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Book Information:

Author: Ivan Cherednik
Title: Double affine Hecke algebras
Additional book information: London Mathematical Society, Lecture Note Series, 319, xii+434 pp., ISBN 978-0-521-609180, US$79.00

[1] Ivan Cherednik, A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), no. 2, 411–431. MR 1128220, https://doi.org/10.1007/BF01243918
[2] Ivan Cherednik, Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald’s operators, Internat. Math. Res. Notices 9 (1992), 171–180. MR 1185831, https://doi.org/10.1155/S1073792892000199
[3] Ivan Cherednik, Induced representations of double affine Hecke algebras and applications, Math. Res. Lett. 1 (1994), no. 3, 319–337. MR 1302647, https://doi.org/10.4310/MRL.1994.v1.n3.a4
[4] Ivan Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191–216. MR 1314036, https://doi.org/10.2307/2118632
[5] Ivan Cherednik, Macdonald’s evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), no. 1, 119–145. MR 1354956, https://doi.org/10.1007/BF01231441
[6] Ivan Cherednik, Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices 10 (1995), 483–515. MR 1358032, https://doi.org/10.1155/S1073792895000341
[7] Ivan Cherednik, Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, Quantum many-body problems and representation theory, MSJ Mem., vol. 1, Math. Soc. Japan, Tokyo, 1998, pp. 1–96. MR 1724948
[8] Ivan Cherednik and Viktor Ostrik, From double Hecke algebra to Fourier transform, Selecta Math. (N.S.) 9 (2003), no. 2, 161–249. MR 1993484, https://doi.org/10.1007/s00029-003-0329-3
[9] V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
[10] Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, Mathematical Surveys and Monographs, vol. 58, American Mathematical Society, Providence, RI, 1998. MR 1629472
[11] I. B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992), no. 1, 1–60. MR 1163666
[12] G. J. Heckman, An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103 (1991), no. 2, 341–350. MR 1085111, https://doi.org/10.1007/BF01239517
[13] N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of 𝔭-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48. MR 0185016
[14] Michio Jimbo and Tetsuji Miwa, Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, vol. 85, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1995. MR 1308712
[15] V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nuclear Phys. B 247 (1984), no. 1, 83–103. MR 853258, https://doi.org/10.1016/0550-3213(84)90374-2
[16] V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1993. MR 1245942
[17] Gail Letzter, Quantum zonal spherical functions and Macdonald polynomials, Adv. Math. 189 (2004), no. 1, 88–147. MR 2093481, https://doi.org/10.1016/j.aim.2003.11.007
[18] George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, https://doi.org/10.1090/S0894-0347-1989-0991016-9
[19] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
[20] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996), Exp. No. 797, 4, 189–207. Séminaire Bourbaki, Vol. 1994/95. MR 1423624
[21] I. G. Macdonald, Orthogonal polynomials associated with root systems, Sém. Lothar. Combin. 45 (2000/01), Art. B45a, 40. MR 1817334
[22] I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, Cambridge, 2003. MR 1976581
[23] Masatoshi Noumi, Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math. 123 (1996), no. 1, 16–77. MR 1413836, https://doi.org/10.1006/aima.1996.0066
[24] S. N. M. Ruijsenaars, Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Comm. Math. Phys. 110 (1987), no. 2, 191–213. MR 887995
[25] F. A. Smirnov, Form factors in completely integrable models of quantum field theory, Advanced Series in Mathematical Physics, vol. 14, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. MR 1253319

Review Information:

Reviewer: Eric M. Opdam
Affiliation: KdV Institute for Mathematics, University of Amsterdam, The Netherlands
Email: e.m.opdam@uva.nl
Reviewer: Jasper V. Stokman
Affiliation: KdV Institute for Mathematics, University of Amsterdam, The Netherlands
Email: j.v.stokman@uva.nl

Journal: Bull. Amer. Math. Soc. 46 (2009), 143-150
MSC (2000): Primary 32G34, 33D80; Secondary 33D52, 20C08
DOI: https://doi.org/10.1090/S0273-0979-08-01208-1
Published electronically: September 15, 2008
Additional Notes: The work of J. V. Stokman was supported by a VIDI-grant of the Netherlands Organization for Scientific Research (NWO)
Review copyright: © Copyright 2008 American Mathematical Society