formula, reducing the question to the construction of the stable transfer factors when the $L$-morphism is the diagonal embedding from $mathrm{GL}_{2}(mathbb{C})$ to finitely many copies of $mathrm{GL}_{2}(mathbb{C})$ under mild assumptions on the residue characteristic of $F$. With the assumptions on the residue characteristic, the reduction formula works uniformly over any local fields of characteristic zero, except that for $p$-adic situation we need to exclude the twisted Steinberg representations.
Let $mathsf k$ be a local field. Let $I_ u$ and $I_{ u}$ be smooth principal series representations of $mathrm{GL}_n(mathsf k)$ and $mathrm{GL}_{n-1}(mathsf k)$ respectively. The Rankin-Selberg integrals yield a continuous bilinear map $I_ utimes I_{ u}rightarrow mathbb C$ with a certain invariance property. We study integrals over a certain open orbit that also yield a continuous bilinear map $I_ utimes I_{ u}rightarrow mathbb C$ with the same invariance property, and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant. Similar results are also obtained for Rankin-Selberg integrals for $mathrm{GL}_n(mathsf k)times mathrm{GL}_n(mathsf k)$.
Let $F$ be a quadratic extension of $mathbb{Q}_p$. We prove that smooth irreducible supersingular representations with central character of $mathrm{GL}_2(F)$ are not of finite presentation.
The standard $L$-functions of $mathrm{GL}_{2n}$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existance or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Binyong Sun, by establishing the existence of certain cohomological test vectors. In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $Lambda_{s,chi}$, which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard $L$-function $L(s,piotimeschi)$ as a meromorphic function of $sin mathbb{C}$. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector, and hence recovers a non-vanishing result of Binyong Sun via a completely different method. Our main result indicates a complete solution to (2), which will be presented in a paper of Dihua Jiang, Binyong Sun and Fangyang Tian with full details and with applications to the global period relations for the twisted standard $L$-functions at critical places.
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