Ivan Cherednik

Macdonald's evaluation conjectures and difference Fourier transform

In the previous authors 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

Quantum Knizhnik-Zamolodchikov equations and affine root systems

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Elliptic quantum many-body problem and double affine Knizhnik-Zamolodchikov equation

-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.

Inverse Harish-Chandra transform and difference operators

Quantum (difference) Knizhnik-Zamolodchikov equations [S1, FR] are generalized for theR-matrices from [Ch1] with the arguments in arbitrary root systems (and their formal counterparts). In particular, QKZ equations with certain boundary conditions are introducted. The self-consistency of the equations from [FR] and the cross-derivative integrability conditions for ther-matrix KZ equations from [Ch2] are obtained as corollaries. A difference counterpart of the quantum many-body problem connected with Macdonalds operators is defined as an application.

Difference Macdonald-Mehta Conjecture

The elliptic-matrix quantum Olshanetsky-Perelomov problem is introduced for arbitrary root systems by means of an elliptic version of the Dunkl operators. Its equivalence with the double affine generalization of the Knizhnik-Zamolodchikov equation (in the induced representations) is established.

DAHA and iterated torus knots

We apply a new technique based on double affine Hecke algebras to the Harish-Chandra theory of spherical zonal functions. The formulas for the Fourier transforms of the multiplications by the coordinates are obtained as well as a simple proof of the Harish-Chandra inversion theorem using the Opdam transform.

Difference-elliptic operators and root systems

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.

Diagonal coinvariants and double affine Hecke algebras

The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which incudes the polynomiality, duality and other properties of the DAHA superpolynomials. Presumably they coincide with the reduced stable Khovanov-Rozansky polynomials in the case of non-negative coefficients. The new theory matches well the classical theory of algebraic knots and (unibranch) plane curve singularities; the Puiseux expansion naturally emerges. The corresponding DAHA superpolynomials are expected to coincide with the reduced ones in the Oblomkov-Shende-Rasmussen Conjecture upon its generalization to arbitrary dominant weights. For instance, the DAHA uncolored superpolynomials at a=0, q=1 are conjectured to provide the Betti numbers of the Jacobian factors of the corresponding singularities.

Hankel transform via double Hecke algebra

Recently a new technique in the harmonic analysis on symmetric spaces was suggested based on certain remarkable representations of affine and double affine Hecke algebras in terms of Dunkl and Demazure operators instead of Lie groups and Lie algebras. In the classic case it resulted (among other applications) in a new theory of radial part of Laplace operators and their deformations including a related concept of the Fourier transform. In the present paper we demonstrate that the new technique works well even in the most general difference-elliptic case conjecturally corresponding to the

Irreducibility of perfect representations of double affine Hecke algebras

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