On singular Calogero-Moser spaces

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).Comment: 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

The Calogero-Moser partition for G(m,d,n)

We show that it is possible to deduce the Calogero-Moser partition of the irreducible representations of the complex reflection groups G(m,d,n) from the corresponding partition for G(m,1,n). This confirms, in the case W = G(m,d,n), a conjecture of Gordon and Martino relating the Calogero-Moser partition to Rouquier families for the corresponding cyclotomic Hecke algebra.Comment: 23 pages; minor revision of section 7; to appear in Nagoya Journal of Mathematic

Endomorphisms of Verma modules for rational Cherednik algebras

We study the endomorphism algebra of Verma modules for rational Cherednik algebras at t=0. It is shown that, in many cases, these endomorphism algebras are quotients of the centre of the rational Cherednik algebra. Geometrically, they define Lagrangian subvariaties of the generalized Calogero-Moser space. In the introduction, we motivate our results by describing them in the context of derived intersections of Lagrangians.Comment: This paper constitutes what used to be those sections on the endomorphisms of Verma modules in the paper "Rational Cherednik algebras and Schubert cells" arXiv:1210.3870, in order to make the latter paper more concise. The main result has been strengthen somewha

James Nayler and the Lamb\u27s War

James Nayler was perhaps the most articulate theologian and political spokesman of the earliest Quaker movement. He was part of a West Yorkshire group of radicals who added revolutionary impetus to George Fox\u27s apocalyptic preaching of Christ\u27s coming in the bodies of common men and women. With other Quaker leaders, Nayler insisted upon disestablishment of the Church, abolition of tithes, and disenfranchisement of the clergy, in order that Christ might rule in England, through human conscience. For early Friends, Christ\u27s sovereignty in the conscience was less a principle of individual freedom to dissociate religiously than a basis for collective practices of revolutionary worship, moral reform, social equality, and economic justice. All these were features of the nonviolent struggle Nayler called the \u27Lamb\u27s War\u27. His meteoric career is outlined in this study, a movement from apocalyptic prophet, to stigmatized Christ-figure, to withdrawn quietist

Counting resolutions of symplectic quotient singularities

Let $\Gamma$ be a finite subgroup of $\mathrm{Sp}(V)$. In this article we count the number of symplectic resolutions admitted by the quotient singularity $V / \Gamma$. 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 $\mathbb{Q}$-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 $\Gamma$ for which it is known that $V / \Gamma$ admits a symplectic resolution. As a consequence of our results, we confirm a conjecture of Ginzburg and Kaledin.Comment: 13 pages, final versio