Principal bundles over a smooth real projective curve of genus zero

Abstract

Abstract Let H 0 denote the kernel of the endomorphism, defined by , of the real algebraic group given by the Weil restriction of . Let X be a nondegenerate anisotropic conic in . The principal -bundle over the complexification defined by the ample generator of Pic(), gives a principal H 0-bundle over X through a reduction of structure group. Given any principal G-bundle EG over X, where G is any connected reductive linear algebraic group defined over ℝ, we prove that there is a homomorphism such that EG is isomorphic to the principal G-bundle obtained by extending the structure group of using . The tautological line bundle over the real projective line and the principal -bundle together give a principal -bundle F on , Given any principal G-bundle EG over , where G is any connected reductive linear algebraic group defined over ℝ, we prove that there is a homomorphism such that EG is isomorphic to the principal G-bundle obtained by extending the structure group of F using .