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 aWe 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. much is known about complete reducibility over an arbitrary k except a few general results and important examples in [3], [6], [7, Sec. 7], [8], [38], [36, Thm. 1.8], [37, Sec. 4]. Let k s be a separable closure of k. Recall that if k is perfect, we have k s = k. The following result [6, Thm. 1.1] shows that if k is perfect and G is connected, most results in this paper just reduce to the algebraically closed case. Proposition 1.2. Let k be a field. Let G be connected. Then a k-subgroup H of G is G-cr over k if and only if H is G-cr over k s .

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

“…One can obtain more examples satisfying Theorem 1.2 using nonseparable subgroups in [35,Sec. 3,4,5] for G = E 6 , E 7 , and E 8 ; see [36].…”

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

“…Note that for a subgroup H of G, N G (H)(k) induces an automorphism group of ∆(G) stabilizing ∆(G) H . Thus, combining the center conjecture with Proposition 4.2 we obtain 3 was an essential tool to prove various theoretical results on complete reducibility over nonperfect k in [33] and [36]. We have asked the following in [33,Rem.…”

“…Throughout, G is always a connected reductive k-group. In this paper, we continue our study of rationality problems for complete reducibility of subgroups of G [36], [33]. By a subgroup of G we mean a (possibly non-k-defined) closed subgroup of G. Following Serre [25,Sec.…”

“…We say that a subgroup H of G is G-cr if it is G-cr over k. Not much is known on complete reducibility over k (especially for nonperfect k) except a few theoretical results and important examples; see [3,Sec. 5], [1], [36], [33]. In [35, Now we move on to a problem with a slightly different flavor.…”

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields IV: An $F_4$ example

“…Throughout, G is always a connected reductive k-group (unless stated otherwise). In this paper, we continue the investigation of rationality problems for complete reducibility of subgroups of G; see [29], [30], [32]. By a subgroup of G we mean a (possibly non-k-defined) closed subgroup of G. Following Serre [23,Sec.…”