Certain Fourier Operators on $\mathrm{GL}_1$ and Local Langlands Gamma functions

TL;DR

This paper develops a local theory for Fourier operators on $ ext{GL}_1$ related to automorphic representations, connecting them to local Langlands gamma functions and generalizing classical Poisson summation formulae within the Langlands program.

Contribution

It introduces distribution kernels for Fourier operators on $ ext{GL}_1$, representing them as convolution operators, and shows these kernels' Mellin transforms are the local Langlands gamma functions, linking harmonic analysis and number theory.

Findings

Kernel functions represent Fourier operators as Hankel transforms.

Local gamma functions are identified as Mellin transforms of kernel functions.

Results connect local harmonic analysis with the Langlands gamma functions.

Abstract

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 $k\backslash\mathbb{A}$. 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 $k_\nu$ of $k$, we define distribution kernel functions $k_{\sigma_\nu,\rho,\psi_\nu }(x)$ on $\mathrm{GL}_1$ that represent the $(\sigma_\nu,\rho)$-Fourier operators $\mathcal{F}_{\sigma_\nu,\rho,\psi_\nu}$ as convolution integral operators, i.e. generalized Hankel transforms, and the local Langlands $\gamma$-functions $\gamma(s,\sigma_\nu,\rho,\psi_\nu)$ 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}.