In type $A$ it is possible to realise the flag variety $\mathcal{F}$ of $\text{SL}_{n}(\mathbb{C})$ via Nakajima's quiver varieties: consider the vectors $v=(1,2,\ldots, n-1), w=(0,\ldots,0,n)$. Then, we can identify the Nakajima quiver variety $M(v,w) = T^{\ast}\mathcal{F}$, the cotangent bundle of $\mathcal{F}$. Hence, we can realise the flag variety as the zero section.

Question: Is it possible to realise the flag varieties of other symmetric Kac-Moody algebras via quiver varieties? To make things easier, let's restrict to finite type; and, if needs be, even further, to $D_{4}$.

I'm aware that there is not a reasonable description of most (all?) Nakajima quiver varieties outside of type $A$, and was wondering whether this is just type $A$ phenomena (similar in spirit, perhaps, to Ginzburg's Lagrangian construction of representations of $U(\mathfrak{sl}_{n})$, which doesn't work outside of type $A$ for explicit reasons).

Cheers, George

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    $\begingroup$ I think the short answer is almost surely not. I don't think I have a truly killer argument that the other $T^*G/B$ is not a quiver variety, but it just feels all wrong. You would need to find a representation of a Lie group where the image of $U(\mathfrak{g})$ in endomorphisms of the representation was canonically isomorphic to $\mathbb{C}[W]$; I have no idea what that would be. $\endgroup$
    – Ben Webster
    Apr 13, 2013 at 19:14


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