Quiver Varieties and Branching

TL;DR

This paper explores the geometric representation theory of affine Kac-Moody groups via quiver varieties, confirming a conjecture relating intersection cohomology and weight spaces for special cases.

Contribution

It develops the theory of branching in quiver varieties and verifies a conjecture connecting intersection cohomology sheaves to representation weight spaces for SL(l).

Findings

Confirmed the conjecture for G=SL(l).

Developed the theory for branching in quiver varieties.

Connected geometric and algebraic structures in affine Kac-Moody representations.

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 $\R^4/\Z_r$ 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 $\algsl(r)_\aff$, 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 $G=\SL(l)$.