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Fischer decomposition in symplectic harmonic analysis

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Abstract

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 \(\mathfrak {sl}(2,\mathbb {C}) \oplus \mathfrak {sl}(2,\mathbb {C}) = \mathfrak {so}(4,\mathbb {C})\).

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References

  1. Abreu Blaya, R., Bory Reyes, J., Brackx, F., De Schepper, H., Sommen, F.: Cauchy integral formulae in Hermitian Quaternionic Clifford analysis. Complex Anal. Oper. Theory 6(5), 971–985 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abreu Blaya, R., Bory Reyes, J., Brackx, F., De Schepper, H., Sommen, F.: Matrix Cauchy and Hilbert transforms in Hermitean quaternionic Clifford analysis. Comp. Var. Elliptic Equ. 58(8), 1057–1069 (2013)

    Article  MATH  Google Scholar 

  3. Brackx, F., Bureš, J., De Schepper, H., Eelbode, D., Sommen, F., Souček, V.: Fundaments of Hermitean Clifford analysis—Part I: complex structure. Complex Anal. Oper. Theory 1(3), 341–365 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brackx, F., Bureš, J., De Schepper, H., Eelbode, D., Sommen, F., Souček, V.: Fundaments of Hermitean Clifford analysis—Part II: splitting of \(h\)-monogenic equations. Comp. Var. Elliptic Equ. 52(10–11), 1063–1079 (2007)

    Article  MATH  Google Scholar 

  5. Brackx, F., De Schepper, H., Eelbode, D., Lávička, R., Souček, V.: Fundaments of Quaternionic Clifford analysis—Part I: quaternionic structure. Adv. Appl. Clifford Alg. (accepted)

  6. Brackx, F., De Schepper, H., Eelbode, D., Souček, V.: The Howe dual pair in Hermitean Clifford analysis. Rev. Mat. Iberoamericana 26(2), 449–479 (2010)

    Article  MATH  Google Scholar 

  7. Brackx, F., De Schepper, H., Sommen, F.: The Hermitean Clifford analysis toolbox. Adv. Appl. Clifford Algebras 18(3–4), 451–487 (2008)

    Article  MATH  Google Scholar 

  8. Brackx, F., De Schepper, H., Souček, V.: Fischer decompositions in Euclidean and Hermitian Clifford analysis. Arch. Math. 46(5), 301–321 (2010)

    MATH  MathSciNet  Google Scholar 

  9. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Research Notes in Mathematics 76. Pitman, Boston (1982)

    MATH  Google Scholar 

  10. Colombo, F., Sabadini, I., Sommen, F., Struppa, D.C.: Analysis of Dirac Systems and Computational Algebra. Birkhäuser, Boston (2004)

    Book  MATH  Google Scholar 

  11. Damiano, A., Eelbode, D., Sabadini, I.: Quaternionic Hermitian spinor systems and compatibility conditions. Adv. Geom. 11, 169–189 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Delanghe, R., Sommen, F., Souček, V.: Clifford Algebra and Spinor-Valued Functions. Kluwer Academic Publishers, Dordrecht (1992)

    Book  MATH  Google Scholar 

  13. Eelbode, D.: A Clifford algebraic framework for \({\mathfrak{sp}}(m)\)-invariant differential operators. Adv. Appl. Clifford Algebras 17, 635–649 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Eelbode, D.: Irreducible \({\mathfrak{sl}}(m)\)-modules of Hermitean monogenics. Complex Var. Elliptic Equ. 53(10), 975–987 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fischer, E.: Über die Differentiationsprozesse der Algebra. J. für Math. 148, 1–78 (1917)

    MATH  Google Scholar 

  16. Fulton, W., Harris, J.: Representation Theory: A First Course. Springer, New York (1991)

    MATH  Google Scholar 

  17. Gilbert, J., Murray, M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  18. Gürlebeck, K., Habetha, K., Sprössig, W.: Holomorphic Functions in the Plane and \(n\)-dimensional Space, Translated from the 2006 German Original. Birkhäuser Verlag, Basel (2008)

    Google Scholar 

  19. Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1998)

    Google Scholar 

  20. Howe, R., Tan, E.-C., Willenbring, J.F.: Stable branching rules for classical symmetric pairs. Trans. Am. Math. Soc. 357, 1601–1626 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Peña-Peña, D., Sabadini, I., Sommen, F.: Quaternionic Clifford analysis: the Hermitian setting. Complex Anal. Oper. Theory 1, 97–113 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rocha-Chavez, R., Shapiro, M., Sommen, F.: Integral Theorems for Functions and Differential Forms in \(\mathbb{C}_m\). Research Notes in Mathematics 428. Chapman&Hall / CRC, New York (2002)

    MATH  Google Scholar 

  23. Sabadini, I., Sommen, F.: Hermitian Clifford analysis and resolutions. Math. Method Appl. Sci. 25(16–18), 1395–1414 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

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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Authors and Affiliations

  1. Clifford Research Group, Faculty of Engineering and Architecture, Ghent University Building S22, Galglaan 2, 9000 , Gent, Belgium

    F. Brackx & H. De Schepper

  2. University of Antwerp, Middelheimlaan 2, Antwerpen, Belgium

    D. Eelbode

  3. Charles University Prague, Faculty of Mathematics and Physics, Mathematical Institute Sokolovská 83, 186 75 , Praha, Czech Republic

    R. Lávička & V. Souček

Authors
  1. F. Brackx
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  2. H. De Schepper
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  3. D. Eelbode
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  4. R. Lávička
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  5. V. Souček
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Correspondence to H. De Schepper.

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Brackx, F., De Schepper, H., Eelbode, D. et al. Fischer decomposition in symplectic harmonic analysis. Ann Glob Anal Geom 46, 409–430 (2014). https://doi.org/10.1007/s10455-014-9431-3

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  • DOI: https://doi.org/10.1007/s10455-014-9431-3

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