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Cherednik and Hecke algebras of varieties with a finite group action
"... Let h be a finite dimensional complex vector space, and G be a finite subgroup of GL(h). To this data one can attach a family of algebras Ht,c(h, G), called the rational Cherednik algebras (see [EG]); for t = 1 it provides the universal deformation of G ⋉ D(h) (where D(h) is the algebra of different ..."
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Let h be a finite dimensional complex vector space, and G be a finite subgroup of GL(h). To this data one can attach a family of algebras Ht,c(h, G), called the rational Cherednik algebras (see [EG]); for t = 1 it provides the universal deformation of G ⋉ D(h) (where D(h) is the algebra of differential operators on h).
Morita Equivalence of Cherednik Algebras
 J. Reine Angew. Math
"... Abstract. We classify the rational Cherednik algebras Hc(W) (and their spherical subalgebras) up to isomorphism and Morita equivalence in case when W is the symmetric group and c is a generic parameter value. 1. ..."
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Abstract. We classify the rational Cherednik algebras Hc(W) (and their spherical subalgebras) up to isomorphism and Morita equivalence in case when W is the symmetric group and c is a generic parameter value. 1.
Generalized Jack polynomials and the representation theory of rational Cherednik algebras
 Selecta Math. (N.S
"... Abstract. We apply the DunklOpdam operators and generalized Jack polynomials to study category Oc for the rational Cherednik algebra of type G(r, p, n). We determine the set of aspherical values and, in case p = 1, answer a question of Iain Gordon on the ordering of category Oc. 1. ..."
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Abstract. We apply the DunklOpdam operators and generalized Jack polynomials to study category Oc for the rational Cherednik algebra of type G(r, p, n). We determine the set of aspherical values and, in case p = 1, answer a question of Iain Gordon on the ordering of category Oc. 1.
Induced and simple modules of double affine Hecke algebras
 Duke Math. J
"... Abstract. We classify the simple integrable modules of double affine Hecke algebras via perverse sheaves. We get also some estimate for the JordanHölder multiplicities of induced modules. Contents 1. ..."
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Abstract. We classify the simple integrable modules of double affine Hecke algebras via perverse sheaves. We get also some estimate for the JordanHölder multiplicities of induced modules. Contents 1.
A remark on rational Cherednik algebras and differential operators on the cyclic quiver
 Glasgow Math. J
"... Abstract. We show that the spherical subalgebra Uk,c of the rational Cherednik algebra associated to Sn ≀ Cℓ, the wreath product of the symmetric group and the cyclic group of order ℓ, is isomorphic to a quotient of the ring of invariant differential operators on a space of representations of the cy ..."
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Abstract. We show that the spherical subalgebra Uk,c of the rational Cherednik algebra associated to Sn ≀ Cℓ, the wreath product of the symmetric group and the cyclic group of order ℓ, is isomorphic to a quotient of the ring of invariant differential operators on a space of representations of the cyclic quiver of size ℓ. This confirms a version of [EG, Conjecture 11.22] in the case of cyclic groups. The proof is a straightforward application of work of Oblomkov, [O], on the deformed Harish–Chandra homomorphism, and of Crawley–Boevey, [CB1] and [CB2], and Gan and Ginzburg, [GG], on preprojective algebras. 1.
Quasi–invariants of complex reflection groups
"... We introduce quasiinvariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasiinvariants of a given multiplicity is not, in general, an algebra but a module Qk over the coordinate ring of some (singular) affine variety Xk. We extend the ma ..."
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We introduce quasiinvariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasiinvariants of a given multiplicity is not, in general, an algebra but a module Qk over the coordinate ring of some (singular) affine variety Xk. We extend the main results of [BEG] to this setting: in particular, we show that the variety Xk and the module Qk are CohenMacaulay, and the rings of differential operators on Xk and Qk are simple rings, Morita equivalent to the Weyl algebra An(C), where n = dim Xk. Our approach relies on representation theory of complex Cherednik algebras introduced in [DO] and is parallel to that of [BEG]. As an application, we prove the existence of shift operators for an arbitrary complex reflection group, confirming a conjecture of Dunkl and Opdam [DO]. Another result is a proof of a conjecture of Opdam [O2], concerning certain operations (KZ twists) on the set of irreducible representations of W.
Zhedanov’s algebra AW(3) and the double affine Hecke algebra in the rank one case. II. The spherical subalgebra
 SIGMA Symmetry Integrability Geom. Methods Appl. 4 (2008), Paper 052
"... This paper builds on the previous paper by the author, where a relationship between Zhedanov’s algebra AW(3) and the double affine Hecke algebra (DAHA) corresponding to the AskeyWilson polynomials was established. It is shown here that the spherical subalgebra of this DAHA is isomorphic to AW(3) wi ..."
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This paper builds on the previous paper by the author, where a relationship between Zhedanov’s algebra AW(3) and the double affine Hecke algebra (DAHA) corresponding to the AskeyWilson polynomials was established. It is shown here that the spherical subalgebra of this DAHA is isomorphic to AW(3) with an additional relation that the Casimir operator equals an explicit constant. A similar result with qshifted parameters holds for the antispherical subalgebra. Some theorems on centralizers and centers for the algebras under consideration will finally be proved as corollaries of the characterization of the spherical and antispherical subalgebra. 1
On category O for the rational Cherednik algebra of G(m,1,n): the almost semisimple case
, 2008
"... We determine the structure of category O for the rational Cherednik algebra of G(m, 1, n) in the case where the KZ functor satisfies a condition called separating simples. We show that this condition holds whenever all but one of the simple objects in O has the maximum possible GelfandKirillov dim ..."
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We determine the structure of category O for the rational Cherednik algebra of G(m, 1, n) in the case where the KZ functor satisfies a condition called separating simples. We show that this condition holds whenever all but one of the simple objects in O has the maximum possible GelfandKirillov dimension (plus a mild genericity condition on the parameters). Our proof involves calculating the blocks of the ArikiKoike algebra in a special case.
Rational Cherednik algebras and diagonal coinvariants of G(m
 Department of Mathematics, University of Glasgow
"... Abstract. We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group W = G(m, p, n) and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category O for the rational ..."
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Abstract. We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group W = G(m, p, n) and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category O for the rational