$p\nmid |\pi_1(G^{{\mathrm{der}}})|$, and as a result Thm. \[thm\_intro:Kottwitz\_conj\] (for which one might also need certain results on the endoscopic transfer of stable Bernstein center, cf. [@Haines14]). But, we can already use our partial results to extend the scope of some previous results, for example, we can relax the ramification condition imposed on the PEL datum in the main result of [@Scholze13].

Finally, we make some comments on the various assumptions appearing in this article. The running assumption, which will be effective except in some general discussions, is that $G_{{{\mathbb Q}_p}}$ is quasi-split and splits over a tamely ramified extension of ${{\mathbb Q}_p}$ (as mentioned above, for the latter condition, in fact, we only need some less restrictive one: see Thm. \[thm:LR-Satz5.3\] and Prop. \[prop:existence\_of\_elliptic\_tori\_in\_special\_parahorics\] for a precise condition). Equally universal assumption, although it is not needed for the important Theorem \[thm\_intro:1st\_Main\_thm\], (1), is that ${\mathbf{K}}_p$ is special maximal parahoric. These two assumptions are somewhat forced on us because we follow closely the original line of arguments [@LR87] for our proofs. We however remark that the two conditions that $G_{{{\mathbb Q}_p}}$ splits over a tamely ramified extension and the level subgroup is special maxima parahoric are imposed only via Lemma \[lem:LR-Lemma5.11\], 