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 $\mathfrak{g}$.

💡 Research Summary

The paper investigates a distinguished locus (M^{d}) inside the moduli stack of (G)-Higgs bundles on a smooth projective curve (\Sigma), where the centraliser dimension of the Higgs field is constantly equal to an integer (d) larger than the rank of the complex reductive group (G). This locus lies deep inside the singular part of the Hitchin fibration. The authors develop a comprehensive framework that combines the geometry of Lie‑theoretic sheets, non‑abelian gerbes, and a generalized cameral construction to describe both the non‑abelian structure of the Hitchin map on (M^{d}) and its “abelianisation” when possible.

1. Sheets and centraliser stratification.
The set (\mathfrak g_{d}) of Lie algebra elements whose centraliser has dimension (d) decomposes into finitely many irreducible components called sheets. Each sheet (S) is a (G)-stable locally closed subvariety; when (S) is normal (the authors assume non‑singular sheets, which holds for all classical Lie algebras and most exceptional ones) the quotient stack (