Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Singular polynomials for finite reflection groups

Authors: C. F. Dunkl, M. F. E. de Jeu and E. M. Opdam
Journal: Trans. Amer. Math. Soc. 346 (1994), 237-256
MSC: Primary 33D80; Secondary 20C15, 20F55
MathSciNet review: 1273532
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Dunkl operators involve a multiplicity function as parameter. For generic values of this function the simultaneous kernel of these operators, acting on polynomials, is equal to the constants. For special values, however, this kernel is larger. We determine these singular values completely and give partial results on the representations of $ G$ that occur in this kernel.

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

  • [B] E. Brieskorn, Die Fundamentalgruppe des Raumes der regulären Orbits einer komplexen Spiegelungs gruppe, Invent. Math. 12 (1971), 57-61. MR 0293615 (45:2692)
  • [CIK] C. W. Curtis, N. Iwahori, and R. W. Kilmoyer, Hecke algebras and the characters of parabolic type of finite groups with $ BN$-pairs, Publ. Math. Inst. Hautes Etudes Sci. 40 (1971), 81-116.
  • [D1] C. F. Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33-60. MR 917849 (89b:42016)
  • [D2] -, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183. MR 951883 (90k:33027)
  • [D3] -, Operators commuting with Coxeter group actions on polynomials, Invariant Theory and Tableaux (D. Stanton, ed.), Springer-Verlag, 1990, pp. 107-117. MR 1035491 (91g:20060)
  • [D4] -, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213-1227. MR 1145585 (93g:33012)
  • [D5] -, Differential-difference operators and monodromy representations of Hecke algebras, Pacific J. Math. 159 (1993), 271-298. MR 1214073 (94h:32040)
  • [DJ1] R. Dipper and G. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986), 20-52. MR 812444 (88b:20065)
  • [DJ2] -, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 54 (1987), 57-82. MR 872250 (88m:20084)
  • [Fe] W. Feit, The representation theory of finite groups, North-Holland, 1982. MR 661045 (83g:20001)
  • [Fu] W. Fulton, Algebraic curves, Addison-Wesley, 1989. MR 1042981 (90k:14023)
  • [GU] A. Gyoja and K. Uno, On the semisimplicity of Hecke algebras, J. Math. Soc. Japan 41 (1989), 75-79. MR 972165 (90a:20097)
  • [H] G. J. Heckman, A remark on the Dunkl differential-difference operators, Harmonic Analysis on Reductive Groups (W. Barker and P. Sally, eds.), Birkhaüser, Basel, 1991, pp. 181-191. MR 1168482 (94c:20075)
  • [M] I. G. Macdonald, The Poincaré series of a Coxeter group, Math. Ann. 199 (1972), 161-174. MR 0322069 (48:433)
  • [O] E. M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Comput. Math. 85 (1993), 333-373. MR 1214452 (95j:33044)
  • [S] T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159-198. MR 0354894 (50:7371)
  • [Y] H. Yamane, Irreducible projective modules of the Hecke algebras of finite Coxeter groups, J. Algebra 127 (1989), 373-384. MR 1028459 (90k:20070)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 33D80, 20C15, 20F55

Retrieve articles in all journals with MSC: 33D80, 20C15, 20F55

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society