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Macdonald’s evaluation conjectures and difference Fourier transform. (English) Zbl 0854.22021
Invent. Math. 122, No. 1, 119-145 (1995); Erratum ibid. 125, No. 2, 391 (1996).
It is known that in the theory of Macdonald’s orthogonal symmetric polynomials depending on the (difference Fourier transform) parameters \(q\), \(t\) and generalizing the characters of compact simple Lie groups three main conjectures have been formulated (Macdonald, 1988). One of them, the norm conjecture, has been proved recently (Cherednik, 1995) in the framework of a new approach (Cherednik, 1992) to the Macdonald’s theory which is based on double affine Hecke algebras and related difference operators. The proof of the other two conjectures, the duality and evaluation conjectures, as well as the recurrence relations and other basic results on Macdonald’s polynomials at roots of unity are presented in the paper.
Reviewer: A.A.Bogush (Minsk)

22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A40 Character groups and dual objects
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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