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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?

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  • $\begingroup$ What do you mean by $x^L$? $\endgroup$
    – YCor
    Jul 29 at 20:25
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    $\begingroup$ The left-invariant vector field generated by $x$. $\endgroup$ Jul 29 at 20:33
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    $\begingroup$ Thanks. One particular case: if $G=\mathrm{SL}_2(\mathbf{R})$ then conjugacy classes are manifolds of dimension $\le 2$ so their cohomology vanish in degree 3. $\endgroup$
    – YCor
    Jul 29 at 20:53

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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.

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