The moduli space for a flat G-bundle over the two-torus is completely determined by its holonomy representation. When G is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product of two tori mod the action of the Weyl group, or equivalently to the conjugacy classes of commuting pairs of elements in G. Since the component group for a non-simply connected group is given by some finite dimensional subgroup in the centralizer of an n-tuple, we use diagram automorphisms of the extended Dynkin diagram to prove properties of centralizers of pairs of elements in G.
Kris A Nairn. "Centralizers of Commuting Elements in Compact Lie Groups." J. Gen. Lie Theory Appl. 10 (1) 1 - 5, 2016. https://doi.org/10.4172/1736-4337.1000246