Affine Hecke Algebras and Orthogonal Polynomials (Cambridge Tracts in Mathematics, Series Number 157)

A satisfactory and coherent theory of orthogonal polynomials in several variables, attached to root systems, and depending on two or more parameters, has developed in recent years. This comprehensive account of the subject provides a unified foundation for the theory to which I.G. Macdonald has been a principal contributor. The first four chapters lead up to Chapter 5 which contains all the main results. Publisher Description (unedited Publisher Data) In Recent Years There Has Developed A Satisfactory And Coherent Theory Of Orthogonal Polynomials In Several Variables, Attached To Root Systems, And Depending On Two Or More Parameters. These Polynomials Include As Special Cases: Symmetric Functions; Zonal Spherical Functions On Real And P-adic Reductive Lie Groups; The Jacobi Polynomials Of Heckman And Opdam; And The Askey-wilson Polynomials, Which Themselves Include As Special Or Limiting Cases All The Classical Families Of Orthogonal Polynomials In One Variable. This First Comprehensive And Organised Account Of The Subject Aims To Provide A Unified Foundation For This Theory, To Which The Author Has Been A Principal Contributor. It Is An Essentially Self-contained Treatment, Accessible To Graduate Students Familiar With Root Systems And Weyl Groups. The First Four Chapters Are Preparatory To Chapter V, Which Is The Heart Of The Book And Contains All The Main Results In Full Generality. Introduction -- Affine Root Systems -- The Extended Affine Weyl Group -- The Braid Group -- The Affine Hecke Algebra -- Orthogonal Polynomials -- The Rank 1 Case -- Bibliography -- Index. I.g. Macdonald. Includes Bibliographical References And Index. In recent years there has developed a satisfactory and coherent theory of orthogonal polynomials in several variables, attached to root systems, and depending on two or more parameters. These polynomials include as special cases: symmetric functions; zonal spherical functions on real and p-adic reductive Lie groups; the Jacobi polynomials of Heckman and Opdam; and the Askey–Wilson polynomials, which themselves include as special or limiting cases all the classical families of orthogonal polynomials in one variable. This book, first published in 2003, is a comprehensive and organised account of the subject aims to provide a unified foundation for this theory, to which the author has been a principal contributor. It is an essentially self-contained treatment, accessible to graduate students familiar with root systems and Weyl groups. The first four chapters are preparatory to Chapter V, which is the heart of the book and contains all the main results in full generality. There has recently developed a satisfactory and coherent theory, created by the author, of orthogonal polynomials in several variables, attached to root systems, and depending on two or more parameters. This Tract, the first comprehensive and organised account of the subject, provides a unified foundation for this theory Let E be an affine space over a field K: that is to say, E is a set on which a K-vector space V acts faithfully and transitively.