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A solution to quantum Knizhnik-Zamolodchikov equations and its application to eigenvalue problems of the Macdonald type. (English) Zbl 0889.17009

The quantum Knizhnik-Zamolodchikov (QKZ) equation was reconstructed in the framework of the extended affine Weyl groups and the extended affine Hecke algebras for arbitrary root systems. Cherednik and Kato made a bridge between the QKZ equations and eigenvalue problems of the Macdonald type: a weighted symmetric sum of the solutions to the former gives rise to a solution of the latter. However, Cherednik and Kato did not obtain concrete solutions.
The aim of this paper is (1) to give a solution of Cherednik’s \(A_{n-1}\)-type QKZ equation through integral representations, especially using \(q\)-Selberg-type integrals, and (2) to apply this result to investigate eigenvalue problems of the Macdonald type. It is supposed throughout the paper that \(q\) is a real number with \(0<q<1\).
Reviewer: A.Klimyk (Kyïv)

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
35Q40 PDEs in connection with quantum mechanics
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[1] K. Aomoto and Y. Kato, Connection formula of symmetric \(A\)-type Jackson integrals , Duke Math. J. 74 (1994), no. 1, 129-143. · Zbl 0802.33016 · doi:10.1215/S0012-7094-94-07406-1
[2] H. Awata, S. Odake, and J. Shiraishi, Integral representations of the Macdonald symmetric functions , preprint (q-alg/9506006). · Zbl 0849.05069
[3] I. Cherednik, Quantum Knizhnik-Zamolodchikov equations and affine root systems , Comm. Math. Phys. 150 (1992), no. 1, 109-136. · Zbl 0849.17025 · doi:10.1007/BF02096568
[4] I. Cherednik, Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators , Internat. Math. Res. Notices (1992), no. 9, 171-180. · Zbl 0770.17004 · doi:10.1155/S1073792892000199
[5] I. Cherednik, Induced representations of double affine Hecke algebras and applications , Math. Res. Lett. 1 (1994), no. 3, 319-337. · Zbl 0837.20052 · doi:10.4310/MRL.1994.v1.n3.a4
[6] I. Cherednik, Difference-elliptic operators and root systems , Internat. Math. Res. Notices (1995), no. 1, 43-58 (electronic). · Zbl 0824.17029 · doi:10.1155/S1073792895000043
[7] I. B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations , Comm. Math. Phys. 146 (1992), no. 1, 1-60. · Zbl 0760.17006 · doi:10.1007/BF02099206
[8] M. Jimbo, A \(q\)-analogue of \(U(\mathfrak g\mathfrak l(N+1))\), Hecke algebra, and the Yang-Baxter equation , Lett. Math. Phys. 11 (1986), no. 3, 247-252. · Zbl 0602.17005 · doi:10.1007/BF00400222
[9] S. Kato, \(R\)-matrix arising from affine Hecke algebras and its application to Macdonald’s difference operators , Comm. Math. Phys. 165 (1994), no. 3, 533-553. · Zbl 0820.17023 · doi:10.1007/BF02099422
[10] T. H. Koornwinder, Askey-Wilson polynomials for root systems of type \(BC\) , Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) ed. D. St. P. Richards, Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 189-204. · Zbl 0797.33014
[11] G. Lusztig, Affine Hecke algebras and their graded version , J. Amer. Math. Soc. 2 (1989), no. 3, 599-635. JSTOR: · Zbl 0715.22020 · doi:10.2307/1990945
[12] I. G. MacDonald, A new class of symmetric functions , Actes Séminaire Lotharingien, Publ. Inst. Rech. Math. Adv., Strasbourg, 1988, pp. 131-171.
[13] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials , Astérisque (1996), no. 237, Exp. No. 797, 4, 189-207, Séminaire Bourbaki, 47 (1994-95). · Zbl 0883.33008
[14] K. Mimachi, Connection problem in holonomic \(q\)-difference system associated with a Jackson integral of Jordan-Pochhammer type , Nagoya Math. J. 116 (1989), 149-161. · Zbl 0688.39002
[15] K. Mimachi and Y. Yamada, Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials , Comm. Math. Phys. 174 (1995), no. 2, 447-455. · Zbl 0842.17045 · doi:10.1007/BF02099610
[16] K. Mimachi and Y. Yamada, Singular vectors of Virasoro algebra in terms of Jack symmetric polynomials , Sūrikaisekikenkyūsho Kōkyūroku (1995), no. 919, 68-78. · Zbl 0900.17008
[17] T. Ōshima and H. Sekiguchi, Commuting families of differential operators invariant under the action of a Weyl group , J. Math. Sci. Univ. Tokyo 2 (1995), no. 1, 1-75. · Zbl 0863.43007
[18] A. N. Varchenko and V. O. Tarasov, Jackson integral representations of solutions of the quantized Knizhnik-Zamolodchikov equation , St. Petersburg Math. J. 6 (1975), 275-313. · Zbl 0824.33012
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