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

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 $H$-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, 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.