The Fourier transform for triples of quadratic spaces
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866929625494454272 |
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| author | Getz, Jayce R. Hsu, Chun-Hsien |
| author_facet | Getz, Jayce R. Hsu, Chun-Hsien |
| contents | Let $V_1,V_2,V_3$ be a triple of even dimensional vector spaces over a number field $F$ equipped with nondegenerate quadratic forms $\mathcal{Q}_1,\mathcal{Q}_2,\mathcal{Q}_3$, respectively. Let $Y \subset \prod_{i=1}^3 V_i$ be the closed subscheme consisting of $(v_1,v_2,v_3)$ such that $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. One has a Poisson summation formula for this scheme under suitable assumptions on the functions involved, but the relevant Fourier transform was previously only defined as a correspondence. In the current paper we employ a novel global-to-local argument to prove that this Fourier transform is well-defined on the Schwartz space of $Y(\mathbb{A}_F).$ To execute the global-to-local argument, we introduce boundary terms and thereby extend the Poisson summation formula to a broader class of test functions. This is the first time a summation formula with boundary terms has been proven for a spherical variety that is not a Braverman-Kazhdan space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2009_11490 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The Fourier transform for triples of quadratic spaces Getz, Jayce R. Hsu, Chun-Hsien Number Theory Representation Theory 11F70 (Primary) 11F27, 11F66 (Secondary) Let $V_1,V_2,V_3$ be a triple of even dimensional vector spaces over a number field $F$ equipped with nondegenerate quadratic forms $\mathcal{Q}_1,\mathcal{Q}_2,\mathcal{Q}_3$, respectively. Let $Y \subset \prod_{i=1}^3 V_i$ be the closed subscheme consisting of $(v_1,v_2,v_3)$ such that $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. One has a Poisson summation formula for this scheme under suitable assumptions on the functions involved, but the relevant Fourier transform was previously only defined as a correspondence. In the current paper we employ a novel global-to-local argument to prove that this Fourier transform is well-defined on the Schwartz space of $Y(\mathbb{A}_F).$ To execute the global-to-local argument, we introduce boundary terms and thereby extend the Poisson summation formula to a broader class of test functions. This is the first time a summation formula with boundary terms has been proven for a spherical variety that is not a Braverman-Kazhdan space. |
| title | The Fourier transform for triples of quadratic spaces |
| topic | Number Theory Representation Theory 11F70 (Primary) 11F27, 11F66 (Secondary) |
| url | https://arxiv.org/abs/2009.11490 |