Quiver Varieties and Branching

Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group Gaff [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of G(cpt)-instantons on R(4)/Z(r) correspond to weight spaces of representations of the Langlands dual group G(aff)(V) at level r. When G = SL(l), the Uhlenbeck compactification is the quiver variety of type sl(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).