Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module

We investigate certain singular categories of Harish-Chandra bimodules realized as the category of -presentable modules in the principal block of the Bernstein-Gelfand-Gelfand category . This category is equivalent to the module category of a properly stratified algebra. We describe the socles and endomorphism rings of standard objects in this category. Further, we consider translation and shuffling functors and their action on the standard modules. Finally, we study a graded version of this category; in particular, we give a graded version of the properly stratified structure, and use graded versions of translation functors to categorify a parabolic Hecke module.