The $\rho$-Fourier transform

TL;DR

This paper constructs a $ ho$-Fourier transform for reductive groups over various fields, confirming key conjectures and introducing a new spectral approach.

Contribution

It develops the $ ho$-Fourier transform and Schwartz space constructions for arbitrary fields, advancing the conjectural framework of Braverman, Kazhdan, and Ngô.

Findings

Constructed the $ ho$-Fourier transform over arbitrary fields.

Established the Schwartz space over non-Archimedean fields.

Provided an approximation to the Schwartz space in the Archimedean case.

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.