Uhlenbeck spaces via affine Lie algebras

Let $G$ be an almost simple simply connected group over $\text{BC}$, and let ${\text{Bun}}_G^a({\text{BP}}^2,{\text{BP}}^1)$ be the moduli scheme of principal $G$-bundles on the projective plave ${\text{BP}}^2$, of second Chern class $a$, trivialized along a line ${\text{BP}}^1\subset {\text{BP}}^2$. We define the Uhlenbeck compactification ${\text{fU}}_G^a$ of ${\text{Bun}}_G^a({\text{BP}}^2,{\text{BP}}^1)$, which classifies, roughly, pairs $({\text{F}}_G,D)$, where $D$ is a 0-cycle on ${\text{BA}}^2={\text{BP}}^2-{\text{BP}}^1$ of degree $b$, and ${\text{F}}_G$ is a point of ${\text{Bun}}_G^{a-b}({\text{BP}}^2,{\text{BP}}^1)$, for varying $b$. In addition, we calculate the stalks of the Intersection Cohomology sheaf of ${\text{fU}}_G^a$. To do that we give a geometric realization of Kashiwara’s crystals for affine Kac-Moody algebras.