Double Affine Hecke Algebras and Macdonald's Conjectures

Abstract

In this paper we prove the main results about the structure of double affine Hecke algebras announced in [C1], [C2]. The technique is based on the realization of these algebras in terms of Demazure-Lusztig operators [BGG], [D], [L2], [LS], [C3] and rather standard facts from the theory of affine Weyl groups. In particular, it completes the proof (partially published in [C2]) of the Macdonald scalar product conjecture (see [Ml], (12,6')), including the famous Macdonald constant-term conjecture (the q, t-case). We mainly follow the Opdam paper [0] where the Macdonald-Mehta conjectures in the degenerate (differential) case were deduced from certain properties of the Heckman-Opdam operators [HO] and the existence of the shift operators. Heckman's interpretation of these operators via the so-called Dunkl operators (see [He] and also [C5]) was important to our approach. We note that the HO operators are closely related to the so-called quantum many-body problem (Calogero, Sutherland, Moser, Olshanetsky, Perelomov), the conformal field theory (Knizhnik-Zamolodchikov equations), the harmonic analysis on symmetric spaces (Harish-Chandra, Helgason etc.), and (last but not the least) the classic theory of the hypergeometric functions. Establishing the connection between the difference counterparts of Heckman-Opdam operators introduced in [C4] and the Macdonald theory [Ml], [M2] including the construction of the difference shift operators is the main thrust of this paper. Once the connection is established it is not very difficult to calculate the scalar squares of the Macdonald polynomials and to prove the constant-term conjecture from his fundamental paper [M3]. To simplify the exposition, we discuss the reduced root systems only and impose the relation q = tk for k E Z+ (to avoid infinite products in the definition of Macdonald's pairing). The purpose of this work is to present a concrete application of the new technique. Arbitrary q, t can be handled in a