Spectral mod p Satake isomorphism for GLₙ

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