The article (really a small research monograph) deals with some special systems of \(q\)-difference equations which the authors call bispectral quantum Knizhnik-Zamolodchikov (KZ) equations and which they construct on the basis of double affine Hecke algebras of type \(GL_N\). These equations include, besides Cherednik’s quantum affine KZ equations associated to principal series representations of the underlying affine Hecke algebra, a compatible system of \(q\)-difference equations acting on the central character of the principal series representations.
The main considerations and arguments of the authors are connected with the construction of a meromorphic self-dual solution \(\Phi\) of the bispectral quantum KZ equations which, upon suitable specializations of the central character, reduces to symmetric self-dual Laurent polynomial solutions of quantum KZ equations.
Further, an explicit correspondence between solutions of bispectral quantum KZ equations and solutions of a particular bispectral problem for Ruijsenaars’ commuting trigonometric \(q\)-difference operators is presented; under this correspondence, \(\Phi\) becomes a self-dual Harish-Chandra series solution \(\Phi^+\) of the bispectral problem. Specializing the central character, as above, reduces from \(\Phi^+\) to the symmetric self-dual Macdonald polynomials.
Below the content of this article is presented: 1. Introduction; 2. The double affine Hecke algebra (The extended affine Weyl group, The extended affine Hecke algebra and Cherednik’s basis representation, intertwiners); 3. The bispectral quantum KZ equations (Construction of the cocycle, bispectral quantum KZ equations); 4. The explicit form of the bispectral quantum KZ equations (Formal principal series, the cocycle values, the \(q\)-connection matrices relation to quantum KZ equations); 5. Solutions of the bispectral quantum KZ equations (The leading term, the basic asymptotically free solution \(\Phi_\kappa\), duality, singularities, evaluation formula, consistency of the bispectral quantum KZ equations); 6. The correspondence with bispectral problems (The monodromy cocycle, the correspondence, bispectral Harish-Chandra series, specialized central character and Harish-Chandra series); 7. Polynomial theory (polynomial solutions of the quantum KZ equations, duality, relation to the basic asymptotically free solution, relation to symmetric self-dual Macdonald polynomials); Appendix: Holonomic systems of \(q\)-difference equations; References (48 items).