Receding polar regions of a spherical building and the center conjecture

Cocharacter-closure and spherical buildings

“…For definitions and further details, see [16]; in particular, note that the spherical building of a reductive algebraic group is thick. The conjecture was proved by Mühlherr-Tits [14], Leeb-Ramos-Cuevas [12] and Ramos-Cuevas [16], and a uniform proof for chamber subcomplexes has also now been given by Mühlherr-Weiss [15]. The condition that Σ v,ks is a subcomplex is satisfied in the theory of complete reducibility for subgroups of G and Lie subalgebras of Lie(G), and our results yield applications to complete reducibility (see Theorem 1.4 below).…”

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

“…Since This so-called centre conjecture of Tits was proved by case-by-case analyses by Tits, Mühlherr, Leeb, and Ramos-Cuevas [15], [19], [21]. Recently, a uniform proof was given in [20]. In relation to the theory of complete reducibility, Serre showed [23]:…”

“…In particular, Question 1.2 has an affirmative answer if k is perfect. Proposition 1.4 depends on the recently proved and deep centre conjecture of Tits (see Conjecture 5.1) in spherical buildings [23], [26], [20]. The centre conjecture (theorem) has been used to study complete reducibility over k, see [1], [30].…”

Recovering the Lie algebra from its extremal geometry

“…The root set is J = {1, n} for A n , J = {2} for (BC) n , D n , E 6 and G 2 , J = {1} for E 7 and F 4 , and J = {8} for E 8 . (We use the Bourbaki labelling of the Dynkin diagrams as in [CI1,CI2,MW]). The root shadow space of type X n,J , where J is the root set, is simply called the root shadow space of ∆ and we denote it by RSh(∆).…”