Orthogonal polynomials associated with root systems

Details Orthogonal polynomials associated with root systems Orthogonal polynomials associated with root systems

2000-11-08

arxiv.org/abs/math/0011046v1

S\'eminaire Lotharingien Combin. 45 (2000), Article B45a, 40 pp

Mathematics

Quantum Algebra

40 pages, AmS-TeX. This is the 1987 preprint of the same title that has been circulated privately only as a handwritten manusc

Scientific paper

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 $q,t_1,t_2,...,t_r$, 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.

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