Twisting functors on 𝒪

This paper studies twisting functors on the main block of the Bernstein-Gelfand-Gelfand category O \mathcal {O} and describes what happens to (dual) Verma modules. We consider properties of the right adjoint functors and show that they induce an auto-equivalence of derived categories. This allows us to give a very precise description of twisted simple objects. We explain how these results give a reformulation of the Kazhdan-Lusztig conjectures in terms of twisting functors.