Harmonic Analysis on Inhomogeneous Nilpotent Lie Groups

Let G be a semi-direct product of a normal, vector subgroup by a connected, simply connected nilpotent Lie group. A detailed study of the coadjoint orbits of G in the dual space g(& lowast;) of its Lie algebra g is motivated by classical harmonic analysis on solvable Lie groups, culminating in the work of Auslander and Kostant, and by more recent work on generalized continuous wavelets. We apply a procedure for matrix reduction to construct a stratification of the space of coadjoint orbits, where each layer of the stratification has an explicit fiber bundle structure, and provides a criterion for the property of regularity for a coadjoint orbit. Examination of the Zariski open layer ohm(0) then yields an algebraic characterization for regularity, and for both regularity and integrality, of every orbit in ohm(0) . When the criterion for collective regularity holds, we construct a simple and explicit topological cross-section for the coadjoint orbits in ohm(0) . When a criterion fails, then the corresponding property fails for a dense G(delta) set in ohm(0).