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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asked Oct 11, 2012 at 16:14
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2$\begingroup$ 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. $\endgroup$– Michael BächtoldCommented Oct 11, 2012 at 20:09
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2 Answers
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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.
answered Oct 14, 2012 at 17:21
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$\begingroup$ 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 $\frak{U(g)}$. $\endgroup$ Commented Oct 16, 2012 at 13:00
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$\begingroup$ 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. $\endgroup$ Commented Oct 16, 2012 at 16:31
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1$\begingroup$ 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. $\endgroup$– joshCommented Feb 3, 2015 at 15:31
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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).
answered Oct 15, 2012 at 8:11