Duke Mathematical Journal

Positivity of Dunkl’s intertwining operator

Margit Rösler
Source: Duke Math. J. Volume 98, Number 3 (1999), 445-463.
First Page: Show Hide
Primary Subjects: 33C80
Secondary Subjects: 33C67, 44A15
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077228355
Mathematical Reviews number (MathSciNet): MR1695797
Zentralblatt MATH identifier: 0947.33013
Digital Object Identifier: doi:10.1215/S0012-7094-99-09813-7

References

[BF] T. H. Baker and P. J. Forrester, Nonsymmetric Jack polynomials and integral kernels, Duke Math. J. 95 (1998), no. 1, 1–50.
Mathematical Reviews (MathSciNet): MR2000b:33006
Zentralblatt MATH: 0948.33012
Digital Object Identifier: doi:10.1215/S0012-7094-98-09501-1
Project Euclid: euclid.dmj/1077229503
[BeF] Christian Berg and Gunnar Forst, Potential theory on locally compact abelian groups, Springer-Verlag, New York, 1975.
Mathematical Reviews (MathSciNet): MR58:1204
Zentralblatt MATH: 0308.31001
[C] Ivan Cherednik, A unification of Knizhnik-Zamolodchikov and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1991), no. 2, 411–431.
Mathematical Reviews (MathSciNet): MR93b:17040
Zentralblatt MATH: 0742.20019
Digital Object Identifier: doi:10.1007/BF01243918
[dJ] M. F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), no. 1, 147–162.
Mathematical Reviews (MathSciNet): MR94m:22011
Zentralblatt MATH: 0789.33007
Digital Object Identifier: doi:10.1007/BF01244305
[D1] Charles F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167–183.
Mathematical Reviews (MathSciNet): MR90k:33027
Zentralblatt MATH: 0652.33004
Digital Object Identifier: doi:10.2307/2001022
[D2] 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.
Mathematical Reviews (MathSciNet): MR91g:20060
Zentralblatt MATH: 0719.33008
[D3] Charles F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), no. 6, 1213–1227.
Mathematical Reviews (MathSciNet): MR93g:33012
Zentralblatt MATH: 0827.33010
[D4] 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.
Mathematical Reviews (MathSciNet): MR94g:33011
Zentralblatt MATH: 0789.33008
[D5] Charles F. Dunkl, Intertwining operators associated to the group $S\sb 3$, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3347–3374.
Mathematical Reviews (MathSciNet): MR97b:22009
Zentralblatt MATH: 0857.22008
Digital Object Identifier: doi:10.2307/2155014
[DJO] 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.
Mathematical Reviews (MathSciNet): MR96b:33012
Zentralblatt MATH: 0829.33010
Digital Object Identifier: doi:10.2307/2154950
[FD] J. M. G. Fell and R. S. Doran, Representations of $\sp *$-algebras, locally compact groups, and Banach $\sp *$-algebraic bundles. Vol. 1, Pure and Applied Mathematics, vol. 125, Academic Press Inc., Boston, MA, 1988, Basic Representation Theory of Groups and Algebras.
Mathematical Reviews (MathSciNet): MR90c:46001
Zentralblatt MATH: 0652.46050
[GS] Ĭ. Ī. Gīhman and A. V. Skorohod, The theory of stochastic processes. II, Grundlehren Math. Wiss., vol. 218, Springer-Verlag, New York, 1975.
Mathematical Reviews (MathSciNet): MR51:11656
Zentralblatt MATH: 0305.60027
[He] G. J. Heckman, An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103 (1991), no. 2, 341–350.
Mathematical Reviews (MathSciNet): MR92i:33012
Zentralblatt MATH: 0721.33009
Digital Object Identifier: doi:10.1007/BF01239517
[Hu] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990.
Mathematical Reviews (MathSciNet): MR92h:20002
Zentralblatt MATH: 0725.20028
[Ka] Tosio Kato, A short introduction to perturbation theory for linear operators, Springer-Verlag, New York, 1982.
Mathematical Reviews (MathSciNet): MR83m:47015
Zentralblatt MATH: 0493.47008
[Ki] Alexander A. Kirillov, Jr., Lectures on affine Hecke algebras and Macdonald's conjectures, Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 251–292.
Mathematical Reviews (MathSciNet): MR99c:17015
Zentralblatt MATH: 0884.05100
Digital Object Identifier: doi:10.1090/S0273-0979-97-00727-1
[LV] Luc Lapointe and Luc Vinet, Exact operator solution of the Calogero-Sutherland model, Comm. Math. Phys. 178 (1996), no. 2, 425–452.
Mathematical Reviews (MathSciNet): MR97c:81217
Zentralblatt MATH: 0859.35103
Digital Object Identifier: doi:10.1007/BF02099456
Project Euclid: euclid.cmp/1104286659
[O1] E. M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compositio Math. 85 (1993), no. 3, 333–373.
Mathematical Reviews (MathSciNet): MR95j:33044
Zentralblatt MATH: 0778.33009
[O2] Eric M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), no. 1, 75–121.
Mathematical Reviews (MathSciNet): MR98f:33025
Zentralblatt MATH: 0836.43017
Digital Object Identifier: doi:10.1007/BF02392487
[P] Alexios P. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69 (1992), no. 5, 703–705.
Mathematical Reviews (MathSciNet): MR93g:58063
Zentralblatt MATH: 0968.37521
Digital Object Identifier: doi:10.1103/PhysRevLett.69.703
[R] Margit Rösler, Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), no. 3, 519–542.
Mathematical Reviews (MathSciNet): MR99k:33048
Zentralblatt MATH: 0908.33005
Digital Object Identifier: doi:10.1007/s002200050307
[RV1] Margit Rösler and Michael Voit, Markov processes related with Dunkl operators, Adv. in Appl. Math. 21 (1998), no. 4, 575–643.
Mathematical Reviews (MathSciNet): MR2000j:60019
Zentralblatt MATH: 0919.60072
Digital Object Identifier: doi:10.1006/aama.1998.0609
[RV2] Margit Rösler and Michael Voit, Biorthogonal polynomials associated with reflection groups and a formula of Macdonald, J. Comput. Appl. Math. 99 (1998), no. 1-2, 337–351.
Mathematical Reviews (MathSciNet): MR2000e:33019
Zentralblatt MATH: 0928.33012
Digital Object Identifier: doi:10.1016/S0377-0427(98)00168-X
[X1] Yuan Xu, Integration of the intertwining operator for $h$-harmonic polynomials associated to reflection groups, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2963–2973.
Mathematical Reviews (MathSciNet): MR97m:33004
Zentralblatt MATH: 0881.33010
Digital Object Identifier: doi:10.1090/S0002-9939-97-03986-5
[X2] Yuan Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canad. J. Math. 49 (1997), no. 1, 175–192.
Mathematical Reviews (MathSciNet): MR98g:33017
Zentralblatt MATH: 0872.33008
[X3] Yuan Xu, Intertwining operator and $h$-harmonics associated with reflection groups, Canad. J. Math. 50 (1998), no. 1, 193–209.
Mathematical Reviews (MathSciNet): MR2000d:33008
Zentralblatt MATH: 0905.33005
[vD] J. F. van Diejen, Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement, Comm. Math. Phys. 188 (1997), no. 2, 467–497.
Mathematical Reviews (MathSciNet): MR98j:33007
Zentralblatt MATH: 0917.33008
Digital Object Identifier: doi:10.1007/s002200050174

2012 © Duke University Press

Duke Mathematical Journal

Duke Mathematical Journal

Turn MathJax Off
What is MathJax?