Algebraic Families of Groups and Commuting Involutions

TL;DR

This paper constructs an algebraic family of groups over the complex projective line that interpolates between two commuting real forms of a complex algebraic group, providing a new framework for understanding their relationships.

Contribution

It introduces a novel algebraic family of groups parametrized by the complex projective line that connects two commuting real forms of a complex algebraic group.

Findings

Constructed an algebraic family over the complex projective line.

Established a real structure interpolating between two real forms.

Provides a new perspective on the relationship between commuting involutions.

Abstract

Let $G$ be a complex affine algebraic group, and let $\sigma_1$ and $\sigma_2$ 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 $G^{\sigma_1}$ and $G^{\sigma_2}$.