admissible pairs:

\[thm\_intro:1st\_Main\_thm\] Let $p>2$ be a rational prime. Let $(G,X)$ be a Shimura datum satisfying the Serre condition. Assume that $G$ is of classical Lie type, and that $G_{{{\mathbb Q}_p}}$ is quasi-split and splits over a tamely ramified extension of ${{\mathbb Q}_p}$. Let ${\mathbf{K}}_p$ be a parahoric subgroup of $G({{\mathbb Q}_p})$. Then, we have the followings.

\(1) Any admissible morphism $\phi:{\mathfrak{P}}\rightarrow {\mathfrak{G}}_G$ is *special*, namely there exists a special Shimura sub-datum $(T,h)$ and $g\in G({\overline{\mathbb Q}})$ such that ${\mathrm{Int}}g\circ\phi=i\circ\psi_{T,h}$, where $i:{\mathfrak{G}}_T\rightarrow {\mathfrak{G}}_G$ is the canonical morphism of neutral Galois gerbs induced by the inclusion $T{\hookrightarrow}G$. If ${\mathbf{K}}_p$ is special maximal parahoric, then every such morphism $i\circ\psi_{T,h}$ is admissible. Every admissible pair $(\phi,\epsilon)$ is conjugate to $(i\circ\psi_{T,h},\epsilon'\in T({\mathbb Q}))$ for a special Shimura sub-datum $(T,h)$.

\(2) Suppose that ${\mathbf{K}}_p$ is special maximal parahoric. For any $\gamma_0\in G({\mathbb Q})$ that is elliptic over ${\mathbb R}$, there exists an admissible pair $(\phi,\gamma_0)$ if and only if there exists $\epsilon\in G({\mathbb Q})$ stably conjugate to $\gamma_0$ and satisfying condition $\ast(\epsilon)$ of . If the latter condition holds, there exists a ${\mathbf{K}}_p$-effective admissible pair $(\phi,\epsilon)$ with $\epsilon$ stably conjugate to $\gamma_0^t$ for some $t\in{\mathbb N}$.

The tame condition on $G_{{{\mathbb Q}_p}}$ can be relaxed 