Koszul duality of translation--and Zuckerman functors

TL;DR

This paper explores Koszul duality in category O, demonstrating that translation and Zuckerman functors are dual, and provides new insights and proofs in representation theory.

Contribution

It presents a new presentation of the Koszul duality functor and proves the duality between translation and Zuckerman functors, confirming a conjecture.

Findings

Translation and Zuckerman functors are Koszul dual to each other.

A new presentation of the Koszul duality functor is provided.

A short proof of the Enright-Shelton equivalence is given.

Abstract

We review Koszul duality in representation theory of category $ \cal O $, especially we give a new presentation of the Koszul duality functor. Combining this with work of Backelin, we show that the translation and Zuckerman functors are Koszul dual to each other, thus verifying a conjecture of Bernstein, Frenkel and Khovanov. Finally we use Koszul duality to give a short proof of the Enright-Shelton equivalence.