The $ρ$-Schwartz space
Jayce R. Getz, Armando Gutiérrez Terradillos, Farid Hosseinijafari, Aaron Slipper, Guodong Xi, HaoYun Yao, Alan Zhao
TL;DR
The paper advances Braverman–Kazhdan–Ngō’s conjecture by constructing a ρ-Schwartz space attached to a reductive group G over a local field, proving a large portion of the conjecture in the non-Archimedean setting and providing Archimedean approximations via spectral methods. It develops a spectral framework anchored in the Harish-Chandra Plancherel formula, the Paley–Wiener theorems, and a robust local Langlands correspondence to define a ρ-Fourier transform whose kernel encodes L- and γ-factors. A key innovation is the asymptotic ρ-Schwartz space S_ (G(F)) and its refinement to the actual ρ-Schwartz space in the non-Archimedean case, together with a detailed treatment of zeta integrals, poles, and holomorphicity properties across Temp_Ind(G). The results unify harmonic-analytic and Langlands-theoretic tools to realize a Schwartz space compatible with a Fourier transform and compatible with tempered representation theory, with potential applications to global theory via BK-lifting and Ngo Hankel constructions.
Abstract
Let $G$ be a reductive group over a local field $F$ and let $ρ:{}^LG \to \mathrm{GL}_{V_ρ}(\mathbb{C})$ be a representation of its $L$-group satisfying suitable assumptions. Braverman, Kazhdan and Ngô conjectured that one has a Schwartz space $\mathcal{S}_ρ(G(F))$ of functions on $G(F)$ that admits a Fourier transform and satisfies certain desiderata. We prove a large portion of this conjecture. More precisely, we construct these Schwartz spaces over non-Archimedean local fields and construct an approximation to them over Archimedean fields. Our methods are spectral in nature.