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ON RELATIVE COMPLETE REDUCIBILITY

Attenborough, C and Bate, M and Gruchot, M and Litterick, A and Röhrle, G (2020) 'ON RELATIVE COMPLETE REDUCIBILITY.' Quarterly Journal of Mathematics, 71 (1). 321 - 334. ISSN 0033-5606

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Abstract

© 2020 The author(s) 2020. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com. 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 [M. Bate, B. Martin, G. Röhrle, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z.269 (2011), no. 1, 809-832], 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 [M. Bate, B. Martin, G. Röhrle, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z.269 (2011), no. 1, 809-832].

Item Type: Article
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Elements
Date Deposited: 11 Sep 2019 11:48
Last Modified: 12 May 2020 21:15
URI: http://repository.essex.ac.uk/id/eprint/25324

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