Higgs bundles, abelian gerbes and cameral data

We study the Hitchin map for $G_{\mathbb{R}}$-Higgs bundles on a smooth curve, where $G_{\mathbb{R}}$ is a quasi-split real form of a complex reductive algebraic group $G$. By looking at the moduli stack of regular $G_{\mathbb{R}}$-Higgs bundles, we prove it induces a banded gerbe structure on a slightly larger stack, whose band is given by sheaves of tori. This characterization yields a cocyclic description of the fibres of the corresponding Hitchin map by means of cameral data. According to this, fibres of the Hitchin map are categories of principal torus bundles on the cameral cover. The corresponding points inside the stack of $G$-Higgs bundles are contained in the substack of points fixed by an involution induced by the Cartan involution of $G_{\mathbb{R}}$. We determine this substack of fixed points and prove that stable points are in correspondence with stable $G_{\mathbb{R}}$-Higgs bundles.