Let $M$ be a (compact) $G$-homogeneous space with fibre group $H$, and let ${\cal E}$ be a $G$-equivariant $k$-dimensional vector bundle over $M$ with corresponding representation $\pi:H \to $R$^k$. What I would like to know is whether all the jet bundles (see [here](http://en.wikipedia.org/wiki/Jet_bundle) for a definition of jet bundle) of $\cal E$ are also $G$-equivariant, and if so, can one construct their corresponding representations from $\pi$?



Jets of Equivariant Vector Bundles
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](http://www.mat.univie.ac.at/~esiprpr/esi722.pdf) ESI preprint. For applications to invariant differential operators see [Differential opperators on homogeneous spaces](http://www.math.okstate.edu/~zierau/papers/ICE/ICE-course.pdf) by L. Barchini and R. Zierau. 


