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Title: The Jordan–Chevalley decomposition for 𝐺-bundles on elliptic curves
We study the moduli stack of degree 0 0 semistable G G -bundles on an irreducible curve E E of arithmetic genus 1 1 , where G 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 H of G G (the E E -pseudo-Levi subgroups), where each stratum is computed in terms of H 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 E has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra g \mathfrak {g} (respectively, algebraic group G G ). We also provide a Tannakian description of these moduli stacks and use it to show that if E E is not a supersingular elliptic curve, the moduli of framed unipotent bundles on E E are equivariantly isomorphic to the unipotent cone in G G . Finally, we classify the E E -pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples.  more » « less
Award ID(s):
2202363
PAR ID:
10415288
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Representation Theory of the American Mathematical Society
Volume:
26
Issue:
39
ISSN:
1088-4165
Page Range / eLocation ID:
1268 to 1323
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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