  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
homomorphism 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.
