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 $\mathrm{\mid }\mathrm{\Delta }{\mathrm{\mid }}^f$ defined by Pappas-Rapoport and prove that the face map is surjective.