Quiver Varieties and Branching

Abstract

Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group G_\aff [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of $G_{\mathrm{cpt}}$-instantons on $\R^4/\Z_r$ correspond to weight spaces of representations of the Langlands dual group G_\aff^\vee at level $r$. When G = \SL(l), the Uhlenbeck compactification is the quiver variety of type \algsl(r)_\aff, and their conjecture follows from the author's earlier result and I. Frenkel's level-rank duality. They further introduce a convolution diagram which conjecturally gives the tensor product multiplicity [Braverman A., Finkelberg M., Private communication, 2008]. In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for G=\SL(l).Comment: 37 page