Tilting modules in category O and sheaves on moment graphs

We describe tilting modules of the deformed category O over a semisimple Lie algebra as certain sheaves on a moment graph associated to the corresponding block of O. We prove that they map to Braden–MacPherson sheaves constructed along the reversed Bruhat order under Fiebigʼs localization functor. By this means, we get character formulas for tilting modules and explain how Soergelʼs result about the Andersen filtration gives a Koszul dual proof of the semisimplicity of subquotients of the Jantzen filtration.