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12
Macdonald's Polynomials And Representations Of Quantum Groups
 Math. Res. Let
, 1994
"... this paper we present a formula for Macdonald's polynomials which arises from the representation theory of the quantum group U q (gl n ). This formula expresses Macdonald's polynomials as vectorvalued characters  (weighted) traces of intertwining operators between certain modules over U ..."
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this paper we present a formula for Macdonald's polynomials which arises from the representation theory of the quantum group U q (gl n ). This formula expresses Macdonald's polynomials as vectorvalued characters  (weighted) traces of intertwining operators between certain modules over U q (gl n ). This result was announced in [EK]. It is an interesting problem to find relation between this construction and a recent paper of Noumi ([No]) which gives interpretation of Macdonald's polynomials for special values of k as zonal spherical functions on a homogeneous space for a quantum group
On Inner Product In Modular Tensor Categories. II Inner Product On Conformal Blocks.
 I & II, math.QA/9508017 and qalg/9611008
, 1995
"... this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras or, equivalently, from WessZuminoWitten model of conformal field theory. We briefly recall construction of this category, first suggested by Moore and Seiberg (see [MS1,2]) and la ..."
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Cited by 43 (0 self)
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this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras or, equivalently, from WessZuminoWitten model of conformal field theory. We briefly recall construction of this category, first suggested by Moore and Seiberg (see [MS1,2]) and later refined by Kazhdan and Lusztig ([KL14]) and Finkelberg ([F]) in Section 9. In particular, spaces of homomorphisms in this category are the spaces of conformal blocks of WZW model. Thus, the general theory developed in Section 2 of [K] gives us an inner product on the space of conformal blocks, and so defined inner product is modular invariant. This definition is constructive: we show how it can be rewritten so that it only involves Drinfeld associator, or, equivalently, asymptotics of solutions of KnizhnikZamolodchikov equations. Since there are integral formulas for the solutions of KZ equations, this shows that the inner product on the space of conformal blocks can be written explicitly in terms of certain integrals. In the case g = sl 2 these integrals can be calculated (see [V]), using Selberg integral, and the answer is written in terms of \Gammafunctions. Thus, in this case we can write explicit formulas for inner product on the space of conformal blocks. These expressions are 1991 Mathematics Subject Classification. Primary 81R50, 05E35, 18D10; Secondary 57M99
Three formulas for eigenfunctions of integrable Schrödinger operators
, 1995
"... Abstract. We give three formulas for meromorphic eigenfunctions (scattering states) of Sutherland’s integrable Nbody Schrödinger operators and their generalizations. The first is an explicit computation of the Etingof–Kirillov traces of intertwining operators, the second an integral representation ..."
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Abstract. We give three formulas for meromorphic eigenfunctions (scattering states) of Sutherland’s integrable Nbody Schrödinger operators and their generalizations. The first is an explicit computation of the Etingof–Kirillov traces of intertwining operators, the second an integral representation of hypergeometric type, and the third of Bethe ansatz type. The last two formulas are degenerations of elliptic formulas obtained previously in connection with the Knizhnik–Zamolodchikov– Bernard equation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctions are parametrized by a “Hermite–Bethe ” variety, a generalization of the spectral variety of the Lamé operator. We also give the qdeformed version of our first formula. In the scalar slN case, this gives common eigenfunctions of the commuting Macdonald–Rujsenaars difference operators. 1.
Algebraic integrability of Schrödinger operators and representations of Lie algebras
, 1994
"... hepth 9403135 ..."
RepresentationTheoretic Proof Of The Inner Product And Symmetry Identities For Macdonald's Polynomials
 Compositio Math
, 1996
"... This paper is a continuation of our papers [EK1, EK2]. In [EK2] we showed that for the root system An\Gamma1 one can obtain Macdonald's polynomials  a new interesting class of symmetric functions recently defined by I. Macdonald [M1]  as weighted traces of intertwining operators between cer ..."
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Cited by 12 (1 self)
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This paper is a continuation of our papers [EK1, EK2]. In [EK2] we showed that for the root system An\Gamma1 one can obtain Macdonald's polynomials  a new interesting class of symmetric functions recently defined by I. Macdonald [M1]  as weighted traces of intertwining operators between certain finitedimensional representations of U q sl n . The main goal of the present paper is to use this construction to give a representationtheoretic proof of Macdonald's inner product and symmetry identities for the root system An\Gamma1 . Macdonald's inner product identities (see [M2]) have been proved by combinatorial methods by Macdonald (unpublished) for the root system An\Gamma1 and by Cherednik in the general case; symmetry identities for the root system An\Gamma1 have also been proved by Macdonald ([Macdonald, private communication]). The paper is organized as follows. In Section 1 we briefly list the basic definitions. In Section 2 we define Macdonald's polynomials P and recall the construction of
Central Elements For Quantum Affine Algebras And Affine Macdonald's Operators
"... We describe a generalization of Drinfeld's description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald's difference operators. ..."
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We describe a generalization of Drinfeld's description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald's difference operators.
Integral formulas for wave functions of quantum manybody problems and representations of gln
"... hepth 9405038 We derive explicit integral formulas for eigenfunctions of quantum integrals of the CalogeroSutherlandMoser operator with trigonometric interaction potential. In particular, we derive explicit formulas for Jack’s symmetric functions. To obtain such formulas, we use the representatio ..."
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hepth 9405038 We derive explicit integral formulas for eigenfunctions of quantum integrals of the CalogeroSutherlandMoser operator with trigonometric interaction potential. In particular, we derive explicit formulas for Jack’s symmetric functions. To obtain such formulas, we use the representation of these eigenfunctions by means of traces of intertwining operators between certain modules over the Lie algebra gln, and the realization of these modules on functions of many variables.
Cental elements for quantum affine . . .
, 1994
"... We describe a generalization of Drinfeld’s description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald’s difference operators. ..."
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We describe a generalization of Drinfeld’s description of the center of a quantum group to the case of quantum affine algebras. We use the obtained central elements to construct the affine analogue of Macdonald’s difference operators.