On singular Calogero-Moser spaces

Abstract

Using combinatorial properties of complex reflection groups, we show that the generalised Calogero-Moser space associated to the centre of the corresponding rational Cherednik algebra is singular for all values of its deformation parameter c if and only if the group is different from the wreath product $S_n\wr C_m$ and the binary tetrahedral group. 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 outside of these cases; conversely we show that there exists a symplectic resolution for the binary tetrahedral group (Hilbert schemes provide resolutions for the wreath product case).

Conjecture 1.3 of version 1 is proved as Corollary 4.2. Inconsistent use of notation in the proof of Lemma 3.3 corrected (thanks to Ulrich Thiel for pointing this out)

References in corpus (3)

• Termination of (many) 4-dimensional log flips
• On the cyclotomic Hecke algebras of complex reflection groups
• Calogero-Moser space, reduced rational Cherednik algebras, and two-sided cells

Cited by in corpus (22)

• A symplectic resolution for the binary tetrahedral group
• 4-dimensional symplectic contractions
• CHAMP: A Cherednik Algebra Magma Package
• Counting resolutions of symplectic quotient singularities
• Cellules de Calogero-Moser
• Poisson traces, D-modules, and symplectic resolutions
• Restricted rational Cherednik algebras
• Highest weight theory for finite-dimensional graded algebras with triangular decomposition
• The combinatorics of ${\mathbb{C}}^{\ast }$-fixed points in generalized Calogero-Moser spaces and Hilbert schemes
• Cox rings of some symplectic resolutions of quotient singularities
• A counter-example to Martino's conjecture about generic Calogero-Moser families
• On the (non)existence of symplectic resolutions for imprimitive symplectic reflection groups
• The Calogero-Moser partition and Rouquier families for complex reflection groups
• On the Kummer construction
• Finite dimensional representations of the rational Cherednik algebra for $G_4$
• All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces
• Symplectic resolutions for Higgs moduli spaces
• Hecke algebras and symplectic reflection algebras
• Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution
• Crepant resolutions of stratified varieties via gluing
• Some applications of CHEVIE to the theory of algebraic groups
• A Casselman-Osborne theorem for rational Cherednik algebras