On the functor of Arakawa, Suzuki and Tsuchiya

TL;DR

This paper provides a detailed proof of a correspondence between modules of the trigonometric Cherednik algebra and affine Lie algebra modules, extending previous work and relating it to degenerate affine Hecke algebra modules.

Contribution

It offers a detailed proof of the Arakawa-Suzuki-Tsuchiya functor using affine Lie algebras and connects it to Cherednik's earlier module correspondence for degenerate affine Hecke algebras.

Findings

Established a detailed proof of the module correspondence.

Related the construction to modules of degenerate affine Hecke algebra.

Extended the functor to work with affine Lie algebra 0 extbackslash mathfrak{gl}_m.

Abstract

Arakawa, Suzuki and Tsuchiya constructed a correspondence between certain modules of the trigonometric Cherednik algebra $\mathfrak{C}_N$ depending on a parameter $\kappa\in\mathbb{C}$, and certain modules of the affine Lie algebra $\widehat{\mathfrak{sl}}_m$ of level $\kappa-m$. We give a detailed proof of this correspondence by working with the affine Lie algebra $\widehat{\mathfrak{gl}}_m$ alongside of $\widehat{\mathfrak{sl}}_m$. We also relate this construction to a correspondence between certain modules of the degenerate affine Hecke algebra $\mathfrak{H}_N$ and all modules of $\mathfrak{sl}_m$ or $\mathfrak{gl}_m$. The latter correspondence was constructed earlier by Cherednik.