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Wall-crossing functors and \({\mathcal{D}}\)-modules. (English) Zbl 0910.05068

Summary: We study translation functors and wall-crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using \({\mathcal{D}}\)-modules. This functorial machinery is then used to prove the endomorphism theorem and the structure theorem; two important results which were established earlier by W. Soergel in a totally different way. Other applications to the category \({\mathcal{O}}\) of Bernstein-Gelfand-Gelfand are given, and some conjectural relationships between Koszul duality, Verdier duality and convolution functors are discussed. A geometric interpretation of tilting modules is given.

MSC:

05E99 Algebraic combinatorics
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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