Wall-crossing functors and 𝒟-modules

Beilinson, Alexander; Ginzburg, Victor

doi:10.1090/S1088-4165-99-00063-1

Wall-crossing functors and $\mathcal {D}$-modules
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by Alexander Beilinson and Victor Ginzburg

Represent. Theory 3 (1999), 1-31

DOI: https://doi.org/10.1090/S1088-4165-99-00063-1

Published electronically: January 11, 1999

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Abstract:

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 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.

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