Intertwining operators associated to the group $S_ 3$
HTML articles powered by AMS MathViewer
- by Charles F. Dunkl PDF
- Trans. Amer. Math. Soc. 347 (1995), 3347-3374 Request permission
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
-
N. Bailey, Generalized hypergeometric series, Cambridge Univ. Press, Cambridge, 1935.
- R. J. Beerends, A transmutation property of the generalized Abel transform associated with root system $A_2$, Indag. Math. (N.S.) 1 (1990), no. 2, 155–168. MR 1060823, DOI 10.1016/0019-3577(90)90002-5
- Robert Carroll, Transmutation theory and applications, North-Holland Mathematics Studies, vol. 117, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 105. MR 807146
- Charles F. Dunkl, Operators commuting with Coxeter group actions on polynomials, Invariant theory and tableaux (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 19, Springer, New York, 1990, pp. 107–117. MR 1035491
- Charles F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), no. 6, 1213–1227. MR 1145585, DOI 10.4153/CJM-1991-069-8
- Charles F. Dunkl, Hankel transforms associated to finite reflection groups, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 123–138. MR 1199124, DOI 10.1090/conm/138/1199124
- C. F. Dunkl, M. F. E. de Jeu, and E. M. Opdam, Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), no. 1, 237–256. MR 1273532, DOI 10.1090/S0002-9947-1994-1273532-6
- Charles F. Dunkl and Donald E. Ramirez, Topics in harmonic analysis, The Appleton-Century Mathematics Series, Appleton-Century-Crofts [Meredith Corporation], New York, 1971. MR 0454515
- Kenneth I. Gross and Donald St. P. Richards, Total positivity, finite reflection groups, and a formula of Harish-Chandra, J. Approx. Theory 82 (1995), no. 1, 60–87. MR 1343132, DOI 10.1006/jath.1995.1068
- Gerrit J. Heckman, A remark on the Dunkl differential-difference operators, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 181–191. MR 1168482
- G. J. Heckman, An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103 (1991), no. 2, 341–350. MR 1085111, DOI 10.1007/BF01239517
- Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
- M. F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), no. 1, 147–162. MR 1223227, DOI 10.1007/BF01244305
- E. M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), no. 3, 333–373. MR 1214452
Additional Information
- © Copyright 1995 American Mathematical Society
- 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