Complete reducibility and separable field extensions

Abstract

Let G be a connected reductive linear algebraic group. The aim of this note is to settle a question of J-P. Serre concerning the behaviour of his notion of G-complete reducibility under separable field extensions. Part of our proof relies on the recently established Tits Centre Conjecture for the spherical building of the reductive group G.

Semisimplification for subgroups of reductive algebraic groups

“…To see this, one can first extend the field from to and then show that the R-parabolic subgroup obtained from either of the constructions is -defined (cf. [3,Proof of Theorem 1.1] and [11,Section 4]), and hence coincides with -this implies that ( ), and hence , normalizes . Remark 5.3.…”

The center conjecture for thick spherical buildings

“…Next we exhibit an element representing the vertices of the fundamental Weyl chamber △, i.e. elements of R + · v i : 1-vertex: v 1 ( 1, 1, 1, 1, 1, 1, −2, −2) 2-vertex: v 2 (−1, 1, 1, 1, 1, 1, −1, −1) 3-vertex: v 3 ( 0, 0, 1, 1, 1, 1, −1, −1) 4-vertex: v 4 ( 1, 1, 1, 3, 3, 3, −3, −3) 5-vertex: v 5 ( 1, 1, 1, 1, 3, 3, −2, −2) 6-vertex: v 6 ( 1, 1, 1, 1, 1, 3, −1, −1) 7-vertex: v 7 ( 1, 1, 1, 1, 1, 1, 0, 0)…”

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

“…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.…”

“…However most studies assume k = k and G is connected; see [5], [18], [31] for example. We say that H < G is G-cr when it is G-cr over k. Not 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.…”