  In this paper we investigate the possibility of constructing a complete
quantization procedure consisting of geometric and deformation quantization.
The latter assigns a noncommutative algebra to a symplectic manifold, by
deforming the ordinary pointwise product of functions, whereas geometric
quantization is a prescription for the construction of a Hilbert space and a
few quantum operators, starting from a symplectic manifold. We determine under
which conditions it is possible to define a representation of the deformed
algebra on this Hilbert space, thereby to extend the small class of quantizable
observables in geometric quantization to all smooth functions, as well as to
give a natural representation of the algebra. In particular we look at the
special cases of a cotangent bundle and a K\"ahler manifold.
