Spectral mod p Satake isomorphism for GL_n

Let $K/\mathbb{Q}_p$ 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 $(φ,Γ)$-modules are labeled by Serre weights of $\mathrm{GL}_n(k)$. Let $σ$ be a non-Steinberg Serre weight and