In:
The Quarterly Journal of Mathematics, Oxford University Press (OUP), Vol. 71, No. 1 ( 2020-03-13), p. 321-334
Abstract:
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] .
Type of Medium:
Online Resource
ISSN:
0033-5606
,
1464-3847
DOI:
10.1093/qmathj/haz044
Language:
English
Publisher:
Oxford University Press (OUP)
Publication Date:
2020
detail.hit.zdb_id:
1476942-6
SSG:
17,1
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