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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.

Hecke algebras. --- Orthogonal polynomials. --- 512.55 --- 517.518 --- Hecke algebras --- Orthogonal polynomials --- Fourier analysis --- Functions, Orthogonal --- Polynomials --- Algebras, Hecke --- Group algebras --- 517.518 Metric theory of functions --- Metric theory of functions --- 512.55 Rings and modules --- Rings and modules

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This monograph provides a comprehensive introduction to the Kazhdan-Lusztig theory of cells in the broader context of the unequal parameter case. Serving as a useful reference, the present volume offers a synthesis of significant advances made since Lusztig’s seminal work on the subject was published in 2002. The focus lies on the combinatorics of the partition into cells for general Coxeter groups, with special attention given to induction methods, cellular maps and the role of Lusztig's conjectures. Using only algebraic and combinatorial methods, the author carefully develops proofs, discusses open conjectures, and presents recent research, including a chapter on the action of the cactus group. Kazhdan-Lusztig Cells with Unequal Parameters will appeal to graduate students and researchers interested in related subject areas, such as Lie theory, representation theory, and combinatorics of Coxeter groups. Useful examples and various exercises make this book suitable for self-study and use alongside lecture courses.

Mathematics. --- Group theory. --- Group Theory and Generalizations. --- Hecke algebras. --- Coxeter groups. --- Representations of algebras. --- Algebra --- Coxeter's groups --- Real reflection groups --- Reflection groups, Real --- Group theory --- Algebras, Hecke --- Group algebras --- Groups, Theory of --- Substitutions (Mathematics)

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The definition of Rouquier for the families of characters introduced by Lusztig for Weyl groups in terms of blocks of the Hecke algebras has made possible the generalization of this notion to the case of complex reflection groups. The aim of this book is to study the blocks and to determine the families of characters for all cyclotomic Hecke algebras associated to complex reflection groups. This volume offers a thorough study of symmetric algebras, covering topics such as block theory, representation theory and Clifford theory, and can also serve as an introduction to the Hecke algebras of complex reflection groups.

Hecke algebras --- Representations of groups --- Cyclotomy --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Algebra --- Cyclotomy. --- Hecke algebras. --- Representations of groups. --- Group representation (Mathematics) --- Groups, Representation theory of --- Algebras, Hecke --- Equations, Cyclotomic --- Mathematics. --- Group theory. --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Math --- Science --- Group algebras --- Number theory --- Equations, Abelian --- Group theory

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This is an essentially self-contained monograph in an intriguing field of fundamental importance for Representation Theory, Harmonic Analysis, Mathematical Physics, and Combinatorics. It is a major source of general information about the double affine Hecke algebra, also called Cherednik's algebra, and its impressive applications. Chapter 1 is devoted to the Knizhnik-Zamolodchikov equations attached to root systems and their relations to affine Hecke algebras, Kac-Moody algebras, and Fourier analysis. Chapter 2 contains a systematic exposition of the representation theory of the one-dimensional DAHA. It is the simplest case but far from trivial with deep connections in the theory of special functions. Chapter 3 is about DAHA in full generality, including applications to Macdonald polynomials, Fourier transforms, Gauss-Selberg integrals, Verlinde algebras, and Gaussian sums. This book is designed for mathematicians and physicists, experts and students, for those who want to master the double Hecke algebra technique. Visit http://arxiv.org/math.QA/0404307 to read Chapter 0 and selected topics from other chapters.

Hecke algebras. --- Affine algebraic groups. --- Harmonic analysis. --- Knizhnik-Zamolodchikov equations. --- Orthogonal polynomials. --- Fourier analysis --- Functions, Orthogonal --- Polynomials --- KZ equations --- Zamolodchikov equations, Knizhnik --- -Mathematical physics --- Quantum field theory --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Algebraic groups, Affine --- Group schemes (Mathematics) --- Algebras, Hecke --- Group algebras