Cherednik algebras and differential operators on quasi-invariants.
type: Item
instance of: scholarly article
title: Cherednik algebras and differential operators on quasi-invariants.
zbMATH Open document ID: 1067.16047
DOI: 10.1215/S0012-7094-03-11824-4
author: Victor Ginzburg
author: Yuri Berest
author: Pavel Etingof
published in: Duke Mathematical Journal
publication date: 2004-03-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0111005
review text: Let \(W\) be a finite Coxeter group generated by reflections in a finite-dimensional complex vector space \(\mathfrak h\). The rational Cherednik algebras associated to \(W\) is a family of associative algebras \(\{H_c(W)\}\) parametrized by the set of \(W\)-invariant complex multiplicities \(c\colon R\to\mathbb{C}\) on the system of roots \(R\subset{\mathfrak h}^*\) of \(W\). Specifically, for a fixed \(c\colon\alpha\to c_\alpha\), the algebra \(H_c=H_c(W)\) is generated by the vectors of \(\mathfrak{h,h}^*\), and the elements of \(W\) subject to the following relations \[ wxw^{-1}=w(x),\quad wyw^{-1}=w(y),\quad\forall y\in{\mathfrak h},\;x\in{\mathfrak h}^*,\;w\in W; \] \[ x_1x_2=x_2x_1,\quad y_1y_2=y_2y_1,\quad\forall y_1,y_2\in{\mathfrak h},\;x_1,x_2\in{\mathfrak h}^*; \] \[ yx-xy=\langle y,x\rangle-\sum_{\alpha\in R_+}c_\alpha\langle y,\alpha\rangle\langle\alpha^\vee,x\rangle s_\alpha,\quad\forall y\in{\mathfrak h},\;x\in{\mathfrak h}^*. \] Here, as usual, one writes \(\alpha^\vee\in{\mathfrak h}\) for the coroot, \(s_\alpha\in W\) for the reflection corresponding to the root \(\alpha\in R\), \(R_+\subset R\) for a choice of `positive' roots in \(R\), and \(\langle\cdot,\cdot\rangle\) for the canonical pairing between \(\mathfrak h\) and \({\mathfrak h}^*\). Furthermore, each \(H_c\) contains a distinguished subalgebra \(B_c:=eH_ce\), where \(e:=\tfrac 1{|W|}\sum w\) is the symmetrizing idempotent in \(\mathbb{C} W\subset H_c\). We call \(B_c=B_c(W)\) the spherical algebra associated to \((W,c)\). In this paper, the authors develop representation theory of the rational Cherednik algebras \(H_c\). The authors show that, for integral values of \(c\), the algebra \(H_c\) is simple and Morita equivalent to \({\mathcal D}({\mathfrak h})\#W\), the cross product of \(W\) with the algebra of polynomial differential operators on \(\mathfrak h\). For the algebra, \(Q_c\), of quasi-invariant polynomials on \(\mathfrak h\) introduced by \textit{O. A. Chalykh, M. Feigin} an…
Mathematics Subject Classification ID: 16S32
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 14L24
zbMATH DE Number: 2059609
zbMATH Keywords: Cherednik algebras
zbMATH Keywords: Hecke algebras
zbMATH Keywords: Coxeter groups
zbMATH Keywords: root systems
zbMATH Keywords: algebras of differential operators
zbMATH Keywords: algebras of quasi-invariants
zbMATH Keywords: spherical algebras
reviewed by: Chen Chengdong
describes a project that uses: VKCURVE
MaRDI profile type: Publication
arXiv ID: math/0111005
cites work: Q5563439
cites work: Q4387366
cites work: Q4804798
cites work: Q4237137
cites work: Q4398664
cites work: Darboux transformations for multidimensional Schrödinger operators
cites work: Commutative rings of partial differential operators and Lie algebras
cites work: Integrability in the theory of Schrödinger operator and harmonic analysis
cites work: When rings of differential operators are maximal orders
cites work: Q4024104
cites work: Q2785535
cites work: Q4888783
cites work: \({\mathcal D}_ n\)-modules with support on a curve
cites work: Differential-difference operators and monodromy representations of Hecke algebras
cites work: Singular Polynomials for Finite Reflection Groups
cites work: DUNKL OPERATORS FOR COMPLEX REFLECTION GROUPS
cites work: Q4408657
cites work: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism.
cites work: Q4531670
cites work: Q4818704
cites work: On primitive ideals.
cites work: On the category \(\mathcal O\) for rational Cherednik algebras.
cites work: BABY VERMA MODULES FOR RATIONAL CHEREDNIK ALGEBRAS
cites work: Projective modules in the category \(\mathcal O\) for the Cherednik algebra.
cites work: Differential Operators on some Singular Surfaces
cites work: Q4012124
cites work: Q4029513
cites work: Q4722269
cites work: Q5460223
cites work: Invariant Differential Operators and an Homomorphism of Harish-Chandra
cites work: Semi-simplicity of invariant holonomic systems on a reductive Lie algebra
cites work: Q4328336
cites work: Q3786586
cites work: Fixed rings of finite automorphism groups of associative rings
cites work: Some Rings of Differential Operators Which are Morita Equivalent to the Weyl Algebra A 1
cites work: Invariants under tori of rings of differential operators and related topics
cites work: Q4694963
cites work: Commutative subalgebras of the ring of differential operators on a curve
cites work: An example of a ring Morita equivalent to the Weyl algebra \(A_ 1\).
cites work: Differential Operators on an Affine Curve
cites work: Homological properties of the enveloping algebra U(Sl2)
cites work: Invariants of finite groups and their applications to combinatorics
cites work: Q4236280
cites work: Q3975997
cites work: Algebraic integrability for the Schrödinger equation and finite reflection groups
cites work: Invariant Differential Operators on a Reductive Lie Algebra and Weyl Group Representations
cites work: Collisions of Calogero-Moser particles and an adelic Grassmannian (with an appendix by I. G. Macdonald)
OpenAlex ID: W2001613603
description: scientific article