## Koszul duality of translation- and Zuckerman functors.(English)Zbl 1107.17004

In this paper, it is shown that the two extant categorizations of the Temperley-Lieb algebra [using Zuckerman and projective functors, respectively] are, as expected, Koszul dual to each other. To this extent, the Koszul duality is reviewed (in representation theory of the category $$\mathcal{O}$$ of $$\mathfrak{g}$$-modules over a complex semisimple Lie algebra in the sense of I. N. Bernstein, I. M. Gel’fand and S. I. Gel’fand [Funkts. Anal. Prilozh. 12, No. 3, 66–67 (1978; Zbl 0402.14005)], and a new presentation of the duality functor is given. It is shown that the translation and Zuckerman functors are Koszul dual to each other (theorem 4.1), i.e., the Beilinson-Ginzburg-Soergel Koszul duality [A. Beilinson, V. Ginzburg and W. Soergel, J. Am. Math. Soc. 9, No. 2, 473–527 (1996; Zbl 0864.17006)] interchanges derived functors of translation and Zuckerman functors. This result is applied to provide an alternative proof of the Enright-Shelton equivalence.

### MSC:

 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B35 Universal enveloping (super)algebras

### Keywords:

Koszul duality; Zuckerman functor

### Citations:

Zbl 0402.14005; Zbl 0864.17006
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