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Orthogonal polynomials associated with root systems. (English) Zbl 1032.33010
Summary: 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 ed., Oxford University Press (1995; Zbl 0824.05059), Chapter VI].

MSC:
33C80Connections of hypergeometric functions with groups and algebras
05E05Symmetric functions and generalizations
33C52Orthogonal polynomials and functions associated with root systems