This paper, the first part of a two-part series, aims to develop a unified theory of unitary representations for connected Lie groups. The authors generalize and rigorize the physicist's concept of quantizing a function, leading to a theory that constructs many irreducible unitary representations. This theory generalizes the Borel-Weil theorem for compact groups and Kirillov's results for nilpotent groups, and it also provides a criterion for solvable groups of type I. For semi-simple groups, it appears to generate enough representations to decompose the regular representation. The theory is grounded in differential geometry, focusing on the 2-form of a symplectic manifold and its curvature as the connection of a line bundle. The Hilbert space is identified with the sections of this line bundle, and the Lie algebra of functions on the manifold acts on these sections. This mapping of functions to operators is the essence of quantization. The paper also discusses the concept of polarizing a symplectic manifold, which is crucial for extracting the Hilbert space and Lie algebra. In Part I, pre-quantization is considered, while the extraction of the Hilbert space and Lie algebra is addressed in Part II. A key result (Theorem 5.4.1) shows that a representation arises from a symplectic homogeneous space when the Lie algebra of Hamiltonian vector fields can be lifted to quantizable functions. This condition is satisfied when the space corresponds to or covers an orbit in the dual of the Lie algebra, generalizing Kirillov's work and providing a characterization of compact Kähler homogeneous spaces. Additionally, a corollary (Corollary 1 to Theorem 5.7.1) generalizes Borel-Weil results for compact groups, stating that the 2-form on the orbit is integral if and only if it is the differential of a character on the isotropy group. Part I focuses on the differential geometry foundations, including basic facts and proofs.This paper, the first part of a two-part series, aims to develop a unified theory of unitary representations for connected Lie groups. The authors generalize and rigorize the physicist's concept of quantizing a function, leading to a theory that constructs many irreducible unitary representations. This theory generalizes the Borel-Weil theorem for compact groups and Kirillov's results for nilpotent groups, and it also provides a criterion for solvable groups of type I. For semi-simple groups, it appears to generate enough representations to decompose the regular representation. The theory is grounded in differential geometry, focusing on the 2-form of a symplectic manifold and its curvature as the connection of a line bundle. The Hilbert space is identified with the sections of this line bundle, and the Lie algebra of functions on the manifold acts on these sections. This mapping of functions to operators is the essence of quantization. The paper also discusses the concept of polarizing a symplectic manifold, which is crucial for extracting the Hilbert space and Lie algebra. In Part I, pre-quantization is considered, while the extraction of the Hilbert space and Lie algebra is addressed in Part II. A key result (Theorem 5.4.1) shows that a representation arises from a symplectic homogeneous space when the Lie algebra of Hamiltonian vector fields can be lifted to quantizable functions. This condition is satisfied when the space corresponds to or covers an orbit in the dual of the Lie algebra, generalizing Kirillov's work and providing a characterization of compact Kähler homogeneous spaces. Additionally, a corollary (Corollary 1 to Theorem 5.7.1) generalizes Borel-Weil results for compact groups, stating that the 2-form on the orbit is integral if and only if it is the differential of a character on the isotropy group. Part I focuses on the differential geometry foundations, including basic facts and proofs.