The Jordan--Chevalley decomposition for G-bundles on elliptic curves

Dragos Fratila IRMA 7 rue René Descartes, Strasbourg, France Sam Gunningham Department of Mathematics, King’s College London, London, UK Penghui Li YMSC, Tsinghua University, Beijing, China

Algebraic Geometry mathscidoc:2205.45011

2020.7
We study the moduli stack of degree 0 semistable G-bundles on an irreducible curve E of arithmetic genus 1, where G is a connected reductive group. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups H of G (the E-pseudo-Levi subgroups), where each stratum is computed in terms of bundles on H together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framingat a fixed basepoint. In the case where E has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra g (respectively, group G). We also provide a Tannakian description of these moduli stacks and use it to show that if E is an ordinary elliptic curve, the collection of framed unipotent bundles on E is equivariantly isomorphic to the unipotent cone in G. Finally, we classify the E-pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples.
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@inproceedings{dragos2020the,
  title={The Jordan--Chevalley decomposition for G-bundles on elliptic curves},
  author={Dragos Fratila, Sam Gunningham, and Penghui Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220518211416857883272},
  year={2020},
}
Dragos Fratila, Sam Gunningham, and Penghui Li. The Jordan--Chevalley decomposition for G-bundles on elliptic curves. 2020. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220518211416857883272.
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