A Bijection Between the Adjoint and Coadjoint Orbits of a Semidirect Product

Differential Geometry and its Applications, Volume 62, February 2019, Pages 267-282 We prove that there exists a geometric bijection between the sets of adjoint and coadjoint orbits of a semidirect product, provided a similar bijection holds for particular subgroups. We also show that under certain conditions the homotopy types of any two orbits in bijection with each other are the same. We apply our theory to the examples of the affine group and the Poincar\'{e} group, and discuss the limitations and extent of this result to other groups.