Conjugacy classes in affine Kac-Moody groups and principal \(G\)-bundles over elliptic curves (Q734750)

The paper under review presents a new proof for the result of \textit{Y. Laszlo} [Ann. Inst. Fourier 48, No.~2, 413--424 (1998; Zbl 0901.14019)] and \textit{R. Friedman} and \textit{J. W. Morgan} [J. Differ. Geom. 56, No.~2, 301--379 (2000; Zbl 1033.14016)] to the effect that if \(G\) is a connected simple complex algebraic group, then the connected components of the moduli space of semistable \(G\)-bundles over an elliptic curve are weighted projective spaces or quotients of such spaces by the action of a finite group. The present proof of this fact relies on invariant theory for affine Kac-Moody groups. More specifically, if \(\widetilde{G}\) denotes the holomorphic affine Kac-Moody group associated with \(G\), one provides a detailed description of the action of the Coxeter element of \(\widetilde{G}\) on the root system of \(G\), which also shows that the holomorphic principal bundle of the Coxeter element is minimally unstable.