Kähler quantization and reduction

Abstract

Exploiting a notion of Kähler structure on a stratified space introduced elsewhere we show that, in the Kähler case, reduction after quantization is equivalent to quantization after reduction in a certain weak sense. Key tools developed for that purpose are stratified polarizations and stratified prequantum modules, the latter generalizing prequantum bundles. These notions encapsulate, in particular, the behaviour of a polarization and that of a prequantum bundle across the strata. Our main result says that, for a positive Kähler manifold with a hamiltonian action of a compact Lie group, when suitable additional conditions are imposed, reduction after quantization is weakly equivalent to quantization after reduction in the sense that not only the reduced and unreduced quantum phase spaces correspond as complex vector spaces (whether or not the natural inner products correspond is unknown whence the qualifier “weakly”) but the (invariant) unreduced and reduced quantum observables as well. Over a stratified space, the appropriate quantum phase space is a costratified Hilbert space in such a way that the costratified structure reflects the stratification. Examples of stratified Kähler spaces arise from representations of compact Lie groups and from the closures of holomorphic nilpotent orbits including angular momentum zero reduced spaces. For illustration, we carry out Kähler quantization on various spaces of that kind including singular Fock spaces.