Double Affine Hecke Algebras

Ivan Cherednik

Contents

0 Introduction

• Universality of Hecke algebras
• KZ and Kac-Moody algabras
• Double Hecke algebras
• DAHA in harmonic analysis
• Applications

1 KZ and QMBP

• Soliton connection
• Affine QZ equation
• Isomorphism theorems for AKZ
• Isomorphisms for QAKZ
• DAHA and Macdonald polynomials
• Abstract KZ and elliptic QMBP
• Harish-Chandra inversion
• Factorization of r-matrices
• Coinvariant, integral formulas

2 One-dimensional DAHA

• Overview
• Euler's integral and Gaussioan sum
• Imaginary integration
• Jackson and Gaussian sums
• Nonsymmetric Hankel transform
• Polynomial representation
• Four corollaries
• DAHA-Fourier transforms
• Finite dimensional modules
• Classification, Verlinde algebras
• Little double Hecke algebra
• DAHA and p-adic theory
• Degenerate DAHA

3 General theory

• Progenitors
• Affine Weyl groups
• Double Hecke algabras
• Macdonald polynomials
• Polynomial Fourier transforma
• Jackson integrals
• Semisimple representations
• The GL-case
• Spherical representations
• Induced and cospherical
• Gaussian and self-duality
• DAHA and double polynomials