Let $G$ be a complex semisimple group and $\mathcal{O} \subset G$ a conjugacy class, i.e. $\mathcal{O} = \{gag^{-1} : g \in G\}$ for some $a \in G$. Let $\Omega$ be the Cartan 3-form on $G$ defined by
$$
\Omega(x^L, y^L, z^L) = \langle x, [y, z] \rangle,\quad (x,y,z \in \mathrm{Lie}(G))
$$
where $\langle,\rangle$ is the Killing form, and $x^L, y^L, z^L$ are the left-invariant vector fields generated by $x,y,z$.


**Question:** Is the pullback of $\Omega$ to $\mathcal{O}$ trivial in cohomology?



Is the restriction of the Cartan 3-form on conjugacy classes exact?
Yes, it is exact, and there is in fact a canonical 2-form on each conjugacy class whose derivative is your $\Omega$. This was an important observation when studying D-branes in WZW models, see, e.g. <https://arxiv.org/abs/hep-th/0008038> or <https://arxiv.org/abs/hep-th/0205233>.


