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Category \({\mathcal O}\), perverse sheaves and modules over coinvariants of the Weyl group. (Kategorie \({\mathcal O}\), perverse Garben und Moduln über den Koinvarianten zur Weylgruppe.) (German) Zbl 0747.17008

Summary: 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. As a byproduct we describe the intersection cohomology of Schubert varieties as modules over the 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.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
14M15 Grassmannians, Schubert varieties, flag manifolds
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
17B20 Simple, semisimple, reductive (super)algebras
22E46 Semisimple Lie groups and their representations
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