  We give a systematic presentation of the stability theory in the
non-algebraic Kaehlerian geometry. We introduce the concept of "energy complete
Hamiltonian action". To an energy complete Hamiltonian action of a reductive
group G on a complex manifold one can associate a G-equivariant maximal weight
function and prove a Hilbert criterion for semistability. In other words, for
such actions, the symplectic semistability and analytic semistability
conditions are equivalent.
