Intertwining operators of double affine Hecke algebras

Continuing [C3,C4], we study the intertwining operators of double affine Hecke algebras H. They appeared in several papers (especially in [C2,C4,C6]). However for the first time here we apply them systematically to create the nonsymmetric [M3,C4] and symmetric [M2] Macdonald polynomials for arbitrary root systems and to start the theory of induced and co-spherical H-modules. The importance of this technique was clearly demonstrated in recent papers by F. Knop and S. Sahi [Kn],[KS],[S]. Using the intertwiners of the double affine Hecke algebras in the case of GL (dual to those considered in [C1,C2]) they proved the q, t-integrality conjecture by I. Macdonald [M1] and managed to establish the positivity of the coefficients of the Macdonald polynomials in the differential case. As to the integrality, we mention another approach based on the so-called Vinet operators (see [LV] and a recent work by Kirillov, Noumi), and the results by Garsia, Remmel, and Tesler. We do not try in this paper to get the best possible estimates for the denominators of the Macdonald polynomials (generally speaking, the problem looks more complicated than in the stable GL-case). However even rather straightforward analysis of the intertwiners gives a lot. For instance, it is enough to ensure the existence of the restricted Macdonald polynomials at roots of unity from [C3,C4], where we used less convenient methods based directly on the definition or on the recurrence relations. The technique of intertwiners combined with the (projective) action of GL(2,Z) from [C3] gives another proof of the norm and the evaluation formulas (see [C4]). Here the H-embedding of the space of nonsymmetric polynomials into the space of functions on the affine Weyl group W̃ ([C4], Proposition 5.2) plays a key role. The latter representation when restricted to the affine Hecke subalgebra turns into the classical one from [IM] as t is a power of p and q → 0 (W̃ is identified with the set of double cosets of the corresponding p-adic group with respect to the Iwahori subroup).