Functions on Irreducible Components of the Emerton-Gee Stack

Abstract

Let $K/{\mathbb{Q}}_p$ be a finite unramified extension, and let ${\mathcal{X}}_n$ denote the Emerton-Gee stack parametrizing étale $(\ddot{I}†,\hat{I}{“}_K)$-modules of rank $n$. It is known since the work of Emerton-Gee that the irreducible components of the reduced special fiber of $\mathcal{X}$ are labeled by Serre weights $\ddot{I}ƒ$ of ${\mathrm{GL}}_n(k)$. If such a component is denoted $\mathcal{X}(\ddot{I}ƒ)$, we prove that $\mathcal{O}(\mathcal{X}(\ddot{I}ƒ))\cong \mathbb{F}[x_1,x_2,\dots ,x_{n-1},x_n^{\pm 1}]$ when $\ddot{I}ƒ$ is sufficiently generic.

To appear in Transactions of the American Mathematical Society