Knizhnik--Zamolodchikov functor for degenerate double affine Hecke algebras : algebraic theory

TL;DR

This paper develops an algebraic version of the Knizhnik--Zamolodchikov functor for degenerate double affine Hecke algebras, establishing key properties and comparing it with existing monodromy functors.

Contribution

It introduces an algebraic KZ functor for degenerate double affine Hecke algebras and proves its double centraliser property and kernel characterization.

Findings

Proves the double centraliser property for the algebraic KZ functor.

Characterizes the kernel of the functor.

Establishes results for quiver double Hecke algebras, including degenerate double affine Hecke algebras.

Abstract

In this article, we define an algebraic version of the Knizhnik--Zamolodchikov functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by Varagnolo--Vasserot. We prove the double centraliser property for our functor and give a characterisation of its kernel. We establish these results for a family of algebras, called quiver double Hecke algebras, which includes the degenerate double affine Hecke algebras as special cases.