Conjugacy classes in affine Kac–Moody groups and principal G-bundles over elliptic curves

For a simple complex Lie group G the connected components of the moduli space of semistable G -bundles over an elliptic curve are weighted projective spaces or quotients of weighted projective spaces by a finite group action. In this note we will provide a new proof of this result using the invariant theory of affine Kac–Moody groups, in particular the action of the (twisted) Coxeter element on the root system of G .