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 ${\mathrm{Bun}}_G$, 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 ${\mathrm{Bun}}_{{\mathrm{PGL}}_2}$. We compare these formulas to the known results in the theory of Eisenstein series. Moreover, our methods allow to construct eigenforms explicitly.

48 pages, 8 figures. Added connection to Eisenstein series (Section 4.7) and explicit bases of generalized eigenspaces in terms of Eisenstein series (Section 4.5)