Skip to main content
Affine Hecke Algebras and Orthogonal Polynomials
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 51
  • Cited by
    This book has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Isachenkov, Mikhail and Schomerus, Volker 2018. Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory. Journal of High Energy Physics, Vol. 2018, Issue. 7,

    Baseilhac, Pascal and Martin, Xavier 2018. A bispectral q-hypergeometric basis for a class of quantum integrable models. Journal of Mathematical Physics, Vol. 59, Issue. 1, p. 011704.

    van Diejen, Jan Felipe Emsiz, Erdal and Zurrián, Ignacio Nahuel 2018. Completeness of the Bethe Ansatz for an Open $$\varvec{q}$$q-Boson System with Integrable Boundary Interactions. Annales Henri Poincaré, Vol. 19, Issue. 5, p. 1349.

    Angiono, Iván and Yamane, Hiroyuki 2018. Bruhat order and nil-Hecke algebras for Weyl groupoids. Journal of Algebra and Its Applications, Vol. 17, Issue. 09, p. 1850166.

    Lemay, Jean-Michel and Vinet, Luc 2018. Bivariate Bannai-Ito polynomials. Journal of Mathematical Physics, Vol. 59, Issue. 12, p. 121703.

    Orr, Daniel and Shimozono, Mark 2018. Specializations of nonsymmetric Macdonald–Koornwinder polynomials. Journal of Algebraic Combinatorics, Vol. 47, Issue. 1, p. 91.

    Brubaker, Ben Buciumas, Valentin Bump, Daniel and Friedberg, Solomon 2018. Hecke modules from metaplectic ice. Selecta Mathematica, Vol. 24, Issue. 3, p. 2523.

    Feigin, Evgeny and Makedonskyi, Ievgen 2018. Semi-infinite Plücker Relations and Weyl Modules. International Mathematics Research Notices,


    Baseilhac, Pascal Vinet, Luc and Zhedanov, Alexei 2017. The q-Onsager algebra and multivariable q-special functions. Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue. 39, p. 395201.

    Anker, Jean-Philippe 2017. Analytic, Algebraic and Geometric Aspects of Differential Equations. p. 3.


    van Diejen, Jan Felipe and Emsiz, Erdal 2016. Spectrum and Eigenfunctions of the Lattice Hyperbolic Ruijsenaars–Schneider System with Exponential Morse Term. Annales Henri Poincaré, Vol. 17, Issue. 7, p. 1615.

    Bardy-Panse, Nicole Gaussent, Stéphane and Rousseau, Guy 2016. Iwahori–Hecke algebras for Kac–Moody groups over local fields. Pacific Journal of Mathematics, Vol. 285, Issue. 1, p. 1.

    Braverman, Alexander Kazhdan, David and Patnaik, Manish M. 2016. Iwahori–Hecke algebras for p-adic loop groups. Inventiones mathematicae, Vol. 204, Issue. 2, p. 347.

    Elliot, Ross and Gukov, Sergei 2016. Exceptional knot homology. Journal of Knot Theory and Its Ramifications, Vol. 25, Issue. 03, p. 1640003.

    Lenart, Cristian Naito, Satoshi Sagaki, Daisuke Schilling, Anne and Shimozono, Mark 2016. A Uniform Model for Kirillov–Reshetikhin Crystals II. Alcove Model, Path Model, and $P=X$. International Mathematics Research Notices, p. rnw129.

    Cherednik, Ivan 2016. DAHA-Jones polynomials of torus knots. Selecta Mathematica, Vol. 22, Issue. 2, p. 1013.

    Chari, Vyjayanthi and Ion, Bogdan 2015. BGG reciprocity for current algebras. Compositio Mathematica, Vol. 151, Issue. 07, p. 1265.

    Balagović, Martina 2015. Degeneration of Trigonometric Dynamical Difference Equations for Quantum Loop Algebras to Trigonometric Casimir Equations for Yangians. Communications in Mathematical Physics, Vol. 334, Issue. 2, p. 629.

  • Export citation
  • Recommend to librarian
  • Recommend this book

    Email your librarian or administrator to recommend adding this book to your organisation's collection.

    Affine Hecke Algebras and Orthogonal Polynomials
    • Online ISBN: 9780511542824
    • Book DOI:
    Please enter your name
    Please enter a valid email address
    Who would you like to send this to *
  • Buy the print book

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.


‘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

Refine List
Actions for selected content:
Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Send to Kindle
  • Send to Dropbox
  • Send to Google Drive
  • Send content to

    To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to .

    To send content items to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

    Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

    Find out more about the Kindle Personal Document Service.

    Please be advised that item(s) you selected are not available.
    You are about to send

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
[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)
[B1] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris (1968)
[B2] Brieskorn, E. and Saito, K., Artin-gruppen und Coxeter-gruppen, Inv. Math.. 17 (1972) 245–271
[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
[C2] Cherednik, I., Double affine Hecke algebras and Macdonald's conjectures, Ann. Math.. 141 (1995) 191–216
[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
[D1] Dyson, F. J., 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] 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
[M8] Macdonald, I. G., Symmetric functions and orthogonal polynomials, University Lecture Series vol. 12, American Mathematical Society (1998)
[M9] Moody, R. V., A new class of Lie algebras, J. Algebra 10 (1968) 211–230
[M10] Moody, R. V., 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)
[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


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed