Graphs of Hecke operators in mixed ramification

TL;DR

This paper explores Hecke operators on moduli spaces of ramified G-bundles using combinatorial Hecke graphs, introducing a new notion of ramification and simplifying complex cases to more manageable ones.

Contribution

It introduces a general notion of -ramification and shows that Hecke operator actions in complex ramification settings can be reduced to simpler cases.

Findings

Hecke operator actions in deep cusps mimic simpler ramification cases.

Reduction to cases with divisors supported at no more than two points.

Computed dimensions of spaces of Hecke eigenforms for G=PGL_2.

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.