I don't understand your terminology, but I'm gonna try to answer your question anyway. Let $M=G/H$ and let $\mathbb{E}$ be a representation of $H$. By $E$ I denote the associated homogeneous vector bundle $G/H \times_H \mathbb{E}$. The jet space of $E$ is also a homogeneous vector bundle which is induced from the $(\mathfrak{g},H)$-representation $J_{eH}(E)$, i.e. from the representation which is induced on the fiber over identity coset. There is a duality between $J_{eH}(E)$ and $\mathfrak{U(g)\otimes_{U(h)}} \mathbb{E}^*$.

Some details can be found in the appendix of this ESI preprint. For applications to invariant differential operators see Differential opperators on homogeneous spaces by L. Barchini and R. Zierau.