Journal of Lie Theory 34 (2024), No. 4, 873--910
Copyright Heldermann Verlag 2024
Harmonic Analysis on Inhomogeneous Nilpotent Lie Groups
Didier Arnal
Institut de Mathématiques de Bourgogne, Université de Bourgogne-Franche-Compté, Dijon, France
didier.arnal@u-bourgogne.fr
Bradley Currey
Dept. of Mathematics and Statistics, Saint Louis University, St. Louis, U.S.A.
bradley.currey@slu.edu
[Abstract-pdf]
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 $\mathfrak{g}^*$ of its Lie algebra $\mathfrak{g}$ 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 $\Omega_0$ then yields an algebraic characterization for regularity, and for both regularity and integrality, of every orbit in $\Omega_0$. When the criterion for collective regularity holds, we construct a simple and explicit topological cross-section for the coadjoint orbits in $\Omega_0$. When a criterion fails, then the corresponding property fails for a dense $\mathcal G_\delta$ set in $\Omega_0$.
Keywords: Inhomogeneous nilpotent Lie group, semi-direct product, coadjoint orbit.
MSC: 22Exx, 22E25, 22E27.
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