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Register a new userArchimedean Non-vanishing, Cohomological Test Vectors, and Standard $L$-functions of $mathrm{GL}_{2n}$: Complex Case
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The purpose of this paper is to study the local zeta integrals of Friedberg-Jacquet at complex place and to establish similar results to our recent work in the reall case joint with C. Cheng and D. Jiang. In this paper, we will (1) give a necessary and sufficient condition on an irreducible essentially tempered cohomological representation $pi$ of $mathrm{GL}_{2n}(mathbb{C})$ with a non-zero Shalika model; (2) construct a new twisted linear period $Lambda_{s,chi}$; (3) give a necessary and sufficient condition on the character $chi$ such that there exists a uniform cohomological test vector $vin V_pi$ (which we construct explicitly) for $Lambda_{s,chi}$. As a consequence, we obtain the non-vanishing of local Friedberg-Jacquet integral at complex place. All of the above are essential preparations for attacking a global period relation problem.
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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.
We study simultaneous non-vanishing of $L(tfrac{1}{2},di)$ and $L(tfrac{1}{2},gotimes di)$, when $di$ runs over an orthogonal basis of the space of Hecke-Maass cusp forms for $SL(3,mathbb{Z})$ and $g$ is a fixed $SL(2,mathbb{Z})$ Hecke cusp form of weight $kequiv 0 pmod 4$.
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 $mathrm{GL}_n(mathsf k)times mathrm{GL}_n(mathsf k)$.
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}.
In this paper we establish a new case of Langlands functoriality. More precisely, we prove that the tensor product of the compatible system of Galois representations attached to a level-1 classical modular form and the compatible system attached to an n-dimensional RACP automorphic representation of GL_n of the adeles of Q is automorphic, for any positive integer n, under some natural hypotheses (namely regularity and irreducibility).
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