Bessel functions for root systems via the trigonometric setting

International Mathematics Research Notices

Volumn , Issue 9, 2005, Pages 551-585

  Saïd, Salem Ben ^a   Ørsted, Bent ^a  

^a AARHUS UNIVERSITY   (Denmark)

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EID: 16244370437     PISSN: 10737928     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. F. Calogero, Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191-2196.
2. I. Cherednik, A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), no. 2, 411-431.
3. J. L. Clerc, Une formule asymptotique du type Mehler-Heine pour les zonales d'un espace riemannien symétrique, Studia Math. 57 (1976), no. 1, 27-32 (French).
4. M. F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), no. 1, 147-162.
5. _, Paley-Wiener theorems for the Dunkl transform, preprint, 2004, http://arxiv.org/abs/math.CA/0404439.
6. A. H. Dooley and J. W. Rice, On contractions of semisimple Lie groups, Trans. Amer. Math. Soc. 289 (1985), no. 1, 185-202.
7. J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), no. 2, 259-268.
8. C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167-183.
9. C. F. Dunkl, M. F. E. de Jeu, and E. Opdam, Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), no. 1, 237-256.
10. C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and Its Applications, vol. 81, Cambridge University Press, Cambridge, 2001.
11. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions. Vols. I, II, McGraw-Hill Book Company, New York, 1953.
12. J. Faraut, J. Hilgert, and G. Ólafsson, Spherical functions on ordered symmetric spaces, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 3, 927-965.
13. Harish-Chandra, Spherical functions on a semisimple Lie group. I, Amer. J. Math. 80 (1958), 241-310.
14. _, Spherical functions on a semisimple Lie group, II, Amer. J. Math. 80 (1958), 553-613.
15. G. Heckman and H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces, Perspectives in Mathematics, vol. 16, Academic Press, California, 1994.
16. G. J. Heckman, Root systems and hypergeometric functions, II, Compositio Math. 64 (1987), no. 3, 353-373.
17. _, A remark on the Dunkl differential-difference operators, Harmonic Analysis on Reductive Groups (Brunswick, Me, 1989), Progr. Math., vol. 101, Birkhäuser Boston, Massachusetts, 1991, pp. 181-191.
18. _, Dunkl operators, Astérisque 245 (1997), 223-246, Séminaire Bourbaki 1997, Exp. No. 828.
19. G. J. Heckman and E. M. Opdam, Root systems and hypergeometric functions. I, Compositio Math. 64 (1987), no. 3, 329-352.
20. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, vol. 80, Academic Press, New York, 1978.
21. J. Hilgert and G. Ólafsson, Causal Symmetric Spaces. Geometry and Harmonic Analysis, Perspectives in Mathematics, vol. 18, Academic Press, California, 1997.
22. C. Itzykson and J. B. Zuber, The planar approximation, II, J. Math. Phys. 21 (1980), no. 3, 411-421.
23. C. Meaney, The inverse Abel transform for SU(p, q), Ark. Mat. 24 (1986), no. 1, 131-140.
24. G. Ólafsson and A. Pasquale, A Palsy-Wiener theorem for the ⊖-hypergeometric transform: the even multiplicity case, J. Math. Pures Appl. (9) 83 (2004), no. 7, 869-927.
25. M. A. Olshanetsky and A. M. Perelomov, Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Invent. Math. 37 (1976), no. 2, 93-108.
26. E. M. Opdam, Root systems and hypergeometric functions. III, Compositio Math. 67 (1988), no. 1, 21-49.
27. _, Root systems and hypergeometric functions. IV, Compositio Math. 67 (1988), no. 2, 191-209.
28. _, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), no. 1, 1-18.
29. _, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), no. 3, 333-373.
30. _, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), no. 1, 75-121.
31. _, Lecture Notes on Dunkl Operators for Real and Complex Reflection Groups, MSJ Memoirs, vol. 8, Mathematical Society of Japan, Tokyo, 2000.
32. T. Ōshima and J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980), no. 1, 1-81.
33. A. Pasquale, A theory of ⊖-spherical functions, Habilitation thesis, Technische Universität Clausthal, Clausthal-Zellerfeld, 2002.
34. M. Rösler, Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), no. 3, 519-542.
35. R. J. Stanton, Mean convergence of Fourier series on compact Lie groups, Trans. Amer. Math. Soc. 218 (1976), 61-87.
36. C. Torossian, Une application des opérateurs de Cherednik à l'isomorphisme d'Harish-Chandra pour les espaces symétriques [An application of the Cherednik operators to the Harish-Chandra isomorphism for symmetric spaces], C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 2, 139-144 (French).