The $ρ$-Fourier transform
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916029672718336 |
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| author | Getz, Jayce R. Terradillos, Armando Gutiérrez Hosseinijafari, Farid Slipper, Aaron Xi, Guodong Yao, HaoYun Zhao, Alan |
| author_facet | Getz, Jayce R. Terradillos, Armando Gutiérrez Hosseinijafari, Farid Slipper, Aaron Xi, Guodong Yao, HaoYun Zhao, Alan |
| contents | 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 $ρ$-Fourier transform on $L^2(G(F))$ and a $ρ$-Schwartz space $\mathcal{S}_ρ(G(F))<L^2(G(F))$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00182 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $ρ$-Fourier transform Getz, Jayce R. Terradillos, Armando Gutiérrez Hosseinijafari, Farid Slipper, Aaron Xi, Guodong Yao, HaoYun Zhao, Alan Number Theory Representation Theory Primary 11F70 Secondary 22E30, 22E35 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 $ρ$-Fourier transform on $L^2(G(F))$ and a $ρ$-Schwartz space $\mathcal{S}_ρ(G(F))<L^2(G(F))$ 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. |
| title | The $ρ$-Fourier transform |
| topic | Number Theory Representation Theory Primary 11F70 Secondary 22E30, 22E35 |
| url | https://arxiv.org/abs/2512.00182 |