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

Frăţilă, Dragoş; Gunningham, Sam; Li, Penghui

doi:10.1090/ert/631

The Jordan–Chevalley decomposition for $G$-bundles on elliptic curves
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by Dragoş Frăţilă, Sam Gunningham and Penghui Li

Represent. Theory 26 (2022), 1268-1323

DOI: https://doi.org/10.1090/ert/631

Published electronically: December 21, 2022

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Abstract:

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 in arbitrary characteristic. 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 $H$-bundles 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 framing at 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 $\mathfrak {g}$ (respectively, algebraic group $G$).

We also provide a Tannakian description of these moduli stacks and use it to show that if $E$ is not a supersingular elliptic curve, the moduli of framed unipotent bundles on $E$ are 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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