Tilting exercises

This is a geometry-oriented review of the basic formalism of tilting objects (originally due to Ringel, see [Ri], §5). In the first section we explain that tilting extensions form a natural framework for the gluing construction from [B1] and [MV]. We show that in case of a stratification with contractible strata, the homotopy category of complexes of tilting perverse sheaves is equivalent to the derived category of sheaves smooth along the stratification. Thus tilting objects play the role similar to projective or injective ones (with advantage of being self-dual and having local origin). In the second section we discuss tilting perverse sheaves smooth along the Schubert stratification of the flag space (or, equivalently, tilting objects in the Bernstein-Gelfand-Gelfand category O). In this case a Radon transform interchanges tilting, projective, and injective modules. As a corollary, we give a short proof of Soergel’s Struktursatz [S1], and describe the Serre functor for D(O) (as conjectured by M. Kapranov).