# 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.

nakajima@math.kyoto-u.ac.jp