  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 quiver
variety $M_\theta(\delta)$ of type $A_{l-1}^{(1)}$.
  We consider a functor between the categories of $H_\kappa$-modules with
different parameters, called the shift functor, and give the condition when it
is an equivalence of categories.
  We also consider a functor from the category of $H_\kappa$-modules with good
filtration to the category of coherent sheaves on $M_\theta(\delta)$. We prove
that the image of the regular representation of $H_\kappa$ by this functor is
the tautological bundle on $M_\theta(\delta)$. As a corollary, we determine the
characteristic cycles of the standard modules. It gives an affirmative answer
to a conjecture given in [Gordon, arXiv:math/0703150v1] in the case of
${\mathbb Z}_l$.
