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

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.