Voronoi-Hankel Transforms

Abstract

Let $\pi$ be a generic irreducible representation of ${\mathrm{GL}}_n(\mathbf{F})$ for a local field $\mathbf{F}$. We introduce the Voronoi--Hankel transform ${\mathcal{V}\mathcal{H}}_{\pi }$ for $\pi$, and prove it is equivalent to the $\pi$-Fourier transform introduced by Jiang--Luo. We give an extension ${\widetilde{\mathcal{V}\mathcal{H}}}_{\pi }$ of ${\mathcal{V}\mathcal{H}}_{\pi }$ 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.

arXiv commentWe add an appendix co-authored with Dihua Jiang and Zhilin Luo