Macdonald's Evaluation Conjectures and Difference Fourier Transform

In the previous author's paper the Macdonald norm conjecture (including the famous constant term conjecture) was proved. This paper contains the proof of the remaining two (the duality and evaluation conjectures). The evaluation theorem is in fact a $q,t$-generalization of the classic Weyl dimension formula. As to the duality theorem, it states that the generalized trigonometric-difference zonal Fourier transform is self-dual (at least formally). We define this transform in terms of double affine Hecke algebras related to elliptic braid groups. The duality appeared to be directly connected with the transposition of the periods of an elliptic curve.Comment: 16 pg., AMSTe

Difference Macdonald-Mehta Conjecture

In the paper we formulate and verify a difference counterpart of the Macdonald-Mehta conjecture and its generalization for the Macdonald polynomials. Namely, we determine the Fourier transforms of the polynomials multiplied by the Gaussian, which is closely connected with the new difference Harish-Chandra theory.Comment: AMSTe

Macdonald's Evaluation Conjectures, Difference Fourier Transform, and Applications

This paper contains the proof of Macdonald's duality and evaluation conjectures, the definition of the difference Fourier transform, the recurrence theorem generalizing Pieri rules, and the action of GL(2,Z) on the Macdonald polynomials at roots of unity.Comment: AMSTe