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La formule de Plancherel pour les groupes de Lie presque algébriques réels

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Noncommutative Harmonic Analysis

Part of the book series: Progress in Mathematics ((PM,volume 220))

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Abstract

In his Maryland lectures in 1983, M. Duflo stated a concrete Plancherel formula for real almost algebraic groups. The aim of this article is to sketch a proof of this formula in the philosophy of the orbit method and following the lines of the one given by M. Duflo and M. Vergne for simply connected semisimple Lie groups.

The main ingredients of the proof are:

  • Harish-Chandra’s descent method which, interpreting the Plancherel formula as an equality of semi-invariant generalized functions, allows one to reduce it to a neighbourhood of zero in the Lie algebra of the centralizer of any elliptic element,

  • character formula for representations constructed by M. Duflo (recently proved by the authors),

  • Poisson-Plancherel formula near any elliptic element s in a good position: this formula, generalizes the classical Poisson summation formula and states that the Fourier transform of an invariant distribution, which is the sum of a series of Harish-Chandra type orbital integrals of elliptic elements in the Lie algebra of the centralizer of s, is a generalized function supported on the subset of admissible strongly regular forms contained in the dual of this Lie algebra.

In order to illustrate the main steps of the proof, we treat the example of the semidirect product of the universal covering of SL2(ℝ) by the three-dimensional Heisenberg group.

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Author information

Authors and Affiliations

  1. Département de Mathématiques Faculté des Sciences de Tunis, Campus universitaire, 1060, Tunis, Tunisie

    M. S. Khalgui

  2. UMR CNRS 6086 “Groupes de Lie et géometrie”, Université de Poitiers Mathématiques, SP2MI, BP 30179, 86962, Chasseneuil Cedex, France

    P. Torasso

Authors
  1. M. S. Khalgui
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  2. P. Torasso
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Editor information

Editors and Affiliations

  1. Institut de Mathématiques de Luminy, UPR 9016 CNRS, 13288, Marseille Cedex 9, France

    Patrick Delorme

  2. Centre de Mathématiques, Ecole Polytechnique, 91128, Palaiseau Cedex, France

    Michèle Vergne

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Khalgui, M.S., Torasso, P. (2004). La formule de Plancherel pour les groupes de Lie presque algébriques réels. In: Delorme, P., Vergne, M. (eds) Noncommutative Harmonic Analysis. Progress in Mathematics, vol 220. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8204-0_9

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  • DOI: https://doi.org/10.1007/978-0-8176-8204-0_9

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6489-7

  • Online ISBN: 978-0-8176-8204-0

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