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First account of a theory, created by Macdonald, of a class of orthogonal polynomial, which is related to mathematical physics.
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.
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.
One of the most classical areas of algebra, the theory of symmetric functions and orthogonal polynomials, has long been known to be connected to combinatorics, representation theory and other branches of mathematics. Written by perhaps the most famous author on the topic, this volume explains some of the current developments regarding these connections. It is based on lectures presented by the author at Rutgers University. Specifically, he gives recent results on orthogonal polynomials associated with affine Hecke algebras, surveying the proofs of certain famous combinatorial conjectures.
In the first part we study Double Affine Hecke algebra of type An-1 which is important tool in the theory of orthogonal polynomials. We prove that the spherical subalgebra eH(t, 1)e of the Double Affine Hecke algebra H(t, 1) of type An-1 is an integral Cohen-Macaulay algebra isomorphic to the center Z of H(t, 1), and H(t, 1)e is a Cohen-Macaulay eH(t, 1)e-module with the property H(t, 1) = EndeH(t, tl)(H(t, 1)e). This implies the classification of the finite dimensional representations of the algebras. In the second part we study the algebraic properties of the five-parameter family H(tl, t2, t3, t4; q) of double affine Hecke algebras of type CVC1, which control Askey- Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is an affine cubic surface C, obtained from a projective one by removing a triangle consisting of smooth points. Moreover, any such surface is obtained as the spectrum of the center of H for some values of parameters. We prove that the only fiat de- formations of H come from variations of parameters. This explains from the point of view of noncommutative geometry why one cannot add more parameters into the theory of Askey-Wilson polynomials. We also prove several results on the universality of the five-parameter family H(tl, t2, t3, t4; q) of algebras.
This new book presents research in orthogonal polynomials and special functions. Recent developments in the theory and accomplishments of the last decade are pointed out and directions for research in the future are identified. The topics covered include matrix orthogonal polynomials, spectral theory and special functions, Asymptotics for orthogonal polynomials via Riemann-Hilbert methods, Polynomial wavelets and Koornwinder polynomials.
Kazhdan and Lusztig classified the simple modules of an affine Hecke algebra Hq (q E C*) provided that q is not a root of 1 (Invent. Math. 1987). Ginzburg had some very interesting work on affine Hecke algebras. Combining these results simple Hq-modules can be classified provided that the order of q is not too small. These Lecture Notes of N. Xi show that the classification of simple Hq-modules is essentially different from general cases when q is a root of 1 of certain orders. In addition the based rings of affine Weyl groups are shown to be of interest in understanding irreducible representations of affine Hecke algebras. Basic knowledge of abstract algebra is enough to read one third of the book. Some knowledge of K-theory, algebraic group, and Kazhdan-Lusztig cell of Cexeter group is useful for the rest
This book is a volume of the Springer Briefs in Mathematical Physics and serves as an introductory textbook on the theory of Macdonald polynomials. It is based on a series of online lectures given by the author at the Royal Institute of Technology (KTH), Stockholm, in February and March 2021. Macdonald polynomials are a class of symmetric orthogonal polynomials in many variables. They include important classes of special functions such as Schur functions and Hall–Littlewood polynomials and play important roles in various fields of mathematics and mathematical physics. After an overview of Schur functions, the author introduces Macdonald polynomials (of type A, in the GLn version) as eigenfunctions of a q-difference operator, called the Macdonald–Ruijsenaars operator, in the ring of symmetric polynomials. Starting from this definition, various remarkable properties of Macdonald polynomials are explained, such as orthogonality, evaluation formulas, and self-duality, with emphasis on the roles of commuting q-difference operators. The author also explains how Macdonald polynomials are formulated in the framework of affine Hecke algebras and q-Dunkl operators.
There has been revived interest in recent years in the study of special functions. Many of the latest advances in the field were inspired by the works of R. A. Askey and colleagues on basic hypergeometric series and I. G. Macdonald on orthogonal polynomials related to root systems. Significant progress was made by the use of algebraic techniques involving quantum groups, Hecke algebras, and combinatorial methods. The CRM organized a workshop for key researchers in the field to present an overview of current trends. This volume consists of the contributions to that workshop. Topics include basic hypergeometric functions, algebraic and representation-theoretic methods, combinatorics of symmetric functions, root systems, and the connections with integrable systems.
Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. Algebras of Iwahori-Hecke type are one of the tools and were, probably, first considered in the context of representation theory of finite groups of Lie type. This volume consists of notes of the courses on Iwahori-Hecke algebras and their representation theory, given during the CIME summer school which took place in 1999 in Martina Franca, Italy.