Algebraic Families of Groups and Commuting Involutions

Abstract

Let $G$ be a complex affine algebraic group, and let $σ_1$ and $σ_2$ be commuting anti-holomorphic involutions of $G$. We construct an algebraic family of algebraic groups over the complex projective line and a real structure on the family that interpolates between the real forms $G^{σ_1}$ and $G^{σ_2}$.