  We construct a local model for Hilbert-Siegel moduli schemes with
$\Gamma_1(p)$-level bad reduction over $\text{Spec }\mathbb{Z}_{q}$, where $p$
is a prime unramified in the totally real field and $q$ is the residue
cardinality over $p$. Our main tool is a variant over the small Zariski site of
the ring-equivariant Lie complex $_A\underline{\ell}_G^{\vee}$ defined by
Illusie in his thesis, where $A$ is a commutative ring and $G$ is a scheme of
$A$-modules. We use it to calculate the $\mathbb{F}_{q}$-equivariant Lie
complex of a Raynaud group scheme, then relate the integral model and the local
model.

Algebraic Geometry;Number Theory