Macdonald ’ s Evaluation Conjectures , Difference Fourier Transform , and applications

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. The evaluation conjecture (now a theorem) is in fact a q, t-generalization of the classic Weyl dimension formula. One can expect interesting applications of this theorem since the so-called q-dimensions are undoubtedly important. To demonstrate deep relations to the representation theory we prove the Recurrence Theorem connected with decomposing of the tensor products of represenations of compact Lie groups in terms of irreducible ones. The arising q, t-multiplicities are in fact the coefficients of our difference operators (one needs the duality to establish this). It is likely that we can incorporate the Kac-Moody case as well. The necessary technique was developed in [C4]. The duality theorem (in its complete form) states that the difference zonal q, t-Fourier transform is self-dual (its reproducing kernel is symmetric). In this paper we introduce the transform formally in terms of double affine Hecke algebras. The self-duality is directly related to the interpretation of these algebras via the so-called elliptic braid groups (the Fourier involution turns into the transposition of the periods of an elliptic curve). It is not very surprising since this interpretation is actually the monodromy representation of the double affine (elliptic) Knizhnik-Zamolodchikov equation from [C6].