From Double Hecke algebra to Fourier transform

Double Affine Hecke Algebras and Difference Fourier Transforms

In the paper, we introduce and calculate difference Fourier transforms on representations of the double affine Hecke algebras in polynomilas, polynomials multiplied by the Gaussian, and various spaces of delta-functions including finite-dimensional ones, give a general description of the semisimple representations with a special consideration of the GL-case, and then gradually restrict ourselves with spherical, pseudo-unitary, and Fourier-invariant representations. The latter generalize the Verlinde algebras and lead to new Gauss-Selberg sums and Macdonald's eta-type identities.

椭圆Hecke代数的Langlands对偶

A Langlands duality of elliptic Hecke algebras

本文通过椭圆Hecke代数与动力学参数建立联络，建立椭圆affine Hecke代数表示范畴与Langlands对偶根datum对应范畴之间的傅里叶-穆卡伊 functor。

Hankel transform via double Hecke algebra

We use a degeneration of the 1D double affine Hecke algebra and the Dunkl operator to study systematically nonsymmetric Bessel functions and their truncations.

Difference Fourier transforms for nonreduced root systems

In the first part of the paper kernels are constructed which meromorphically extend the Macdonald-Koornwinder polynomials in their degrees. In the second part of the paper the kernels associated with rank one root systems are used to define nonsymmetric variants of the spherical Fourier transform on the quantum SU(1,1) group. Related Plancherel and inversion formulas are derived using double affine Hecke algebra techniques.

Introduction to double Hecke algebras

This paper is based on the introduction to the monograph ``Double affine Hecke algebras'' to be published by Cambridge University Press. The connections with Knizhnik-Zamolodchikov equations, Kac-Moody algebras, tau-function, harmonic analysis on symmetric spaces, and special functions are discussed. The rank one case is considered in detail including the classification of Verlinde algebras and their deformations, Gauss-Selberg integrals and Gaussian sums, a topological interpretation of DAHA, a relation of the rational DAHA to sl(2), and applications to the diagonal coinvariants. The last thr…

洗牌积、退化仿射Hecke代数与量子Toda格子

Shuffle Products, Degenerate Affine Hecke Algebras, and Quantum Toda Lattice

本文从代数角度重新识别GL_N的量子Toda格子与截断移位Yangian，并通过有限Miura变换和mirabolic约简方法提出新的代数构造。