Global symbolic calculus of pseudo-differential operators on homogeneous vector bundles

A symbolic calculus for a pseudo-differential operators acting on sections of a homogeneous vector bundle over a compact homogeneous space $G\mathrm{/}H$ with compact $G$ and $H$ is developed. We realize the symbol of a pseudo-differential operator as a linear operator acting on corresponding irreducible unitary representations of $H$ valued in the algebra $C^{\mathrm{\infty }}(G)$ of smooth functions. We write down how left invariant vector fields of $SU(2)$ act on the sections of homogeneous vector bundles associated to the fibration $\mathbb{T}\hookrightarrow SU(2)\to \mathbb{C}{P}^1$, which is known as the Hopf fibration. Lastly, we outline how functional calculus of a pseudo-differential operator can be computed using our calculus.