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)

 
 

 

Intertwining operators associated to the group $ S\sb 3$


Author: Charles F. Dunkl
Journal: Trans. Amer. Math. Soc. 347 (1995), 3347-3374
MSC: Primary 22E30; Secondary 20B30, 33C50, 33C80
DOI: https://doi.org/10.1090/S0002-9947-1995-1316848-8
MathSciNet review: 1316848
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For any finite reflection group $ G$ on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections in $ G$. There exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients, for all but a singular set of parameter values (containing only certain negative rational numbers). This paper constructs an integral transform implementing the intertwining operator for the group $ {S_3}$, the symmetric group on three objects, for parameter value $ \geqslant \frac{1} {2}$. The transform is realized as an absolutely continuous measure on a compact subset of $ {M_2}({\mathbf{R}})$, which contains the group as a subset of its boundary. The construction of the integral formula involves integration over the unitary group $ U(3)$.


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

  • [Ba] N. Bailey, Generalized hypergeometric series, Cambridge Univ. Press, Cambridge, 1935.
  • [Be] R. J. Beerends, A transmutation property of the generalized Abel transform associated with root system $ {A_2}$, Indag. Math. (N.S.) 1 (1990), 155-168. MR 1060823 (91e:22013)
  • [C] R. Carroll, Transmutation theory and applications, North-Holland, Amsterdam, 1985. [Du1] C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183. MR 807146 (87i:35203)
  • [Du2] -, Operators commuting with Coxeter group actions on polynomials, Invariant Theory and Tableaux, (D. Stanton, ed.), Springer-Verlag, New York-Berlin-Heidelberg, 1990, pp. 107-117. MR 1035491 (91g:20060)
  • [Du3] -, Integral transforms with reflection group invariance, Canad. J. Math. 43 (1991), 1213-1227. MR 1145585 (93g:33012)
  • [Du4] -, Hankel transforms associated to finite reflection groups, Contemporary Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 123-138. MR 1199124 (94g:33011)
  • [DJO] C. F. Dunkl, M. de Jeu, and E. Opdam, Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), 237-256. MR 1273532 (96b:33012)
  • [DR] C. F. Dunkl and D. E. Ramirez, Topics in harmonic analysis, Appleton-Century-Crofts, New York, 1971. MR 0454515 (56:12766)
  • [GR] K. Gross and D. Richards, Total positivity, finite reflection groups, and a formula of Harish-Chandra, J. Approx. Theory (to appear) MR 1343132 (96i:22022)
  • [Hec1] G. Heckman, A remark on the Dunkl differential-difference operators, Harmonic Analysis on Reductive Groups, (W. Barker and P. Sally, eds.), Birkhäuser, Basel, 1991, pp. 181-191. MR 1168482 (94c:20075)
  • [Hec2] -, An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103 (1991), 341-350. MR 1085111 (92i:33012)
  • [Hel] S. Helgason, Groups and geometric analysis, Academic Press, New York, 1984. MR 754767 (86c:22017)
  • [J] M. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), 147-162. MR 1223227 (94m:22011)
  • [O] E. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), 333-373. MR 1214452 (95j:33044)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E30, 20B30, 33C50, 33C80

Retrieve articles in all journals with MSC: 22E30, 20B30, 33C50, 33C80


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1316848-8
Keywords: Dunkl operators, intertwining operator, reflection groups, integral transform
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society