On the quantum Ruijs']jsenaars model, some quantum homogeneous spaces and Hecke algebras

The aim of the article is to derive in the explicit form the radial components of Casimir elements of U-q(gl(n)) corresponding to a quantum analogue of the homogeneous space GL(n)/SO(n). They coincide with the Macdonald-Ruijsenaars difference operators (MRDOs), if one starts from a special set of Casimir elements from the center of U-q(gl(n)). The derivation is essentially based on Cherednik's approach to MRDOs by means of affine Hecke algebras. From the other side, MRDOs coincide with commuting Hamiltonians of quantum trigonometric n-particle Ruijsenaars model.