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The Local Langlands Correspondence for GL sub(n) over p-adic fields
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Journal Article
The Local Langlands Correspondence for GL sub(n) over p-adic fields
2013
Overview
We extend our methods from Scholze (Invent. Math. 2012, doi:10.1007/s00222-012-0419-y) to reprove the Local Langlands Correspondence for GL sub(n) over p-adic fields as well as the existence of -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.
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