Quiver Varieties and Branching

Braverman-Finkelberg [4] recently propose the geometric Satake correspondence for the affine Kac-Moody group Gaff . They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of Gcpt-instantons on R /Zr correspond to weight spaces of representations of the Langlands dual group G∨aff 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 [5]. In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for G = SL(l).