Kategorie $\mathcal{O}$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe

We give a description of ``the algebra of category $\mathcal{O}$'' which is explicit enough to prove that the structure of the direct summands of $\mathcal{O}$ depends only on the integral Weyl group and the singularity of the central character, as well as to establish a weak version of the duality conjectures of Beilinson and Ginsburg [BGi]. As a byproduct we describe the intersection cohomology of Schubert varieties as modules over global cohomology ring. These are certain indecomposable graded self-dual modules over the coinvariant algebra of the Weyl group, via the Borel picture for the global cohomology ring of a flag manifold. They play a central role in this article and should have an interesting future.