## 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^{∨}(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 S_{σ,ρ}(A^{×}) and (σ, ρ)-Fourier operator F_{σ,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 S_{σ,ρ}(A^{×}) and (σ, ρ)-Fourier operator F_{σ,ρ}. More precisely, over any local field k_{ν} of k, we define distribution kernel functions k_{σν,ρ,ψν (x)} on GL_{1} that represent the (σ_{ν}, ρ)-Fourier operators F_{σν,ρ,ψν} as convolution integral operators, i.e., generalized Hankel transforms, and the local Langlands γ -functions γ (s, σ_{ν}, ρ,ψ_{ν}) 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 language | English (US) |
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Pages (from-to) | 339-374 |

Number of pages | 36 |

Journal | Pacific Journal of Mathematics |

Volume | 318 |

Issue number | 2 |

DOIs | |

State | Published - 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.

## Keywords

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

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