Affine Hecke algebras and orthogonal polynomials. (English) Zbl 0883.33008

Macdonald, I. G.

Séminaire Bourbaki. Volume 1994/95. Exposés 790-804. Paris: Société Mathématique de France, Astérisque. 237, 189-207, Exp. No. 797 (1996).

The paper is a review on the Macdonald orthogonal polynomials. They are orthogonal polynomials in several variables and depend on the additional two or more parameters. To each root system there corresponds at least one family of such orthogonal polynomials and they possess the symmetry of the corresponding Weyl group. For certain limiting values of the parameters, these polynomials give zonal spherical functions for semisimple Lie groups over the field of real numbers or over their \(p\)-adic counterpart. Some years ago the author of the paper showed how to construct these polynomials and formulated various conjectures about them. The author emphasizes that afterwards these conjectures have been proved in full generality by Cherednik using the theory of affine Hecke algebras. In the paper, the author gives a somewhat different proof for the formula for the scalar product of the polynomials.
For the entire collection see [Zbl 0851.00039].

Reviewer: A.Klimyk (Kiev)

MSC:

33C52	Orthogonal polynomials and functions associated with root systems
33D52	Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
20C08	Hecke algebras and their representations
17B22	Root systems

Keywords:

Macdonald orthogonal polynomials; Weyl group; root system; affine Hecke algebra