Conjugation of semisimple subgroups over real number fields of bounded degree

TL;DR

This paper proves that conjugate semisimple subgroups over algebraic closures or real numbers are actually conjugate over bounded finite extensions of the base field, with bounds independent of the specific subgroups.

Contribution

It establishes uniform bounds on the degree of field extensions needed for conjugacy of semisimple subgroups over various fields, including real number fields.

Findings

Conjugate semisimple subgroups over algebraic closures are conjugate over bounded degree extensions.

Over real number fields, conjugate subgroups are conjugate over bounded real extensions.

Bounds are independent of the specific subgroups involved.

Abstract

Let $G$ be a linear algebraic group over a field $k$ of characteristic 0. We show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over an algebraic closure of $k$ are actually conjugate over a finite field extension of $k$ of degree bounded independently of the subgroups. Moreover, if $k$ is a real number field, we show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over the field of real numbers $\mathbb{R}$ are actually conjugate over a finite real extension of $k$ of degree bounded independently of the subgroups.