On the Face Map of the Admissible Set With Iwahori Level

To each face $\mathcal{F}$ of the coweight polytope $\mathcal{P}_{\mu}$, we associate a subset $\text{Adm}(\mu)_{\mathcal{F}}$ of the $\mu$-admissible set $\text{Adm}(\mu)$, which we refer to as a face of $\text{Adm}(\mu)$. This gives rise to a face decomposition of $\text{Adm}(\mu)$. As an application, we give a complete description of the fibers of the face map $|\Delta|^f$ defined by Pappas-Rapoport and prove that the face map is surjective.