Relative complete reducibility and normalised subgroups

TL;DR

This paper extends the concept of complete reducibility in algebraic groups to a relative setting, linking subgroup properties to automorphism group representations, and generalizes key results to Lie algebras and non-algebraically closed fields.

Contribution

It introduces a relative notion of $G$-complete reducibility with respect to a subgroup $K$, and establishes equivalences with automorphism group representations, broadening fundamental results to new contexts.

Findings

Characterization of relative $G$-complete reducibility via automorphism groups

Generalization of classical results from absolute to relative setting

Extension of results to Lie algebras and non-algebraically closed fields

Abstract

We study a relative variant of Serre's notion of $G$-complete reducibility for a reductive algebraic group $G$. We let $K$ be a reductive subgroup of $G$, and consider subgroups of $G$ which normalise the identity component $K^{\circ}$. We show that such a subgroup is relatively $G$-completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ}$ is completely reducible. This allows us to generalise a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$, as well as 'rational' versions over non-algebraically closed fields.