Oppositeness in Buildings and Simple Modules for Finite Groups of Lie Type

In the building of a finite group of Lie type, we consider the incidence relations defined by oppositeness of flags. Such a relation gives rise to a homo-morphism of permutation modules (in the defining characteristic) whose image is a simple module for the group. The p-rank of the incidence relation is then the dimension of this simple module. We give some general reductions towards the determination of the character of the simple module. Its highest weight is identified and the problem is reduced to the case of a prime field. The reduced problem can be approached through the representation theory of algebraic groups and the methods are illustrated for some examples.