Uhlenbeck spaces via affine lie algebras

Abstract

Let G be an almost simple simply connected group over ℂ, and let BunGa (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 UGa of BunGa (ℙ2, ℙ1), which classifies, roughly, pairs (ℱG, D), where D is a 0-cycle on A2 = P2 - P1 of degree b, and ℱG is a point of BunGa−b (ℙ2, ℙ1), for varying b. In addition, we calculate the stalks of the Intersection Cohomology sheaf of UGa. To do that we give a geometric realization of Kashiwara’s crystals for affine Kac-Moody algebras.