Revisiting the moduli space of semistable G-bundles over elliptic curves

TL;DR

This paper characterizes the moduli space of semistable G-bundles over elliptic curves, showing it is isomorphic to a quotient of a power of the elliptic curve by a Weyl group, extending previous results.

Contribution

It generalizes Laszlo's result to all connected components and provides a purely algebraic geometric proof applicable in any characteristic.

Findings

Moduli space is isomorphic to a power of the elliptic curve modulo a Weyl group.

Generalizes previous descriptions to arbitrary topological types.

Proof works in all characteristics.

Abstract

We show that the moduli space of semistable G-bundles on an elliptic curve for a reductive group G is isomorphic to a power of the elliptic curve modulo a certain Weyl group which depend on the topological type of the bundle. This generalises a result of Laszlo to arbitrary connected components and recovers the global description of the moduli space due to Friedman--Morgan--Witten and Schweigert. The proof is entirely in the realm of algebraic geometry and works in arbitrary characteristic.