I feel a strong relations among classical compact Lie groups G (in particular, $U(n)$ and $SO(n)$), their representations $V$, their cohomologies, and the associated bundles on some manifold $M$.
The last object needs some clarification: by that I mean given a smooth manifold $M$ with its principle $G$-bundle $P_G$ that associates to the tangent bundle $TM$ (namely its frame bundle), one can form the bundle associates to $V$ by changing fibre: $$P_V := P_G \times _G V.$$
I cannot find any reference or literature. Probably the most relevant thing I have come across is Atiyah's Classical Groups and Classical Differential Operators, but I don't have access to it.
Questions
- Given a representation $V$ of a classical compact Lie group $G$, what can we say about the (Lie group) cohomology? For example, is there a natural cohomology class that corresponds to $V$?
- What can we say about the bundle associated to $V$? What are its Chern classes and Chern roots? Are they related to the character of $V$? How does its characterizing map $M \to BG$ look like?
- Does $\{P_V | V \in Rep(G)\}$ exhaust all possible G-bundles on $M$ up to isomorphism?