{"paper":{"title":"The $\\rho$-Fourier transform","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.NT","authors_text":"Aaron Slipper, Alan Zhao, Armando Guti\\'errez Terradillos, Farid Hosseinijafari, Guodong Xi, HaoYun Yao, Jayce R. Getz","submitted_at":"2025-11-28T19:43:22Z","abstract_excerpt":"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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.00182","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2512.00182/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}