Stages and Semidirect Products with Cocycles

This chapter is concerned with two major themes. The first theme, which is presented in §10.1 and §10.2, deals with the semidirect product M of a group G with an Abelian group A, where the construction of the semidirect product itself involves an A-valued cocycle of G. In this context, A (which is N in the general theory) is still a normal subgroup and the reduction by stages program is fully carried out. In particular, the stages hypothesis holds and so the reduction by stages program for the action of M on a symplectic manifold can be implemented.We focus on the case of the action of M on T*M (by the lift of right translation) so that the final reduced spaces will be the coadjoint orbits of M.