*Nikolai IORGOV*

Bogolyubov Institute for Theoretical Physics,

14b Metrologochna Str., 03143 Kyiv, UKRAINE

E-mail: iorgov@aep1.bitp.kiev.ua
**Quantum Ruijsenaars system and quantum symmetric spaces**

**Abstract:**

The aim of the talk is to present in an explicit form the radial components
of Casimir elements of *U*_{q}(gl(*n*)) corresponding
to the quantum analogues of homogeneous spaces *GL*(*n*)/*SO*(*n*)
and *GL*(2*n*)/*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
*U*_{q}(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.