Coherent sheaves with parabolic structure and construction of Hecke eigensheaves for some ramified local systems

The aim of these notes is to generalize Laumon’s construction [18] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article “On the geometric Langlands conjecture” by Frenkel, Gaitsgory and Vilonen [10] carry over to our situation. We show that our sheaves descend to the moduli space of parabolic bundles if the rank is $\le 3$ and that the general case can be deduced form a generalization of the vanishing conjecture of [10].