  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) non-critical singular blocks of (a deformed version of)
category $\mathcal{O}$, using indecomposable special modules over the structure
algebra of the corresponding Bruhat graph.
