Categorification of a parabolic Hecke module via sheaves on moment graphs

TL;DR

This paper extends the categorification of parabolic Hecke modules using sheaves on moment graphs, providing a combinatorial description of projective objects in singular blocks of category O.

Contribution

It introduces modified translation functors to extend Fiebig's framework to the singular setting, enabling new categorification and combinatorial descriptions.

Findings

Extended Fiebig's translation functors to singular cases

Categorified parabolic Hecke modules via sheaves on moment graphs

Provided combinatorial descriptions of projective objects in singular blocks

Abstract

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.