COMMUTING DIFFERENCE-OPERATORS WITH POLYNOMIAL EIGENFUNCTIONS

We present explicit generators D,..., D(n) of an algebra of commuting difference operators in n variables with trigonometric coefficients. T he algebra depends, apart from two scale factors, on five parameters. The operators are simultaneously diagonalized by Koornwinder's multiva riable generalization of the Askey-Wilson polynomials. For special val ues of the parameters and via limit transitions, one obtains different operators for the Macdonald polynomials that are associated with (adm issible pairs of) the classical root systems: A(n-1), B(n), C(n), D(n) and BC(n). By sending the step size of the differences to zero, the d ifference operators reduce to known hypergeometric diffrential operato rs. This limit corresponds to sending q --> 1; the eigenfunctions redu ce to the multivariable Jacobi polynomials of Heckman and Opdam. Physi cally the algebra can be interpreted as an integrable quantum system t hat generalizes the (trigonometric) Calogero-Moser systems related to classical root systems.