On relative complete reducibility
Abstract
Let be a reductive subgroup of a reductive group over an algebraically closed field . The notion of relative complete reducibility, introduced in previous work of Bate-Martin-Roehrle-Tange, gives a purely algebraic description of the closed -orbits in , where acts by simultaneous conjugation on -tuples of elements from . This extends work of Richardson and is also a natural generalization of Serre's notion of -complete reducibility. In this paper we revisit this idea, giving a characterization of relative -complete reducibility which directly generalizes equivalent formulations of -complete reducibility. If the ambient group 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.
Cite
@article{arxiv.1806.03067,
title = {On relative complete reducibility},
author = {Christopher Attenborough and Michael Bate and Maike Gruchot and Alastair Litterick and Gerhard Roehrle},
journal= {arXiv preprint arXiv:1806.03067},
year = {2019}
}
Comments
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