Local model of Hilbert-Siegel moduli schemes in $Gamma_1(p)$-level

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 $_Aunderline{ell}_G^{vee}$ 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.