Eigenforms and graphs of Hecke operators with wild ramification

TL;DR

This paper analyzes the structure of Hecke operators with ramification on moduli of bundles over function fields, providing bounds, formulas, and explicit constructions of eigenforms by reducing to the unramified case.

Contribution

It introduces a combinatorial framework to understand ramified Hecke operators, enabling explicit eigenform construction and dimension formulas in complex ramification scenarios.

Findings

Reduced ramified Hecke operator study to unramified case using combinatorics

Derived bounds and exact formulas for Hecke eigenspaces with ramification

Constructed eigenforms explicitly in the context of $ ext{Bun}_{ ext{PGL}_2}$

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 $\mathrm{Bun}_{\mathrm{PGL}_2}$. Moreover, our methods allow to construct eigenforms explicitly.