Graphs of Hecke operators in mixed ramification

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 ${\mathrm{B}\mathrm{u}\mathrm{n}}_G$ 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 $G={\mathrm{P}\mathrm{G}\mathrm{L}}_2$ and computing the dimensions of spaces of Hecke eigenforms for generic eigenvalues.