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On relative complete reducibility

Attenborough, Christopher and Bate, Michael and Gruchot, Maike and Litterick, Alastair and Roehrle, Gerhard (2020) 'On relative complete reducibility.' Quarterly Journal of Mathematics, 71 (1). pp. 321-334. ISSN 0033-5606

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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 Gn, 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.

Item Type: Article
Additional Information: 10 pages; v2 15 pages; substantially revised and expanded version: most results are generalized from the case of a general linear group to an arbitrary connected reductive algebraic group. List of authors expanded. To appear in Quarterly Journal of Mathematics
Uncontrolled Keywords: reductive algebraic groups; G-complete reducibility; closed G-orbits
Divisions: Faculty of Science and Health
Faculty of Science and Health > Mathematical Sciences, Department of
SWORD Depositor: Elements
Depositing User: Elements
Date Deposited: 11 Sep 2019 11:48
Last Modified: 06 Jan 2022 14:04

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