Conjugation of semisimple subgroups over real number fields of bounded degree

Abstract

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