Vector bundles and equivariant vector spaces It seems commonly accepted that most of the results of equivariant geometry for vector spaces yield analog result for vector bundles.
In so far as I understand it, the reason for that is the interplay between maximal tori in equivariant geometry and the splitting principle for vector bundles.


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*What is a good/rigorous reference on this relation?

*Is there any restriction, e.g. on the arguments used in the proof, so that the results for equivariant vector bundles automatically holds for vector spaces?

 A: I'm not sure about a reference, but the relationship is quite easy to see if you look at things from a homotopy theoretic point of view. Let $G$ compact connected Lie group and fix a unitary representation $\rho:G\to U$. Let
$$\chi:G\overset{\rho}{\to} U \to U(1)$$
be the multiplicative group character of the representation. Then the delooping of $\chi$ gives a map which I'll suggestively call $c_1$:
$$c_1:B G\to BU \to B U(1)\;.$$
If you precompose the first map in the composition with a classifying map, $M\to BG \to BU$ and pull back the universal complex vector bundle over $BU$, you get exactly the vector bundle associated to the principal $G$ bundle (classified by $M\to BG$) and the representation $\rho$. The second map $BU\to BU(1)\simeq K(\mathbb{Z},2)$ is the universal Chern character, so $c_1$ is the universal Chern character for vector bundles with $G$-structure.
The splitting principal comes from delooping the inclusion of the maximal torus
$$i:BT\simeq BU(1)\times BU(1)\times...\times BU(1)\to BG\;,$$
and examining the induced map on cohomology 
$$i^*:H^*(BG;\mathbb{Z})\to H^*(BU(1)\times ... BU(1);\mathbb{Z})\simeq H^*(BU(1);\mathbb{Z})\otimes... H^*(BU(1);\mathbb{Z})\;,$$
where the last isomorphism can be deduced from the Kunneth formula (along with the fact that $BU(1)\sim \mathbb{C} P^{\infty}$ has torsion free cohomology). 
Since $\chi$ is a multiplicative character, it is constant on conjugacy classes and therefore determined by its restriction to a maximal torus. Let $\chi^T$ denote this restriction and write $\chi^T=(\lambda_1,\lambda_2,...,\lambda_k)$, where $\lambda_i:U(1)\to U(1)$ is the restriction to one of the factors of $T$.
Then map $i^*$ sends this class to the class
$$i^*(c_1)=[B(\lambda_1)]+[B(\lambda_2)]+...+[B(\lambda_k)]\;,$$
where $B(\lambda_i):BU(1)\to BU(1)\simeq K(\mathbb{Z},2)$ is the just the delooping of the weight $\lambda_i$ and the brackets denote its homotopy class. Since the classes $[B(\lambda_i)]$ came from decomposing $c_1$ via the splitting principal, we have that by definition, the classes $[B(\lambda_i)]$ are the Chern roots.
