Functions on Irreducible Components of the Emerton-Gee Stack

Let $K\mathrm{/}{\mathbb{Q}}_p$ be a finite unramified extension, and let ${\mathcal{X}}_n$ denote the Emerton-Gee stack parametrizing étale $(\phi ,{\mathrm{\Gamma }}_K)$-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 $\sigma$ of ${\mathrm{GL}}_n(k)$. If such a component is denoted $\mathcal{X}(\sigma )$, we prove that $\mathcal{O}(\mathcal{X}(\sigma ))\cong \mathbb{F}[x_1,x_2,\dots ,x_{n-1},x_n^{\pm 1}]$ when $\sigma$ is sufficiently generic.