Quiver Varieties and Branching

Abstract

Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group $G_\aff$ [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of $G_{\mathrm{cpt}}$-instantons on ${\mathbb{R}}^4/{\mathbb{Z}}_r$ 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 $\algsl(r)_\aff$, 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 $G=\SL(l)$.

37 pages

References in corpus (3)

• Quiver varieties and tensor products
• Geometric construction of representations of affine algebras
• Instantons and affine algebras I: The Hilbert scheme and vertex operators

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• Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras
• Examples of quantum cluster algebras associated to partial flag varieties
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• Dynamical Weyl groups and equivariant cohomology of transversal slices in affine Grassmannians
• Eight dimensional physics and the Langlands program - A short note