whose action on $X$ is locally free. One can compute the cohomology of the orbit space $X/S$ by means of the following version of the Gysin sequence (cf. [@Co Lemma 2.2]): $$\ldots \to H^j(X/S) \to H^{j+2}(X/S) \to H^{j+2}(X) \to H^{j+1}(X/S) \to \ldots .$$ It follows that $H^*(X/S)\simeq H^*(\bC P^n)$ by an isomorphism of vector spaces. Consequently, by Ex. \[exodd\], the canonical action of $T/S$ on $X/S$ is equivariantly formal, hence, by Cor. \[cmcrit\], the ring $H^*_{T/S}(X/S)$ is Cohen-Macaulay. To conclude the proof, it only remains to notice that we have the ring isomorphism $$H^*_T(X)\simeq H^*_{T/S}(X/S).$$

The case when $G$ is not simple
-------------------------------

From now on, $G$ is just simply connected, not necessarily being simple. Our main tool in dealing with this situation is the following result, see [@Ko Prop. 18]:

[(A. Kollross)]{}\[andreas\] If $G$ is simply connected then the $H$-action on $G/K$ is a direct product of actions of one of the following types:

-   the action of $H'\times L^{n-1} \times K'$ on $L^n$ defined by $$\begin{aligned}
    {}&(h, g_1, \ldots, g_{n-1}, k)\cdot(x_1, \ldots, x_n)\\{}&=
    (hx_1g_1^{-1}, g_1x_2g_2^{-1}, \ldots, g_{n-2}x_{n-1}g_{n-1}^{-1}, g_{n-1}x_n k^{-1}),\end{aligned}$$

-   the action of $H'\times L^{n-1}$ on $L^{n-1}\times L/K'$ defined by $$\begin{aligned}
    {}&(h, g_1, \ldots, g_{n-1})\cdot(x_1, \ldots, x_{n-1}, x_nK')\\{}&=
    