The Local Langlands correspondence for $\GL_n$ over $p$-adic fields

TL;DR

This paper provides a new proof of the Local Langlands Correspondence for GL_n over p-adic fields, establishing the existence of associated Galois representations and local-global compatibility without relying on Henniart's numerical correspondence.

Contribution

It offers a novel proof of the Local Langlands Correspondence for GL_n over p-adic fields using inertia-invariant nearby cycles, bypassing previous numerical methods.

Findings

Established the Local Langlands Correspondence for GL_n over p-adic fields.

Proved existence of associated mbda-adic Galois representations for regular algebraic automorphic representations.

Demonstrated local-global compatibility in the context of automorphic and Galois representations.

Abstract

We reprove the Local Langlands Correspondence for $\GL_n$ over $p$-adic fields as well as the existence of $\ell$-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor. In contrast to the proofs of the Local Langlands Correspondence given by Henniart and Harris-Taylor our proof completely by-passes the numerical Local Langlands Correspondence of Henniart. Instead, we make use of a previous result describing the inertia-invariant nearby cycles in certain regular situations.