  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.
