2018/10/29 by Maike Gruchot, Gruchot, Maike, Alastair Litterick +3
Mathematics · #14L24 #20G15 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1810.12096
openalex publication_date 2018/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∘. 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∘ 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.