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

ABSTRACT

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.

KEYWORDS:

• Algebraic groups
• complete reducibility
• geometric invariant theory
• rationality
• spherical buildings

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20G15
• 20G05
• 14L24
• 20E42