Graphs of Hecke operators in mixed ramification
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914563689021440 |
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| author | Kashyap, Rudrendra Zveryk, Vladyslav |
| author_facet | Kashyap, Rudrendra Zveryk, Vladyslav |
| contents | We study Hecke operators on moduli spaces of ramified $G$-bundles using the combinatorial language of Hecke graphs. We introduce a general notion of $\mathcal H$-ramification in the spirit of parahoric ramification, which depends on a choice of a divisor and subgroups of $G$ at every point of the divisor. Building on our previous work, we prove that, under mild regularity conditions, the action of a Hecke operator in the deep cusp of $\mathrm{Bun}_G$ in a highly complex ramification mimics an action in a much simpler ramification. This reduces the study to a smaller number of cases which, in particular, involve divisors supported at no more than two points. We demonstrate our methods by computing various examples for $G=\mathrm{PGL}_2$ and computing the dimensions of spaces of Hecke eigenforms for generic eigenvalues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13824 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Graphs of Hecke operators in mixed ramification Kashyap, Rudrendra Zveryk, Vladyslav Algebraic Geometry Number Theory Representation Theory We study Hecke operators on moduli spaces of ramified $G$-bundles using the combinatorial language of Hecke graphs. We introduce a general notion of $\mathcal H$-ramification in the spirit of parahoric ramification, which depends on a choice of a divisor and subgroups of $G$ at every point of the divisor. Building on our previous work, we prove that, under mild regularity conditions, the action of a Hecke operator in the deep cusp of $\mathrm{Bun}_G$ in a highly complex ramification mimics an action in a much simpler ramification. This reduces the study to a smaller number of cases which, in particular, involve divisors supported at no more than two points. We demonstrate our methods by computing various examples for $G=\mathrm{PGL}_2$ and computing the dimensions of spaces of Hecke eigenforms for generic eigenvalues. |
| title | Graphs of Hecke operators in mixed ramification |
| topic | Algebraic Geometry Number Theory Representation Theory |
| url | https://arxiv.org/abs/2605.13824 |