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  1. Home
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  3. Affine Hecke Algebras and Orthogonal Polynomials
  • Cited by 106
Publisher:
Cambridge University Press
Online publication date:
July 2009
Print publication year:
2003
Online ISBN:
9780511542824
DOI:
https://doi.org/10.1017/CBO9780511542824
Subjects:
Mathematics (general), Mathematics, Logic, Categories and Sets, Algebra
Series:
Cambridge Tracts in Mathematics (157)
Information

Book description

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.

Reviews

‘This is a beautiful book, treating in a concise and clear way the recent developments concerning the connection between orthogonal polynomials in several variables and root systems in two or more parameters.’

Source: Zentralblatt für Mathematik

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Contents

Contents
References
Bibliography
Bibliography
[A1] G. E. Andrews, Problems and prospects for basic hypergeometric functions. In Theory and Applications of Special Functions, ed. R. Askey, Academic Press, New York (1975)
[A2] Askey, R. and Wilson, J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs of the American Mathematical Society 319 (1985)
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[B3] Bruhat, F. and Tits, J., Groupes réductifs sur un corps local: I. Données radicielles valuées, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, no. 41 (1972)
[C1] Cherednik, I., Double affine Hecke algebras, Knizhnik — Zamolodchikov equations, and Macdonald's operators, International Mathematics Research Notices 9 (1992). 171–179
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[C3] Cherednik, I., Non-symmetric Macdonald polynomials, International Mathematics Research Notices 10 (1995) 483–515
[C4] Cherednik, I., Macdonald's evaluation conjectures and difference Fourier transform, Inv. Math.. 122 (1995) 119–145
[C5] Cherednik, I., Intertwining operators of double affine Hecke algebras, Sel. Math. new series 3 (1997) 459–495
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[G2] Gustafson, R. A., A generalization of Selberg's beta integral, Bulletin of the American Mathematical Society 22 (1990) 97–105
[H1] Heckman, G. J. and Opdam, E. M., Root systems and hypergeometric functions I, Comp. Math.. 64 (1987) 329–352
[H2] Heckman, G. J., Root systems and hypergeometric functions II, Comp. Math. 64 (1987) 353–373
[I1] B. Ion, Involutions of double affine Hecke algebras, preprint (2001)
[K1] V. G. Kac, Infinite Dimensional Lie Algebras, Birkhäuser, Boston (1983)
[K2] Kirillov, A. A., Lectures on affine Hecke algebras and Macdonald's conjectures, Bulletin of the American Mathematical Society 34 (1997) 251–292
[K3] Koornwinder, T., Askey-Wilson polynomials for root systems of type BC, Contemp. Math. 138 (1992) 189–204
[L1] Lusztig, G., Affine Hecke algebras and their graded version, Journal of the American Mathematical Society 2 (1989) 599–635
[M1] I. G. Macdonald, Spherical functions on a group of p-adic type, Publications of the Ramanujan Institute, Madras (1971)
[M2] Macdonald, I. G., Affine root systems and Dedekind's η-function, Inv. Math. 15 (1972) 91–143
[M3] Macdonald, I. G., The Poincaré series of a Coxeter group, Math. Annalen 199 (1972) 161–174
[M4] Macdonald, I. G., Some conjectures for root systems, SIAM Journal of Mathematical Analysis 13 (1982) 988–1007
[M5] Macdonald, I. G., Orthogonal polynomials associated with root systems, preprint (1987); Séminaire Lotharingien de Combinatoire 45 (2000) 1–40
[M6] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press (1995)
[M7] Macdonald, I G., Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996) 189–207
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[M11] W. G. Morris, Constant Term Identities for Finite and Affine Root Systems: Conjectures and Theorems, Ph. D. thesis, Madison (1982)
[N1] Noumi, M., Macdonald — Koornwinder polynomials and affine Hecke rings, Sūriseisekikenkyūsho Kōkyūroku 919 (1995) 44–55 (in Japanese)
[O1] Opdam, E. M., Root systems and hypergeometric functions III, Comp. Math. 67 (1988) 21–49
[O2] Opdam, E. M., Root systems and hypergeometric functions IV, Comp. Math. 67 (1988) 191–209
[O3] Opdam, E. M., Some applications of hypergeometric shift operators, Inv. Math. 98 (1989) 1–18
[O4] Opdam, E. M., Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995) 75–121
[R1] Rogers, L. J., On the expansion of some infinite products, Proc. London Math. Soc. 24 (1893) 337–352
[R2] Rogers, L. J., Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894) 318–343
[R3] Rogers, L. J., Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1895) 15–32
[S1] Sahi, S., Nonsymmetric Koornwinder polynomials and duality, Arm. Math. 150 (1999) 267–282
[S2] Sahi, S., Some properties of Koornwinder polynomials, Contemp. Math. 254 (2000) 395–411
[S3] Stanley, R., Some combinatorial properties of Jack symmetric functions, Adv. in Math. 77 (1989) 76–115
[S4] J. V. Stokman, Koornwinder polynomials and affine Hecke algebras, preprint (2000)
[V1] H. van der Lek, The Homotopy Type of Complex Hyperplane Arrangements, Thesis, Nijmegen (1983)
[V2] Diejen, J., Self-dual Koornwinder-Macdonald polynomials, Inv. Math. 126 (1996) 319–339

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