Semidirect products and the Pukanszky condition

TL;DR

This paper explores the geometric structure of coadjoint orbits in semidirect product Lie groups, using symplectic induction to analyze their properties and conditions like Pukanszky's, with applications to physics.

Contribution

It introduces a new perspective on coadjoint orbits of semidirect product groups via symplectic induction and examines polarizations related to Pukanszky's condition.

Findings

Coadjoint orbits are obtained by symplectic induction from smaller groups.

Analysis of polarizations related to semidirect products.

Applications to physical systems discussed.

Abstract

We study the general geometrical structure of the coadjoint orbits of a semidirect product formed by a Lie group and a representation of this group on a vector space. The use of symplectic induction methods gives new insight into the structure of these orbits. In fact, each coadjoint orbit of such a group is obtained by symplectic induction on some coadjoint orbit of a "smaller" Lie group. We study also a special class of polarizations related to a semidirect product and the validity of Pukanszky's condition for these polarizations. Some examples of physical interest are discussed using the previous methods.