Bulletin of the London Mathematical Society Advance Access originally published online on March 11, 2009
Bulletin of the London Mathematical Society 2009 41(2):315-326; doi:10.1112/blms/bdp019
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© 2009 London Mathematical Society
On singular Calogero–Moser spaces
School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh
James Clerk Maxwell Building
Kings Buildings
Mayfield Road
Edinburgh
EH9 3JZ
United Kingdom
Received 27 November 2007. Revision received 2 September 2008.
Using combinatorial properties of complex reflection groups we show that, if the group W is different from the wreath product 
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 
x */W when W is a complex reflection group different from and the binary tetrahedral group (where 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 
. 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 the work of Chlouveraki, are consistent with a conjecture of Gordon and Martino on block partitions in the restricted rational Cherednik algebra.
2000 Mathematics Subject Classification 16S38 (primary), 16Rxx, 14E15 (secondary).
The research described here was done at the University of Edinburgh with the financial support of the EPSRC. This material will form part of the author's PhD thesis for the University of Edinburgh.