Quantization of Kaehler manifolds I: geometric interpretation of Berezin's quantization

Abstract. We give a geometric interpretation of Berezin 's symbolic calculus on Kahler manifolds in the framework of geometric quantization. Berezin's covariant symbols are defined in terms of coherent states and we study a function 6 on the manifold which is the central object of the theory. When this function is constant Berezin's quantization rule coincides with the prescription of geometric quantization for the quantizable functions. It is defined on a larger class of functions. We show in the compact homogeneous case how to extend Berezin's procedure to a dense subspace of the algebra of smooth functions.