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Gwyn Bellamy

Publications and source records attributed to Gwyn Bellamy.

47 records · Page 3Linked to original sources
arXiv2013

Hamiltonian reduction and nearby cycles for Mirabolic D-modules

We study holonomic D-modules on SL_n(C)xC^n, called mirabolic modules, analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus. We introduce an analogue of the Verdier specialization functor for representations of Cherednik algebras which agrees, on category O, with the restriction functor of Bezrukavnikov and Etingof. In type A, we also consider a Verdier specialization functor on mirabolic D-modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic D-modules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof.

math.RT↗
arXiv2013

On the (non)existence of symplectic resolutions for imprimitive symplectic reflection groups

We study the existence of symplectic resolutions of quotient singularities V/G where V is a symplectic vector space and G acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the form $K \rtimes S_2$ where $K < \SL_2(\C)$, for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for $\dim V \neq 4$, we classify all quotient singularities $V/G$ admitting a projective symplectic resolution which do not decompose as a product of smaller-dimensional quotient singularities, except for at most four explicit singularities, that occur in dimensions at most 10, for whom the question of existence remains open.

math.SG↗
arXiv2013

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).

math.RT↗
arXiv2013

Affinity of Cherednik algebras on projective space

We give sufficient conditions for the affinity of Etingof's sheaves of Cherednik algebras on projective space. To do this we introduce the notion of pull-back of modules under certain flat morphisms.

math.RT↗
arXiv2012

On Deformation Quantizations of Hypertoric varieties

Based on a construction by Kashiwara and Rouquier, we present an analogue of the Beilinson- Bernstein localization theorem for hypertoric varieties. In this case, sheaves of differential operators are replaced by sheaves of W-algebras. As a special case, our result gives a localization theorem for rational Cherednik algebras associated to cyclic groups.

math.RT↗
arXiv2011

On the smoothness of centres of rational Cherednik algebras in positive characteristic

In this article we study rational Cherednik algebras at $t = 1$ in positive characteristic. We study a finite dimensional quotient of the rational Cherednik algebra called the restricted rational Cherednik algebra. When the corresponding pseudo-reflection group belongs to the infinite series $G(m,d,n)$, we describe explicitly the block decomposition of the restricted algebra. We also classify all pseudo-reflection groups for which the centre of the corresponding rational Cherednik algebra is regular for generic values of the deformation parameter.

math.RA↗
arXiv2011

A new linear quotient of C^4 admitting a symplectic resolution

We show that the quotient C^4/G admits a symplectic resolution for G = (Q_8 x D_8)/(Z/2) < Sp(4,C). Here Q_8 is the quaternionic group of order eight and D_8 is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements -1 of each. It is equipped with the tensor product of the defining two-dimensional representations of Q_8 and D_8. This group is also naturally a subgroup of the wreath product group of Q_8 by S_2. We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C^4/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V / G admitting symplectic resolutions.

math.SG↗
arXiv2011

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.

math.RT↗
arXiv2009

Cuspidal representations of rational Cherednik algebras at t=0

We study those finite dimensional quotients of the rational Cherednik algebra at t=0 that are supported at a point of the centre. It is shown that each such quotient is Morita equivalent to a certain cuspidal quotient of a rational Cherednik algebra associated to a parabolic subgroup of W.

math.RT↗
arXiv2008

Factorization in generalized Calogero-Moser spaces

Using a recent construction of Bezrukavnikov and Etingof we prove that there is a factorization of the Etingof-Ginzburg sheaf on the generalized Calogero-Moser space associated to a complex reflection group. In the case W = S_n, this confirms a conjecture of Etingof and Ginzburg.

math.RT↗