  We study the action of Bogoliubov transformations on admissible generalized
one-particle density matrices arising in Hartree-Fock-Bogoliubov theory. We
show that the orbits of this action are reductive homogeneous spaces, and we
give several equivalences that characterize when they are embedded submanifolds
of natural ambient spaces. We use Lie theoretic arguments to prove that these
orbits admit an invariant symplectic form. If, in addition, the operators in
the orbits have finite spectrum, or infinite spectrum and trivial kernel, then
we obtain that the orbits are actually K\"ahler homogeneous spaces.
