The global nilpotent variety is Lagrangian

Abstract

The purpose of this paper is to present a short elementary proof of a theorem due to G. Faltings and G. Laumon, which says that the global nilpotent cone is a Lagrangian substack in the cotangent bundle of the moduli space of G-bundles on a complex compact curve. This result plays a crucial role in the geometric Langlands program (see [BD]) since it insures that the $\mathscr{D}$-modules on the moduli space of G-bundles whose characteristic variety is contained in the global nilpotent cone are automatically holonomic and, in particular, have finite length.