A Bijection Between the Adjoint and Coadjoint Orbits of a Semidirect Product
Abstract 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 Poincare group, and discuss the limitations and extent of this result to other groups.