Journal of Fourier Analysis and Applications

, Volume 25, Issue 6, pp 2837–2876 | Cite as

The Plancherel Formula for an Inhomogeneous Vector Group

  • Didier ArnalEmail author
  • Bradley Currey
  • Béchir Dali


We give a concrete realization of the Plancherel measure for a semi-direct product \(N \rtimes H\) where N and H are vector groups for which the linear action of H on N is almost everywhere regular. A procedure using matrix reductions produces explicit (orbital) parameters by which a continuous field of unitary irreducible representations is realized and the almost all of the dual space of \(N \rtimes H\) naturally has the structure of a smooth manifold. Using the simplest possible field of positive semi-invariant operators, the Plancherel measure is obtained via an explicit volume form on a smooth cross-section \(\Sigma \) for almost all H-orbits. The associated trace characters are also shown to be tempered distributions.


Semi-direct product Regular representation Plancherel formula 

Mathematics Subject Classification

22Exx 22E27 22E45 22D10 22D30 



  1. 1.
    Arnal, D., Currey, B., Dali, B.: Canonical coordinates for a class of solvable Lie groups. Monatsh. Math. 166(1), 19–55 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arnal, D., Currey, B., Oussa, V.: Characterization of regularity for a connected abelian action. Monatsh. Math. 180(1), 1–37 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arnal, D., Dali, B., Currey, B., Oussa, V.: Regularity of Abelian Linear Actions. Commutative and Noncommutative Harmonic Analysis and Applications, vol. 603, pp. 89–109. American Mathematical Society, Providence, RI (2013)zbMATHGoogle Scholar
  4. 4.
    Bernat, P., Conze, N., Duflo, M., Lévy-Nahas, M., Rais, M., Renouard, P., Vergne, M.: Représentations des groupes de Lie résolubles. Monographies de la S.M.F. Dunod, Paris (1972)zbMATHGoogle Scholar
  5. 5.
    Bruna, J., Cufii, J., Führ, H., Miró, M.: Characterizing abelian admissible groups. J. Geom. Anal. 25, 1045–1074 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Currey, B.: Explicit orbital parameters and the Plancherel formula for exponential Lie groups. Pac. J. Math. 219(1), 97–138 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Currey, B., Führ, H., Taylor, K.F.: Integrable wavelet transforms with abelian dilation groups. J. Lie Theory 26(2), 567–595 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dixmier, J., Malliavin, P.: Factorisations de fonctions et de vecteurs indéfiniment différentiables. Bull. Sci. Math. 102, 305–330 (1978)zbMATHGoogle Scholar
  9. 9.
    Duflo, M., Moore, C.: On the regular representation of a non-unimodular locally compact group. J. Funct. Anal. 21, 209–243 (1976)CrossRefGoogle Scholar
  10. 10.
    Duflo, M., Rais, M.: Sur l’analyse harmonique sur les groupes de Lie résolubles. Ann. Sci. École Norm. Sup. 9(1), 107–144 (1976)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gröchenig, K., Kaniuth, E., Taylor, K.F.: Compact open sets in duals and projections in the \(L^1\)-algebras of certain semi-direct product groups. Math. Proc. Camb. Philos. Soc. 111, 545–556 (1992)CrossRefGoogle Scholar
  12. 12.
    Harish-Chandra, : Plancherel formula for complex semi-simple Lie groups. Proc. Natl. Acad. Sci. U.S.A. 12, 813–818 (1951)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Herb, R., Wolf, J.: The Plancherel theorem for general semi-simple Lie groups. Comput. Math. 57(3), 813–818 (1951)Google Scholar
  14. 14.
    Kato, T.: Perturbation theory for linear operators, Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, (1995). xxii+619 pp. ISBN: 3-540-58661-X 47A55 (46-00 47-00)Google Scholar
  15. 15.
    Kleppner, A., Lipsman, R.L.: The Plancherel formula for group extensions. I, II. Ann. Sci. École Norm. Sup. 5(3), 459–516 (1972)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Larson, D., Schulz, E., Speegle, D., Taylor, K.F.: Explicit cross-sections of singly-generated group actions, Harmonic Analysis and Applications. In: Heil, C. (ed.) Applied and Numerical Harmonic Analysis. Birkhauser, Boston (2006)Google Scholar
  17. 17.
    Mackey, G.W.: Induced representations of groups. Am. J. Math. 73, 576–592 (1951)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mackey, G.W.: Induced representations of locally compact groups. Ann. Math. 58, 193–221 (1953)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Oussa, V.: Sampling and interpolation on some nilpotent Lie groups. Forum Math. 28(2), 255–273 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Oussa, V.: Regular sampling on metabelian nilpotent Lie groups: the multiplicity-free case. In: Oussa, V. (ed.) Frames and other bases in abstract and function spaces, pp. 377–411. Birkhäuser/Springer, Cham (2017)CrossRefGoogle Scholar
  21. 21.
    Pukanszky, L.: Leçons sur les Représentations des Groupes. Dunod, Paris (1967)zbMATHGoogle Scholar
  22. 22.
    Vergne, M.: A Poisson-Plancherel formula for semi-simple Lie groups. Ann. Math. 115(3), 639–666 (1982)MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Inst. de Mathematiques de Bourgogne, UMR CNRS 5584Université de Bourgogne-Franche ComtéDijonFrance
  2. 2.Department of Mathematics and Computer ScienceSt. Louis UniversitySt. LouisUSA
  3. 3.Département de Mathématiques, Faculté des Sciences de Bizerte, Laboratoire AGTS LR11ES53Université de CarthageBizerteTunisia

Personalised recommendations