TITLE: Representation viewpoint on Chern–Weil (cohomology computations done with rep theory?)
QUESTION [12 upvotes]: $\DeclareMathOperator\Sym{Sym}$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.

What is the representation theoretic viepoint on the isomorphism $H^{*}(BG;\mathbb{R}) = (\Sym^{\bullet} \mathfrak{g^*})^G$?
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?

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?

REPLY [10 votes]: The construction you describe appears in Tamvakis' The connection between representation theory and Schubert calculus (Enseign. Math. 50 (2004), 267-2860). Basically, instead of working with representations of $GL(n)$ he works with representations of Vec, which he then applies to the tautological bundle over the Grassmannian. The main result is that this map corresponds (part of) the basis of irreps with the basis of Schubert classes.