Vol. 318, No. 2, 2022

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Certain Fourier operators on $\operatorname{GL}_1$ and local Langlands gamma functions

Dihua Jiang and Zhilin Luo

Vol. 318 (2022), No. 2, 339–374
Abstract

For a split reductive group G over a number field k, let ρ be an n-dimensional complex representation of its complex dual group G(). For any irreducible cuspidal automorphic representation σ of G(𝔸), where 𝔸 is the ring of adeles of k, in [Jiang and Luo 2021], the authors introduce the (σ,ρ)-Schwartz space 𝒮σ,ρ(𝔸×) and (σ,ρ)-Fourier operator σ,ρ, and study the (σ,ρ,ψ)-Poisson summation formula on GL 1, under the assumption that the local Langlands functoriality holds for the pair (G,ρ) at all local places of k, where ψ is a nontrivial additive character of k𝔸. Such general formulas on  GL 1, as a vast generalization of the classical Poisson summation formula, are expected to be responsible for the Langlands conjecture [Langlands 1970] on global functional equation for the automorphic L-functions L(s,σ,ρ). In order to understand such Poisson summation formulas, we continue with Jiang and Luo [2021] and develop a further local theory related to the (σ,ρ)-Schwartz space 𝒮σ,ρ(𝔸×) and (σ,ρ)-Fourier operator σ,ρ. More precisely, over any local field kν of k, we define distribution kernel functions kσν,ρ,ψν(x) on GL 1 that represent the (σν,ρ)-Fourier operators σν,ρ,ψν as convolution integral operators, i.e., generalized Hankel transforms, and the local Langlands γ-functions γ(s,σν,ρ,ψν) as Mellin transform of the kernel functions. As a consequence, we show that any local Langlands γ-functions are the gamma functions in the sense of I. Gelfand, M. Graev, and I. Piatetski-Shapiro [Gelfand et al. 2016] and of A. Weil [1995].

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Keywords
invariant distribution, Fourier operator, Hankel transforms, representation of real and $p$-adic reductive groups, Langlands local gamma functions
Mathematical Subject Classification
Primary: 11F66, 43A32, 46S10
Secondary: 11F70, 22E50, 43A80
Milestones
Received: 11 January 2022
Revised: 20 April 2022
Accepted: 7 May 2022
Published: 20 August 2022
Authors
Dihua Jiang
School of Mathematics
University of Minnesota
Minneapolis, MN
United States
Zhilin Luo
Department of Mathematics
University of Chicago
Chicago, IL
United States