Complete reducibility and separable field extensions

TL;DR

This paper investigates how the concept of G-complete reducibility for connected reductive algebraic groups behaves under separable field extensions, utilizing recent advances like the Tits Centre Conjecture.

Contribution

It provides a definitive answer to Serre's question about the stability of G-complete reducibility under separable extensions, leveraging the Tits Centre Conjecture.

Findings

G-complete reducibility is preserved under separable field extensions

Utilizes the Tits Centre Conjecture in the proof

Clarifies the behavior of reductive groups over different fields

Abstract

Let G be a connected reductive linear algebraic group. The aim of this note is to settle a question of J-P. Serre concerning the behaviour of his notion of G-complete reducibility under separable field extensions. Part of our proof relies on the recently established Tits Centre Conjecture for the spherical building of the reductive group G.