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

ABSTRACT

Let k be a separably closed field. Let G be a reductive algebraic k-group. We study Serre’s notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We show that a regular reductive k-subgroup of G is G-completely reducible over k. We present examples where the number of overgroups of irreducible subgroups and the number of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite.

KEYWORDS:

• Algebraic groups
• complete reducibility
• pseudo-reductivity
• spherical buildings

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20G15
• 20G05
• 14L24
• 20E42

Acknowledgments

The author would like to thank Benjamin Martin, Don Taylor, Philippe Gille, and Brian Conrad for helpful discussions. He is also grateful for detailed comments from an anonymous referee.