The global nilpotent variety is Lagrangian. (English) Zbl 1116.14007

Summary: The purpose of this paper is to present a short elementary proof of a theorem due to G. Faltings [J. Algebr. Geom. 2, 507–568 (1993; Zbl 0790.14019)] and G. Laumon [Duke Math. J. 57, 647–671 (1988; Zbl 0688.14023)], 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 [A. A. Beilinson and V. G. Drinfeld, in: Algebraic and geometric methods in mathematical physics. Proc. 1st Ukrainian-French-Romanian summer school, Kaciveli, Ukraine 1993. Math. Phys. Stud. 19, 3–7 (1996; Zbl 0864.14007)] since it insures that the \(\mathcal D\)-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.


14D20 Algebraic moduli problems, moduli of vector bundles
53D12 Lagrangian submanifolds; Maslov index
Full Text: DOI arXiv