D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras

Details D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras D-modules on the affine Grassmannian and representations of affine Kac-Moody algebras

2003-03-13

arxiv.org/abs/math/0303173v3

Mathematics

Algebraic Geometry

Revised version; to appear in Duke Math. Journal

Scientific paper

Let ${\mathfrak g}$ be a simple Lie algebra. For a level $\kappa$ (thought of as a symmetric ${\mathfrak g}$-invariant form of ${\mathfrak g}$), let $\hat{\mathfrak g}_\kappa$ be the corresponding affine Kac-Moody algebra. Let $Gr_G$ be the affine Grassmannian of ${\mathfrak g}$, and let $D_\kappa(Gr_G)-mod$ be the category of $\kappa$-twisted right D-modules on $Gr_G$. By taking global sections of a D-module, we obtain a functor $\Gamma:D_\kappa(Gr_G)-mod\to {\mathfrak g}_\kappa-mod$. It is known that this functor is exact and faithful when $\kappa$ is negative or irrational. In this paper, we show that the functor $\Gamma$ is exact and faithful also when $\kappa$ is the critical level.

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