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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
Article
  • 60 Downloads

Abstract

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.

Keywords

Semi-direct product Regular representation Plancherel formula 

Mathematics Subject Classification

22Exx 22E27 22E45 22D10 22D30 

Notes

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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

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