The Plancherel Formula for an Inhomogeneous Vector Group

We give a concrete realization of the Plancherel measure for a semi-direct product N (sic) 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 (sic) 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.