will further assume that ${\mathbf{K}}_p$ is special maximal parahoric. When one works with this general level subgroups, the first notable change occurs in definitions, especially that of admissible morphism, where instead of a single affine Deligne-Lusztig variety $X(\{\mu_X\},b)_{{\mathbf{K}}_p}$ which defined the set $X_p(\phi)$, one needs to use a *finite union* of *generalized* affine Deligne-Lusztig varieties $X(w,b)_{{\mathbf{K}}_p}$: $$X(\{\mu_X\},b)_{{\mathbf{K}}_p}:=\bigsqcup_{w\in{\mathrm{Adm}}_{{\tilde{\mathbf{K}}}_p}(\{\mu_X\})} X(w,b)_{{\mathbf{K}}_p}$$ Here, ${\mathrm{Adm}}_{{\tilde{\mathbf{K}}}_p}(\{\mu_X\})$ is a certain subset of the double coset ${\tilde{\mathbf{K}}}_p\backslash G({\mathfrak{k}})/{\tilde{\mathbf{K}}}_p$ determined by the datum $(G,X)$. When ${\mathbf{K}}_p$ is hyperspecial, this specializes to the previous definition. Meanwhile, the conjecture (\[eqn:LRconj-ver1\]) itself was extended by Rapoport [@Rapoport05] to cover general parahoric levels. Secondly, to work beyond the restriction of simply-connected derived group, we use the definition of admissible morphism generalized by Kisin for this purpose (cf. Def. \[defn:admissible\_morphism\], [@Kisin17 (3.3.6)]).

To keep statements short, we will call the following condition the *Serre condition for the Shimura datum $(G,X)$*[^4] : *the center $Z(G)$ of $G$ splits over a CM field and the weight homomorphism $w_X$ is defined over ${\mathbb Q}$.* Note that this condition holds if the Shimura datum is of Hodge type.

Our main results in the first part are generalizations (sometimes, including improvements) of the key properties of admissible morphisms and 