Factorisation of Macdonald polynomials

1. Macdonald polynomials Macdonald polynomials P λ (x; q, t) are orthogonal symmetric polynomials which are the natural multivariable generalisation of the continuous q-ultraspherical polyno-mials C n (x; β|q) [2] which, in their turn, constitute an important class of hyper-geometric orthogonal polynomials in one variable. Polynomials C n (x; β|q) can be obtained from the general Askey-Wilson polynomials [3] through a specification of their four parameters (see, for instance, [9]), so that C n (x; β|q) depend only on one parameter β, apart from the degree n and the basic parameter q. In an analogous way, the Macdonald polynomials P λ (x; q, t) with one parameter t could be obtained as a limiting case of the 5-parameter Koornwinder's multivariable generalisation of the Askey-Wilson polynomials [10]. The main reference for the Macdonald polynomials is the book [17], Ch. VI, where they are called symmetric functions with two parameters. Let K = Q(q, t) be the field of rational functions in two indeterminants q, t; K[x] = K[x 1 ,. .. , x n ] be the ring of polynomials in n variables x = (x 1 ,. .. , x n) with coefficients in K; and K[x] W be the subring of all symmetric polynomials. The Macdonald polynomials P λ (x) = P λ (x; q, t) are symmetric polynomials labelled by the sequences λ = {0 ≤ λ 1 ≤ λ 2 ≤. .. ≤ λ n } of integers (dominant weights). They form a K basis of K[x] W and are uniquely characterised as joint eigenvectors of the commuting q-difference