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

TL;DR

This paper explores the structure of Whittaker functors within category al, demonstrating their relation to translation functors and their properties as quotient functors that commute with projective functors.

Contribution

It establishes a new perspective on Whittaker functors as compositions of translation to the wall and known equivalences, revealing their quotient nature and compatibility with projective functors.

Findings

Whittaker functor can be constructed via translation to the wall and an established equivalence.

The functor is a quotient functor that commutes with all projective functors.

The approach links Whittaker modules with singular blocks of al.

Abstract

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.