On the Face Map of the Admissible Set With Iwahori Level

TL;DR

This paper introduces a face decomposition of the $ ext{Adm}()$ set associated with coweight polytopes, providing a complete description of the face map fibers and proving its surjectivity.

Contribution

It defines a face decomposition of the admissible set and characterizes the fibers of the face map, establishing its surjectivity.

Findings

Complete description of the fibers of the face map.

Proof that the face map is surjective.

Introduction of a face decomposition of the admissible set.

Abstract

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.