To define a Nakajima quiver variety associated to a quiver $Q = (Q_0,Q_1)$ (vertices and arrows), one first doubles it to $Q^\heartsuit$ by attaching an extra vertex to every old vertex in $Q_0$. Then a weight space of a representation should arise as the top homology of a moduli space ${\mathcal M}(v,w)$ of representations of $Q^\heartsuit$, where $v$ is the dimension vector on the original vertices and $w$ on the new vertices. (I am following Ginzburg's lectures on this topic.)

Am I right in thinking that the relevant weight space is the $\sum_i w_i \omega_i - \sum_i v_i \alpha_i$ of the $\sum_i w_i \omega_i$ representation, where the $(\omega_i)$ are the fundamental weights and the $\alpha_i$ are the simple roots?

**Edit:**I tracked down this formula in Proposition 6.9 of this.~~If so, can I~~Can I therefore think of the second set of vertices as Langlands dual to the first set, in any useful way?In particular, has anyone looked into attaching the second set of vertices to one another, to give a second copy of $Q$? (I really want to say "the Langlands dual of $Q$" but I guess we're automatically in the simply-laced case.)

amoduli space". – Allen Knutson Dec 1 '10 at 5:31