Representation viewpoint on Chern–Weil (cohomology computations done with rep theory?)

Question

$\DeclareMathOperator\Sym{Sym}$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?

Answer 1

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 establishes an isomorphism between $K(X) \otimes \mathbb{R}$ and periodic even cohomology $H^{2 \bullet}(X, \mathbb{R})$. The Atiyah-Segal 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.

Answer 2

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.