2015/12/15 by Tomohiro Uchiyama, Uchiyama, Tomohiro
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1512.04616
openalex publication_date 2015/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be a separably closed field. Let G be a reductive algebraic k-group. In this paper, 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 also show that a regular reductive k-subgroup of G is G-completely reducible over k. Various open problems concerning complete reducibility are discussed. 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.