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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2602.00274 |
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| _version_ | 1866914296488787968 |
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| author | Früh, Alexander |
| author_facet | Früh, Alexander |
| contents | 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 $\mathfrak{g}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00274 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The singular Hitchin fibration, cameral data, and representation theory Früh, Alexander Representation Theory Algebraic Geometry 14D20 (Primary) 14D23, 14L35, 17B35 (Secondary) 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 $\mathfrak{g}$. |
| title | The singular Hitchin fibration, cameral data, and representation theory |
| topic | Representation Theory Algebraic Geometry 14D20 (Primary) 14D23, 14L35, 17B35 (Secondary) |
| url | https://arxiv.org/abs/2602.00274 |