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44
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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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.
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.
Differential operators and Cherednik algebras
, 2007
"... Abstract. We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction [GS1]; the other involving quantum hamiltonian reduction of an algebra of differential operators [GG]. In the present ..."
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Cited by 14 (4 self)
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Abstract. We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction [GS1]; the other involving quantum hamiltonian reduction of an algebra of differential operators [GG]. In the present paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on Dmodules with the shift functor for the Cherednik algebra. That enables us to give a direct and relatively short proof of the key result [GS1, Theorem 1.4] without recourse to Haiman’s deep results on the n! theorem [Ha1]. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from [GG]. Contents
HeckeClifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type
, 2009
"... We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skewpolynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomialClifford subalgebras. There is ..."
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Cited by 12 (2 self)
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We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skewpolynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomialClifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Vermatype representations via Dunkl operators for these algebras.
Double affine Hecke algebras for the spin symmetric group
"... Abstract. We introduce a new class (in two versions, ¨ Hc and ¨ H − ) of rational double affine Hecke algebras (DaHa) associated to the spin symmetric group. We establish the basic properties of the algebras, such as PBW and Dunkl representation, and connections to Nazarov’s degenerate affine Hecke ..."
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Cited by 12 (6 self)
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Abstract. We introduce a new class (in two versions, ¨ Hc and ¨ H − ) of rational double affine Hecke algebras (DaHa) associated to the spin symmetric group. We establish the basic properties of the algebras, such as PBW and Dunkl representation, and connections to Nazarov’s degenerate affine HeckeClifford algebra and to a new degenerate affine Hecke algebra introduced here. We formulate a Morita equivalence between the two versions of rational DaHa’s. The trigonometric generalization of the above constructions is also formulated and its relation to the rational counterpart is established. 1.
On category O for the rational Cherednik algebra of G(m,1,n): the almost semisimple case
, 2008
"... We determine the structure of category O for the rational Cherednik algebra of G(m, 1, n) in the case where the KZ functor satisfies a condition called separating simples. We show that this condition holds whenever all but one of the simple objects in O has the maximum possible GelfandKirillov dim ..."
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Cited by 11 (1 self)
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We determine the structure of category O for the rational Cherednik algebra of G(m, 1, n) in the case where the KZ functor satisfies a condition called separating simples. We show that this condition holds whenever all but one of the simple objects in O has the maximum possible GelfandKirillov dimension (plus a mild genericity condition on the parameters). Our proof involves calculating the blocks of the ArikiKoike algebra in a special case.
Rational Cherednik algebras and diagonal coinvariants of G(m
 Department of Mathematics, University of Glasgow
"... Abstract. We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group W = G(m, p, n) and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category O for the rational ..."
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Abstract. We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group W = G(m, p, n) and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category O for the rational
Deformation quantization modules I. Finiteness and duality.
, 2009
"... Consider a ring K, a topological space X and a sheaf A on X of K[[�]]algebras. Assuming A �adically complete and without �torsion, we first show how to deduce a coherency theorem for complexes of Amodules from the corresponding property for complexes of A /�Amodules. We apply this result to pro ..."
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Cited by 9 (0 self)
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Consider a ring K, a topological space X and a sheaf A on X of K[[�]]algebras. Assuming A �adically complete and without �torsion, we first show how to deduce a coherency theorem for complexes of Amodules from the corresponding property for complexes of A /�Amodules. We apply this result to prove that, under a natural properness condition, the convolution of two coherent kernels over deformation quantization algebroids on complex Poisson manifolds is coherent. We also construct the dualizing complexes for such algebroids
Supports of representations of the rational Cherednik algebra of type A, arXiv:1012.2585
 P.S. : Université Paris 7, UMR CNRS 7586, F75013 Paris, E.V. : Université Paris 7, UMR CNRS 7586, F75013 Paris, Email address: shan@math.jussieu.fr, vasserot@math.jussieu.fr
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