Uhlenbeck Spaces for ${\mathbb{A}}^2$ and Affine Lie Algebra ${\widehat{\mathfrak{sl}}}_n$

Abstract

We introduce an Uhlenbeck closure of the space of based maps from projective line to the Kashiwara flag scheme of an untwisted affine Lie algebra. For the algebra ${\widehat{\mathfrak{sl}}}_n$ this space of based maps is isomorphic to the moduli space of locally free parabolic sheaves on $P^1\times P^1$ trivialized at infinity. The Uhlenbeck closure admits a resolution of singularities: the moduli space of torsion free parabolic sheaves on $P^1\times P^1$ trivialized at infinity. We compute the Intersection Cohomology sheaf of the Uhlenbeck space using this resolution of singularities. The moduli spaces of parabolic sheaves of various degrees are connected by certain Hecke correspondences. We prove that these correspondences define an action of ${\widehat{\mathfrak{sl}}}_n$ in the cohomology of the above moduli spaces.