Let F/Q be a CM field in which p splits completely and r : Gal(Q/F) → GLn(Fp) a continuous automorphic Galois representation. We assume that r|Gal(Qp/Fw) is an ordinary representation at a place w above p. In this talk, we discuss a problem about local-global compatibility in the mod p Langlands program for GLn(Qp). It is expected that if r|Gal(Qp/Fw) is tamely ramified, then it is determined by the set of modular Serre weights and the Hecke action on its constituents. However, this is not true if r|Gal(Qp/Fw) is wildly ramified, and the question of determining r|Gal(Qp/Fw) from a space of mod p automorphic forms lies deeper than the weight part of Serre’s conjecture. We define a local invariant associated to r|Gal(Qp/Fw) in terms of Fontaine-Laffaille theory, and discuss a way to prove that the local invariant associated to r|Gal(Qp/Fw) can be obtained in terms of a refined Hecke action on a space of mod p algebraic automorphic forms on a compact unitary group cut out by the maximal ideal of a Hecke algebra associated to r, that is a candidate on the automorphic side corresponding to r|Gal(Qp/Fw) for mod p Langlands program.
The talk is based on a joint work with Zicheng Qian.