Projective and Whittaker functors on category $\mathcal{O}$

We show that the Whittaker functor on a regular block of the BGG-category $\mathcal{O}$ of a semisimple complex Lie algebra can be obtained by composing a translation to the wall functor with Soergel and Mili\v{c}i\'{c}'s equivalence between the category of Whittaker modules and a singular block of $\mathcal{O}$. We show that the Whittaker functor is a quotient functor that commutes with all projective functors and endomorphisms between them.Comment: 14 page