# Complete reducibility: Variations on a theme of Serre

Gruchot, Maike and Litterick, Alastair and Roehrle, Gerhard (2022) 'Complete reducibility: Variations on a theme of Serre.' Manuscripta Mathematica, 168 (3-4). pp. 439-451. ISSN 0025-2611

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Gruchot2021_Article_CompleteReducibilityVariations.pdf - Published Version
In this note, we unify and extend various concepts in the area of $G$-complete reducibility, where $G$ is a reductive algebraic group. By results of Serre and Bate--Martin--R\"{o}hrle, the usual notion of $G$-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of $G$. We show that other variations of this notion, such as relative complete reducibility and $\sigma$-complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.