  In this article we develop tools to compute the Geometric Quantization of a
symplectic manifold with respect to a regular Lagrangian foliation via sheaf
cohomology and obtain important new applications in the case of real
polarizations. The starting point is the definition of representation spaces
due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to
Guillemin and Sternberg very few examples of explicit computations of real
polarizations are known. The computation of Geometric Quantization for
Gelfand-Cetlin systems is based on a theorem due to \'Sniatycki for fibrations
which identifies the representation space with the set of Bohr-Sommerfeld
leaves determined by the integral action coordinates.
  In this article we check that the associated sheaf cohomology apparatus of
Geometric Quantization satisfies Mayer-Vietoris and K\"unneth formulae. As a
consequence, a new short proof of this classical result for fibrations due to
\'Sniatycki is obtained. We also compute Geometric Quantization with respect to
any generic regular Lagrangian foliation on a 2-torus and the case of the
irrational flow. In the way, we recover some classical results in the
computation of foliated cohomology of these polarizations.
