Categorification of a parabolic Hecke module via sheaves on moment graphs

We investigate certain categories, associated by Fiebig with the geometric representation of a Coxeter system, via sheaves on Bruhat graphs. We modify Fiebig's definition of translation functors in order to extend it to the singular setting and use it to categorify a parabolic Hecke module. As an application we obtain a combinatorial description of indecomposable projective objects of (truncated) noncritical singular blocks of (a deformed version of) category O, using indecomposable special modules over the structure algebra of the corresponding Bruhat graph.