Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields III

Let G be a reductive group over a nonperfect field k. We study rationality problems for Serre's notion of complete reducibility of subgroups of G. In our previous work, we constructed examples of subgroups H of G that are G-completely reducible but not G-completely reducible over k (and vice ve...

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Bibliographic Details
Published inCommunications in algebra Vol. 47; no. 12; pp. 4928 - 4944
Main Author Uchiyama, Tomohiro
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 02.12.2019
Taylor & Francis Ltd
Subjects
Algebraic groups
complete reducibility
conjugacy classes
Fields (mathematics)
geometric invariant theory
Group theory
rationality
spherical buildings
Subgroups
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Summary:Let G be a reductive group over a nonperfect field k. We study rationality problems for Serre's notion of complete reducibility of subgroups of G. In our previous work, we constructed examples of subgroups H of G that are G-completely reducible but not G-completely reducible over k (and vice versa). In this article, we give a theoretical underpinning of those constructions. Then using Geometric Invariant Theory, we obtain a new result on the structure of G(k)-(and G-) orbits in an arbitrary affine G-variety. We discuss several related problems to complement the main results.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2019.1602873