reath product of a finite subgroup of $SL_2({\mathbb{C}})$ with the symmetric group ${{W}}$, see [@wang Theorem 4.2]. We believe that our methods will generalise to the symplectic reflection algebras $H_{c}(G)=H_{1,c}(G)$ associated with $(({\mathbb{C}}^2)^n, G)$ to give non-commutative deformations of those Hilbert schemes. Even when there is no crepant resolution of such a singularity (by [@GK] this happens for Weyl groups $G$ of types other than A and B) the ${\mathbb{Z}}$-algebra associated to $H_c(G)$ will still contain interesting information, as [@gordc] demonstrates. For a Weyl group, the analogue of Theorem \[morrat-intro\] is at least known for sufficiently large values of the defining parameter $c$ [@BEGfd Proposition 4.3], but little is known for small values of $c$.

The translation principle obviously holds for factors of enveloping algebras of semisimple Lie algebras and we can prove an analogue of Theorem \[mainthm-intro\] in this case. However, the proof uses nontrivial Lie theoretic results, notably the Beilinson-Bernstein equivalence of categories, and it is unclear whether this approach carries information that cannot be obtained from that equivalence. It would be interesting to see if the recent work [@BK; @tan] on the Beilinson-Bernstein equivalence for quantised enveloping algebras can be understood in a ${\mathbb{Z}}$-algebra framework.

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The paper is organised as follows. In Section \[sect-rationalchered\] we recall the needed facts about rational Cherednik algebras, while in Section \[shift\] we prove Theorem \[morrat-intro\]. In Section \[sect-haiman\] we describe some of Haiman’s work on Hilbert schemes, adapted to the variety $\operatorname{Hilb(n)}$, and use it to describe various Poincaré series that will be needed in the proof of Theorem \[mainthm-intro\]. Section \[zalg\] proves the results about ${\mathbb{Z}}$-algebras that were mentioned earlier in this introduction. Section \[sect-filt\] is the heart of the paper: in it we prove Theorem \[mainthm-i