Abstract

We present explicit generators D1,⋯,Dn of an algebra of commuting difference operators in n variables with trigonometric coefficients. The algebra depends, apart from two scale factors, on five parameters. The operators are simultaneously diagonalized by Koornwinder's multivariable generalization of the Askey-Wilson polynomials. For special values of the parameters and via limit transitions, one obtains difference operators for the Macdonald polynomials that are associated with (admissible pairs of) the classical root systems: An,Bn,Cn,Dn and BCn. By sending the step size of the differences to zero, the difference operators reduce to known hypergeometric differential operators. This limit corresponds to sending q→1; the eigenfunctions reduce to the multivariable Jacobi polynomials of Heckman and Opdam. Physically the algebra can be interpreted as an integrable quantum system that generalizes the (trigonometric) Calogero-Moser systems related to classical root systems.