Cherednik algebras and Zhelobenko operators

TL;DR

This paper explores the relationship between intertwining operators in Cherednik algebras and Zhelobenko operators for affine Lie algebras, using a functor connecting modules of these algebraic structures.

Contribution

It establishes a correspondence between Cherednik algebra operators and Zhelobenko operators via a functor linking affine Lie algebra modules and Cherednik modules.

Findings

Intertwining operators in Cherednik algebras correspond to Zhelobenko operators.

The functor of Arakawa, Suzuki, and Tsuchiya is used to connect modules.

The correspondence enhances understanding of the structure of Cherednik algebras.

Abstract

We study canonical intertwining operators between modules of the trigonometric Cherednik algebra, induced from the standard modules of the degenerate affine Hecke algebra. We show that these operators correspond to the Zhelobenko operators for the affine Lie algebra $\widehat{\mathfrak{sl}}_m$. To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain $\widehat{\mathfrak{sl}}_m$-modules to modules of the Cherednik algebra.