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.
Categories: math.AG math.NT