The $\rho$-Fourier transform

Abstract

Let $G$ be a reductive group over a local field $F$ and let $\rho :{}^{L}G\to {\mathrm{GL}}_{V_{\rho }}(\mathbb{C})$ be a representation of its $L$-group satisfying suitable assumptions. Braverman, Kazhdan and Ngô conjectured that one has a $\rho$-Fourier transform satisfying certain desiderata. We give a uniform construction of the Fourier transform for arbitrary local fields. This complements work of L.~Lafforgue in the non-Archimedean case, and proves a portion of the conjectures of Braverman, Kazhdan and Ngô. Our methods are spectral in nature.

arXiv commentAdded references to related work of L.~Lafforgue. Added a proof of compatiblity with the Fourier transform in the special case of matrices. Revised section 6 in view of the recent work of Hsu and Yao https://arxiv.org/abs/2607.28507