Let $G$ be a real or complex Lie group with Lie algebra $\mathfrak g$, and let $\mathbb C[\mathfrak g]$ be the algebra of $\mathbb C$-valued polynomials on $\mathfrak g$. Denote by $$\mathbb C[\mathfrak g]^G=\{f\in \mathbb C[\mathfrak g] | ~f(Ad_g x)=f(x),~ \forall g \in G,~ \forall x\in \mathfrak g\}$$ the subalgebra of fixed points in $\mathbb C[\mathfrak g]$ under the adjoint action of $G$.

Now, given principal $G$-bundle $P$ on a manifold $M$, the Chern-Weil homomorphism is an homomorphism of $\mathbb C$-algebras $\mathbb C[\mathfrak g]^G \to H^*_{dR}(M, \mathbb C)$, which is given by $f\mapsto f(F_\nabla)$ for some connection $\nabla$ on the given bundle.

Suppose $G$ is compact, then according to Wikipedia, we have an isomorphism of $\mathbb C$-algebras $$\mathbb C[\mathfrak g]^G \cong H^*_{dR}(BG, \mathbb C) $$

Question: Is this isomorphism given by the Chern-Weil homomorphism?

If so, a problem is that $BG$ may fail to be a smooth manifold, so additional conditions should be needed. And, what $G$-bundles on $BG$ should be taken into consideration? And, can we construct explicitly the inverse homomorphism?

If not, how to understand this isomorphism?

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    $\begingroup$ Yes. $BG$ isn't a smooth manifold, but it's a union of them. $\endgroup$ – Ben Webster Jan 19 '17 at 18:54
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    $\begingroup$ $BG$ is a smooth stack, and Chern-Weil theory can meaningfully be understood as describing its de Rham cohomology. See Freed and Hopkins arxiv.org/abs/1301.5959 for details. $\endgroup$ – Qiaochu Yuan Jan 19 '17 at 19:05
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    $\begingroup$ A down-to-earth description of $BG$ as a simplicial manifold, its simplicial de-Rham-cohomology and characteristic classes can be found in Dupont's textbook "Curvature and Characteristic Classes" . $\endgroup$ – ThiKu Jan 20 '17 at 11:02

Is this isomorphism given by the Chern-Weil homomorphism?

Yes, see Theorem 7.20 in the paper of Freed and Hopkins, which computes the de Rham complex of BG as C[g]G equipped with the zero differential.

Some important remarks:

1) BG is the stack of principal G-bundles with connection. Connections are necessary to define the Chern-Weil homomorphism.

2) The statement already holds on the level of chain complexes, not just individual cohomology groups.

3) BG is not a smooth manifold, but a stack (namely, a simplicial presheaf) on the site of smooth manifolds.


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