Parabolic projective functors in type A

We classify projective functors on the regular block of Rocha-Caridi's parabolic version of the BGG category OO in type A. In fact, we show that, in type A  , the restriction of an indecomposable projective functor from OO to the parabolic category is either indecomposable or zero. As a consequence, we obtain that projective functors on the parabolic category OO in type A are completely determined, up to isomorphism, by the linear transformations they induce on the level of the Grothendieck group, which was conjectured by Stroppel in [39].