Kaehler spaces, nilpotent orbits, and singular reduction

Abstract

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.