Quiver Varieties and Branching

Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac–Moody group $G_\mathrm{aff}$ [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 $\mathbb R^4/\mathbb Z_r$ correspond to weight spaces of representations of the Langlands dual group $G_{\mathrm{aff}}^\vee$ at level $r$. When $G=\operatorname{SL}(l)$, the Uhlenbeck compactification is the quiver variety of type $\mathfrak{sl}(r)_{\mathrm{aff}}$, 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 $G=\operatorname{SL}(l)$.