Macdonald's evaluation conjectures and difference Fourier transform

Generalizing the characters of compact simple Lie groups, Ian Macdonald introduced in [M1,M2] and other works remarkable orthogonal symmetric polynomials dependent on the parameters q, t. He came up with three main conjectures formulated for arbitrary root systems. A new approach to the Macdonald theory was suggested in [C1] on the basis of (double) affine Hecke algebras and related difference operators. In [C2] the norm conjecture (including the celebrated constant term conjecture [M3]) was proved for all (reduced) root systems. This paper contains the proof of the remaining two (the duality and evaluation conjectures), the recurrence relations, and basic results on Macdonald’s polynomials at roots of unity. In the next paper the same questions will be considered for the non-symmetric polynomials