arXiv subjects

Gwyn Bellamy

Publications and source records attributed to Gwyn Bellamy.

At least 19 recordsLinked to original sources

arXiv2026

Crepant partial resolutions of the nilpotent cone via Hamiltonian reduction

We show that the nilpotent cone associated to a simply connected semisimple algebraic group, together with all its crepant projective partial resolutions, can be constructed as Hamiltonian reductions of the affine closure of the cotangent bundle of base affine space for suitable choices of stability parameter of a maximal torus. We also realize the base change of the universal Poisson deformation of each of these crepant projective partial resolutions as the GIT quotient of the affine closure of the cotangent bundle of base affine space at the same stability parameter.

math.AG↗

arXiv2026

Namikawa--Weyl groups of symplectic quotient singularities

We classify the Namikawa--Weyl groups associated to symplectic quotient singularities $V/G$ when $G$ is a symplectic reflection group. Our classification shows that every irreducible Weyl group can be realized as a factor of the Namikawa--Weyl group of $V/G$ for a suitable $G$.

math.SG↗

arXiv2026

Two invariant subalgebras of rational Cherednik algebras

Originally motivated by connections to integrable systems, two natural subalgebras of the rational Cherednik algebra have been considered in the literature. The first is the subalgebra of all degree zero elements and the second is the Dunkl angular momentum subalgebra. In this article, we study the ring-theoretic and homological properties of these algebras. Our approach is to realise them as rings of invariants under the action of certain reductive subgroups of $\rm SL_2$. This allows us to describe their centres. Moreover, we show that they are Auslander-Gorenstein and Cohen-Macaulay and, at $t = 0$, give rise to prime PI-algebras whose PI-degree we compute. Since the degree zero subalgebra can be realized as the ring of invariants for the maximal torus $\rm T \subset SL_2$ and the action of this torus on the rational Cherednik algebra is Hamiltonian, we also consider its (quantum) Hamiltonian reduction with respect to $\rm T$. At $t = 1$, the quantum Hamiltonian reduction of the spherical subalgebra is a filtered quantization of the quotient of the minimal nilpotent orbit closure $\overline{\mathcal O}_{\min}$ in ${\mathfrak gl}(n)$ by the reflection group $W$. At $t = 0$, we get a graded Poisson deformation of the symplectic singularity $\overline{\mathcal O}_{\min}/W$.

math.QA↗

arXiv2025

Module structure of Weyl algebras

The seminal paper "J.T. Stafford, Module structure of Weyl algebras, J. London Math. Soc. (2) 18 (1978), no. 3, 429--442" was a major step forward in our understanding of Weyl algebras. Beginning with Serre's Theorem on free summands of projective modules and Bass' Stable Range Theorem in commutative algebra, we attempt to trace the origins of this work and explain how it led to Stafford's construction of non-holonomic simple modules over Weyl algebras. We also describe Bernstein-Lunt's geometric construction of infinite families of non-holonomic simple modules. We recall more recent developments related to Weyl algebras, especially that of parametrizing right ideals in the first Weyl algebra and its relation to Calogero-Moser spaces. Finally, we revisit Stafford's results in the context of quantized symplectic singularities, where they lead naturally to open problems on the behaviour of simple modules.

math.RA↗

arXiv2025

The Procesi bundle over the $Γ$-fixed points of the Hilbert scheme of points in $\mathbb{C}^2$

For $Γ$ a finite subgroup of $\mathrm{SL}_2(\mathbb{C})$ and $n \geq 1$, we study the fibers of the Procesi bundle over the $Γ$-fixed points of the Hilbert scheme of $n$ points in the plane. For each irreducible component of this fixed point locus, our approach reduces the study of the fibers of the Procesi bundle, as an $(\mathfrak{S}_n \times Γ)$-module, to the study of the fibers of the Procesi bundle over an irreducible component of dimension zero in a smaller Hilbert scheme. When $Γ$ is of type $A$, our main result shows, as a corollary, that the fiber of the Procesi bundle over the monomial ideal associated with a partition $λ$ is induced, as an $(\mathfrak{S}_n \times Γ)$-module, from the fiber of the Procesi bundle over the monomial ideal associated with the core of $λ$. We give different proofs of this corollary in two edge cases, using only representation theory and symmetric functions.

math.AG↗

arXiv2025

Non-homogeneous Koszul duality in representation theory

Motivated by the representation theory of symplectic reflection algebras, deformed preprojective algebras, and graded Hecke algebras, we consider filtered algebras $U$ whose associated graded is Koszul. The Koszul dual of $U$, as defined by Positselski, is a curved dg-algebra. We establish an exact equivalence between the unbounded derived category of $U$ and an explicit quotient of the homotopy category of injective modules over the dual curved dg-algebra. This recovers a special case of a result of Positselski. In the case where $U$ has finite global dimension, the quotient is trivial and hence the unbounded derived category of $U$ is equivalent to the homotopy category of injective modules over the dual curved dg-algebra.

math.RT↗

arXiv2025

Birational geometry of quiver varieties and other GIT quotients

We prove that all projective crepant resolutions of Nakajima quiver varieties satisfying natural conditions are also Nakajima quiver varieties. More generally, we classify the small birational models of many Geometric Invariant Theory (GIT) quotients by introducing a sufficient condition for the GIT quotient of an affine variety $V$ by the action of a reductive group $G$ to be a relative Mori Dream Space. Two surprising examples illustrate that our new condition is optimal. When the condition holds, we show that the linearisation map identifies a region of the GIT fan with the Mori chamber decomposition of the relative movable cone of $V /\!/_θ G$. If $V/\!/_θ G$ is a crepant resolution of $Y\!\!:= V/\!/_{0} G$, then every projective crepant resolution of $Y$ is obtained by varying $θ$. Under suitable conditions, we show that this is the case for quiver varieties and hypertoric varieties. Similarly, for any finite subgroup $Γ\subset \mathrm{SL}(3,\mathbb{C})$ whose nontrivial conjugacy classes are all junior, we obtain a simple geometric proof of the fact that every projective crepant resolution of $\mathbb{C}^3/Γ$ is a fine moduli space of $θ$-stable $Γ$-constellations.

math.AG↗

arXiv2025

Singularities in Calogero--Moser Varieties

In this article we describe completely the singularities appearing in Calogero--Moser varieties associated (at any parameter) to the wreath product symplectic reflection groups. We do so by parameterizing the symplectic leaves in the variety, describing combinatorially the resulting closure relation and computing a transverse slice to each leaf. We also show that the normalization of the closure of each symplectic leaf is isomorphic to a Calogero--Moser variety for an associated (explicit) subquotient of the symplectic reflection group. This confirms a conjecture of Bonnafé for these groups. We use the fact that the Calogero--Moser varieties associated to wreath products can be identified with certain Nakajima quiver varieties. In particular, our result identifying the normalization of the closure of each symplectic leaf with another quiver variety holds for arbitrary quiver varieties.

math.AG↗

arXiv2025

Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties

We consider the category of modules over sheaves of Deformation-Quantization (DQ) algebras on bionic symplectic varieties. These spaces are equipped with both an elliptic $\mathbb{G}_m$-action and a Hamiltonian $\mathbb{G}_m$-action, with finitely many fixed points. On these spaces one can consider geometric category $\mathcal{O}$: the category of (holonomic) modules supported on the Lagrangian attracting set of the Hamiltonian action. We show that there exists a local generator in geometric category $\mathcal{O}$ whose dg endomorphism ring, cohomologically supported on the Lagrangian attracting set, is derived equivalent to the category of all DQ-modules. This is a version of Koszul duality generalizing the equivalence between D-modules on a smooth variety and dg-modules over the de Rham complex.

math.AG↗

arXiv2024

Symplectic resolutions of quiver varieties

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; we study their étale local structure and find their symplectic leaves. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT.

math.AG↗

arXiv2024

Categorical Cell Decomposition of Quantized Symplectic Algebraic Varieties

We prove a new symplectic analogue of Kashiwara's Equivalence from D-module theory. As a consequence, we establish a structure theory for module categories over deformation quantizations that mirrors, at a higher categorical level, the Bialynicki-Birula stratification of a variety with an action of the multiplicative group. The resulting categorical cell decomposition provides an algebro-geometric parallel to the structure of Fukaya categories of Weinstein manifolds. From it, we derive concrete consequences for invariants such as K-theory and Hochschild homology of module categories of interest in geometric representation theory.

math.AG↗

arXiv2024

The semi-invariant ring as the Cox ring of a GIT quotient

We study GIT quotients $X_θ=V\!/\!\!/\!_θG$ whose linearisation map defines an isomorphism between the group of characters of $G$ and the Picard group of $X_θ$ modulo torsion. Our main result establishes that the Cox ring of $X_θ$ is isomorphic to the semi-invariant ring of the $θ$-stable locus in $V$. This applies to quiver flag varieties, Nakajima quiver varieties, hypertoric varieties, and crepant resolutions of threefold Gorenstein quotient singularities with fibre dimension at most one. As an application, we present a simple, explicit calculation of the Cox ring of the Hilbert scheme of $n$-points in the affine plane.

math.AG↗

arXiv2024

Pull-back and push-forward functors for holonomic modules over Cherednik algebras

In this article we continue the study of holonomic modules over sheaves of Cherednik algebras, initiated by the third author in [Tho18]. Under mild assumptions on the parameters, we first develop a theory of b-functions to prove that push-forward along open embeddings preserves holonomicity. This implies that pull-back along closed embeddings also preserves holonomicity. We use these facts to show that both push-forward and pull-back under any melys morphism preserves holonomicity. Since duality preserves holonomicity, we deduce that extraordinary push-forward and extraordinary pull-back also exist for holonomic modules. As a consequence, we give a general classification of irreducible holonomic modules similar to the classification of irreducible holonomic $\mathscr{D}$-modules as minimal extensions of integrable connections on locally closed subsets. Finally, we prove that Ext-groups between holonomic modules are finite-dimensional and explore applications of our work to the classification of aspherical parameters and existence of finite-dimensional modules for sheaves of Cherednik algebras.

math.QA↗

arXiv2023

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

We demonstrate that the linear quotient singularity for the exceptional subgroup G in Sp(4,C) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of the quotient singularity C4/G as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the n-pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2n - 6; for example, we show that there are 1684 projective crepant resolutions when n = 6. We also prove that the resulting affine cones are not quotient singularities for n >= 6.

math.AG↗

arXiv2023

Coulomb Branches have symplectic singularities

We show that Coulomb branches for $3$-dimensional $\mathcal{N}=4$ supersymmetric gauge theories have symplectic singularities. This confirms a conjecture of Braverman-Finkelberg-Nakajima.

math.AG↗

arXiv2023

The Rank One property for free Frobenius extensions

A conjecture by the second author, proven by Bonnafé-Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $\mathbb{Q}$ when restricted to each block of the algebra. In this paper, we show that if $H$ is a prime algebra that is a free Frobenius extension over a regular central subalgebra $R$, and the centre of $H$ is normal Gorenstein, then each central quotient $A$ of $H$ by a maximal ideal $\mathfrak{m}$ of $R$ satisfies the rank one property with respect to the Cartan matrix of $A$. Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Skylanin algebras. In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture.

math.RT↗

arXiv2022

On Parabolic Subgroups of Symplectic Reflection Groups

Using Cohen's classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group are themselves symplectic reflection groups. This is the symplectic analogue of Steinberg's Theorem for complex reflection groups. Using computational results required in the proof, we show the non-existence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open. Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.

math.GR↗

arXiv2022

Cellularity of endomorphism algebras of tilting objects

We show that, in a highest weight category with duality, the endomorphism algebra of a tilting object is naturally a cellular algebra. Our proof generalizes a recent construction of Andersen, Stroppel, and Tubbenhauer. This result raises the question of whether all cellular algebras can be realized in this way. The construction also works without the presence of a duality and yields standard bases, in the sense of Du and Rui, which have similar combinatorial features to cellular bases. As an application, we obtain standard bases -- and thus a general theory of "cell modules" -- for Hecke algebras associated to finite complex reflection groups (as introduced by Broué, Malle, and Rouquier) via category $\mathcal{O}$ of the rational Cherednik algebra. For real reflection groups these bases are cellular.

math.RT↗