%0 Journal Article %1 nakajima_quiver_2008 %A Nakajima, Hiraku %D 2008 %J 0809.2605 %K Mathematical Theory {Algebra,Quiver} {Physics,Quantum} {varieties,Representation} %T Quiver varieties and branching %U http://arxiv.org/abs/0809.2605 %X Braverman-Finkelberg arXiv:0711.2083 recently propose the geometric Satake correspondence for the affine Kac-Moody group \$G\_\textbackslashaff\$. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of \$G\_\textbackslashmathrmcpt\$-instantons on \$\textbackslashRˆ4/\textbackslashZ\_r\$ correspond to weight spaces of representations of the Langlands dual group \$G\_\textbackslashaffˆ\textbackslashvee\$ at level \$r\$. When \$G = \textbackslashSL(l)\$, the Uhlenbeck compactification is the quiver variety of type \$\textbackslashalgsl(r)\_\textbackslashaff\$, 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 (paper in preparation). In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for \$G=\textbackslashSL(l)\$.

@article{nakajima_quiver_2008, abstract = {{Braverman-Finkelberg} {arXiv:0711.2083} recently propose the geometric Satake correspondence for the affine {Kac-Moody} group {\$G\_{\textbackslash}aff\$.} They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of {\$G\_{{\textbackslash}mathrm{cpt}}\$-instantons} on {\${\textbackslash}R{\textasciicircum}4/{\textbackslash}Z\_r\$} correspond to weight spaces of representations of the Langlands dual group {\$G\_{\textbackslash}aff{\textasciicircum}{\textbackslash}vee\$} at level \$r\$. When {\$G} = {{\textbackslash}SL(l)\$,} the Uhlenbeck compactification is the quiver variety of type \${\textbackslash}algsl(r)\_{\textbackslash}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 (paper in preparation). In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for {\$G={\textbackslash}SL(l)\$.}}, added-at = {2009-05-11T21:36:02.000+0200}, author = {Nakajima, Hiraku}, biburl = {https://www.bibsonomy.org/bibtex/278a67021af621ef554dd2f67e994de02/tbraden}, interhash = {2faad178e8c7217c9f8d91a11c32308f}, intrahash = {78a67021af621ef554dd2f67e994de02}, journal = {0809.2605}, keywords = {Mathematical Theory {Algebra,Quiver} {Physics,Quantum} {varieties,Representation}}, month = {September}, timestamp = {2009-05-11T21:36:02.000+0200}, title = {Quiver varieties and branching}, url = {http://arxiv.org/abs/0809.2605}, year = 2008 }