significantly (see Thm. \[thm:LR-Satz5.3\] for a precise condition). These statements (and their counterpart results in [@LR87]) are found respectively in Theorem \[thm:LR-Satz5.3\], Lemma \[lem:LR-Lemma5.2\], \[lem:LR-Lemma5.23\] (for (1)) and Theorem \[thm:LR-Satz5.21\] (for (2)). The statement (1) and the first claim of (2) are generalizations of the same results in *loc. cit.*, while the second claim of statement (2) is new. We also remark that the condition $\ast(\epsilon)$ in (2) is our generalization of the original condition in [@LR87 Satz 5.21] of the same name $\ast(\epsilon)$, in our general set-up that the level subgroup is parahoric and $G_{{{\mathbb Q}_p}}$ is not necessarily unramified nor $G^{{\mathrm{der}}}$ is simply connected.

The statement (1) is a fundamental fact about admissible morphisms, and underlies the CM lifting theorem [@Kisin17 Thm. 0.4] (in the hyperspecial level case) that every isogeny class in ${\mathscr{S}}_{{\mathbf{K}}_p}(G,X)({\bar{\mathbb{F}}_{p}})$ contains a point that is the reduction of a special(=CM) point.

The following theorem is the first version of the aforementioned effectivity criterion of Kottwitz triple which is established for the first time here (even in the original set-up of [@LR87], cf. Remark \[rem:comments\_on\_Milne92\]).

\[thm\_intro:2st\_Main\_thm\] \[Thm. \[thm:LR-Satz5.25\]\] Under the same assumptions as in (2) of Thm. \[thm\_intro:1st\_Main\_thm\], for every Kottwitz triple $(\gamma_0;(\gamma_l)_{l\neq p},\delta)$ with 