Advertisement

Generalized Fourier Transforms Associated with the Oscillator Representation

  • Jing-Song HuangEmail author
  • Ling Zhou
Research Article
  • 13 Downloads

Abstract

By identifying the Fourier transform \(\mathcal {F}\) with an operator in the oscillator representation of the metaplectic group \({\widetilde{Sp}(2n,\mathbb {R})}\), the twofold cover of the symplectic group \(Sp(2n,\mathbb {R})\), we study generalized Fourier transforms inspired by the work of De Bie, Oste and Van der Jeugt. We obtain several families of operators \(\mathcal {T}\)’s that have the important properties similar to \(\mathcal {F}\) using various dual pair correspondences.

Keywords

Fourier transform Oscillator representation Dual pair correspondence 

Notes

References

  1. 1.
    Cohen, H.: A conceptual breakthrough in sphere packing. Not. AMS 64(2), 102–115 (2017)zbMATHGoogle Scholar
  2. 2.
    De Bie, H., Oste, R., Van der Jeugt, J.: Generalized Fourier transforms arising from the enveloping algebras of \(sl(2)\) and \(osp(1|2)\). Int. Math. Res. Not. 2015, 4649–4705 (2016)CrossRefzbMATHGoogle Scholar
  3. 3.
    Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants. Springer, New York (2006)zbMATHGoogle Scholar
  4. 4.
    Howe, R.: On some results of Strichartz and of Rallis and Schiffman. J. Funct. Anal. 32, 297–303 (1979)CrossRefzbMATHGoogle Scholar
  5. 5.
    Howe, R., Tan, E.C.: Non-Abelian Harmonic Analysis—Applications of \(SL(2,{\mathbb{R}})\). Springer, New York (1992)CrossRefzbMATHGoogle Scholar
  6. 6.
    Kirillov, A.: An Introduction to Lie Groups and Lie Algebras. Cambridge University Press, New York (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Knapp, A.W.: Lie Groups Beyond an Introduction, 1st edn. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  8. 8.
    Kudla, S.S.: Seesaw Dual Reductive Pair. Princeton University Press, Princeton (1986)zbMATHGoogle Scholar
  9. 9.
    Kobayashi, T., Mano, G.: The Schrödinger Model for the Minimal Representation of the Indefinite Orthogonal Group \(O(p, q)\). American Mathematical Society, Providence (2008)zbMATHGoogle Scholar
  10. 10.
    Repka, J.: Tensor Products of Unitary Representations of \(SL_2({\mathbb{R}})\). Am. J. Math. 100, 747–774 (1978)CrossRefzbMATHGoogle Scholar
  11. 11.
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHong Kong University of Science and TechnologyKowloonChina
  2. 2.Department of MathematicsOhio State UniversityColumbusUSA

Personalised recommendations