Uhlenbeck spaces via affine lie algebras

Let G be an almost simple simply connected group over ℂ, and let Bun_G ^a (ℙ², ℙ¹) be the moduli scheme of principalG-bundles on the projective plane ℙ², of second Chern class a, trivialized along a line ℙ¹ ⊂ ℙ². We define the Uhlenbeck compactification U_G ^a of Bun_G ^a (ℙ², ℙ¹), which classifies, roughly, pairs (ℱ_G, D), where D is a 0-cycle on A² = P² - P¹ of degree b, and ℱ_G is a point of Bun_G ^a−b (ℙ², ℙ¹), for varying b. In addition, we calculate the stalks of the Intersection Cohomology sheaf of U_G ^a. To do that we give a geometric realization of Kashiwara’s crystals for affine Kac-Moody algebras.