Functions on Irreducible Components of the Emerton-Gee Stack

Let $K / \mathbb{Q}_p$ be a finite unramified extension, and let $\mathcal{X}_n$ denote the Emerton-Gee stack parametrizing étale $(\varphi,\Gamma_K)$-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 $\operatorname{GL}_n(k)$.

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.