Cohen-Macaulay, see [@Go-Ma Cor. 1.2]. Thus, to prove Thm. \[thm:realmain\], there are two steps to be performed: show first that any Hermann action is Cohen-Macaulay and second that the Cohen-Macaulay property is preserved under orbit equivalence. The first goal is achieved by the following result:

\[main\] If $G$, $K$, and $H$ are as above, then the (Hermann) action of $K$ on $G/H$ by left translations is Cohen-Macaulay.

A big part of the paper is devoted to the proof of this theorem, see Sect. \[semisimple\] (also note that the result remains true even if $G$ is not necessarily semisimple, see Rem. \[nonsemi\]). It is worth recalling at this point a related result, obtained by the first-named author of this paper in [@Go], which says that the $K$-action on $G/K$ by left translations is equivariantly formal; recently, a conceptually different proof has been obtained by the second-named author in [@No]. Thm. \[main\] above is a generalization of this (taking into account Prop. \[characef\] below). Pairs $(G,K)$ with the property that the action of $K$ on $G/K$ is equivariantly formal have also been investigated in [@Go-No], [@Ca2], [@Ca1], [@Ca-Fo], and again [@No].

The second of the aforementioned goals is to show that the 