Affine Hecke Algebras and Orthogonal Polynomials


Affine Hecke Algebras and Orthogonal Polynomials

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.


 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.' Zentralblatt für Mathematik

[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] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs of the American Mathematical Society 319 (1985)
[B1] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris (1968)
[B2] E. Brieskorn and K. Saito, Artin-gruppen und Coxeter-gruppen, Inv. Math.. 17 (1972) 245–271
[B3] F. Bruhat and J. Tits, Groupes réductifs sur un corps local: I. Données radicielles valuées, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, no. 41 (1972)
[C1] I. Cherednik, Double affine Hecke algebras, Knizhnik — Zamolodchikov equations, and Macdonald's operators, International Mathematics Research Notices 9 (1992). 171–179
[C2] I. Cherednik, Double affine Hecke algebras and Macdonald's conjectures, Ann. Math.. 141 (1995) 191–216
[C3] I. Cherednik, Non-symmetric Macdonald polynomials, International Mathematics Research Notices 10 (1995) 483–515
[C4] I. Cherednik, Macdonald's evaluation conjectures and difference Fourier transform, Inv. Math.. 122 (1995) 119–145
[C5] I. Cherednik, Intertwining operators of double affine Hecke algebras, Sel. Math. new series 3 (1997) 459–495
[D1] F. J. Dyson, Statistical theory of the energy levels of complex systems I, J. Math. Phys.. 3 (1962) 140–156
[G1] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press (1990)
[G2] R. A. Gustafson, A generalization of Selberg's beta integral, Bulletin of the American Mathematical Society 22 (1990) 97–105
[H1] G. J. Heckman and E. M. Opdam, Root systems and hypergeometric functions I, Comp. Math.. 64 (1987) 329–352
[H2] G. J. Heckman, 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] A. A. Kirillov, Lectures on affine Hecke algebras and Macdonald's conjectures, Bulletin of the American Mathematical Society 34 (1997) 251–292
[K3] T. Koornwinder, Askey-Wilson polynomials for root systems of type BC, Contemp. Math. 138 (1992) 189–204
[L1] G. Lusztig, Affine Hecke algebras and their graded version, Journal of the American Mathematical Society 2 (1989) 599–635
[M10] R. V. Moody, Euclidean Lie algebras, Can. J. Math.. 21 (1969) 1432–1454
[M11] W. G. Morris, Constant Term Identities for Finite and Affine Root Systems: Conjectures and Theorems, Ph. D. thesis, Madison (1982)
[M1] I. G. Macdonald, Spherical functions on a group of p-adic type, Publications of the Ramanujan Institute, Madras (1971)
[M2] I. G. Macdonald, Affine root systems and Dedekind's η-function, Inv. Math. 15 (1972) 91–143
[M3] I. G. Macdonald, The Poincaré series of a Coxeter group, Math. Annalen 199 (1972) 161–174
[M4] I. G. Macdonald, Some conjectures for root systems, SIAM Journal of Mathematical Analysis 13 (1982) 988–1007
[M5] I. G. Macdonald, 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] I G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996) 189–207
[M8] I. G. Macdonald, Symmetric functions and orthogonal polynomials, University Lecture Series vol. 12, American Mathematical Society (1998)
[M9] R. V. Moody, A new class of Lie algebras, J. Algebra 10 (1968) 211–230
[N1] M. Noumi, Macdonald — Koornwinder polynomials and affine Hecke rings, Sūriseisekikenkyūsho Kōkyūroku 919 (1995) 44–55 (in Japanese)
[O1] E. M. Opdam, Root systems and hypergeometric functions III, Comp. Math. 67 (1988) 21–49
[O2] E. M. Opdam, Root systems and hypergeometric functions IV, Comp. Math. 67 (1988) 191–209
[O3] E. M. Opdam, Some applications of hypergeometric shift operators, Inv. Math. 98 (1989) 1–18
[O4] E. M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995) 75–121
[R1] L. J. Rogers, On the expansion of some infinite products, Proc. London Math. Soc. 24 (1893) 337–352
[R2] L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894) 318–343
[R3] L. J. Rogers, Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1895) 15–32
[S1] S. Sahi, Nonsymmetric Koornwinder polynomials and duality, Arm. Math. 150 (1999) 267–282
[S2] S. Sahi, Some properties of Koornwinder polynomials, Contemp. Math. 254 (2000) 395–411
[S3] R. Stanley, 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] J. van Diejen, Self-dual Koornwinder-Macdonald polynomials, Inv. Math. 126 (1996) 319–339