On the Localisation Theorem for rational Cherednik algebra modules

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, $U_{\text{c}}(W)$, via a functor, ${\mathbb{T}}_{\text{c},\theta }$, which takes invariant global sections of certain twisted sheaves on some Nakajima quiver variety $Y_{\theta }$. A parameter for the Cherednik algebra, $\text{c}$, is considered `good' if there exists a choice of GIT parameter $\theta$, such that ${\mathbb{T}}_{\text{c},\theta }$ is exact and `bad' otherwise. By calculating the Kirwan--Ness strata for $\theta =\pm (1,\dots ,1)$ 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.