Soergel category O

Moment graphs are labeled graphs associated to varieties with a nice torus action and can be used to give a combinatorial algorithm to compute torus-equivariant intersection cohomology. I will describe an analogue of the "mixed derived category" of P. Achar and S. Riche in this setting, which leads to a notion of "mixed perverse sheaves" on moment graphs, even those that do not arise from torus-equivariant geometry. This construction is motivated by the desire to define a"Soergel category O" in its natural generality. I will discuss how Soergel category O provides a natural homological home for the p-Kazhdan--Lusztig basis of the Hecke algebra, whose importance in modular representation theory is beginning to be understood.