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Macdonald's Evaluation Conjectures and Difference Fourier Transform

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arxiv q-alg/9412016 v1 pith:SVFPGXSY submitted 1994-12-30 q-alg math.QA

Ivan Cherednik This is my paper · ORCID
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keywords dualityevaluationtransformconjectureconjecturesellipticfouriermacdonald
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abstract

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.

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Macdonald deformation of Vogel's universality and link hyperpolynomials

    hep-th 2025-05 conditional novelty 6.0 of 10

    For the adjoint square in ADE Lie algebras, products of Macdonald dimensions with deformed Littlewood-Richardson coefficients are universal, yielding universal formulas for T[2,2n] link hyperpolynomials.

  2. A basic triad in Macdonald theory

    hep-th 2024-11 conditional novelty 6.0 of 10

    At t=q^{-m}, the Noumi-Shiraishi series reproduces the Baker-Akhiezer function, completing a triad with the Macdonald polynomials.

  3. Non-commutative creation operators for symmetric polynomials

    hep-th 2025-08 unverdicted novelty 5.0 of 10

    Non-commutative creation operators B̂_m are built for symmetric polynomials in matrix and Fock representations of W_{1+∞} and affine Yangian algebras.

  4. On Refined Vogel's universality

    hep-th 2025-04 conditional novelty 5.0 of 10

    For the simply laced simple Lie algebras (A_n, D_n, E_6, E_7, E_8), the adjoint Macdonald dimension is captured by one universal rational function of Vogel's parameters.

  5. Integrable systems inspired by DAHA and DIM algebra: type $C^\vee C$ versus type $A$

    hep-th 2026-07 accept novelty 4.5 of 10

    Type C∨C DAHA and Koornwinder systems mirror type-A Macdonald structures for Hamiltonians, recursions, evaluations and dualities, but lack a usable Noumi-Shiraishi-style universal series and SL(2,Z)-type twisting auto...

  6. Vogel's universality and Macdonald dimensions

    hep-th 2025-07 conditional novelty 4.0 of 10

    The paper gives a single rational formula for adjoint Macdonald dimensions that unifies the simply laced Lie algebras A_n, D_n, E6, E7, E8, plus explicit mixed-root-system dimension formulas.

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