Spectral mod p Satake isomorphism for GL_n

Let K/Qₚ 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 \'etale (φ,Γ)-modules are labeled by Serre weights of GLₙ(k). Let σ be a non-Steinberg Serre weight and C_σ be the corresponding irreducible component. Motivated by the categorical p-adic local Langlands program, we construct a natural injective map O(C_σ) H(σ) from the ring of global functions on C_σ to the Hecke algebra of σ compatible with the mod p Satake isomorphism by Herzig and Henniart--Vign\'eras in a suitable sense. For sufficiently generic σ, 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.