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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,
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
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.
THE HILBERT SCHEME OF A PLANE CURVE SINGULARITY AND THE HOMFLY HOMOLOGY OF ITS LINK
, 2012
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Combinatorics, Symmetric Functions, and Hilbert Schemes
 CDM 2002: Current Developments in Mathematics, Intl
, 2003
"... We survey the proof of a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for Macdonald's symmetric functions, and the "n!" and "(n + 1) " conjectures relating Macdon ..."
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Cited by 30 (0 self)
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We survey the proof of a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for Macdonald's symmetric functions, and the "n!" and "(n + 1) " conjectures relating Macdonald polynomials to the characters of doublygraded Sn modules. To make the treatment selfcontained, we include background material from combinatorics, symmetric function theory, representation theory and geometry. At the end we discuss future directions, new conjectures and related work of Ginzburg, Kumar and Thomsen, Gordon, and Haglund and Loehr.
Almostcommuting variety, Dmodules, and Cherednik
"... We study a scheme M closely related to the set of pairs of n × nmatrices with rank 1 commutator. We show that M is a reduced complete intersection with n+ 1 irreducible components, which we describe. There is a distinguished Lagrangian subvariety Mnil ⊂ M. We introduce a category, C, of Dmodules w ..."
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Cited by 28 (5 self)
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We study a scheme M closely related to the set of pairs of n × nmatrices with rank 1 commutator. We show that M is a reduced complete intersection with n+ 1 irreducible components, which we describe. There is a distinguished Lagrangian subvariety Mnil ⊂ M. We introduce a category, C, of Dmodules whose characteristic variety is contained in Mnil. Simple objects of that category are
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,