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Nonsymmetric Macdonald polynomials. (English) Zbl 0886.05121
Let $$R$$ be a root system of rank $$n$$. I. G. Macdonald [Sém. Bourbaki 1994/95, Exp. No. 797, Astérisque 237, 189-207 (1996; Zbl 0883.33008)] introduced nonsymmetric polynomials as eigenfunctions of the difference Dunkl operators. In the present paper the author studies the Macdonald nonsymmetric polynomials using the double affine Hecke algebras. He defines the Fourier pairing and establishes a duality theorem on Macdonald polynomials. The author proves a recurrence result for the product of a Macdonald polynomial with a monomial. An evaluation theorem of the Macdonald polynomials is proved by using the recurrence formula. The author further describes a projective action of $$\text{SL}_2(\mathbb{Z})$$ on the Macdonald polynomials.
 05E35 Orthogonal polynomials (combinatorics) (MSC2000) 33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) 33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, $$p$$-adic groups, Hecke algebras, and related topics