$\newcommand{\Z}{\mathbb Z}\newcommand{\HdR}{H_{\mathrm{dR}}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\g}{\mathfrak g}$I have a specific question about invariant polynomials for some Lie groups, motivated by a question from differential cohomology. I'll begin with the specific question and then provide the general context.
Specific question: if $G$ is a Lie group, let $\Sym^\bullet(\g^*)$ denote the algebra of $G$-invariant polynomials on $\g$, where $G$ acts on $\g$ by the adjoint action. This is contravariant in $G$. Are the following three maps isomorphisms, and if so, where is a good reference to read about/cite this fact?
- $\Sym^\bullet(\mathfrak{gl}(n, \mathbb R))^{\mathit{GL}(n, \mathbb R)}\to\Sym^\bullet(\mathfrak o(n))^{O(n)}$ induced by the inclusion $O(n)\hookrightarrow\mathit{GL}(n, \mathbb R)$.
- Letting $\mathit{GL}_0(n, \mathbb R)\subset \mathit{GL}(n, \mathbb R)$ denote the connected component of the identity, the map $\Sym^\bullet(\mathfrak{gl}_0(n, \mathbb R))^{\mathit{GL}_0(n, \mathbb R)}\to\Sym^\bullet(\mathfrak{so}(n))^{\mathit{SO}(n)}$ induced by the inclusion $\mathit{SO}(n)\hookrightarrow\mathit{GL}_0(n, \mathbb R)$.
- $\Sym^\bullet(\mathfrak{gl}(n, \mathbb C))^{\mathit{GL}(n, \mathbb C)}\to\Sym^\bullet(\mathfrak u(n))^{U(n)}$ induced by the inclusion $U(n)\hookrightarrow\mathit{GL}(n, \mathbb C)$.
This question came up for me in the context of Chern-Weil theory. This is a machine that takes in a $G$-invariant polynomial and a principal $G$-bundle $P\to M$ with connection and builds a closed differential form on $M$, hence a de Rham cohomology class (which doesn't depend on the connection). Letting $G$ be $O(n)$, $\mathit{SO}(n)$, or $U(n)$ gives rise to the usual Pontrjagin, Euler, and Chern classes of vector bundles through an associated vector bundle construction: when $G$ is compact, the Chern-Weil map $\Sym^\bullet(\g^*)\to H^{2\bullet}(BG;\mathbb R)$ is an isomorphism. Using the compact groups amounts to choosing a metric on your vector bundle, but this is OK: the maps $BO(n)\hookrightarrow B\mathit{GL}(n, \mathbb R)$ (etc.) are homotopy equivalences, so picking a metric doesn't change what characteristic classes are available.
Cheeger-Simons lifted this construction to differential cohomology: if you have an integer lift of your de Rham cohomology class, you obtain a characteristic class of principal $G$-bundles with connection in differential cohomology $\check c(P, \nabla)\in \check H{}^\bullet(M;\Z)$, hence also differential Pontrjagin, Euler, and Chern classes for vector bundles with connection (which can depend on the connection!).
However, there is a subtlety here. If we use the compact groups, we know the Chern-Weil map is an isomorphism, so we obtain Pontrjagin, Chern, and Euler classes — but for vector bundles with a metric, and only for connections which are compatible with the metric. If we use the noncompact groups, we get characteristic classes for vector bundles with arbitrary connection, but a priori we don't know that we get all Pontrjagin, Euler, and Chern classes: differential cohomology is not homotopy-invariant in general.
This seems like the kind of invariant-theoretic computation that has been done before, and if so, I want to attribute it correctly, but I haven't been able to find a reference.
