Arakawa-Suzuki functors for Whittaker modules

TL;DR

This paper constructs exact functors linking Whittaker modules of type A Lie algebras to finite-dimensional modules of graded affine Hecke algebras, generalizing classical dualities and preserving module structures.

Contribution

It introduces a new family of exact functors from Whittaker modules to graded affine Hecke algebra modules, extending previous duality results.

Findings

Functors map standard modules to standard modules or zero

Simple modules are preserved or mapped to zero

All simple modules of the Hecke algebra are images of simple Whittaker modules

Abstract

In this paper we construct a family of exact functors from the category of Whittaker modules of the simple complex Lie algebra of type $A_n$ to the category of finite-dimensional modules of the graded affine Hecke algebra of type $A_\ell$. Using results of Backelin and of Arakawa-Suzuki, we prove that these functors map standard modules to standard modules (or zero) and simple modules to simple modules (or zero). Moreover, we show that each simple module of the graded affine Hecke algebra appears as the image of a simple Whittaker module. Since the Whittaker category contains the BGG category $\mathcal{O}$ as a full subcategory, our results generalize results of Arakawa-Suzuki, which in turn generalize Schur-Weyl duality between finite-dimensional representations of $SL_n(\mathbb{C})$ and representations of the symmetric group $S_n$.