PRINCIPAL BUNDLES OVER A SMOOTH REAL PROJECTIVE CURVE OF GENUS ZERO

Abstract. Let H0 denote the kernel of the endomorphism, defined by z 7− → (z/z)2, of the real algebraic group given by the Weil restriction of C∗. Let X be a nondegenerate anisotropic conic in P2R. The principal C∗–bundle over the complexification XC, defined by the ample generator of Pic(XC), gives a principal H0–bundle FH0 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 ρ: H0 − → G such that EG is isomorphic to the principal G–bundle obtained by extending the structure group of FH0 using ρ. The tautological line bundle over the real projective line P1R, and the principal Z/2Z– bundle P1C − → P1R, together give a principal Gm × (Z/2Z)–bundle F on P1R. Given any principal G–bundle EG over P1R, where G is any connected reductive linear algebraic group defined over R, we prove that there is a homomorphism ρ: Gm × (Z/2Z) − → G such that EG is isomorphic to the principalG–bundle obtained by extending the structure group of F using ρ. 1