Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 003, 37 pages      arXiv:0809.2605
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Quiver Varieties and Branching

Hiraku Nakajima a, b
a) Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
b) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Received September 15, 2008, in final form January 05, 2009; Published online January 11, 2009

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 R4/Zr correspond to weight spaces of representations of the Langlands dual group Gaff 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).

Key words: quiver variety; geometric Satake correspondence; affine Lie algebra; intersection cohomology.

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