On Singular Calogero-moser Spaces

Using combinatorial properties of complex reflection groups we show that if the group W is different from the wreath product Sn ≀ Z/mZ and the binary tetrahedral group (labelled G(m, 1, n) and G4 respectively in the Shephard-Todd classification), then the generalised Calogero-Moser space Xc associated to the centre of the rational Cherednik algebra H0,c(W ) is singular for all values of the parameter c. This result and a theorem of Ginzburg and Kaledin imply that there does not exist a symplectic resolution of the singular symplectic variety h× h∗/W when W is a complex reflection group different from Sn ≀ Z/mZ and the binary tetrahedral group (where h is the reflection representation associated to W ). Conversely it has been shown by Etingof and Ginzburg that Xc is smooth for generic values of c when W ∼= Sn ≀ Z/mZ. We show that this is also the case when W is the binary tetrahedral group. A theorem of Namikawa then implies the existence of a symplectic resolution in this case, completing the classification. Finally, we note that the above results together with work of Chlouveraki are consistent with a conjecture of Gordon and Martino on block partitions in the restricted rational Cherednik algebra.