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ON RELATIVE COMPLETE REDUCIBILITY
Quarterly Journal of Mathematics ( IF 0.704 ) Pub Date : 2020-02-05 , DOI: 10.1093/qmathj/haz044
Christopher Attenborough; Michael Bate; Maike Gruchot; Alastair Litterick; Gerhard Röhrle

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 |$G^n$|⁠, 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 [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].
更新日期:2020-04-17

 

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