On singular Calogero–Moser spaces

By G. Bellamy. 2007

63 citations recorded by the source corpus.

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 the work of Chlouveraki, are consistent with a conjecture of Gordon and Martino on block partitions in the restricted rational Cherednik algebra.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d2bebef52602ede70a3eb3341c4c0c34e0e06c31

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In Algebra and Number Theory

1. I. Gordon
2. K. Goodearl

In Geometry and Topology

1. I. Gordon
2. Stephen Griffeth
3. V. Ginzburg
4. Pavel Etingof
5. J. T. Stafford
6. E. Gorsky
7. I. Losev
8. Y. Berest

In Mathematical Physics

1. I. Gordon
2. Stephen Griffeth
3. V. Ginzburg
4. Pavel Etingof
5. J. T. Stafford
6. E. Gorsky
7. I. Losev
8. Y. Berest

Citations in the corpus

• Symplectic reflection algebras (2007, background): “e-by-case, that this is impossible for all the complex reflection groups except G(ℓ,1,n) and G4. 6.8. Application: Symplectic resolutions. Theorem 6.6 has a pleasant consequence. Theorem ([69], [142], [7]). Let G ≤GL(h) be a complex reflection group acting on V = h⊕h∗. Then V/G has a symplectic resolu”