in 1938, erdos, ko and rado proved a foundational result on the size of intersecting families of sets. ever since, there has been a rich body of results proving similar theorems in different contexts. notably, in geometries over finite fields like projective and polar spaces, such results were obtained by several groups of researchers. i will explain a very successful technique that can be used to prove these results, and indicate its shortcomings. we will show how a generalization of this problem recovers all known results and how algebraic combinatorics such as iwahori-hecke algebras and their representations come into play. the latter is based on joint work with jan de beule and klaus metsch.