I'm trying to understand how these two perspectives on vector bundle with a $G$-action come together.
Perspective 1:
Let $P \to X$ be a principal $G$-bundle. The associated bundle construction gives an exact tensor functor from finite linear representation of $G$ to vector bundle with $G$-action: $P \times_G(-): Rep(G) \to Vect^G(X)$. How does the essential image of this functor looks like? Is it faithful? full?
Perspective 2:
Let $V \to X$ be a vector bundle with $G$-action. Under some suitable conditions on $G$ (finite will obviously do but I'm pretty sure weaker assumptions will do - perhaps semisimple is enough) we have the following characterization of $E$. Let $\{V_j\}$ be the trivial vector bundles with $G$ action over $X$ corresponding to the irreducible reprepsentations of $G$.
$$V \cong \bigoplus_j V_j \otimes Hom_{G}(V_j,V)$$
Where $G$ acts on $Hom_G(V_j,V)$ trivially. The fact that $Hom_G(V_j,V)$ is a vector bundle follows from the averaging projection operator on sections of any vector bundle with $G$-action. This is a fulll description of the objects in $Vect^G(X)$ (for when $G$ is nice enough so that it holds).
One way to spell out my confusion is this:
- Is a vector bundle with $G$-action the same as a reduction of structure group from $GL(V)$ to $G$?
I'm pretty convinced that being a $G$-vector bundle is weaker than having structure group $G$. For example if $G$ is finite then a principal $G$ bundle will always be flat and so will any associated bundle while it looks like $G$-vector bundles mat be non-flat. I don't understand really how these POV's come together. In particular:
- When is a $G$-vector bundle an associated bundle of some principal bundle?
- Let $\rho : G \to GL(V)$ be a representation. How does the associated bundle $P\times_{\rho}V$ decompose via perspective 2?
- For $G$ finite: Is every $G$-vector bundle flat (locally constant)?
If you associate a bundle $P[V]$ to a principal $G$-bundle $P\to M$ with the help of representation $G\to GL(V)$, you also construct a mapping $\tau^V: P\times_M P[V] \to V$ which encodes the "associated bundle structure". It can be paraphrased as: it gives the coordinates of a point in $P[V]$ with respect to a frame in $P$. See 18.7 and the paragraph "Notation" after it of here. In 19.9 you find: "Recognizing induced connections". I hope that this source answers all your questions.