  In this work, we introduce a global theory of subelliptic pseudo-differential
operators on arbitrary homogeneous vector bundles over orientable compact
homogeneous manifolds. We will show that a global pseudo-differential calculus
can be associated to the operators acting on any pair of homogeneous
vector-bundles with base space $M,$ if the compact Lie group $G$ that acts on
$M=G/K$ is endowed with a (Riemannian or) sub-Riemannian structure. This is
always possible if we choose on $G$ a sub-Laplacian associated to a H\"ormander
system of vector-fields or we fix the Laplace-Beltrami operator on $G$. We
begin with developing a global subelliptic symbolic calculus for vector-valued
pseudo-differential operators on $G$ and then, we show that this vector-valued
calculus induces a pseudo-differential calculus on homogeneous vector bundles,
which, among other things, is stable under the action of the complex functional
calculus. We prove global versions of the Calder\'on-Vaillancourt theorem,
Fefferman theorem and also, of the G\r{a}rding inequality. We present
applications of the obtained G\r{a}rding inequality to the wellposedness of
evolution problems. We characterise the classes of pseudo-differential
operators on homogeneous vector bundles in the sense of H\"ormander (which are
defined by using local coordinate systems) in terms of their global symbols.
Finally, using this formalism, we compute the global symbol of the exterior
derivative, its adjoint, and the symbol of the Dirac operator on the vector
bundle of differential forms. We hope that this work will provide a solid
foundation for further research using the global quantisation of operators on
(vector-bundles over) compact homogeneous manifolds.
