Annals of Global Analysis and Geometry

, Volume 46, Issue 4, pp 409–430 | Cite as

Fischer decomposition in symplectic harmonic analysis

  • F. Brackx
  • H. De Schepper
  • D. Eelbode
  • R. Lávička
  • V. Souček


In the framework of quaternionic Clifford analysis in Euclidean space \(\mathbb {R}^{4p}\), which constitutes a refinement of Euclidean and Hermitian Clifford analysis, the Fischer decomposition of the space of complex valued polynomials is obtained in terms of spaces of so-called (adjoint) symplectic spherical harmonics, which are irreducible modules for the symplectic group Sp\((p)\). Its Howe dual partner is determined to be \(\mathfrak {sl}(2,\mathbb {C}) \oplus \mathfrak {sl}(2,\mathbb {C}) = \mathfrak {so}(4,\mathbb {C})\).


Quaternionic Clifford analysis Symplectic harmonics Fischer decomposition 



The authors kindly acknowledge financial support by the E. Cech Institute, more particularly from grant P201/12/G028 of the Grant Agency of the Czech Republic.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • F. Brackx
    • 1
  • H. De Schepper
    • 1
  • D. Eelbode
    • 2
  • R. Lávička
    • 3
  • V. Souček
    • 3
  1. 1.Clifford Research Group, Faculty of Engineering and ArchitectureGhent University Building S22GentBelgium
  2. 2.University of AntwerpAntwerpenBelgium
  3. 3.Charles University Prague, Faculty of Mathematics and PhysicsMathematical Institute Sokolovská 83PrahaCzech Republic

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