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

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 $M_{\theta }(\delta )$ 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 ${\mathbb{Z}}_l$.

40 pages, To appear in J. Math. Kyoto Univ

References in corpus (1)

• Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras