If $G$ is a compact Lie group, the moduli space of orbits of the coadjoint action of $G$ on $\mathfrak{g}^{*}$ (the dual of the Lie algebra) is given by $\mathfrak{h} / W$ where $\frak{h}$ is a Cartan subalgebra of $\mathfrak{g}$ and $W$ is the Weyl group (for $SU(N)$, it is simply saying that up to conjugation by $SU(N)$, an antihermitian matrix is characterized by its eigenvalues up to permutation).

Gorsky and Nekasov consider an infinite dimensional version of this story, replacing $G$ by the loop group of $G$, i.e. the group of maps from the circle to $G$. The moduli space of the orbits becomes the space of maps from the circle to $\mathfrak{h}/W$.