The singular Hitchin fibration, cameral data, and representation theory

For a complex reductive group $G$, we consider the locus $M^d$ in the moduli stack of $G$-Higgs bundles on which the centraliser dimension of the Higgs field takes a constant value $d> rk(G)$. We describe a non-abelian structure for the Hitchin fibration on $M^d$, under mild conditions on the geometry of the centraliser level set $\mathfrak{g}_d$ in the Lie algebra. If $G$ is a classical group, we also show that the restriction of the Hitchin map to the locus of generically semisimple Higgs bundles in $M^d$ factors through an abelian fibration. The abelianised fibres can be described using a generalisation of the cameral data of Donagi and Gaitsgory. We apply these constructions to $G_\mathbb{R}$-Hitchin fibrations for real forms $G_\mathbb{R}$. In particular we give a cameral description for an abelianisation of the $G_\mathbb{R}$-Hitchin fibration, which extends the known description in the quasi-split case. We determine this explicitly in the examples $G_\mathbb{R} = SU(p,q)$ and $G_{\mathbb{R}} = SO^*(4m+2)$. Our local results also give a connection between the geometry of the Hitchin fibration on $M^d$ and the representation theory of the Lie algebra $\mathfrak{g}$, via the orbit method. As a corollary, we determine an explicit asymptotic relationship between two notions of multiplicity, one attached to an adjoint orbit in $\mathfrak{g}$ and one attached to a primitive ideal of the universal enveloping algebra of $\mathfrak{g}$.