Uhlenbeck Spaces via Affine Lie Algebras

Let G be an almost simple simply connected group over C, and let BunaG(P ,P) be the moduli scheme of principal G-bundles on the projective plave P, of second Chern class a, trivialized along a line P ⊂ P. We define the Uhlenbeck compactification U G of BunaG(P , P), which classifies, roughly, pairs (FG,D), where D is a 0-cycle on A 2 = P − P of degree b, and FG is a point of Bun G (P,P), for varying b. In addition, we calculate the stalks of the Intersection Cohomology sheaf of UaG. To do that we give a geometric realization of Kashiwara’s crystals for affine Kac-Moody algebras.