On the Face Map of the Admissible Set With Iwahori Level

Abstract

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