TITLE: Jets of Equivariant Vector Bundles
QUESTION [5 upvotes]: 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 for a definition of jet bundle) of $\cal E$ are also $G$-equivariant, and if so, can one construct their corresponding representations from $\pi$?

REPLY [2 votes]: See chapter IV and  section 32 (and others) of the book:
Ivan Kolár, Jan Slovák, Peter W. Michor: Natural operations in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, (1993)
(pdf).