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)


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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