Abstract
Let 𝑘 be a non-perfect 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 the first example of a connected non-abelian 𝑘-subgroup 𝐻 of 𝐺 that is 𝐺-completely reducible but not 𝐺-completely reducible over 𝑘, and the first example of a connected non-abelian 𝑘-subgroup
H
′
H^{\prime}
of 𝐺 that is 𝐺-completely reducible over 𝑘 but not 𝐺-completely reducible.
This is new: all previously known such examples are for finite (or non-connected) 𝐻 and
H
′
H^{\prime}
only.