The strong Centre Conjecture: An invariant theory approach

The aim of this paper is to describe an approach to a strengthened form of J. Titsʼ Centre Conjecture for spherical buildings. This is accomplished by generalizing a fundamental result of G.R. Kempf from Geometric Invariant Theory and interpreting this generalization in the context of spherical buildings. We are able to recapture the conjecture entirely in terms of our generalization of Kempfʼs notion of a state. We demonstrate the utility of this approach by proving the Centre Conjecture in some special cases.