Dihua Jiang, Zhilin Luo

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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(C). 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 Fσ,ρ, and study the (σ, ρ,ψ)- Poisson summation formula on GL1, 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 k\A. Such general formulas on GL1, 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 Fσ,ρ. More precisely, over any local field kν of k, we define distribution kernel functions kσν,ρ,ψν (x) on GL1 that represent the (σν, ρ)-Fourier operators Fσν,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.

Original languageEnglish (US)
Pages (from-to)339-374
Number of pages36
JournalPacific Journal of Mathematics
Issue number2
StatePublished - 2022

Bibliographical note

Funding Information:
The research of this paper is supported in part by the NSF grant DMS-1901802. MSC2020: primary 11F66, 43A32, 46S10; secondary 11F70, 22E50, 43A80. Keywords: invariant distribution, Fourier operator, Hankel transforms, representation of real and p-adic reductive groups, Langlands local gamma functions.

Publisher Copyright:
© 2022, Pacific Journal of Mathematics.All Rights Reserved.


  • Fourier operator
  • Hankel transforms
  • Invariant distribution
  • Langlands local gamma functions
  • Representation of real and p-adic reductive groups


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