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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 110 (13 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,
DUNKL OPERATORS FOR COMPLEX REFLECTION GROUPS
, 2001
"... Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameter family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the “rational Cherednik algebra”, and a natural contr ..."
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Cited by 73 (3 self)
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Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameter family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the “rational Cherednik algebra”, and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups G(m, p, N), the set of singular parameters in the parameter family of these structures is described explicitly, using the theory of nonsymmetric Jack polynomials.
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 67 (12 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).
Rational Cherednik algebras and Hilbert schemes
"... Abstract. Let Hc be the rational Cherednik algebra of type An−1 with spherical subalgebra Uc = eHce. Then Uc is filtered by order of differential operators with associated graded ring gr Uc = C[h ⊕ h ∗ ] W, where W is the nth symmetric group. Using the Zalgebra construction from [GS] it is also po ..."
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Cited by 58 (6 self)
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Abstract. Let Hc be the rational Cherednik algebra of type An−1 with spherical subalgebra Uc = eHce. Then Uc is filtered by order of differential operators with associated graded ring gr Uc = C[h ⊕ h ∗ ] W, where W is the nth symmetric group. Using the Zalgebra construction from [GS] it is also possible to associate to a filtered Hc or Ucmodule M a coherent sheaf Φ(M) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of Uc and Hc, and relate it to Hilb(n) and to the resolution of singularities τ: Hilb(n) → h ⊕ h ∗ /W. For example, we prove: • If c = 1/n, so that Lc(triv) is the unique onedimensional simple Hcmodule, then Φ(eLc(triv)) ∼ = OZn, where Zn = τ −1 (0) is the punctual Hilbert scheme. • If c = 1/n + k for k ∈ N then, under a canonical filtration on the finite dimensional module Lc(triv), gr eLc(triv) has a natural bigraded structure which coincides with that on H 0 (Zn, L k), where L ∼ = O Hilb(n)(1); this confirms conjectures of Berest, Etingof and Ginzburg. • Under mild restrictions on c, the characteristic cycle of Φ(e∆c(µ)) equals ∑ λ Kµλ[Zλ], where Kµλ are Kostka numbers and the Zλ are (known) irreducible components of τ −1 (h/W). Contents
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 58 (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.
Baby Verma modules for rational Cherednik algebras
 Bull. London Math. Soc
"... Abstract. Symplectic reflection algebras arise in many different mathematical disciplines: integrable systems, Lie theory, representation theory, differential operators, symplectic geometry. In this paper, we introduce baby Verma modules for symplectic reflection algebras of complex reflection group ..."
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Cited by 57 (9 self)
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Abstract. Symplectic reflection algebras arise in many different mathematical disciplines: integrable systems, Lie theory, representation theory, differential operators, symplectic geometry. In this paper, we introduce baby Verma modules for symplectic reflection algebras of complex reflection groups at parameter t = 0 (the so–called rational Cherednik algebras at parameter t = 0) and present their most basic properties. By analogy with the representation theory of reductive Lie algebras in positive characteristic, we believe these modules are fundamental to the understanding of the representation theory and associated geometry of the rational Cherednik algebras at parameter t = 0. As an example, we use baby Verma modules to solve one problem posed by Etingof and Ginzburg and partially solve another, [5], and give an elementary proof of a theorem of Finkelberg and Ginzburg, [6]. 1. Notation 1.1. Let W be a complex reflection group and h its reflection representation over C. Let S denote the set of complex reflections in W. Let ω be the canonical symplectic form on V = h ⊕ h ∗. For s ∈ S, let ωs by skew–symmetric form which coincides with ω on im(idV − s) and has ker(idV − s) as the radical. Let c: S − → C be a W–invariant function sending s to cs. The rational Cherednik
On the quotient ring by diagonal invariants
 Invent. Math
"... Abstract. For a Weyl group, W, and its reflection representation h, we find the character and Hilbert series for a quotient ring of C[h ⊕ h ∗ ] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman. 1. ..."
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Cited by 50 (4 self)
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Abstract. For a Weyl group, W, and its reflection representation h, we find the character and Hilbert series for a quotient ring of C[h ⊕ h ∗ ] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman. 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 39 (8 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.
Finite dimensional representations of rational Cherednik algebras
"... A complete classification and character formulas for finitedimensional irreducible representations of the rational Cherednik algebra of type A are given. Less complete results for other types are obtained. Links to the geometry of affine flag manifolds and Hilbert schemes are discussed. 1 Main resu ..."
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Cited by 37 (1 self)
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A complete classification and character formulas for finitedimensional irreducible representations of the rational Cherednik algebra of type A are given. Less complete results for other types are obtained. Links to the geometry of affine flag manifolds and Hilbert schemes are discussed. 1 Main results 1.1 Preliminaries Fix a finite Coxeter group W in a complex vector space h. Thus, h is the complexification of a real Euclidean vector space and W is generated by a finite set S ⊂ W of reflections s ∈ S with respect to certain hyperplanes {Hs}s∈S in that Euclidean space. For each s ∈ S, we choose a nonzero linear function αs ∈ h ∗ which vanishes on Hs (the positive root corresponding to s), and let α ∨ s = 2(αs, −)/(αs,αs) ∈ h be the corresponding coroot. The group W acts naturally on the set S by conjugation. Put ℓ: = dim C