The Local Langlands Correspondence for GL n over p-adic fields

Abstract

We extend our methods from Scholze (Invent. Math. 2012, doi: 10.1007/s00222-012-0419-y ) to reprove the Local Langlands Correspondence for GL n over p-adic fields as well as the existence of l-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 (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001). In contrast to the proofs of the Local Langlands Correspondence given by Henniart (Invent. Math. 139(2), 439–455, 2000), and Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001), our proof completely by-passes the numerical Local Langlands Correspondence of Henniart (Ann. Sci. Ec. Norm. Super. 21(4), 497–544, 1988). Instead, we make use of a previous result from Scholze (Invent. Math. 2012, doi: 10.1007/s00222-012-0419-y ) describing the inertia-invariant nearby cycles in certain regular situations.