Eigenforms and graphs of Hecke operators with wild ramification

Hecke operators on moduli of bundles over a global function field become substantially more complicated in the presence of ramification. We show that far enough in the Harder-Narasimhan cone of ${\mathrm{B}\mathrm{u}\mathrm{n}}_G$, this extra complexity has a simple structure, which allows to reduce most of the study to the unramified case. Using the theory of graphs of Hecke operators, we transform this statement into a combinatorial condition. Utilizing the combinatorial language, we obtain tight bounds, and for generic eigenvalues exact formulas for the dimensions of Hecke eigenspaces with arbitrary ramification for ${\mathrm{B}\mathrm{u}\mathrm{n}}_{{\mathrm{P}\mathrm{G}\mathrm{L}}_2}$. Moreover, our methods allow to construct eigenforms explicitly.