Let $G$ a Lie group and $H\subset G$ a Lie subgroup. For simplicity we assume that the adjoint action of $H$ on $\mathfrak g/\mathfrak h$ is faithful.

Let $M$ a differentiable manifold of the same dimension as $G/H$. A $(H,G)$-Cartan geometry on $M$ is defined as a reduction of the structure group of the frame bundle of $M$ to $H$ (more accurately to the adjoint action of $H$ on $\mathfrak g /\mathfrak h$), along with a "Cartan connection one-form" which is a $\mathfrak g$-valued equivariant one-form which is non degenerate. Call the $H$-principal bundle $P$.

Here are two things one can do with Cartan geometries:

Construct the associated (right) principal $G$-bundle $P\times_H G$ on which the Cartan connection can be extended to a $G$-principal connection. For a $G$-module $V$, one can construct the associated bundle $P\times_H G\times_G V\simeq P\times_H V$ (called tractor bundle) which is in particular endowed with a induced $G$-principal connection called the

*tractor connection*.Considering $G\to G/H$ as a reference space with $(H,G)$-Cartan geometry, one can construct the space of $1$-jets of maps from $M$ to $G/H$

*preserving the $H$-structure*(between the chosen points), in other words linear isomorphisms between tangent spaces which preserve the class of "$H$-frames". It is readily constructed as $(P\times G)/H$ for the diagonal action on the right. It is a bundle over $M\times G/H$ with fibre diffeomorphic to $H$; the fibre over $M$ is $G$. It inherits an action of $G$ (naturally)*on the left*through the germs of the associated diffeomorphisms acting on the $1$-jets and is hence a left principal $G$-bundle over $M$.

Now, the two constructed $G$-principal bundles are isomorphic. They can be easily related by the inversion of the component $G$ of $P\times G$ before taking the quotient by $H$. If one is rather willing to "invert" the principal bundle $P$, using the inverse right action, then the first construction can be related to $1$-jets of diffeomorphisms between $M$ and $H\backslash G$.

My question is: **how can the tractor bundle construction be naturally expressed with the $1$-jet interpretation of the "extended" $G$-principal bundle?**
To me the natural idea would be to consider applications $G/H \to V$ which are equivariant under the (left) action of $G$, but it is not a construction on the coset space that is familiar to me.

projectson the $1$-jet space). I will add a first order assumption to the question so as to avoid introducing one more bundle. Thank you for the comment. $\endgroup$