Complete reducibility, Külshammer's question, conjugacy classes: A D4 example

Let k be a nonperfect separably closed field. Let G be a connected reductive algebraic group defined over k. We study rationality problems for Serre's notion of complete reducibility of subgroups of G. In particular, we present a new example of subgroup H of G of type D4 in characteristic 2 suc...

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Bibliographic Details
Published in:Communications in algebra 2018-03, Vol.46 (3), p.1333
Main Author: Uchiyama, Tomohiro
Format: Article
Language:eng ; jpn
Subjects:
Subgroups
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Summary:Let k be a nonperfect separably closed field. Let G be a connected reductive algebraic group defined over k. We study rationality problems for Serre's notion of complete reducibility of subgroups of G. In particular, we present a new example of subgroup H of G of type D4 in characteristic 2 such that H is G-completely reducible but not G-completely reducible over k (or vice versa). This is new: all known such examples are for G of exceptional type. We also find a new counterexample for Külshammer's question on representations of finite groups for G of type D4. A problem concerning the number of conjugacy classes is also considered. The notion of nonseparable subgroups plays a crucial role in all our constructions.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2017.1346106