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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
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.
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 28 (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,
Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities
"... Abstract. We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter t = 0) and the derived category of coherent sheaves on a crepant resolution of the spectrum ..."
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Cited by 25 (3 self)
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Abstract. We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter t = 0) and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras. 1.
Symmetric functions, parabolic category O and the Springer fibre, preprint
"... Abstract. We prove that the center of a regular block of parabolic categoryOfor the Lie algebra gl n (C) is isomorphic to the cohomology algebra of a corresponding Springer fiber. This was conjectured by Khovanov. We also find presentations for the centers of singular blocks. Our approach is based o ..."
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Cited by 22 (2 self)
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Abstract. We prove that the center of a regular block of parabolic categoryOfor the Lie algebra gl n (C) is isomorphic to the cohomology algebra of a corresponding Springer fiber. This was conjectured by Khovanov. We also find presentations for the centers of singular blocks. Our approach is based on a new construction making the direct sum of the centers of all integral blocks of O into a polynomial representation of gl ∞(C), namely, the nth tensor power of its natural module. 1.
On singular CalogeroMoser spaces
"... Abstract. Using combinatorial properties of complex reflection groups we show that if the group W is different from the wreath product Sn ≀ Z/mZ and the binary tetrahedral group (labelled G(m, 1, n) and G4 respectively in the ShephardTodd classification), then the generalised CalogeroMoser space X ..."
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Cited by 17 (6 self)
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Abstract. Using combinatorial properties of complex reflection groups we show that if the group W is different from the wreath product Sn ≀ Z/mZ and the binary tetrahedral group (labelled G(m, 1, n) and G4 respectively in the ShephardTodd classification), then the generalised CalogeroMoser space Xc associated to the centre of the rational Cherednik algebra H0,c(W) is singular for all values of the parameter c. This result and a theorem of Ginzburg and Kaledin imply that there does not exist a symplectic resolution of the singular symplectic variety h × h ∗ /W when W is a complex reflection group different from Sn ≀ Z/mZ and the binary tetrahedral group (where h is the reflection representation associated to W). Conversely it has been shown by Etingof and Ginzburg that Xc is smooth for generic values of c when W ∼ = Sn ≀ Z/mZ. We show that this is also the case when W is the binary tetrahedral group. A theorem of Namikawa then implies the existence of a symplectic resolution in this case, completing the classification. Finally, we note that the above results together with work of Chlouveraki are consistent with a conjecture of Gordon and Martino on block partitions in the restricted rational Cherednik algebra. 1.
Orthogonal functions generalizing Jack polynomials
 Trans. Amer. Math. Soc
"... Abstract. The rational Cherednik algebra H is a certain algebra of differentialreflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S λ of W corresponds to a standard module M(λ) for H. This paper deals with the ..."
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Cited by 16 (9 self)
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Abstract. The rational Cherednik algebra H is a certain algebra of differentialreflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S λ of W corresponds to a standard module M(λ) for H. This paper deals with the infinite family G(r, 1, n) of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra t of H discovered by Dunkl and Opdam. In this case, the irreducible Wmodules are indexed by certain sequences λ of partitions. We first show that t acts in an upper triangular fashion on each standard module M(λ), with eigenvalues determined by the combinatorics of the set of standard tableaux on λ. As a consequence, we construct a basis for M(λ) consisting of orthogonal functions on C n with values in the representation S λ. For G(1, 1, n) with λ = (n) these functions are the nonsymmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of M(λ) in the case in which the orthogonal functions are all welldefined. A consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure. Finally, we show that for a certain choice of parameters there is a cyclic group of automorphisms of H so that the rational Cherednik algebra for G(r, p, n) is the fixed subalgebra. Our results therefore descend to the rational Cherednik algebra for G(r, p, n) by Clifford theory.
Braided doubles and rational Cherednik algebras
 Adv. Math
"... Abstract. We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasi ..."
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Cited by 13 (5 self)
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Abstract. We introduce and study a large class of algebras with triangular decomposition which we call braided doubles. Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. We classify braided doubles in terms of quasiYetterDrinfeld (QYD) modules over Hopf algebras which turn out to be a generalisation of the ordinary YetterDrinfeld modules. To each braiding (a solution to the braid equation) we associate a QYDmodule and the corresponding braided Heisenberg double — this is a quantum deformation of the Weyl algebra where the role of polynomial algebras is played by NicholsWoronowicz algebras. Our main result is that any rational Cherednik algebra canonically embeds into the braided Heisenberg double attached to the corresponding complex reflection group.
Calogero–Moser space, reduced rational Cherednik algeras and twosided cells. arXiv:math.RT/0703153
"... Abstract. We conjecture that the “nilpotent points ” of CalogeroMoser space for reflection groups are parametrised naturally by the twosided cells of the group with unequal parameters. The nilpotent points correspond to blocks of restricted Cherednik algebras and we describe these blocks in the ca ..."
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Cited by 12 (3 self)
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Abstract. We conjecture that the “nilpotent points ” of CalogeroMoser space for reflection groups are parametrised naturally by the twosided cells of the group with unequal parameters. The nilpotent points correspond to blocks of restricted Cherednik algebras and we describe these blocks in the case G = µℓ ≀ Sn and show that in type B our description produces an existing conjectural description of twosided cells. 1.