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arXiv · 2512.00182

The $\rho$-Fourier transform

Jayce R. Getz·Armando Gutiérrez Terradillos·Farid Hosseinijafari·Aaron Slipper·Guodong Xi·HaoYun Yao·Alan Zhao
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Abstract

Let $G$ be a reductive group over a local field $F$ and let $\rho:{}^LG \to \mathrm{GL}_{V_{\rho}}(\mathbb{C})$ be a representation of its $L$-group satisfying suitable assumptions. Braverman, Kazhdan and Ng\^o conjectured that one has a $\rho$-Fourier transform on $L^2(G(F))$ and a $\rho$-Schwartz space $\mathcal{S}_{\rho}(G(F))<L^2(G(F))$ fixed under the Fourier transform that satisfies certain desiderata. We construct the Fourier transform for arbitrary fields. Over non-Archimedean fields we construct the Schwartz space, and in the Archimedean case we construct an approximation to it. This proves a large portion of their conjectures. Our methods are spectral in nature.

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Jayce R. Getz, Armando Gutiérrez Terradillos, Farid Hosseinijafari, Aaron Slipper, Guodong Xi, HaoYun Yao, Alan Zhao. 2025-11-28. The $\rho$-Fourier transform. https://arxiv.org/abs/2512.00182

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