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

TL;DR

This paper explores the representation theory of the rational Cherednik algebra for cyclic groups, connecting it with quiver variety geometry, and determines characteristic cycles of standard modules, confirming a conjecture for cyclic groups.

Contribution

It establishes conditions for the shift functor to be an equivalence and links standard modules to tautological bundles on quiver varieties, confirming a conjecture for cyclic groups.

Findings

The shift functor is an equivalence under certain conditions.

The regular representation maps to the tautological bundle.

Characteristic cycles of standard modules are explicitly determined.

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 ${\mathbb Z}_l$.