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On the category O for rational Cherednik algebras
 Invent. Math
"... Abstract. We study the category O of representations of the rational Cherednik algebra AW attached to a complex reflection group W. We construct an exact functor, called KnizhnikZamolodchikov functor: O → HWmod, where HW is the (finite) IwahoriHecke algebra associated to W. We prove that the Kniz ..."
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Cited by 111 (14 self)
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Abstract. We study the category O of representations of the rational Cherednik algebra AW attached to a complex reflection group W. We construct an exact functor, called KnizhnikZamolodchikov functor: O → HWmod, where HW is the (finite) IwahoriHecke algebra associated to W. We prove that the KnizhnikZamolodchikov functor induces an equivalence between O/Otor, the quotient of O by the subcategory of AWmodules supported on the discriminant,
Cherednik algebras and differential operators on quasiinvariants
 Duke Math. J
"... We develop representation theory of the rational Cherednik algebra Hc associated to a finite Coxeter group W in a vector space h, and a parameter “c. ” We use it to show that, for integral values of “c, ” the algebra Hc is simple and Morita equivalent to D(h)#W, the cross product of W with the algeb ..."
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Cited by 69 (13 self)
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We develop representation theory of the rational Cherednik algebra Hc associated to a finite Coxeter group W in a vector space h, and a parameter “c. ” We use it to show that, for integral values of “c, ” the algebra Hc is simple and Morita equivalent to D(h)#W, the cross product of W with the algebra of polynomial differential operators on h. O. Chalykh, M. Feigin, and A. Veselov [CV1], [FV] introduced an algebra, Qc, of quasiinvariant polynomials on h, such that C[h] W ⊂ Qc ⊂ C[h]. We prove that the algebra D(Qc) of differential operators on quasiinvariants is a simple algebra, Morita equivalent to D(h). The subalgebra D(Qc) W ⊂ D(Qc) of Winvariant operators turns out to be isomorphic to the spherical subalgebra eHce ⊂ Hc. We show that D(Qc) is generated, as an algebra, by Qc and its “Fourier dual ” Q ♭ c, and that D(Qc) is a rankone projective (Qc ⊗ Q ♭ c)module (via multiplicationaction on D(Qc) on opposite sides).
qSchur algebras and complex reflection groups
"... Abstract. We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a qSchur algebra (parameter ̸ ∈ 1 2 + Z), providing thus character formulas for simple modules. We give some generalization to Bn(d). We prove an “abstract ” translation principle. These r ..."
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Cited by 59 (2 self)
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Abstract. We show that the category O for a rational Cherednik algebra of type A is equivalent to modules over a qSchur algebra (parameter ̸ ∈ 1 2 + Z), providing thus character formulas for simple modules. We give some generalization to Bn(d). We prove an “abstract ” translation principle. These results follow from the unicity of certain highest weight categories covering Hecke algebras. We also provide a semisimplicity criterion for Hecke algebras of complex reflection groups. 1.
Representations of rational Cherednik algebras
 CONTEMPORARY MATHEMATICS
"... This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category O. For type A, we explain relations with the Hilbert scheme of points on C². We insist on the analogy with ..."
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Cited by 44 (4 self)
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This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category O. For type A, we explain relations with the Hilbert scheme of points on C². We insist on the analogy with the representation theory of complex semisimple Lie algebras.
Cherednik algebras and Hilbert schemes in characteristic p
, 2006
"... We prove a localization theorem for the type An−1 rational Cherednik algebra Hc = H1,c(An−1) overFp, an algebraic closure of the finite field. In the most interesting special case where c ∈ Fp, we construct an Azumaya algebra Hc on Hilbn A2, the Hilbert scheme of n points in the plane, such that Γ(H ..."
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Cited by 40 (9 self)
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We prove a localization theorem for the type An−1 rational Cherednik algebra Hc = H1,c(An−1) overFp, an algebraic closure of the finite field. In the most interesting special case where c ∈ Fp, we construct an Azumaya algebra Hc on Hilbn A2, the Hilbert scheme of n points in the plane, such that Γ(Hilbn A2, Hc) = Hc. Our localization theorem provides an equivalence between the bounded derived categories of Hcmodules and sheaves of coherent Hcmodules on Hilbn A2, respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the HilbertChow morphism. This provides a link between our results and those of Bridgeland, King and Reid, and Haiman.
Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras
, 2007
"... We relate the representations of the rational Cherednik algebras associated with the complex reflection group µℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Zalgebra construction. This is done so that as the parameters defining the Cherednik algebra va ..."
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Cited by 30 (2 self)
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We relate the representations of the rational Cherednik algebras associated with the complex reflection group µℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Zalgebra construction. This is done so that as the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change. This construction motivates us to use the geometry of the quiver varieties to interpret the ordering function (the cfunction) used to define a highest weight structure on category O of the Cherednik algebra. This interpretation provides a natural partial ordering on O which we expect will respect the highest weight structure. This partial ordering has appeared in a conjecture of Yvonne on the composition factors in O and so our results provide a small step towards a geometric picture for that. We also interpret geometrically another ordering function (the afunction) used in the study of Hecke algebras. (The connection between Cherednik algebras and Hecke algebras is provided by the KZfunctor.) This is related to a conjecture of Bonnafé and Geck on equivalence classes of weight functions for Hecke algebras with unequal parameters since the classes should (and do for type B) correspond to the G.I.T. chambers defining the quiver varieties. As a result anything that can be defined via the quiver varieties,
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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Cited by 27 (8 self)
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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).
Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r, p, n)
 IN PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY. ARXIV:MATH/0612733
"... The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r, p, n). As a first application, we give a selfcontained and elementary proof of the analog for the groups G( ..."
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Cited by 20 (6 self)
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The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r, p, n). As a first application, we give a selfcontained and elementary proof of the analog for the groups G(r, p, n), with r> 1, of Gordon’s theorem (previously Haiman’s conjecture) on the diagonal coinvariant ring. We impose no restriction on p; the result for p ̸ = r has been proved by Vale using a technique analogous to Gordon’s. Because of the combinatorial application to Haiman’s conjecture, the paper is logically selfcontained except for standard facts about complex reflection groups. The main results should be accessible to mathematicians working in algebraic combinatorics who are unfamiliar with the impressive range of ideas used in Gordon’s proof of his theorem.
HarishChandra homomorphisms and symplectic reflection algebras for wreathproducts
, 2005
"... The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreathproduct in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existen ..."
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Cited by 18 (2 self)
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The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreathproduct in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized