  For a stratified symplectic space, a suitable concept of stratified Kaehler
polarization, defined in terms of an appropriate Lie-Rinehart algebra,
encapsulates Kaehler polarizations on the strata and the behaviour of the
polarizations across the strata and leads to the notion of stratified Kaehler
space. This notion establishes an intimate relationship between nilpotent
orbits, singular reduction, invariant theory, reductive dual pairs, Jordan
triple systems, symmetric domains, and pre-homogeneous spaces; in particular,
in the world of singular Poisson geometry, the closures of principal
holomorphic nilpotent orbits, positive definite hermitian JTS's, and certain
pre-homogeneous spaces appear as different incarnations of the same structure.
The space of representations of the fundamental group of a closed surface in a
compact Lie group inherits a (positive) normal (stratified) Kaehler structure,
as does the closure of a holomorphic nilpotent orbit in a semisimple Lie
algebra of hermitian type. The closure of the principal holomorphic nilpotent
orbit arises from a regular semisimple holomorphic orbit by contraction.
Symplectic reduction carries a (positive) Kaehler manifold to a (positive)
normal Kaehler space in such a way that the sheaf of germs of polarized
functions thereupon coincides with the ordinary sheaf of germs of holomorphic
functions. Symplectic reduction establishes a close relationship between
singular reduced spaces and nilpotent orbits of the dual groups.
Projectivization of holomorphic nilpotent orbits yields exotic stratified
Kaehler structures on complex projective spaces and on certain complex
projective varieties including complex projective quadrics. Physical examples
are provided by certain reduced spaces arising from angular momentum zero.
