By Ivan CherednikThis can be a particular, primarily self-contained, monograph in a brand new box of primary significance for illustration concept, harmonic research, mathematical physics, and combinatorics. it's a significant resource of common information regarding the double affine Hecke algebra, also known as Cherednik's algebra, and its awesome functions. bankruptcy 1 is dedicated to the Knizhnik-Zamolodchikov equations hooked up to root platforms and their family to affine Hecke algebras, Kac-Moody algebras, and Fourier research. bankruptcy 2 includes a systematic exposition of the illustration conception of the one-dimensional DAHA. it's the least difficult case yet faraway from trivial with deep connections within the concept of specified features. bankruptcy three is ready DAHA in complete generality, together with functions to Macdonald polynomials, Fourier transforms, Gauss-Selberg integrals, Verlinde algebras, and Gaussian sums. This ebook is designed for mathematicians and physicists, specialists and scholars, all those that are looking to grasp the hot double Hecke algebra procedure.