## Affine Hecke Algebras and Orthogonal PolynomialsIn 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. |

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### Contents

The extended affine Weyl group | 17 |

The braid group | 37 |

The affine Hecke algebra | 55 |

Orthogonal polynomials | 85 |

The rank 1 case | 148 |

170 | |

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### Common terms and phrases

a e S1 adjoint ae R+ affine Hecke algebra affine root system affine Weyl group affine-linear functions antidominant assume braid group 25 braid relations Bruhat ordering Chapter Cherednik ci(X coefficient commutes Coxeter group define denote double affine Hecke double braid group Dually Dynkin diagram e(wo equal extended affine Weyl Finally follows form a K-basis fundamental weight group algebra Hecke relations Hence highest root hyperplane irreducible affine root isomorphism je.J labels k(a Let i e linear combination linearly independent longest element lower terms nonzero norm formula notation obtain oe R+ orthogonal polynomials permutes Proof Let q-binomial theorem re(u re(x reduced expression resp right-hand side roots a e scalar multiple scalar product shortest element subset symmetric polynomials T(wo type Cn vector space vertex Wo-orbit Wo(q XD f zero