%0 Journal Article
%T Quiver Varieties and Branching
%A Hiraku Nakajima
%J Symmetry, Integrability and Geometry : Methods and Applications
%D 2009
%I National Academy of Science of Ukraine
%X 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 Gcpt-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 $sl(r)_{aff}$, 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 $G = SL(l)$.
%K quiver variety
%K geometric Satake correspondence
%K affine Lie algebra
%K intersection cohomology
%U http://dx.doi.org/10.3842/SIGMA.2009.003