Harish–Chandra homomorphisms and symplectic reflection algebras for wreath-products

Etingof, Pavel; Gan, Wee Liang; Ginzburg, Victor; Oblomkov, Alexei

doi:10.1007/s10240-007-0005-9

Cited by 30 publications

(32 citation statements)

References 25 publications

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“…The deformation quantization of the singularity C 2 /Z N is a member of many interesting families of algebras that appear in the mathematical literature such as finite Walgebras [113], symplectic reflection algebras [114,115], and hypertoric enveloping algebras [88]. The right boundary condition N ε produces a left module for the algebraĈ[M C ]…”

Section: Sqed Quantizedmentioning

confidence: 99%

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Bullimore

Dimofte

Gaiotto

et al. 2016

J. High Energ. Phys.

106

229

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Abstract:We introduce several families of N = (2, 2) UV boundary conditions in 3d N = 4 gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs-and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d N = 4 gauge theories and their boundary conditions, we propose a physical origin for symplectic duality -an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.

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Section: Sqed Quantizedmentioning

confidence: 99%

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Bullimore

Dimofte

Gaiotto

et al. 2016

J. High Energ. Phys.

106

229

View full text Add to dashboard Cite

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“…Consider the projective system of algebraic groups H l = Aut(E l ). Since End k (E l ) is a finite-dimensional k-vector space, the projective system above satisfies the Mittag-Leffler condition, i.e., for any integer l there exists an integer N ≥ l such that Im(H k → H l ) = Im(H N → H l ), ∀k > N. 7 In particular, Xŷ is equipped with an action of the formal completionT of T.…”

Section: Proposition 63 Assume That X → Y Is Proper and That Extmentioning

confidence: 99%

“…Moreover, he has (at least partly) described this order combinatorially in [13,Sections 6,7]. In order to formulate the description, recall the bijection τ s [13, 6.2], [16, 7.2.17] between the set P(r, n) of multipartitions, and the set P ν0 (|ν 0 | + rn) of usual partitions of size |ν 0 | + rn having r-core ν 0 .…”

Section: Lagrangian Componentsmentioning

confidence: 99%

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Wreath Macdonald polynomials and the categorical McKay correspondence

Bezrukavnikov

Finkelberg

Vologodsky

2014

Cambridge Journal of Mathematics

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with an Appendix by Vadim Vologodsky To the memory of Andrei ZelevinskyMark Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product S n (Z/rZ) n . He has proven the original conjecture by establishing the geometric statement about the Hilbert scheme, as a byproduct he obtained a derived equivalence between coherent sheaves on the Hilbert scheme and coherent sheaves on the orbifold quotient of A 2n by the symmetric group S n .A short proof of a similar derived equivalence for any symplectic quotient singularity has been obtained by the first author and Kaledin [2] via quantization in positive characteristic. In the present note we prove various properties of these derived equivalences and then deduce generalized Macdonald positivity for wreath products.

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“…This group naturally acts on C 2n (= (C 2 ) ⊕n ) by linear symplectomorphisms. It turns out that the corresponding symplectic reflection algebra eH 1,c e can be presented as a quantum Hamiltonian reduction, [EGGO,L5].…”

Section: Algebras Of Interestmentioning

confidence: 99%

Bernstein inequality and holonomic modules

Losev

2017

Advances in Mathematics

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Abstract. In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that, provided the algebraic fundamental groups of all leaves are finite, the generalized Bernstein inequality holds for the simple modules and turns into equality for holonomic simples. Under the same finiteness assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length, then any holonomic module has finite length. This allows to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions, for quantum Hamiltonian reductions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.

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Harish–Chandra homomorphisms and symplectic reflection algebras for wreath-products

Cited by 30 publications

References 25 publications

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory

Wreath Macdonald polynomials and the categorical McKay correspondence

Bernstein inequality and holonomic modules

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