On relative complete reducibility

TL;DR

This paper explores the concept of relative complete reducibility in algebraic groups, providing new characterizations, extending previous results, and connecting to representation theory, thereby broadening the understanding of group actions and orbit structures.

Contribution

It offers a new characterization of relative G-complete reducibility that generalizes existing formulations and extends prior results, including representation-theoretic criteria for linear groups.

Findings

Provides a characterization of relative G-complete reducibility.

Extends and generalizes previous results by Bate-Martin-Roehrle-Tange.

Connects the concept to representation theory in linear groups.

Abstract

Let $K$ be a reductive subgroup of a reductive group $G$ over an algebraically closed field $k$. The notion of relative complete reducibility, introduced in previous work of Bate-Martin-Roehrle-Tange, gives a purely algebraic description of the closed $K$-orbits in $G^n$, where $K$ acts by simultaneous conjugation on $n$-tuples of elements from $G$. This extends work of Richardson and is also a natural generalization of Serre's notion of $G$-complete reducibility. In this paper we revisit this idea, giving a characterization of relative $G$-complete reducibility which directly generalizes equivalent formulations of $G$-complete reducibility. If the ambient group $G$ is a general linear group, this characterization yields representation-theoretic criteria. Along the way, we extend and generalize several results from the aforementioned work of Bate-Martin-Roehrle-Tange.