Characteristic cycles of standard modules for the rational Cherednik algebra of type $\mathbb{Z}/l\mathbb{Z}$

Abstract

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 $\mathfrak{M}_{\theta}(\delta )$ of type $A^{(1)}_{l-1}$.

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 $\mathfrak{M}_{\theta}(\delta )$. We prove that the image of the regular representation of $H_{\kappa}$ by this functor is the tautological bundle on $\mathfrak{M}_{\theta}(\delta )$. As a corollary, we determine the characteristic cycles of the standard modules. It gives an affirmative answer to a conjecture given in [Go] in the case of $\mathbb{Z}_{l}$.