arXiv · 2503.17958
In this short note, we address a gap in the proof of Sauvageot's density principle, which was pointed out in a paper by Nelson-Venkatesh.
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Yugo Takanashi. 2025-03-28. A note on the Sauvageot density principle. https://doi.org/10.1007/s00229-025-01650-x
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Let $π$ be a generic irreducible representation of $\mathrm{GL}_n(\mathbf{F})$ for a local field $\mathbf{F}$. We introduce the Voronoi--Hankel transform $\mathcal{VH}_π$ for $π$, and prove it is equivalent to the $π$-Fourier transform introduced by Jiang--Luo. We give an extension $\widetilde{\mathcal{VH}}_π$ of $\mathcal{VH}_π$ to some larger class of functions on $\mathbf{F}^{\times}$, and prove the multiplicativity with respect to parabolic induction via the above-mentioned equivalence. For non-archimedean $\mathbf{F}$ we give a systematical study of the relevant kernel function, including the asymptotic behavior at $0$ and $\infty$, 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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