Galois F4-Bundles, Involutions, and Fixed-Point Subvarieties

Let X be a compact Riemann surface of genus g≥2, σX∈Aut(X) (the group of biholomorphic self-maps of X) an involution, and F4(C) (one of the five exceptional complex simple Lie groups) the fixed-point subgroup of the outer involution σ of the complex simple Lie group E6(C) of type E6. We study Galois triples (E,f,ω): a principal F4(C)-bundle E with an isomorphism f:E→∼σX*E (here σX*E denotes the pull-back bundle, whose fibre over x∈X is the fibre of E over σX(x)) satisfying a cocycle condition, together with a nontrivial finite-order automorphism ω commuting with f. The automorphism ω determines a semisimple element ge∈F4(C) and forces a reduction to the centraliser ZF4(ge), the Levi factor of a proper parabolic subgroup (a proper closed subgroup of F4(C) containing a Borel subgroup, together with its natural reductive quotient, the Levi factor); by Ramanathan’s criterion, E is strictly polystable (built, via such a reduction, from stable bundles of equal slope, and in particular not itself stable). The main theorem proves that E(V26), the vector bundle associated with E via the representation V26 (the smallest nontrivial irreducible representation of F4(C)), with fibre V26, decomposes into k eigenspace sub-bundles, each fixed by σX*, where 2≤k≤25 is determined by the eigenvalue pattern of ge in the 26-dimensional representation V26. Structural corollaries show that the decomposition depends only on [ge], stratify the set of Galois triples by semisimple conjugacy classes, and give the splitting Ad(ι*(E))=E(f4)⊕⨁jEj with each summand preserved by σX*. Since Out(F4(C))={1} (i.e., every automorphism of F4(C) is inner, meaning it is conjugated by an element of F4(C) itself), there are exactly two conjugacy classes of involutions, with fixed-point subgroups KI=(Sp(6,C)×SL(2,C))/μ2 and KII=Spin(9,C); the general theorem specialises in each case in decompositions reflecting the product structure of KI and the spinorial geometry of KII.