The Global Jacquet-Langlands Correspondence via Tensor Products
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917257866641408 |
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| author | Yang, Jun |
| author_facet | Yang, Jun |
| contents | We prove that the global Jacquet--Langlands correspondence ${\rm JL}$ for ${\rm GL}(2)$ can be realized via tensor products over Hecke algebras. Let $G$ be a non-split inner form of ${\rm GL}(2)$ over a number field. Using the similitude theta correspondence, the space $L^2(D(\mathbb{A})\times \mathbb{A}^{\times})$ acquires the structure of a $G(\mathbb{A})$-$(G(\mathbb{A})\times {\rm GL}(2,\mathbb{A}))$ bimodule such that
$L^2(G(F)\backslash G(\mathbb{A}),χ)\otimes_{\mathcal{H}(G)}L^2(D(\mathbb{A})\times \mathbb{A}^{\times})~\cong~\oplus_{π\in {\mathcal{A}}(G,χ^{-1})}$ $π\otimes{\rm JL}(π).$
This decomposition into irreducible representations of $G(\mathbb{A})\times {\rm GL}(2,\mathbb{A})$ recovers the full global Jacquet-Langlands correspondence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_08053 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Global Jacquet-Langlands Correspondence via Tensor Products Yang, Jun Representation Theory Number Theory We prove that the global Jacquet--Langlands correspondence ${\rm JL}$ for ${\rm GL}(2)$ can be realized via tensor products over Hecke algebras. Let $G$ be a non-split inner form of ${\rm GL}(2)$ over a number field. Using the similitude theta correspondence, the space $L^2(D(\mathbb{A})\times \mathbb{A}^{\times})$ acquires the structure of a $G(\mathbb{A})$-$(G(\mathbb{A})\times {\rm GL}(2,\mathbb{A}))$ bimodule such that $L^2(G(F)\backslash G(\mathbb{A}),χ)\otimes_{\mathcal{H}(G)}L^2(D(\mathbb{A})\times \mathbb{A}^{\times})~\cong~\oplus_{π\in {\mathcal{A}}(G,χ^{-1})}$ $π\otimes{\rm JL}(π).$ This decomposition into irreducible representations of $G(\mathbb{A})\times {\rm GL}(2,\mathbb{A})$ recovers the full global Jacquet-Langlands correspondence. |
| title | The Global Jacquet-Langlands Correspondence via Tensor Products |
| topic | Representation Theory Number Theory |
| url | https://arxiv.org/abs/2602.08053 |