Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T21:14:22.428Z Has data issue: false hasContentIssue false
  • English
  • Français

Laguerre semigroup and Dunkl operators

Part of: Integral transforms, operational calculus Lie groups Abstract harmonic analysis Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  18 May 2012

Salem Ben Saïd ,
Toshiyuki Kobayashi  and
Bent Ørsted
Salem Ben Saïd
Affiliation:
Université Henri Poincaré Nancy 1, Institut Elie Cartan, B.P. 239, 54506 Vandoeuvre-Les-Nancy, France (email: [email protected])
Toshiyuki Kobayashi
Affiliation:
The University of Tokyo, IPMU, Graduate School of Mathematical Sciences, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan (email: [email protected])
Bent Ørsted
Affiliation:
University of Aarhus, Department of Mathematical Sciences, Building 530, Ny Munkegade, DK 8000, Aarhus C, Denmark (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct a two-parameter family of actions ωk,a of the Lie algebra 𝔰𝔩(2,ℝ) by differential–difference operators on ℝN∖{0}. Here k is a multiplicity function for the Dunkl operators, and a>0 arises from the interpolation of the two 𝔰𝔩(2,ℝ) actions on the Weil representation of Mp(N,ℝ) and the minimal unitary representation of O(N+1,2). We prove that this action ωk,a lifts to a unitary representation of the universal covering of SL (2,ℝ) , and can even be extended to a holomorphic semigroup Ωk,a. In the k≡0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup studied by the second author with G. Mano (a=1) . One boundary value of our semigroup Ωk,a provides us with (k,a) -generalized Fourier transforms ℱk,a, which include the Dunkl transform 𝒟k (a=2) and a new unitary operator ℋk  (a=1) , namely a Dunkl–Hankel transform. We establish the inversion formula, a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty relation for ℱk,a. We also find kernel functions for Ωk,a and ℱk,a for a=1,2 in terms of Bessel functions and the Dunkl intertwining operator.

Keywords

Dunkl operatorsgeneralized Fourier transformCoxeter groupsSchrödinger modelholomorphic semigroupWeil representationHermite semigroupHankel transformsHeisenberg inequalityrational Cherednik algebraminimal representation

MSC classification

Secondary: 33C52: Orthogonal polynomials and functions associated with root systems 22E46: Semisimple Lie groups and their representations 43A32: Other transforms and operators of Fourier type 44A15: Special transforms (Legendre, Hilbert, etc.) 47D03: Groups and semigroups of linear operators
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 4 , July 2012 , pp. 1265 - 1336
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[AAR99]Andrews, G., Askey, R. and Roy, R., Special functions (Cambridge University Press, Cambridge, 1999).Google Scholar
[Ben07]Ben Said, S., On the integrability of a representation of 𝔰𝔩(2,ℝ), J. Funct. Anal. 250 (2007), 249–264.Google Scholar
[BK{\OT1\O }09]Ben Said, S., Kobayashi, T. and Ørsted, B., Generalized Fourier transforms ℱk,a, C. R. Acad. Sci. Paris Ser. I 347 (2009), 1119–1124.CrossRefGoogle Scholar
[B{\OT1\O }06]Ben Said, S. and Ørsted, B., Segal–Bargmann transforms associated with finite Coxeter groups, Math. Ann. 334 (2006), 281–323.Google Scholar
[Che95]Cherednik, I., Macdonald’s evaluation conjectures and difference Fourier transform, Invent. Math. 122 (1995), 119–145.Google Scholar
[CM02]Cherednik, I. and Markov, Y., Hankel transform via double Hecke algebra, in Iwahori–Hecke algebras and their representation theory (Martina-Franca, 1999), Lecture Notes in Mathematics, vol. 1804, eds Baldoni, M. W. and Barbasch, D. (Springer, Berlin, 2002), 1–25.Google Scholar
[DB07]Debnath, L. and Bhatta, D., Integral transforms and their applications, second edition (Chapman & Hall/CRC, Boca Raton, FL, 2007).Google Scholar
[Dun88]Dunkl, C. F., Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33–60.Google Scholar
[Dun89]Dunkl, C. F., Differential–difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167–183.Google Scholar
[Dun91]Dunkl, C. F., Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213–1227.Google Scholar
[Dun92]Dunkl, C. F., Hankel transforms associated to finite reflection groups, in Proceedings of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications (Tampa, FL, 1991), Contemp. Math. 138 (1992), 123–138.Google Scholar
[Dun03]Dunkl, C. F., A Laguerre polynomial orthogonality and the hydrogen atom, Anal. Appl. (Singap.) 1 (2003), 177–188.Google Scholar
[Dun08]Dunkl, C. F., Reflection groups in analysis and applications, Jpn. J. Math. (3rd ser.) 3 (2008), 215–246.CrossRefGoogle Scholar
[DJO94]Dunkl, C. F., de Jeu, M. and Opdam, E., Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), 237–256.CrossRefGoogle Scholar
[DX01]Dunkl, C. F. and Xu, Y., Orthogonal polynomials of several variables (Cambridge University Press, Cambridge, 2001).Google Scholar
[Eti10]Etingof, P., A uniform proof of the Macdonald–Mehta–Opdam identity for finite Coxeter groups, Math. Res. Lett. 17 (2010), 275–282.CrossRefGoogle Scholar
[EG02]Etingof, P. and Ginzburg, V., Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243–348.Google Scholar
[Fol89]Folland, G. B., Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122 (Princeton University Press, Princeton, NJ, 1989).Google Scholar
[FS97]Folland, G. B. and Sitaram, A., The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), 207–238.CrossRefGoogle Scholar
[GY06]Gallardo, L. and Yor, M., A chaotic representation property of the multidimensional Dunkl processes, Ann. Probab. 34 (2006), 1530–1549.CrossRefGoogle Scholar
[GG77]Gelfand, I. M. and Gindikin, S. G., Complex manifolds whose spanning trees are real semisimple Lie groups, and analytic discrete series of representations, Funct. Anal. Appl. 11 (1977), 19–27.Google Scholar
[GR80]Gradshteyn, I. S. and Ryzhik, I. M., Table of integrals, series, and products (Academic Press [Harcourt Brace Jovanovich, Publishers], New York–London–Toronto, 1980), Corrected and enlarged edition edited by Alan Jeffrey. Incorporating the fourth edition edited by Yu. V. Geronimus and M. Yu. Tseytlin. Translated from the Russian.Google Scholar
[Hec91]Heckman, G. J., A remark on the Dunkl differential–difference operators, in Harmonic analysis on reductive groups, Progress in Mathematics, vol. 101, ed. Barker, W. (Birkhäuser, Basel, 1991), 181–191.CrossRefGoogle Scholar
[Hel84]Helgason, S., Groups and geometric analysis (Academic Press, New York, 1984).Google Scholar
[HN93]Hilgert, J. and Neeb, K.-H., Lie semigroups and their applications, Lecture Notes in Mathematics, vol. 1552 (Springer, Berlin, 1993).Google Scholar
[HN00]Hilgert, J. and Neeb, K.-H., Positive definite spherical functions on Olshanski domains, Trans. Amer. Math. Soc. 352 (2000), 1345–1380.CrossRefGoogle Scholar
[How80]Howe, R., On the role of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc. (N.S.) 3 (1980), 821–843.Google Scholar
[How88]Howe, R., The oscillator semigroup, in The mathematical heritage of Hermann Weyl (Durham, NC, 1987), Proceedings of Symposia in Pure Mathematics, vol. 48, ed. Wells, R. O. Jr (American Mathematical Society, Providence, RI, 1988), 61–132.CrossRefGoogle Scholar
[HT92]Howe, R. and Tan, E.-C., Nonabelian harmonic analysis applications of SL(2,R), Universitext (Springer, New York, 1992).Google Scholar
[Hum90]Humphreys, J. E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29 (Cambridge University Press, Cambridge, 1990).Google Scholar
[Jeu93]de Jeu, M. F. E., The Dunkl transform, Invent. Math. 113 (1993), 147–162.Google Scholar
[KN95]Knapp, A. W. and Vogan, D. A. Jr, Cohomological induction and unitary representations (Princeton University Press, Princeton, NJ, 1995).Google Scholar
[Kob98]Kobayashi, T., Discrete decomposability of the restriction of A 𝔮(λ) with respect to reductive subgroups and its applications, Part I. Invent. Math. 117 (1994), 181–205; Part II. Ann. of Math. (2) 147 (1998), 709–729; Part III. Invent. Math. 131 (1998), 229–256.CrossRefGoogle Scholar
[Kob00]Kobayashi, T., Discretely decomposable restriction of unitary representations of reductive Lie groups—examples and conjectures, Adv. Stud. Pure Math. 26 (2000), 99–127.Google Scholar
[KM05]Kobayashi, T. and Mano, G., Integral formulas for the minimal representations for O(p,2), Acta Appl. Math. 86 (2005), 103–113.Google Scholar
[KM07a]Kobayashi, T. and Mano, G., The inversion formula and holomorphic extension of the minimal representation of the conformal group, in Harmonic analysis, group representations, automorphic forms and invariant theory: in honor of Roger Howe, eds Li, J. S., Tan, E. C., Wallach, N. and Zhu, C. B. (World Scientific, Singapore, 2007), 159–223 (cf. math.RT/0607007).Google Scholar
[KM07b]Kobayashi, T. and Mano, G., Integral formula of the unitary inversion operator for the minimal representation of O(p,q), Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), 27–31.Google Scholar
[KM11]Kobayashi, T. and Mano, G., The Schrödinger model for the minimal representation of the indefinite orthogonal group O(p,q), Memoirs of the American Mathematical Societies, vol. 212, no. 1000 (American Mathematical Societies, Providence, RI, 2011), vi + 132 (cf. arXiv:0712.1769).Google Scholar
[K{\OT1\O }03]Kobayashi, T. and Ørsted, B., Analysis on the minimal representation of O(p,q). II. Branching laws, Adv. Math. 180 (2003), 513–550.Google Scholar
[Kos00]Kostant, B., On Laguerre polynomials, Bessel functions, and Hankel transform and a series in the unitary dual of the simply-connected covering group of SL(2,ℝ), Represent. Theory 4 (2000), 181–224.Google Scholar
[Lan85]Lang, S., SL 2(ℝ), Graduate Texts in Mathematics, vol. 105 (Springer, New York, 1985), reprint of the 1975 edition.Google Scholar
[Mac82]Macdonald, I. G., Some conjectures for root systems, SIAM J. Math. Anal. 13 (1982), 988–1007.Google Scholar
[Man08]Mano, G., A continuous family of unitary representations with two hidden symmetries—an example, in Representation theory and analysis on homogeneous spaces, ed. H. Sekiguchi (RIMS Kokyuroku Bessatsu, B7, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008), 137–143 (in Japanese).Google Scholar
[Nel59]Nelson, E., Analytic vectors, Ann. of Math. (2) 70 (1959), 572–615.CrossRefGoogle Scholar
[Ols81]Olshanski, G., Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete series, Funct. Anal. Appl. 15 (1981), 275–285.CrossRefGoogle Scholar
[Opd93]Opdam, E., Dunkl operators, Bessel functions and discriminant of a finite Coxeter group, Compositio Math. 85 (1993), 333–373.Google Scholar
[{\OT1\O }SS09]Ørsted, B., Somberg, P. and Soucek, V., The Howe duality for the Dunkl version of the Dirac operator, Adv. Appl. Clifford Algebr. 19 (2009), 403–415.Google Scholar
[Rao77]Ranga Rao, R., Unitary representations defined by boundary conditions—the case of 𝔰𝔩(2,ℝ), Acta Math. 139 (1977), 185–216.Google Scholar
[Rös99a]Rösler, M., Positivity of Dunkl’s intertwining operator, Duke Math. J. 98 (1999), 445–463.CrossRefGoogle Scholar
[Rös99b]Rösler, M., An uncertainty principle for the Dunkl transform, Bull. Austral. Math. Soc. 59 (1999), 353–360.Google Scholar
[RV98]Rösler, M. and Voit, M., Markov processes related with Dunkl operators, Adv. Appl. Math. 21 (1998), 575–643.Google Scholar
[Sat59]Sato, M., Theory of hyperfunctions I, J. Fac. Sci. Shinshu Univ. Tokyo 8 (1959), 139–193.Google Scholar
[Shi01]Shimeno, N., A note on the uncertainty principle for the Dunkl transform, J. Math. Sci. Univ. Tokyo 8 (2001), 33–42.Google Scholar
[Sta86]Stanton, R. J., Analytic extension of the holomorphic discrete series, Amer. J. Math. 108 (1986), 1411–1424.Google Scholar
[Tri02]Trimèche, K., Paley–Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transforms Spec. Funct. 13 (2002), 17–38.Google Scholar
[Ver79]Vergne, M., On Rossmann’s character formula for discrete series, Invent. Math. 54 (1979), 11–14.Google Scholar
[Vil68]Vilenkin, N. Ja., Special functions and the theory of group representations, Translations of Mathematical Monographs, vol. 22 (American Mathematical Society, Providence, RI, 1968), translated from the Russian by V. N. Singh.Google Scholar
[WG89]Wang, Z. X. and Guo, D. R., Special functions (World Scientific, Teaneck, NJ, 1989), translated from the Chinese by Guo and X. J. Xia.Google Scholar
[Wei64]Weil, A., Sur certains groupes d’opèrateurs unitaires, Acta Math. 111 (1964), 143–211.Google Scholar
[Xu00]Xu, Y., Funk–Hecke formula for orthogonal polynomials on spheres and on balls, Bull. Lond. Math. Soc. 32 (2000), 447–457.CrossRefGoogle Scholar