Quantum Ruijsenaars system and quantum symmetric spaces
The aim of the talk is to present in an explicit form the radial components of Casimir elements of Uq(gl(n)) corresponding to the quantum analogues of homogeneous spaces GL(n)/SO(n) and GL(2n)/Sp(n). These radial components lead to commuting set of Hamiltonians of the quantum trigonometric n-particle Ruijsenaars model. Explicit formulas for invariant vectors with respect to a q-deformation of U(so(n)) in representation spaces of Uq(gl(n)) are presented. They are used to obtain common eigenfunctions (which are, in fact, Macdonald polynomials) of commuting set of Hamiltonians of Ruijsenaars model.