Gaiotto's Lagrangian subvarieties via loop groups

The purpose of this note is to give a simple proof of the fact that a certain substack, defined in [2], of the moduli stack $T^{\ast}Bun_G(\Sigma)$ of Higgs bundles over a curve $\Sigma$, for a connected, simply connected semisimple group $G$, possesses a Lagrangian structure. The substack, roughly speaking, consists of images under the moment map of global sections of principal $G$-bundles over $\Sigma$ twisted by a smooth symplectic variety with a Hamiltonian $G$-action.