The Plancherel Formula for an Inhomogeneous Vector Group

We give a concrete realization of the Plancherel measure for a semi-direct product N ⋊ 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 unita...

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Bibliographic Details
Published in:The Journal of fourier analysis and applications 2019-12, Vol.25 (6), p.2837-2876
Main Authors: Arnal, Didier, Currey, Bradley, Dali, Béchir
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Approximations and Expansions
Fourier Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Operators (mathematics)
Partial Differential Equations
Signal,Image and Speech Processing
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Summary:We give a concrete realization of the Plancherel measure for a semi-direct product N ⋊ 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 ⋊ 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 Σ for almost all H -orbits. The associated trace characters are also shown to be tempered distributions.
ISSN:1069-5869
1531-5851
DOI:10.1007/s00041-019-09684-y