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

Abstract

ABSTRACTLet 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.