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SIGMA 5 (2009), 003, 37 pages arXiv:0809.2605
https://doi.org/10.3842/SIGMA.2009.003
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
Abstract
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 GaffÚ
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.
pdf (563 kb)
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