Geometric quantization: Hilbert space structure on the space of generalized sections

The Hilbert space constructed by KS-geometric quantization consists of the solutions of the equations ▽XFs = 0 for X in the invariant complex polarization F ⊂ (TM)C. It is well known that if the holonomy groups of the restriction ▽F to the leaves of the subbundle D, DC = F ∩ F̄, are nontrivial, there are no smooth global solutions. The supports of the generalized solutions are subsets of the union BS of all D-leaves with trivial holonomy group. It is shown in this paper that for a wide class of polarizations, BS is the union of E-leaves, where EC = F + F̄. We construct a Hilbert space structure on the space of generalized sections such that any quantizable real function with a complete Hamiltonian vector field generates a one-parameter group of unitary operators.