Hecke Eigensheaf

A Hecke eigensheaf is a sheaf or complex on a moduli stack of bundles whose geometric Hecke transforms factor as the sheaf itself times the local systems associated with a fixed Langlands dual group parameter. The eigenisomorphisms must be compatible with tensor products, permutations of modification points, and fusion when points collide. For a family of dual-group representations , the schematic form of the eigenproperty is

where the modifications take place on the punctured curve and is the local system associated with . The symbol denotes exterior tensor product.