Lusztig sheaves, local systems, and the extended Springer resolution

Abstract

The Springer resolution of the nilpotent cone is used to give a geometric construction of the Springer correspondence by applying tools in the derived category of perverse sheaves, particularly the decomposition theorem. In this paper we continue earlier work of the authors on the extended Springer resolution, a map introduced by the first author that factors through the Springer resolution. Our first main result shows that applying the decomposition theorem to this map yields a sum of Lusztig sheaves in all Lie types, each corresponding to a triple of so-called minimal cuspidal datum. We also study the simple perverse sheaves that occur as summands of the resulting sheaf. Our second main result proves that every simple perverse sheaf corresponding to orbits that are Levi-regular must occur, generalizing earlier work in the type A setting to all Lie types.