On the Localisation Theorem for rational Cherednik algebra modules

TL;DR

This paper investigates the conditions under which the localization functor for modules over rational Cherednik algebras is exact, demonstrating that for certain groups all parameters are good and bad parameters form a bounded set.

Contribution

It establishes the boundedness of bad parameters for the localization functor in rational Cherednik algebra modules and confirms all parameters are good for specific groups.

Findings

Bad parameters are bounded in the parameter space.

All parameters are good for the groups S_n, μ_3, and B_2.

Abstract

Let $W$ be a complex reflection group of the form $G(l,1,n)$. Following [BK12, BPW12, Gor06, GS05, GS06, KR08, MN11], the theory of deform quantising conical symplectic resolutions allows one to study the category of modules for the spherical Cherednik algebra, $U_\textbf{c}(W)$, via a functor, $\mathbb T_{\textbf{c},\theta}$, which takes invariant global sections of certain twisted sheaves on some Nakajima quiver variety $Y_\theta$. A parameter for the Cherednik algebra, $\textbf{c}$, is considered `good' if there exists a choice of GIT parameter $\theta$, such that $\mathbb T_{\textbf{c},\theta}$ is exact and `bad' otherwise. By calculating the Kirwan--Ness strata for $\theta=\pm(1,\ldots,1)$ and using criteria of [MN13], it is shown that the set of all bad parameters is bounded. The criteria are then used to show that, for the cases $W=\mathfrak S_n, \mu_3, B_2$, all parameters are good.