Galois E6 -Bundles over a Hyperelliptic Algebraic Curve

Let X be a hyperelliptic algebraic curve and let M(E6) be the moduli space of polystable principal E6-bundles over X. Suppose, in addition, that the outer involution σ of E6 acts as the hyperelliptic involution of X. Then, an automorphism of M(E6) is defined which acts by E → σ ∗(E∗ ), where E is a principal E6-bundle over X seen as a vector bundle through the fundamental irreducible 27-dimensional representation of E6. In this paper, Galois E6-bundles over X are defined and related to the fixed points of the above automorphism of M(E6). If P E6 is the centerless group with Lie algebra e6, then Galois P E6-bundles over X are also defined and related to Galois E6-bundles. Finally, a specific expression for a certain family of Galois E6-bundles over X is given and some implications of the study in terms of representations of the fundamental group π1(X) of the base curve are drawn.