$\DeclareMathOperator\Sym{Sym}$Let $G$ be a compact lie group. Chern–Weil theory tells us that there's a homomorphism:


$$H^{\*}(BG;\mathbb{R}) \to (\Sym^{\bullet} \mathfrak{g^\*})^G$$


which in our case is an isomorphism since $G$ is compact. The procedure I know to prove this is pretty ad-hoc. Starting with a $G$-bundle on a manifold $M$ we pick an arbitrary connection and evaluate invariant polynomials on its curvature form to get characteristic classes.



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> What is the representation theoretic viepoint on the isomorphism $H^{\*}(BG;\mathbb{R}) = (\Sym^{\bullet} \mathfrak{g^\*})^G$?
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> For a finite dimensional lie group $G$ (adding compact here doesn't matter) is there always a canonical way to build $BG$ as a colimit of homogeneous manifolds?
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Recently I found that many computations in algebraic topology can be simplified using representation theory. Here are some more complutations I'd like to be able to understand in representation theoretic terms:


**1. Cohomology ring of a homogeneous space $H^\*(G/H)$.**


**2. Cohomology ring of a parallel curvature cartan geometry $(P, \omega)$ for the pair $(\mathfrak{g},H)$ with curvature form $K \in \operatorname{Hom}(\bigwedge^2\mathfrak{g}/\mathfrak{h}, \mathfrak{g})$.** (Side question: is $K$ some kind of cocycle here?)


Is there a reference for these kinds of computations?



Representation viewpoint on Chern–Weil (cohomology computations done with rep theory?)
I'll assume that $G$ is connected, and maybe even that it's semisimple to be safe. Here's a sketch. Let's take on faith that the odd cohomology vanishes. Recall that the [Chern character](https://en.wikipedia.org/wiki/Chern_class#The_Chern_character) establishes an isomorphism between $K(X) \otimes \mathbb{R}$ and periodic even cohomology $H^{2 \bullet}(X, \mathbb{R})$. The [Atiyah-Segal completion theorem](https://en.wikipedia.org/wiki/Atiyah%E2%80%93Segal_completion_theorem) furthermore gives (maybe up to some mild subtlety about completions) that $K(BG) \otimes \mathbb{R}$ is the completion of the real representation ring


$$K\_G(\text{pt}) \otimes \mathbb{R} \cong R(G) \otimes \mathbb{R}$$


of $G$ at the augmentation ideal $I$ (where the augmentation $R(G) \to \mathbb{Z}$ sends a representation to its dimension). 


Now, what does the representation ring $R(G)$ have to do with $\text{Sym}(\mathfrak{g}^{\ast})^G$? You can think of the latter as functions on the stacky quotient $\mathfrak{g}/G$, which in turn is an infinitesimal version of functions on the stacky quotient $G/G$, or in other words class functions. So the connection has something to do with taking characters. (From the algebraic geometry point of view, $G/G$ is the loop space $L(BG)$, while $\mathfrak{g}/G$ is the "formal" loop space.)


To actually complete the computation from here you can use the fact that if $T$ is a maximal torus and $W$ the Weyl group then we have an isomorphism


$$R(G) \cong R(T)^W.$$


So we reduce, more or less, to a computation on $BT$. If $T$ has rank $n$ then $R(T)$ is Laurent polynomials in $n$ variables while $\text{Sym}(\mathfrak{t}^{\ast})$ is polynomials in $n$ variables; these become identified after completing at the augmentation ideal, and the Weyl group actions match up. Then, by the Chevalley restriction theorem,


$$\text{Sym}(\mathfrak{t}^{\ast})^W \cong \text{Sym}(\mathfrak{g}^{\ast})^G.$$


Actually, as long as we were going to pass to the maximal torus at all, this computation is shorter without passing through K-theory. Namely, we already have a rational isomorphism


$$H^{\bullet}(BG, \mathbb{Q}) \cong H^{\bullet}(BT, \mathbb{Q})^W$$


which is a version of the splitting principle. Then we can apply Chevalley restriction to the RHS. 


