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On relative complete reducibility

Attenborough, Christopher and Bate, Michael and Gruchot, Maike and Litterick, Alastair and Roehrle, Gerhard 'On relative complete reducibility.'

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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 $G^n$, 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.

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: math.GR, math.GR, math.RT, 20G15, 14L24
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Elements
Date Deposited: 11 Sep 2019 11:48
Last Modified: 18 Mar 2020 10:15

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