Principal bundles over a smooth real projective curve of genus zero

Abstract

Let H₀ denote the kernel of the endomorphism, defined by z → (z/z^-)², of the real algebraic group given by the Weil restriction of C*. Let X be a nondegenerate anisotropic conic in P²_R. The principal C*-bundle over the complexification X_C, defined by the ample generator of Pic(X_C), gives a principal H₀-bundle F_H0 over X through a reduction of structure group. Given any principal G-bundle E_G over X, where G is any connected reductive linear algebraic group defined over R, we prove that there is a homomorphism ρ : H₀ → G such that E_G is isomorphic to the principal G-bundle obtained by extending the structure group of F_H0 using ρ. The tautological line bundle over the real projective line P¹ _R, and the principal Z/2Z- bundle P¹ _C → P¹ _R, together give a principal G_m × (Z/2Z)-bundle F on P¹ _R. Given any principal G-bundle E_G over P¹_R, where G is any connected reductive linear algebraic group defined over R, we prove that there is a homomorphism ρ : G_m × (Z/2Z) → G such that E_G is isomorphic to the principal G-bundle obtained by extending the structure group of F using ρ.