Orthogonal Polynomials Associated with Root Systems

Ian G. Macdonald

Séminaire lotharingien de combinatoire, Tome 45 (2000-2001)

Résumé

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,...,t_r, 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 A_n, they conincide with the symmetric polynomials described in I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press (1995), Chapter VI.

Foreword

The text of the paper is that of my 1987 preprint with the above title. It is now in many ways a period piece, and I have thought it best to reproduce it unchanged. I am grateful to Tom Koornwinder and Christian Krattenthaler for arranging for its publication in the Séminaire Lotharingien de Combinatoire.

I should add that the subject has advanced considerably in the intervening years. In particular, the conjectures in Section 12 are now theorems. For a sketch of these later developments the reader may refer to my booklet "Symmetric functions and orthogonal polynomials", University Lecture Series Vol. 12, American Mathematical Society (1998), and the references to the literature given there.

Ian G. Macdonald, November 2000