Inventiones mathematicae

, Volume 103, Issue 1, pp 341–350 | Cite as

An elementary approach to the hypergeometric shift operators of Opdam

  • G. J. Heckman


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  1. [B] Beerends, R.: On the Abel transformation and its inversion. Proefschrift Leiden, 1987Google Scholar
  2. [BGA] Bernstein, I.N., Gel'fand, I.M., Gel'fand, S.I.: Schubert cells and the cohomology ofG/P. Russ. Math. Surveys28, 1–26 (1973)Google Scholar
  3. [D] Debiard, A.: Polynômes de Tchébychev et de Jacobi dans un espace Euclidien de dimensionp. C.R. Acad. Sci. Paris296, 529–532 (1983)Google Scholar
  4. [De1] Demazure, M.: Désingularisation des variétés de Schubert généralisés. Ann. Sci. Ec. Norm. Supér7, 53–88 (1974)Google Scholar
  5. [De2] Demazure, M.: Une nouvelle formule des caractères. Bull. Soc. Math.98, 163–172 (1974)Google Scholar
  6. [Du] Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. AMS311, 167–183 (1989)Google Scholar
  7. [Ha] Harish-Chandra: Spherical functions on a semisimple Lie group I, Am. J. Math.80, 553–613 (1958), or the Collected Works, Vol. 2, pp. 409–478Google Scholar
  8. [HO] Heckman, G.J., Opdam, E.M.: Root systems and hypergeometric functions I. Comp. Math.64, 329–352 (1987)Google Scholar
  9. [He1] Heckman, G.J.: Root systems and hypergeometric functions II. Comp. Math.64, 353–373 (1987)Google Scholar
  10. [He2] Heckman, G.J.: Hecke algebras and hypergeometric functions. Invent. Math.100, 403–417 (1990)Google Scholar
  11. [He3] Heckman, G.J.: A remark on the Dunkl differential-difference operators, Proceedings of the Bowdoin conference on Harmonic analysis on reductive groups 1989Google Scholar
  12. [Hel1] Helgason, S.: Differential Geometry, Lie groups and Symmetric Spaces. Academic Press: New York 1978Google Scholar
  13. [Hel2] Helgason, S.: Groups and Geometric Analysis. Academic Press: New York 1984Google Scholar
  14. [K] Koornwinder, T.H.: Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent differential operators, I–IV, Indag. Math.36, 48–66 and 358–381 (1974)Google Scholar
  15. [Ma1] Macdonald, I.G.: Some conjectures for root systems. Siam J. Math. Anal.13, 988–1007 (1982)Google Scholar
  16. [Ma2] Macdonald, I.G.: Orthogonal polynomials associated to root systems. Oxford 1988 (Preprint)Google Scholar
  17. [Ma3] Macdonald, I.G.: Commuting differential operators and zonal spherical functions, Algebraic Groups Utrecht 1986, LNM vol. 1271, pp. 189–200Google Scholar
  18. [Mo] Moser, J.: Three integrable systems connected with isospectral deformation. Adv. Math.16, 197–220 (1975)Google Scholar
  19. [O1] Opdam, E.M.: Root systems and hypergeometric functions III. Comp. Math.67, 21–49 (1988)Google Scholar
  20. [O2] Opdam, E.M.: Root systems and hypergeometric functions IV. Comp. Math.67, 191–209 (1988)Google Scholar
  21. [O3] Opdam, E.M.: Some applications of hypergeometric shift operators. Invent. Math.98, 1–18 (1989)Google Scholar
  22. [O4] Opdam, E.M.: Generalized hypergeometric functions associated with root systems, Proefschrift Leiden 1988Google Scholar
  23. [OP1] Olshanetsky, M.A., Perelomov, A.M.: Completely integrable systems connected with semisimple Lie algebras. Invent Math.37, 93–108 (1976)Google Scholar
  24. [OP2] Olshanetsky, M.A., Perelomov, A.M.: Classical integrable finite dimensional systems related to Lie algebras. Phys. Reps. 71 (1981), 313–400.Google Scholar
  25. [OP3] Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras, Phys. Reps.94, 313–400 (1983)Google Scholar
  26. [R] Ruijsenaars, S.N.M.: Finite-dimensional soliton systems. In: Kupershmidt, B. (ed.). Integrable and superintegrable systems. World Scientific Singapore 1990Google Scholar
  27. [Se] Sekiguchi, J.: Zonal spherical functions on some symmetric spaces. Publ. RIMS Kyoto Univ.12, 455–459 (1977)Google Scholar
  28. [Sp] Sprinkhuizen-Kuyper, I.G.: Orthogonal polynomials in two variables. A further analysis of the polynomials orthogonal over a region bounded by two lines and a parabola. Siam J. Math. An. 7 (4), 501–518 (1976)Google Scholar
  29. [V] Vretare, L.: Formulas for elementary spherical functions and generalized Jacobi polynomials. Siam J. Math. An.15 (4), 805–833 (1984)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • G. J. Heckman
    • 1
  1. 1.Mathematisch InstituutKatholieke UniversiteitNijmegenThe Netherlands

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