Abstract : One of the many contributions of Kostant is a rare gem which probably has not been sufficiently explored: a sheaf-theoretical model for geometric quantization associated to real polarizations. Kostant's model works very well for polarizations given by fibrations or fibration-like objects (like integrable systems away from singularities). For toric manifolds where the real polarization is determined by the fibers of the moment map, Kostant's model yields a representation space whose dimension is the number of integer points inside the corresponding Delzant polytope. We will discuss extensions of this model to consider almost toric manifolds and integrable systems with non-degenerate singularities where “unexpected” infinities can show up even if the manifold is compact.

Keywords : geometric quantization; Kostant's model; polarization; Bohr-Sommerfeld leaf; Delzant polytope; foliation; toric manifold