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

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

2010-10-07

arxiv.org/abs/1010.1540v1

Mathematics

Algebraic Geometry

37 pages

Scientific paper

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.

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