Graphs of Hecke operators in mixed ramification

Rudrendra Kashyap; Vladyslav Zveryk

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arXiv · 2605.13824

Graphs of Hecke operators in mixed ramification

Rudrendra Kashyap·Vladyslav Zveryk

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Abstract

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. We connect the obtained dimension formulas with the known results from the theory of Eisenstein series.

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Rudrendra Kashyap, Vladyslav Zveryk. 2026-05-13. Graphs of Hecke operators in mixed ramification. https://arxiv.org/abs/2605.13824

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