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.
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}.
We prove a conjecture of the first-named author ([J14]) on the upper bound Fourier coefficients of automorphic forms in Arthur packets of split classical groups over any number field.
Over a $p$-adic local field $F$ of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group $G={mathbb G}_mtimes{mathrm Sp}_{2n}$. It is associated to the Langlands $gamma$-functions attached to any irreducible admissible representations $chiotimespi$ of $G(F)$ and the standard representation $rho$ of the dual group $G^vee({mathbb C})$, and confirms a series of the conjectures in the local theory of the Braverman-Kazhdan proposal for the case under consideration. Meanwhile, we develop a new type of harmonic analysis on ${rm GL}_1(F)$, which is associated to a $gamma$-function $beta_psi(chi_s)$ (a product of $n+1$ certain abelian $gamma$-functions). Our work on ${rm GL}_1(F)$ plays an indispensable role in the development of our work on $G(F)$. These two types of harmonic analyses both specialize to the well-known local theory developed in Tates thesis when $n=0$. The approach is to use the compactification of ${rm Sp}_{2n}$ in the Grassmannian variety of ${rm Sp}_{4n}$, with which we are able to utilize the well developed local theory of Piatetski-Shapiro and Rallis and many other works) on the doubling local zeta integrals for the standard $L$-functions of ${rm Sp}_{2n}$. The method can be viewed as an extension of the work of Godement-Jacquet for the standard $L$-function of ${rm GL}_n$ and is expected to work for all classical groups. We will consider the archimedean local theory and the global theory in our future work.
This article is to understand the critical values of $L$-functions $L(s,Piotimes chi)$ and to establish the relation of the relevant global periods at the critical places. Here $Pi$ is an irreducible regular algebraic cuspidal automorphic representation of $mathrm{GL}_{2n}(mathbb A)$ of symplectic type and $chi$ is a finite order automorphic character of $mathrm{GL}_1(mathbb A)$, with $mathbb A$ is the ring of adeles of a number field $mathrm k$.
We complete the proof of Proposition 5.3 of [GJR04].
We discuss the theory of automorphic descents of Bessel type and its relation to automorphic version of branching problem and its relevant reciprocal branching problem.
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.
Let $G$ be a group and $H$ be a subgroup of $G$. The classical branching rule (or symmetry breaking) asks: For an irreducible representation $pi$ of $G$, determine the occurrence of an irreducible representation $sigma$ of $H$ in the restriction of $pi$ to $H$. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation $sigma$ of $H$, find an irreducible representation $pi$ of $G$ such that $sigma$ occurs in the restriction of $pi$ to $H$. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan-Gross-Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [JZ15]. The method may be applied to other classical groups as well.
Let $pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${rm U}_{mathfrak n}$ defined over a number field $F$. Under the assumption that $pi$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(frac{1}{2},pitimeschi)$, with a certain automorphic character $chi$ of ${rm U}_1$, for the case of ${mathfrak n}=2,3,4$, and for the general ${mathfrak n}geq 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results for the central $L$-values by means of the theory of endoscopy.
