  In this paper, oppositeness in spherical buildings is used to define an
EKR-problem for flags in projective and polar spaces. A novel application of
the theory of buildings and Iwahori-Hecke algebras is developed to prove sharp
upper bounds for EKR-sets of flags. In this framework, we can reprove and
generalize previous upper bounds for EKR-problems in projective and polar
spaces. The bounds are obtained by the application of the Delsarte-Hoffman
coclique bound to the opposition graph. The computation of its eigenvalues is
due to earlier work by Andries Brouwer and an explicit algorithm is worked out.
For the classical geometries, the execution of this algorithm boils down to
elementary combinatorics. Connections to building theory, Iwahori-Hecke
algebras, classical groups and diagram geometries are briefly discussed.
Several open problems are posed throughout and at the end.
