Affine category O, Koszul duality and Zuckerman functors

The parabolic category O for affine glN at level −N−e admits a structure of categorical representation sl˜e with respect to some endofunctors E and F. This contains smaller A that categorifies the higher Fock space. We prove functors F in are Koszul dual Zuckerman functors. key point proof is show functor can be decomposed terms components −N−e−1. To this, we use following fact from [9]: an action sl˜e+1 (canonically defined) subcategory sl˜e. also general statement says situation satisfies list axioms automatically sort functor.