The global nilpotent variety is Lagrangian

The purpose of this note is to present a short elementary proof of a theorem due to Faltings and Laumon, saying 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, due to Beilinson-Drinfeld, since it insures that the D-modules on the moduli space of G-bundles whose characteristic variety is contained in the global nilpotent cone are automatically holonomic, hence, e.g. have finite length.Comment: LaTeX, 9pp. Final version, to appear in Duke Math.

A Tannakian classification of torsors on the projective line

In this small note we present a Tannakian proof of the theorem of Grothendieck-Harder on the classification of torsors under a reductive group on the projective line over a field.Comment: 13 pages; any comments or hints to existing literature welcom

Reduction theory for a rational function field

Let $G$ be a split reductive group over a finite field \Fq. Let F=\Fq(t) and let \A denote the ad\`eles of $F$. We show that every double coset in G(F)\bsl G(\A)/ K has a representative in a maximal split torus of $G$. Here $K$ is the set of integral ad\`elic points of $G$. When $G$ ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.Comment: 10 page

Finiteness Properties of Chevalley Groups over the Laurent Polynomial Ring over a Finite Field

We show that if G is a Chevalley group of rank n and F_q[t,t^{-1}] is the ring of Laurent polynomials over a finite field, then G(F_q[t,t^{-1}]) is of type F_{2n-1}. This bound is optimal because it is known -- and we show again -- that the group is not of type F_{2n}.Comment: 36 pages, 4 figure

Three-point Lie algebras and Grothendieck's dessins d'enfants

We define and classify the analogues of the affine Kac-Moody Lie algebras for the ring corresponding to the complex projective line minus three points. The classification is given in terms of Grothendieck's dessins d'enfants. We also study the question of conjugacy of Cartan subalgebras for these algebras.Comment: 16 page