Quiver Varieties and Geometric Langlands Correspondence for Affine Lie Algebras

Braverman-Finkelberg recently propose the geometric Satake correspondence for the affine Kac-Moody group $G_{\mathrm{aff}}$. They propose that the affine Grassmanian variety for the usual geometric Satake correspondence for the finite dimensional case is replaced by the Uhlenbeck compactification of the framed moduli space of $G_{\mathrm{cpt}}$-instantons on $\mathbf R^4/\mathbf Z_l$. Here the additional parameter $l$ (positive integer) corresponds to the level of representations of the Langlands dual group $G_{\mathrm{aff}}^\vee$. When $G = SL(r)$, the Uhlenbeck compactification is the quiver variety of type $\mathfrak{sl}(l)_{\mathrm{aff}}$, and their conjecture follows from the theory of quiver varieties and I.Frenkel's level-rank duality. In my talks, I will explain Braverman-Finkelberg's conjecture, the relation of quiver varieties to representation theory of affine Lie algebras, and then to that of quantum affine algebras.