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This is an application of the theory of tilting objects to the geometric setting of perverse sheaves. We show that this theory is a natural framework for Beilinson's gluing of perverse sheaves construction. In the special case of Schubert stratification of a flag variety we get a short proof of Soergel's Struktursatz, and describe (following a conjecture of Kapranov) Serre functor for category O. Some of our results were obtained independently by Rouquier.
Beilinson et al. (Thu,) studied this question.