Local-global compatibility for l=p, II

TL;DR

This paper establishes the compatibility between local and global Langlands correspondences for certain automorphic representations over CM or totally real fields, advancing the understanding of Galois representations and automorphic forms.

Contribution

It proves local-global compatibility for l-adic Galois representations associated to automorphic representations, including cases with Shin-regular weight, up to semisimplification.

Findings

Compatibility proven up to semisimplification in all cases.

Compatibility up to Frobenius semisimplification for Shin-regular weights.

Advances the understanding of the Langlands program for automorphic forms.

Abstract

We prove the compatibility at places dividing l of the local and global Langlands correspondences for the l-adic Galois representations associated to regular algebraic essentially (conjugate) self-dual cuspidal automorphic representations of GL_n over an imaginary CM or totally real field. We prove this compatibility up to semisimplification in all cases, and up to Frobenius semisimplification in the case of Shin-regular weight.