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28
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.
Generalized Jack polynomials and the representation theory of rational Cherednik algebras
 Selecta Math. (N.S
"... Abstract. We apply the DunklOpdam operators and generalized Jack polynomials to study category Oc for the rational Cherednik algebra of type G(r, p, n). We determine the set of aspherical values and, in case p = 1, answer a question of Iain Gordon on the ordering of category Oc. 1. ..."
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Cited by 17 (5 self)
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Abstract. We apply the DunklOpdam operators and generalized Jack polynomials to study category Oc for the rational Cherednik algebra of type G(r, p, n). We determine the set of aspherical values and, in case p = 1, answer a question of Iain Gordon on the ordering of category Oc. 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.
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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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.
Quantum toroidal algebras and their representations
, 2008
"... Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine KacMoody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to whom they are related via Sc ..."
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Cited by 12 (3 self)
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Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine KacMoody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to whom they are related via SchurWeyl duality. In this review paper, we give a glimpse on some aspects of their very rich representation theory in the context of general quantum affinizations. We illustrate with several examples. We also announce new results and explain possible further developments, in particular on finite dimensional representations at roots of unity.
CYCLOTOMIC DOUBLE AFFINE HECKE ALGEBRAS AND AFFINE PARABOLIC CATEGORY O
, 2008
"... Using the orbifold KZ connection we construct a functor from an affine ..."
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Using the orbifold KZ connection we construct a functor from an affine
The CalogeroMoser partition and Rouquier families for complex reflection groups
 J. Algebra
"... Abstract. Let W be a complex reflection group. We formulate a conjecture relating blocks of the corresponding restricted rational Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. We verify the conjecture in the case that W is a wreath product of a symmetric group with a cyclic ..."
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Cited by 11 (4 self)
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Abstract. Let W be a complex reflection group. We formulate a conjecture relating blocks of the corresponding restricted rational Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. We verify the conjecture in the case that W is a wreath product of a symmetric group with a cyclic group of order l. 1.
A relation for domino RobinsonSchensted algorithms
"... Abstract. We describe a map relating hyperoctahedral RobinsonSchensted algorithms on standard domino tableaux of unequal rank. Iteration of this map relates the algorithms defined by Garfinkle and Stanton–White and when restricted to involutions, this construction answers a question posed by M. A. ..."
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Cited by 8 (5 self)
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Abstract. We describe a map relating hyperoctahedral RobinsonSchensted algorithms on standard domino tableaux of unequal rank. Iteration of this map relates the algorithms defined by Garfinkle and Stanton–White and when restricted to involutions, this construction answers a question posed by M. A. A. van Leeuwen. The principal technique is derived from operations defined on standard domino tableaux by D. Garfinkle which must be extended to this more general setting.