Let $\mathbb{V}$ be a representation of a Lie group $G$ and let $P \to M$ be a principal $G$-bundle with a principal connection. If $\mathbb{V}$ is finite-dimensional, then one can associate to this data an associated vector bundle $P\times\_G \mathbb{V}$ with linear connection. I thought that basically the same construction should work also when $\mathbb{V}$ is an infinite-dimensional representation, but I haven't found any textbook that would not constrain itself to finite rank. All the textbooks concerning to infinite-dimensional differential geometry that I know of (Michor, Lang, Neeb) doesn't treat associated bundles and induced connections.




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Edit:


I now realize that it may not be as straightforward as it seems on a first glance. I want to, in fact, generalize a slightly more complicated construction -- the so called tractor connection induced by a Cartan connection.


Changing the notation a little bit, given a finite-dimensional Lie group $G$ with a closed subgroup $H$, I need to work with an infinite-dimensional vector space $\mathbb{V}$ which is a representation of $\mathfrak{g}$ and also a representation of $H$ (so I can form associated bundles to $H$-principal bundles) with these two representation being compatible. Practically, I am interested mainly in Harish-Chandra modules and their globalizations. I think I am also fine with just a "sort of connection" working on some dense subbundle of the associated bundle and so $L^2$-globalizations are also OK.


I can briefly describe the construction for $\mathbb{V}$ being finite-dimensional representation of $G$ if it is needed. 



Associated vector bundles of infinite rank and induced connections
If you use the cocycle description of principal bundles (for finite dimensions) as in 18.7.4 of [1](http://www.mat.univie.ac.at/~michor/dgbook.pdf), you can describe the associated bundle using convenient calculus, since then you can flip coordinates freely. 
Inducing connections is then described in 19.8 (for associated fiber bundles) and in 19.10 (for associated vector bundles) of [1](http://www.mat.univie.ac.at/~michor/dgbook.pdf); use also 19.9 ("recognizing induced connections") where the cocycle description is spelled out. 


Idea: You can carry over to infinite dimensions all constructions of finite dimensional differential geometry for which you have direct chart descriptions. But be very careful whenever you have to solve equations (ODE's, implicit functions, etc.)


[1](http://www.mat.univie.ac.at/~michor/dgbook.pdf) 
Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008.
[(pdf)](http://www.mat.univie.ac.at/~michor/dgbook.pdf).


