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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.
Enveloping algebras of Slodowy slices and the Joseph ideal
 J. Eur. Math. Soc. (JEMS
"... Abstract. Let G be a simple algebraic group over an algebraically closed field k of characteristic 0, and g = Lie G. Let (e, h, f) be an sl2triple in g with e being a long root vector in g. Let ( · , · ) be the Ginvariant bilinear form on g with (e, f) = 1 and let χ ∈ g ∗ be such that χ(x) = ( ..."
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Cited by 38 (6 self)
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Abstract. Let G be a simple algebraic group over an algebraically closed field k of characteristic 0, and g = Lie G. Let (e, h, f) be an sl2triple in g with e being a long root vector in g. Let ( · , · ) be the Ginvariant bilinear form on g with (e, f) = 1 and let χ ∈ g ∗ be such that χ(x) = (e, x) for all x ∈ g. Let S be the Slodowy slice at e through the adjoint orbit of e and let H be the enveloping algebra of S; see [30]. In this note we give an explicit presentation of H by generators and relations. As a consequence we deduce that H contains an ideal of codimension 1 which is unique if g is not of type A. Applying Skryabin’s equivalence of categories we then construct an explicit Whittaker model for the Joseph ideal of U(g). Inspired by Joseph’s Preparation Theorem we prove that there exists a homeomorphism between the primitive spectrum of H and the spectrum of all primitive ideals of infinite codimension in U(g) which respects Goldie rank and Gelfand–Kirillov dimension. We study highest weight modules for the algebra H and apply earlier results of Miličić–Soergel and Backelin to express the composition multiplicities of the Verma modules for H in terms of some inverse parabolic Kazhdan–Lusztig polynomials. Our results confirm in the minimal nilpotent case the de Vos–van Driel conjecture on composition multiplicities of Verma modules for finite Walgebras. We also obtain some general results on the enveloping algebras of Slodowy slices and determine the associated varieties of related primitive ideals of U(g). A sequel to this paper will treat modular aspects of this theory. 1.
On category O for the cyclotomic rational Cherednik algebras
, 2011
"... We study equivalences for category Op of the rational Cherednik algebras Hp of type Gℓ(n) = (µℓ) n ⋊ Sn: a highest weight equivalence between Op and Oσ(p) for σ ∈ Sℓ and an action of Sℓ on a nonempty Zariski open set of parameters p; a derived equivalence between Op and Op ′ whenever p and p ′ h ..."
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Cited by 20 (7 self)
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We study equivalences for category Op of the rational Cherednik algebras Hp of type Gℓ(n) = (µℓ) n ⋊ Sn: a highest weight equivalence between Op and Oσ(p) for σ ∈ Sℓ and an action of Sℓ on a nonempty Zariski open set of parameters p; a derived equivalence between Op and Op ′ whenever p and p ′ have integral difference; a highest weight equivalence between Op and a parabolic category O for the general linear group, under a nonrationality assumption on the parameter p. As a consequence, we confirm special cases of conjectures of Etingof and of Rouquier.
CENTER AND REPRESENTATIONS OF INFINITESIMAL HECKE ALGEBRAS OF sl2
"... Abstract. In this paper, we compute the center of the infinitesimal Hecke algebras Hz associated to sl2; then using nontriviality of the center, we study representations of these algebras in the framework of the BGG category O. We also discuss central elements in infinitesimal Hecke algebras over gl ..."
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Abstract. In this paper, we compute the center of the infinitesimal Hecke algebras Hz associated to sl2; then using nontriviality of the center, we study representations of these algebras in the framework of the BGG category O. We also discuss central elements in infinitesimal Hecke algebras over gl n and sp(2n) for all n. We end by proving an analogue of Duflo’s theorem for Hz. 1.
PRIMITIVE IDEALS, NONRESTRICTED REPRESENTATIONS AND FINITE WALGEBRAS
, 2006
"... Abstract. Let G be a simple algebraic group over C and g = Lie G. Let (e, h, f) be an sl2triple in g and ( · , · ) the Ginvariant bilinear form on g such that (e, f) = 1. Let χ ∈ g ∗ be such that χ(x) = (e, x) for all x ∈ g and let Hχ denote the enveloping algebra of the Slodowy slice e+Keradf ..."
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Cited by 13 (3 self)
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Abstract. Let G be a simple algebraic group over C and g = Lie G. Let (e, h, f) be an sl2triple in g and ( · , · ) the Ginvariant bilinear form on g such that (e, f) = 1. Let χ ∈ g ∗ be such that χ(x) = (e, x) for all x ∈ g and let Hχ denote the enveloping algebra of the Slodowy slice e+Keradf. Let I be a primitive ideal of the universal enveloping algebra U(g) whose associated variety is the closure of the coadjoint orbit Oχ. We prove in this note that if I has rational infinitesimal character, then there is a finitedimensional irreducible Hχmodule V such that I = AnnU(g) Qχ ⊗Hχ V), where Qχ is the generalised Gelfand–Graev gmodule associated with the triple (e, h, f). In conjunction with wellknown results of Barbasch and Vogan this implies that all finite Walgebras possess finite dimensional representations. 1.
FINITE DIMENSIONAL REPRESENTATIONS OF WALGEBRAS
, 2009
"... Walgebras of finite type are certain finitely generated associative algebras closely related to universal enveloping algebras of semisimple Lie algebras. In this paper we prove a conjecture of Premet that gives an almost complete classification of finite dimensional irreducible modules for Walge ..."
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Walgebras of finite type are certain finitely generated associative algebras closely related to universal enveloping algebras of semisimple Lie algebras. In this paper we prove a conjecture of Premet that gives an almost complete classification of finite dimensional irreducible modules for Walgebras. Also we get some partial results towards a conjecture by Ginzburg on their finite dimensional bimodules.