Hecke operators and analytic Langlands correspondence for curves over local fields
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866913240371429376 |
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| author | Etingof, Pavel Frenkel, Edward Kazhdan, David |
| author_facet | Etingof, Pavel Frenkel, Edward Kazhdan, David |
| contents | We construct analogues of the Hecke operators for the moduli space of G-bundles on a curve X over a local field F with parabolic structures at finitely many points. We conjecture that they define commuting compact normal operators on the Hilbert space of half-densities on this moduli space. In the case F=C, we also conjecture that their joint spectrum is in a natural bijection with the set of opers on X for the Langlands dual group with real monodromy. This may be viewed as an analytic version of the Langlands correspondence for complex curves. Furthermore, we conjecture an explicit formula relating the eigenvalues of the Hecke operators and the global differential operators studied in our previous paper arXiv:1908.09677. Assuming the compactness conjecture, this formula follows from a certain system of differential equations satisfied by the Hecke operators, which we prove in this paper for G=PGL(n). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_01509 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Hecke operators and analytic Langlands correspondence for curves over local fields Etingof, Pavel Frenkel, Edward Kazhdan, David Algebraic Geometry High Energy Physics - Theory Analysis of PDEs Functional Analysis Representation Theory We construct analogues of the Hecke operators for the moduli space of G-bundles on a curve X over a local field F with parabolic structures at finitely many points. We conjecture that they define commuting compact normal operators on the Hilbert space of half-densities on this moduli space. In the case F=C, we also conjecture that their joint spectrum is in a natural bijection with the set of opers on X for the Langlands dual group with real monodromy. This may be viewed as an analytic version of the Langlands correspondence for complex curves. Furthermore, we conjecture an explicit formula relating the eigenvalues of the Hecke operators and the global differential operators studied in our previous paper arXiv:1908.09677. Assuming the compactness conjecture, this formula follows from a certain system of differential equations satisfied by the Hecke operators, which we prove in this paper for G=PGL(n). |
| title | Hecke operators and analytic Langlands correspondence for curves over local fields |
| topic | Algebraic Geometry High Energy Physics - Theory Analysis of PDEs Functional Analysis Representation Theory |
| url | https://arxiv.org/abs/2103.01509 |