The $\ddot{I}$-Fourier transform

Abstract

Let $G$ be a reductive group over a local field $F$ and let $\ddot{I}:{}^{L}G\to {\mathrm{GL}}_{V_{\ddot{I}}}(\mathbb{C})$ be a representation of its $L$-group satisfying suitable assumptions. Braverman, Kazhdan and Ngô conjectured that one has a $\ddot{I}$-Fourier transform on $L^2(G(F))$ and a $\ddot{I}$-Schwartz space ${\mathcal{S}}_{\ddot{I}}(G(F))<L^2(G(F))$ 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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