The Plancherel Formula for an Inhomogeneous Vector Group

We give a concrete realization of the Plancherel measure for a semi-direct product N⋊H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \rtimes H$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \rtimes H$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} for almost all H-orbits. The associated trace characters are also shown to be tempered distributions.