In this question we consider only cohomology with rational coefficients. All groups will be connected Lie groups. All group actions will be smooth.

Let $M$ be a manifold. Let $G$ be a group acting on $M$. Let $H$ be a normal Lie subgroup of $G$. What can be said about the relation between $H^\ast_{G/H}(M)$ and $H^\ast_G(M)$?

When is $H^\ast_{G/H}(M)$ ring-isomorphic to $H^\ast_{G/H}\otimes_{\mathbb{Q}}H^\ast_G(M)$?

Consider the following particular case: $M$ is the complex projective space $\mathbb{P}^{n-1}$, $G=(\mathbb{C}^*)^n$

acting in the standard way, and $H$ is the sub-torus

$\{(t,\ldots,t):t\in\mathbb{C}^*\}$ of $G$.

Then, $$H^*_G=\mathbb{Q}[\alpha_1,\ldots,\alpha_n],$$ while

$$H^*_{G/H}=\mathfrak{\mathbb{Q}[\alpha_1,\ldots,\alpha_n]}{\alpha_1+\ldots+\alpha_n=0},$$ and

the above ring isomorphism holds.

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