Global $GL_2$ Hecke-Baxter operator
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912577435467776 |
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| author | Gerasimov, Anton A. Lebedev, Dmitry R. Oblezin, Sergey V. |
| author_facet | Gerasimov, Anton A. Lebedev, Dmitry R. Oblezin, Sergey V. |
| contents | We construct a global Hecke-Baxter operator for integrable systems of arithmetic type associated with the group $GL_2$. This is an element of a global Hecke algebra associated with the double coset space $GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$. Eigenvalues of the global Hecke-Baxter operator acting on the $GL_2$-Eisenstein series are given by the corresponding global $L$-factors. This construction generalizes our previous construction of the Hecke-Baxter operators over local completions $\mathbb{R}$ and $\mathbb{Q}_p$ of the number field $\mathbb{Q}$. Presumably, zeroes of the corresponding global $L$-factors should be subjected to an arithmetic version of the Bethe ansatz equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_09979 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global $GL_2$ Hecke-Baxter operator Gerasimov, Anton A. Lebedev, Dmitry R. Oblezin, Sergey V. Representation Theory Number Theory 11F41, 11F70, 11M06, 11M36 We construct a global Hecke-Baxter operator for integrable systems of arithmetic type associated with the group $GL_2$. This is an element of a global Hecke algebra associated with the double coset space $GL_2(\mathbb{Z})\backslash GL_2(\mathbb{R})/O_2$. Eigenvalues of the global Hecke-Baxter operator acting on the $GL_2$-Eisenstein series are given by the corresponding global $L$-factors. This construction generalizes our previous construction of the Hecke-Baxter operators over local completions $\mathbb{R}$ and $\mathbb{Q}_p$ of the number field $\mathbb{Q}$. Presumably, zeroes of the corresponding global $L$-factors should be subjected to an arithmetic version of the Bethe ansatz equations. |
| title | Global $GL_2$ Hecke-Baxter operator |
| topic | Representation Theory Number Theory 11F41, 11F70, 11M06, 11M36 |
| url | https://arxiv.org/abs/2507.09979 |