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 $\mathrm{ Bun}_G$, this extra complexity has a simple structure, which allows to reduce the main 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 $\mathrm{ Bun}_{ \mathrm{ PGL}_2}$. Moreover, our methods allow to construct eigenforms explicitly.
Categories: math.AG math.NT math.RT