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

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.