From Double Hecke algebra to Fourier transform

TL;DR

This paper develops a comprehensive theory of the one-dimensional Double Hecke algebra, exploring its applications to Fourier transforms, special polynomials, and algebraic structures, with a focus on classifying its finite-dimensional representations.

Contribution

It provides a systematic classification of finite-dimensional representations of the Double Hecke algebra for generic q and roots of unity, connecting various mathematical concepts.

Findings

Classification of finite-dimensional representations at generic q

Classification of finite-dimensional representations at roots of unity

Applications to Fourier transform and Macdonald polynomials

Abstract

The paper contains a systematic theory of the one-dimensional Double Hecke algebra, including applications to the difference Fourier transform, Macdonald's polynomials, Gaussian sums at roots of unity, and Verlinde algebras. The main result is the classification of finite-dimensional representations for generic q and at roots of unity.