The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).


Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Lectures on affine Hecke algebras and Macdonald's conjectures

Author: Alexander A. Kirillov Jr.
Journal: Bull. Amer. Math. Soc. 34 (1997), 251-292
MSC (1991): Primary 05E35; Secondary 33D80
MathSciNet review: 1441642
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives a review of Cherednik's results on the
representation-theoretic approach to Macdonald polynomials and related special functions. Macdonald polynomials are a remarkable 2-parameter family of polynomials which can be associated to every root system. As special cases, they include the Schur functions, the $q$-Jacobi polynomials, and certain spherical functions on real and $p$-adic symmetric spaces. They have a number of elegant combinatorial properties, which, however, are extremely difficult to prove. In this paper we show that a natural setup for studying these polynomials is provided by the representation theory of Hecke algebras and show how this can be used to prove some of the combinatorial identities for Macdonald polynomials.

References [Enhancements On Off] (What's this?)

  • [AI] Askey, R. and Ismail, M.E.H., A generalization of ultraspherical polynomials, Studies in Pure Mathematics (P. Erdös, ed.), Birkhäuser, 1982, pp. 55-78. MR 87a:33015
  • [AW] Askey, R. and Wilson, J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs of AMS 319 (1985). MR 87a:05023
  • [B] Bourbaki, N., Groupes et algèbres de Lie, Ch. 4-6, Hermann, Paris, 1969. MR 39:1590
  • [BZ] Bressoud, D. and Zeilberger, D., A proof of Andrews' $q$-Dyson conjecture, Discrete Math. 54 (1985), 201-224. MR 87f:05015
  • [C1] Cherednik, I., Double affine Hecke algebras, Knizhnik- Zamolodchikov equations, and Macdonald's operators, IMRN (Duke M.J.) 9 (1992), 171-180. MR 94b:17040
  • [C2] -, The Macdonald constant term conjecture, IMRN 6 (1993), 165-177. MR 94i:17016
  • [C3] -, A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Inventiones Math. 106 (2) (1991), 411-432. MR 93b:17040
  • [C4] -, Quantum Knizhnik-Zamolodchikov equations and affine root systems, Commun. Math. Phys. 150 (1992), 109-136. MR 94a:17019
  • [C5] -, Integration of Quantum many-body problems by affine Knizhnik-Zamolodchikov equations, Advances in Math. 106 (1994), 65-95. MR 95m:32031
  • [C6] -, Double affine Hecke algebras and Macdonald's conjectures, Annals of Math. 141 (1995), 191-216. MR 96m:33010
  • [C7] -, Macdonald's evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), 119-145. MR 1:354 956
  • [CRM] Calogero, P., Ragnisco, O., and Marchioro, C., Exact solution of the classical and quantal one-dimensional many-body problems with the two-body potential $V_{a}(x)=g^{2} a^{2} /\sinh ^{2}(ax)$, Lett. Nuovo Cimento 13 (1975), 383-387. MR 56:2029
  • [D] van Diejen, J.F., Commuting difference operators with polynomial eigenfunctions, Compos. Math. 95 (1995), 183-233. MR 96i:39023
  • [Dy] Dyson, F., Statistical theory of the energy levels of complex systems I, J. Math. Phys. 3 (1962), 140-156. MR 26:1111
  • [EFK] Etingof, P.I., Frenkel, I.B. and Kirillov, A.A., Jr, Spherical functions on affine Lie groups, Duke Math. J. 80 (1995), 59-90. MR 97e:22018
  • [EK1] Etingof, P.I., and Kirillov, A.A.,Jr, Macdonald's polynomials and representations of quantum groups, Math. Res. Let. 1 (1994), 279-296. MR 96m:17025
  • [EK2] -, Representation-theoretic proof of the inner product and symmetry identities for Macdonald's polynomials, Compositio Math. 102 (1996), 179-202. MR 1:394 525
  • [GG] Garvan, F. and Gonnet, G., Macdonald's constant term conjectures for exceptional root systems, Bull. AMS (N.S.) 24 (2) (1991), 343-347. MR 92b:33054
  • [Go] Good, I.J., Short proof of a conjecture by Dyson, J. Math. Phys. 11 (1970), 1884. MR 41:3290
  • [Gu] Gunson, J., Proof of a conjecture by Dyson in the statistical theory of energy levels, J. Math. Phys. 3 (1962), 752-753. MR 26:5908
  • [Ha] Habsieger, L., La $q$-conjecture de Macdonald-Morris pour $G_{2}$, C.R.Acad. Sci. Paris Sér. 1 Math. 303 (1986), 211-213. MR 87k:17019
  • [HO] Heckman, G.J., Opdam, E.M., Root systems and hypergeometric functions I, Compos. Math. 64 (1987), 329-352. MR 89b:58192a
  • [H1] Heckman, G.J., Root systems and hypergeometric functions II, Compos. Math. 64 (1987), 353-373. MR 89b:58192b
  • [H2] -, A remark on the Dunkl differential-difference operators, Harmonic analysis on reductive groups (W. Barker, P. Sally, eds.), Birkhäuser, 1991, pp. 181-191. MR 94c:20075
  • [H3] -, An elementary approach to the hypergeometric shift operators of Opdam, Invent.Math. 103 (1991), 341-350. MR 92i:33012
  • [Hu1] Humphreys, J.E., Introduction to Lie algebras and representation theory, Springer-Verlag, New York, 1972. MR 48:2197
  • [Hu2] -, Reflection groups and Coxeter groups, Cambridge Univ. Press, Cambridge, 1990. MR 92h:20002
  • [J] Jack, H., A class of symmetric polynomials with a parameter, Proc. Roy. Soc. Edinburgh Sect. A 69 (1970-1971), 1-18. MR 44:6652
  • [K] Kadell, K., A proof of the $q$-Macdonald-Morris conjecture for $BC_{n}$, Mem. Amer. Math. Soc. 108 (1994). MR 94h:33013
  • [Ko1] Koornwinder, T.H., Special functions associated with root systems: recent progress, From Universal Morphisms to Megabytes - a Baayen Space Odyssey (K. R. Apt, A. Schrijver, & N. M. Temme, eds.), CWI, Amsterdam, 1994, pp. 391-404.
  • [Ko2] -, Askey-Wilson polynomials for root systems of type BC, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (D. St. Richards, ed.), Contemp. Math., vol. 138, Amer. Math. Soc., 1992, pp. 189-204. MR 94e:33039
  • [L] Lusztig, G., Affine Hecke algebras and their graded version, J. of the AMS 2 (3) (1989), 599-685. MR 90e:16049
  • [M1] Macdonald, I.G., A new class of symmetric functions, Publ. I.R.M.A. Strasbourg, 372/S-20, Actes 20 Séminaire Lotharingien (1988), 131-171.
  • [M2] -, Orthogonal polynomials associated with root systems, preprint (1988).
  • [M3] -, Some conjectures for root systems, SIAM J. of Math. Analysis 13 (6) (1982), 988-1007. MR 84h:17006a
  • [M4] -, The Poincaré series of a Coxeter group, Math. Annalen 199 (1972), 161-174. MR 48:433
  • [M5] -, Orthogonal polynomials and constant term conjectures, Lectures at Leiden University (May 1994).
  • [M6] -, Symmetric functions and Hall polynomials, 2nd ed., Oxford Univ. Press, 1995. MR 96h:05207
  • [M7] -, Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki, Vol. 1994-95 (797). MR 1:423 624
  • [Ma] Matsuo, A., Integrable connections related to zonal spherical functions, Inv. Math. 110 (1992), 95-121. MR 94g:33013
  • [N] Noumi, M., Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. in Math. 123 (1996), 16-77. MR 1:413 836
  • [O1] Opdam, E.M., Root systems and hypergeometric functions III, Compos. Math. 67 (1988), 21-49. MR 90k:17021
  • [O2] -, Root systems and hypergeometric functions IV, Compos. Math. 67 (1988), 191-209. MR 90c:58079
  • [O3] -, Some applications of hypergeometric shift operators, Inv. Math. 98 (1989), 1-18. MR 91h:33024
  • [OOS] Ochiai, H., Oshima, T., and Sekiguchi, H., Commuting families of symmetric differential operators, Proc. of the Japan Acad. 70, Ser. A (2) (1994), 62-66. MR 1:272 672
  • [OP] Olshanetsky, M.A. and Perelomov, A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404. MR 84k:81007
  • [St] Stanley, R. Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76-115. MR 90g:05020
  • [Su] Sutherland, B., Exact results for quantum many-body problem in one dimension, Phys. Rep. A5 (1972), 1375-1376.
  • [V] Verma, D-N., The role of affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras, Lie groups and their representations (Proceedings of the Summer School on Group Representations), Budapest, 1971, pp. 653-705. MR 53:13425
  • [W] Wilson, K., Proof of a conjecture by Dyson, J. Math. Phys. 3 (1962), 1040-1043. MR 26:2170

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 05E35, 33D80

Retrieve articles in all journals with MSC (1991): 05E35, 33D80

Additional Information

Alexander A. Kirillov Jr.
Affiliation: Department of Mathematics, MIT, Cambridge, Massachusetts 02139

Received by editor(s): January 19, 1995
Received by editor(s) in revised form: April 3, 1997
Additional Notes: The author was supported by Alfred P. Sloan dissertation fellowship.
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society