The center conjecture for thick spherical buildings

Abstract

In this paper we show that a convex subcomplex of a spherical building of type E 6 , E 7 or E 8 is a subbuilding or the automorphisms of the subcomplex fix a point on it. Together with previous results of Mühlherr-Tits, and Leeb and the author, this completes the proof of Tits' Center Conjecture for spherical buildings without factors of type H 4 , in particular, for thick spherical buildings.

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

“…2.4] and [29, Lem. 1.2]), which was recently proved by Tits, Mühlherr, Leeb, and Ramos-Cuevas [20], [22], [23]: Theorem 3.1. Let X be a convex contractible subcomplex of ∆(G).…”

(Non)-completeness of $ℝ$-buildings and fixed point theorems

“…Bernhard Leeb and Carlos Ramos-Cuevas gave an alternative proof for the F 4 case and also proved the E 6 case ( [7]). Finally Carlos Ramos-Cuevas proved the E 7 and E 8 case in [11]. The reducible cases obey the conjecture if their irreducible components do.…”

“…If such a case occurs one can restrict the Weyl group of both the spherical building at infinity and the residue at˛(which is a retract of the building at infinity) to no longer have a direct factor of type H 4 . The convex chamber complex K stays a chamber complex after this restriction of the Weyl group, so we can apply the center conjecture in this case (compare [11], p. 3).…”

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

“…Note that in Propositions 1.3, 1.6 and Theorems 1.5, 4.3 we assume G to be connected. This is because the proofs of these results depend on the following (Theorem 1.7) that is a consequence of the recently proved center conjecture of Tits [15], [23], [25], [ Here is the structure of the paper. In Section 2, we set out the notation.…”