# Associated vector bundles of infinite rank and induced connections

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.

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.

• @Andrew: sure you know these things better than I do, but - as long as $M$ and $G$ are finite-dimensional - what is the problem with taking the same formula as in the finite-dim. case? Feb 13, 2012 at 10:17
• Johannes: That case didn't occur to me! I assumed that $G$ was an infinite dimensional Lie group. Nonetheless, topology is still important and might behave a bit nastily. A standard situation is the space of $L^2$ sections of some fibre bundle. Then you run into the problem that, for example, $S^1$ doesn't act as nicely as it could on $L^2(S^1)$. Then I guess the explanation for the absence of this from the literature is that one can usually decompose the infinite representation into a sum of finite ones of different characters and study that collection instead of the single infinite one. Feb 13, 2012 at 10:47
• r0b0t: Incidentally, what is the actual question here? Feb 13, 2012 at 14:22
• rObOt: Most study of infinite dimensional representations of a semisimple (or reductive) Lie group involves additional hypotheses on the vector space and representation: Banach or Hilbert space, etc. Is it clear what effect this richer structure would have on your question? Feb 13, 2012 at 21:17
• That's what I wanted to avoid as I hoped that these issues are already sorted out somewhere. Looks like I have to get my hands dirty. Thanks for comments. Feb 14, 2012 at 10:52