Voronoi-Hankel Transforms — Draconitic Research

Abstract

Let π be a generic irreducible representation of GL_n(F) for a local field F. We introduce the Voronoi--Hankel transform 𝒱ℋ_π for π, and prove it is equivalent to the π-Fourier transform introduced by Jiang--Luo. We give an extension 𝒱ℋ̃_π of 𝒱ℋ_π to some larger class of functions on F^×, and prove the multiplicativity with respect to parabolic induction via the above-mentioned equivalence. For non-archimedean F we give a systematical study of the relevant kernel function, including the asymptotic behavior at $0$ and ∞, as well as some simple integral representation based on the local Langlands correspondences for the essentially tame supercuspidals. As an application in the non-archimedean case, we give an effective version of the stability theorem for the twisted local gamma factors.

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BibTeX RIS

BibTeX

@misc{shen2026voronoihankel,
  title = {{Voronoi-Hankel Transforms}},
  author = {Zhefeng Shen and Han Wu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.04336},
  eprint = {2606.04336},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.04336},
}

RIS

TY  - UNPB
TI  - Voronoi-Hankel Transforms
AU  - Zhefeng Shen
AU  - Han Wu
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2606.04336
UR  - https://arxiv.org/abs/2606.04336
AB  - Let π be a generic irreducible representation of GL_n(F) for a local field F. We introduce the Voronoi--Hankel transform 𝒱ℋ_π for π, and prove it is equivalent to the π-Fourier transform introduced by Jiang--Luo. We give an extension 𝒱ℋ̃_π of 𝒱ℋ_π to some larger class of functions on F^×, and prove the multiplicativity with respect to parabolic induction via the above-mentioned equivalence. For non-archimedean F we give a systematical study of the relevant kernel function, including the asymptotic behavior at $0$ and ∞, as well as some simple integral representation based on the local Langlands correspondences for the essentially tame supercuspidals. As an application in the non-archimedean case, we give an effective version of the stability theorem for the twisted local gamma factors.
ER  -

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