Graphs of Hecke operators in mixed ramification

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.

36 pages, 10 figures. Added connection to the theory of Eisenstein series (Section 6), remarks about generalized eigenspaces (Theorem 5.4, Remark 5.5), and more preliminaries