Spectral mod p Satake isomorphism for GL_n

Let $K\mathrm{/}{\mathbb{Q}}_p$ be a finite extension with residue field $k$. By a work of Emerton–Gee, irreducible components inside the reduced special fiber of the moduli stack of rank $n$ étale $(\phi ,\mathrm{\Gamma })$-modules are labeled by Serre weights of ${\mathrm{G}\mathrm{L}}_n(k)$. Let $\sigma$ be a non-Steinberg Serre weight and ${\mathcal{C}}_{\sigma }$ be the corresponding irreducible component. Motivated by the categorical $p$-adic local Langlands program, we construct a natural injective map $\mathcal{O}({\mathcal{C}}_{\sigma })\hookrightarrow \mathcal{H}(\sigma )$ from the ring of global functions on ${\mathcal{C}}_{\sigma }$ to the Hecke algebra of $\sigma$ compatible with the mod $p$ Satake isomorphism by Herzig and Henniart–Vignéras in a suitable sense. For sufficiently generic $\sigma$, we prove that it is an isomorphism. As an application, we obtain a natural stratification of the irreducible component whose strata are equipped with a parabolic structure. Our main input is a construction of a morphism from an integral Hecke algebra of a generic tame type to the ring of global functions on a tamely potentially crystalline Emerton–Gee stack.