Hecke eigen-functions for curves over local non-archimedian fields

We give a survey of some recent constrictions and results concerning Hecke operators for the moduli space of G-bundles on a smooth projective curve X defined over a local non-archimedian field F (possibly with level structures). These operators are somewhat analogous to the usual Hecke operators in the case when F is a finite field but describing their common eigen-values is a much more difficult task. We'll discuss on what space the Hecke operators act, formulate some general conjectures about eigen-functions and consider some examples. In the end (if time permits) I would like to discuss the example when X is of genus 0, and we study bundles with triviliazation at two points. This example is closely related to  the usual representation theory of G(F).