Eigenforms and graphs of Hecke operators with wild ramification
Abstract
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 BunG, 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 BunPGL2. Moreover, our methods allow to construct eigenforms explicitly.
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