The singular Hitchin fibration, cameral data, and representation theory
Alexander Früh
TL;DR
This work develops a generalized Hitchin system by restricting to the locus M^d of G-Higgs bundles with fixed centraliser dimension, yielding a non-abelian fibration that is controlled by the geometry of sheets in the Lie algebra. It introduces an abelianisation via cameral data for Dixmier sheets, defines an S-Chevalley base, and constructs a smooth cameral group that realises the abelian part of the fibration, including explicit abelian fibrations in classical groups and for non-quasi-split real forms. The paper provides concrete spectral data descriptions in GL_n and Sp_4, extends cameral descriptions to real Higgs fibrations, and connects the local Hitchin fibration geometry to the orbit method, producing explicit multiplicity relations between adjoint-orbit data and primitive ideals. These results unify geometric, representation-theoretic, and spectral perspectives on Hitchin systems, with applications to real forms and a broader understanding of singular Hitchin fibres. The framework leverages Grothendieck-Springer theory for sheets, Katsylo quotients, and rigidified base stacks to describe both non-abelian fibres and their abelianised counterparts, offering new tools for investigating non-regular Hitchin fibres and their representation-theoretic implications.
Abstract
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 $\mathfrak{g}$.