Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 010, 22 pages      arXiv:1101.5551

The Clifford Deformation of the Hermite Semigroup

Hendrik De Bie a, Bent Ørsted b, Petr Somberg c and Vladimir Souček c
a) Department of Mathematical Analysis, Ghent University, Galglaan 2, 9000 Gent, Belgium
b) Department of Mathematical Sciences, University of Aarhus, Building 530, Ny Munkegade, DK 8000, Aarhus C, Denmark
c) Mathematical Institute of Charles University, Sokolovská 83, 186 75 Praha, Czech Republic

Received September 21, 2012, in final form January 29, 2013; Published online February 05, 2013

This paper is a continuation of the paper [De Bie H., Ørsted B., Somberg P., Souček V., Trans. Amer. Math. Soc. 364 (2012), 3875–3902], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many results obtained in [Ben Saïd S., Kobayashi T., Ørsted B., Compos. Math. 148 (2012), 1265–1336]. We establish the analogues of Bochner's formula and the Heisenberg uncertainty relation in the framework of the (holomorphic) Hermite semigroup, and also give a detailed analytic treatment of the series expansion of the associated integral transform.

Key words: Dunkl operators; Clifford analysis; generalized Fourier transform; Laguerre polynomials; Kelvin transform; holomorphic semigroup.

pdf (474 kb)   tex (28 kb)


  1. Barbasch D., Ciubotaru D., Trapa P.E., Dirac cohomology for graded affine Hecke algebras, Acta Math. 209 (2012), 197-227, arXiv:1006.3822.
  2. Ben Saïd S., Kobayashi T., Ørsted B., Laguerre semigroup and Dunkl operators, Compos. Math. 148 (2012), 1265-1336, arXiv:0907.3749.
  3. Cação I., Constales D., Krausshar R.S., On the role of arbitrary order Bessel functions in higher dimensional Dirac type equations, Arch. Math. (Basel) 87 (2006), 468-477.
  4. Cherednik I., Markov Y., Hankel transform via double Hecke algebra, in Iwahori-Hecke algebras and their representation theory (Martina-Franca, 1999), Lecture Notes in Math., Vol. 1804, Springer, Berlin, 2002, 1-25, math.QA/0004116.
  5. De Bie H., De Schepper N., Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator, Bull. Belg. Math. Soc. Simon Stevin 18 (2011), 193-214.
  6. De Bie H., De Schepper N., Sommen F., The class of Clifford-Fourier transforms, J. Fourier Anal. Appl. 17 (2011), 1198-1231, arXiv:1101.1793.
  7. De Bie H., Ørsted B., Somberg P., Souček V., Dunkl operators and a family of realizations of osp(1|2), Trans. Amer. Math. Soc. 364 (2012), 3875-3902, arXiv:0911.4725.
  8. De Bie H., Xu Y., On the Clifford-Fourier transform, Int. Math. Res. Not. 2011 (2011), no. 22, 5123-5163, arXiv:1003.0689.
  9. Delanghe R., Sommen F., Souček V., Clifford algebra and spinor-valued functions. A function theory for the Dirac operator, Mathematics and its Applications, Vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1992.
  10. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  11. Dunkl C.F., Hankel transforms associated to finite reflection groups, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, Amer. Math. Soc., Providence, RI, 1992, 123-138.
  12. Dunkl C.F., de Jeu M.F.E., Opdam E.M., Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), 237-256.
  13. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  14. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. II, Mc-Graw Hill, New York, 1953.
  15. Frappat L., Sciarrino A., Sorba P., Dictionary on Lie algebras and superalgebras, Academic Press Inc., San Diego, CA, 2000.
  16. Gilbert J.E., Murray M.A.M., Clifford algebras and Dirac operators in harmonic analysis, Cambridge Studies in Advanced Mathematics, Vol. 26, Cambridge University Press, Cambridge, 1991.
  17. Howe R., The oscillator semigroup, in The Mathematical Heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math., Vol. 48, Amer. Math. Soc., Providence, RI, 1988, 61-132.
  18. Humphreys J.E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, Vol. 29, Cambridge University Press, Cambridge, 1990.
  19. Kobayashi T., Mano G., Integral formulae for the minimal representation of O(p,2), Acta Appl. Math. 86 (2005), 103-113.
  20. Kobayashi T., 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, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., Vol. 12, World Sci. Publ., Hackensack, NJ, 2007, 151-208, math.RT/0607007.
  21. Kobayashi T., Mano G., The Schrödinger model for the minimal representation of the indefinite orthogonal group O(p,q), Mem. Amer. Math. Soc. 213 (2011), no. 1000, vi+132 pages, arXiv:0712.1769.
  22. Ol'shanski G.I., Complex Lie semigroups, Hardy spaces and the Gel'fand-Gindikin program, Differential Geom. Appl. 1 (1991), 235-246.
  23. Ørsted B., Somberg P., Souček V., The Howe duality for the Dunkl version of the Dirac operator, Adv. Appl. Clifford Algebr. 19 (2009), 403-415.
  24. Rösler M., A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355 (2003), 2413-2438, math.CA/0210137.
  25. Rösler M., Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Math., Vol. 1817, Springer, Berlin, 2003, 93-135, math.CA/0210366.
  26. Rösler M., Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006.
  27. Szegö G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 23, 4th ed., American Mathematical Society, Providence, R.I., 1975.
  28. Thangavelu S., Xu Y., Convolution operator and maximal function for the Dunkl transform, J. Anal. Math. 97 (2005), 25-55, math.CA/0403049.
  29. Watson G.N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1944.

Previous article  Next article   Contents of Volume 9 (2013)