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.
In this article, we consider Nakajima quiver varieties from the point of view of symplectic algebraic geometry. We prove that they are all symplectic singularities in the sense of Beauville and completely classify which admit symplectic resolutions. Moreover we show that the smooth locus coincides with the locus of canonically θ-polystable points, generalizing a result of Le Bruyn, and we describe the Namikawa Weyl group. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT.We apply this to the G-character variety of a compact Riemann surface of genus g > 0, when G is SL(n, C) or GL(n, C). We show that these varieties are symplectic singularities and classify when they admit symplectic resolutions: they do when g = 1 or (g, n) = (2, 2) (assuming n ≥ 2). This is analogous to the case of a quiver with one vertex, g arrows, and dimension vector (n).We note that our results show that existence of proper and projective symplectic resolutions are equivalent for many of the varieties in question. This does not seem to be known in general.Dedicated, with admiration and thanks, to Victor Ginzburg, on the occasion of his 60th Birthday. Table of Contents 1. Introduction 2. Quiver varieties 3. Canonical Decompositions 4. Smooth vs. stable points 5. The (2, 2) case 6. Factoriality of quiver varieties 7. Namikawa's Weyl group 8. Character varieties
ABSTRACT. The goal of this paper is to compute the cuspidal Calogero-Moser families for all infinite families of finite Coxeter groups, at all parameters. We do this by first computing the symplectic leaves of the associated Calogero-Moser space and then by classifying certain "rigid" modules. Numerical evidence suggests that there is a very close relationship between Calogero-Moser families and Lusztig families. Our classification shows that, additionally, the cuspidal Calogero-Moser families equal cuspidal Lusztig families for the infinite families of Coxeter groups.
Let Γ be a finite subgroup of Sp(V ). In this article we count the number of symplectic resolutions admitted by the quotient singularity V /Γ. Our approach is to compare the universal Poisson deformation of the symplectic quotient singularity with the deformation given by the Calogero-Moser space. In this way, we give a simple formula for the number of Q-factorial terminalizations admitted by the symplectic quotient singularity in terms of the dimension of a certain Orlik-Solomon algebra naturally associated to the Calogero-Moser deformation. This dimension is explicitly calculated for all groups Γ for which it is known that V /Γ admits a symplectic resolution. As a consequence of our results, we confirm a conjecture of Ginzburg and Kaledin.
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