2004/04/17 by Ivan Cherednik, Cherednik, Ivan
Mathematics · Physics and Astronomy · #05E05 #11T24 #14H52 #14J25 #14M12 #14M15 #16S90 #20B30 #20F34 #20F36 #22Exx #33Cxx #33Dxx #55R80 #81Rxx #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math-ph #math.CO #math.GT #math.MP #math.QA #math.RT #msc:05E05 #msc:11T24 #msc:14H52 #msc:14J25 #msc:14M12 #msc:14M15 #msc:16S90 #msc:20B30 #msc:20F34 #msc:20F36 #msc:22Exx #msc:33Cxx #msc:33Dxx #msc:55R80 #msc:81Rxx
paper · pdf · doi:10.48550/arxiv.math/0404307
LaTeX, 93 pgs, 7 figures, a significantly extended variant
openalex publication_date 2004/04/17 · arxiv created 2004/09/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 three sections are devoted to relations of the general DAHAs to the p-adic affine Hecke algebras, trigonometric and rational DAHAs, and applications to the Harish-Chandra theory. The purpose of this introduction is a demonstration that DAHA can be considered as a natural formalization of the concept of the Fourier transform in mathematics and physics.