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.

MSC:

 14D20 Algebraic moduli problems, moduli of vector bundles 53D12 Lagrangian submanifolds; Maslov index
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