Orthogonal Polynomials Associated with Root Systems

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W , such that S (but not necessarily R) is reduced. For each such pair (R, S) we construct a family of W -invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters q, t1, t2, . . . , tr, where r (= 1, 2 or 3) is the number of W -orbits in R. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and p-adic symmetric spaces. Also when R = S is of type An, they conincide with the symmetric polynomials described in I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press (1995), Chapter VI.