Twisted Verma modules

Using principal series Harish-Chandra modules, local cohomology with support in Schubert cells and twisting functors we construct certain modules parametrized by the Weyl group and a highest weight in the subcategory O of the category of representations of a complex semisimple Lie algebra. These are in a sense modules between a Verma module and its dual. We prove that the three different approaches lead to the same modules. Moreover, we demonstrate that they possess natural Jantzen type filtrations with corresponding sum formulae.