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We study the representation theory of the rational Cherednik algebra$H_\kappa = H_\kappa({\mathbb Z}_l)$ for the cyclic group ${\mathbb Z}_l ={\mathbb Z} / l {\mathbb Z}$ and its connection with the geometry of the quivervariety $M_\theta(\delta)$ of type $A_{l-1}^{(1)}$.  We consider a functor between the categories of $H_\kappa$-modules withdifferent parameters, called the shift functor, and give the condition when itis an equivalence of categories.  We also consider a functor from the category of $H_\kappa$-modules with goodfiltration to the category of coherent sheaves on $M_\theta(\delta)$. We provethat the image of the regular representation of $H_\kappa$ by this functor isthe tautological bundle on $M_\theta(\delta)$. As a corollary, we determine thecharacteristic cycles of the standard modules. It gives an affirmative answerto a conjecture given in [Gordon, arXiv:math/0703150v1] in the case of${\mathbb Z}_l$.

Characteristic cycles of standard modules for the rational Cherednik algebra of type Z/lZ