n-Cohomology of simple highest weight modules on walls and purity

Summary LetGPB be respectively a complex connected linear algebraic semisimple group, a parabolic subgroup and a Borel subgroup. The first main result is the following theorem: Let be a pure complex onG/B, smooth with respect to Bruhat cells. Then its restriction to anyP-orbit is pure as well, of the same weight. As a consequence we are able to compute then-cohomology of simple highest weight modules on walls.Written during the author's stay at MSRI, supported by a Stipendium der Clemens-Plassmann-Stiftung