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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 $\pi$?

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The answer to the first question is yes: since $G$ acts by bundle morphisms on $\mathcal{E}$ it will induce an action on sections, which preserves tangency of sections, hence induces an action on jets. Concerning the second questions: if I'm not mistake the representation $\pi:H\to R^k$ determines at least locally up to isomorphism the action of $G$ on the bundle, hence you should also be able to reconstruct the representation of $H$ on the fibers of jets. –  Michael Bächtold Oct 11 '12 at 20:09

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up vote 1 down vote accepted

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.

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So then what is the explicit representation of $H$ on $\frak{U(g)}\otimes _{\frak{U(h)}}$ $E^*$. I'm guessing that its the tensor product of the dual rep of $H$ on $E$ with some rep of $H$ on $\frak{U(g)}$. –  Jean Delinez Oct 16 '12 at 13:00
Yes. I believe that the representation of $H$ on $\mathfrak{U(g)\otimes_{U(h)}\mathbb{E}^*$ is the tensor product of representation on $\mathfrak{U(g)}$ and the dual representation on $\mathbb{E}$. The representation of $H$ on the universal enveloping algebra is the extension of the adjoint representation of $H$ on $\mathfrak{g}$. Unfortunately, I'm quite busy at the moment so I can't check the details and to be honest I do not know of better references other than that I've already given in the answer. Hopefully somebody will fill that gap. –  Vít Tuček Oct 16 '12 at 16:31
Yes, that fills the gap! –  Jean Delinez Oct 16 '12 at 17:32
I have also found the reference "Invariant Differential Operators" by W. Smoke (1967) to be a concise but clear introduction to the basics of invariant differential operators and the relationship with the jet bundle and jet algebras over a homogeneous space. –  josh Feb 3 at 15:31

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).

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