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

An algebraic approach to Erd\H{o}s-Ko-Rado sets of flags in spherical buildings