Principal SO(2n,ℂ)-Bundle Fixed Points over a Compact Riemann Surface

Let X be a compact connected Riemann surface of genus g 2 equipped with a holomorphic involution X , and let G be a semisimple complex Lie group which admits an outer involution .A principal (G, X , )-bundle over X is a pair (E, ), where E is a principal G-bundle over X and :is an automorphism of E which acts as the product by an element of the center of G.In this paper, principal (G, X , )-bundles over X are introduced and the study is particularized to the case of G = SO(2n, C).It is shown that the stability and polystability conditions for a principal (SO(2n, C), X , )bundle coincide with those of the corresponding principal SO(2n, C)-bundle.Finally, the explicit form that a principal (SO(2n, C), X , )-bundle takes is provided, and the stability of these principal (SO(2n, C), X , )-bundles is discussed.