Double Hecke algebras

This paper is based on the introduction to the monograph ”Double affine Hecke algebras” to be published by Cambridge University Press. It is based on a series of lectures delivered by the author in Kyoto (1996–1997), at University Paris 7 (1997–1998), at Harvard University in 2001, and in several other places, including recent talks at the conferences ”Quantum Theory and Symmetries 3” (Cincinnati, 2003), ”Geometric methods in algebra and number theory” (Miami, 2003), and also at RIMS (Kyoto University), MIT, and UC at San Diego in 2004. The connections with Knizhnik–Zamolodchikov equations, Kac–Moody algebras, harmonic analysis on symmetric spaces, special functions are discussed. We demonstrate that the τ–function from soliton theory is a generic solution of the so-called r–matrix KZ with respect to the Sugawara L−1– operators, which is an important part of the theory of integral formulas of the KZ equations. The double affine Hecke algebra (DAHA) of type A1 is considered in detail including the classification of the nonsymmetric Verlinde algebras, their deformations, Gauss–Selberg integrals and Gaussian sums, the topological interpretation, the relation of the rational DAHA to sl(2), and recent applications to the diagonal coinvariants. The last three sections of the paper are devoted to the general DAHA, its origins in the classical p–adic theory of affine Hecke algebras, the trigonometric and rational degenerations, and applications to the Harish-Chandra theory.