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

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.