Science China Mathematics

, Volume 53, Issue 12, pp 3153–3162 | Cite as

Howe duality in Dunkl superspace



In the framework of superspace in Clifford analysis for the Dunkl version, the Fischer decomposition is established for solutions of the Dunkl super Dirac operators. The result is general without restrictions on multiplicity functions or on super dimensions. The Fischer decomposition provides a module for the Howe dual pair G × osp(1|2) on the space of spinor valued polynomials with G the Coxeter group, while the generators of the Lie superspace reveal the naturality of the Fischer decomposition.


Dunkl operators superspace Fischer decomposition Dirac operator 


Primary 30G35 Secondary 31A30 33C52 58C50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brackx F, Delanghe R, Sommen F. Clifford Analysis. Boston-London-Melbourne: Pitman Publishers, 1982MATHGoogle Scholar
  2. 2.
    Brackx F, De Schapper H, Eelbode D, et al. The Howe dual pair in Hermitean Clifford analysis. Rev Mat Iberoamericana, 2010, 26: 449–479MATHGoogle Scholar
  3. 3.
    Cerejeiras P, Kähler U, Ren G B. Clifford analysis for finite reflection groups. Complex Var Elliptic Equ, 2006, 51: 487–495MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    De Bie H, Sommen F. Sperical hamonics and intergration in superspce. J Phys A, 2007, 40: 7193–7212MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    De Bie H. Fourier transform and related integral transform in superspace. J Math Anal Appl, 2008, 345: 147–164MATHMathSciNetGoogle Scholar
  6. 6.
    De Bie H. Schrödinger equation with delta potential in superspace. Phys Lett A, 2008, 372: 4350–4352CrossRefMathSciNetGoogle Scholar
  7. 7.
    De Bie H, Sommen F. A Clifford analysis approach to superspace. Ann Phys, 2007, 322: 2978–2993MATHGoogle Scholar
  8. 8.
    De Bie H, Sommen F. Correct rules for Clifford calculus on superspace. Adv Appl Clifford Algebr, 2007, 17: 357–382MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    De Bie H, Sommen F. Hermite and Gegenbauer polynomials in superspace using Clifford analysis. J Phy A, 2007, 40: 7193–7212MATHGoogle Scholar
  10. 10.
    De Bie H, Sommen F. Fischer decompositions in superspace. In: Function Spaces in Complex and Clifford Analysis. Hanoi: National Univ Publ, 2008, 170–188Google Scholar
  11. 11.
    De Bie H, Sommen F. Fundamental solutions for the super Laplace and Dirac operators and all their natural powers. J Math Anal Appl, 2008, 338: 1320–1328MATHMathSciNetGoogle Scholar
  12. 12.
    Delanghe R, Sommen F, Souček V. Clifford Algebra and Spinor-Valued Functions. Amsterdam: Kluwer Acad Publ, 1992MATHGoogle Scholar
  13. 13.
    van Diejen J F, Vinet L. Calogero-Moser-Sutherland Models. New York: Springer-Verlag, 2000MATHGoogle Scholar
  14. 14.
    Dunkl C F, Xu Y. Orthogonal Polynomials of Several Variables. Cambridge: Cambridge University Press, 2001MATHCrossRefGoogle Scholar
  15. 15.
    Gilbert J, Murray M. Clifford Algebra and Dirac Operators in Harmonic Analysis. Cambridge: Cambridge University Press, 1991CrossRefGoogle Scholar
  16. 16.
    Gürlebeck K, Sprössig W. Quaternionic Analysis and Elliptic Boundary Value Problems. Berlin: Akademie-Verlag, 1989MATHGoogle Scholar
  17. 17.
    Heckman G J. Dunkl operators. Astérisque, 1997, 245: 223–246MathSciNetGoogle Scholar
  18. 18.
    Humphreys J E. Reflection Groups and Coxter Groups. Cambridge: Cambridge Univ Press, 1990Google Scholar
  19. 19.
    Malonek H R, Ren G B. Almansi-type theorems in Clifford analysis. Math Methods Appl Sci, 2002, 25: 1541–1552MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Orsted B, Somberg P, Souček V. The Howe duality for the Dunkl version of the Dirac operator. Adv Appl Clifford Algebr, 2009, 19: 403–415CrossRefMathSciNetGoogle Scholar
  21. 21.
    Ren G B. Almansi decomposition in Dunkl superspace. SubmittedGoogle Scholar
  22. 22.
    Ren G B. Almansi decomposition for Dunkl operators. Sci China Ser A, 2005, 48Suppl: 333–342MATHCrossRefGoogle Scholar
  23. 23.
    Ren G B, Kähler U. Almansi decompositions for polyharmonic, polyheat, and polywave functions. Studia Math, 2006, 172: 91–100MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Saïd S B, Ørsted B. Segal-Bargmann transforms associated with finite Coxter groups. Math Ann, 2006, 334: 281–323MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Department of MathematicsUniversity of AveiroAveiroPortugal

Personalised recommendations