An algebraic approach to the KZ-functor for rational Cherednik algebras associated with cyclic groups

TL;DR

This paper provides an algebraic proof relating the projective object representing the KZ-functor to the coinvariant algebra in rational Cherednik algebras for cyclic groups, identifying specific parameter values and describing the Hecke algebra action.

Contribution

It offers an algebraic approach to understanding the KZ-functor for cyclic groups, including explicit parameter conditions and the action of the cyclotomic Hecke algebra.

Findings

Isomorphism between $P_{KZ}$ and the $ riangle$-module for certain parameters

Explicit characterization of parameter values for the isomorphism

Complete algebraic description of the KZ-functor in this setting

Abstract

In the case of rational Cherednik algebras associated with cyclic groups, we give an alternative proof that the projective object $P_{\text{KZ}}$ representing the KZ-functor is isomorphic to the $\Delta$-module associated with the coinvariant algebra for a subset of parameter values from which all parameter values can be obtained by integral translations. We also specify the exact parameter values for which this isomorphism occurs. Furthermore, we determine the action of the cyclotomic Hecke algebra on $P_{\text{KZ}}$ for these parameter values, thereby giving a complete algebraic description of the KZ-functor in this case.