Triality and automorphisms of principal bundles moduli spaces

Let X be a compact Riemann surface of genus g >= 2 and let G be the group Spin(8, C) or PSO(8, C). Let tau be a choice of a non-trivial outer automorphism of G of order 3, called triality. Suppose that X is equipped with a holomorphic order 3 automorphism alpha. Then alpha acts on the moduli space M(G) of principal G-bundles over X by taking the pull-back, and tau also acts on M(G), by modifying the action of G on the principal bundles through any automorphism of G of order 3 representing tau. The combination of these two actions gives an automorphism of order 3 of the moduli space M(G). In this work, the notion of alpha-trialitarian G-bundle is introduced to describe the fixed points of the automorphism of M(G) mentioned above. In particular, an equivalent notion of stability and polystability for such bundles is proved, and an explicit construction of them is provided.