2000/11/08 by Ian G. Macdonald
Mathematics · #math.QA #math.CA #math.CO
published as Séminaire Lotharingien Combin. 45 (2000), Article B45a, 40 pp · 40 pages, AmS-TeX. This is the 1987 preprint of the same title that has been circulated privately only as a handwritten manuscript. It has now been typed and published in the Séminaire Lotharingien de Combinatoire
arxiv created 2000/11/08 · arxiv updated 2009/11/30
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.