Local-global compatibility for regular algebraic cuspidal automorphic representation when $\ell \ne p$

Abstract

We prove the compatibility of local and global Langlands correspondences for $G{L}_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne and Scholze. More precisely, let $r_p(\pi )$ denote an $n$-dimensional $p$-adic representation of the Galois group of a CM field $F$ attached to a regular algebraic cuspidal automorphic representation $\pi$ of $G{L}_n({\mathbb{A}}_F)$. We show that the restriction of $r_p(\pi )$ to the decomposition group of a place $v\nmid p$ of $F$ corresponds up to semisimplification to $rec({\pi }_v)$, the image of ${\pi }_v$ under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of $.r_p(\pi ){|}_{G_{F_v}}$ is `more nilpotent' than the monodromy of $rec({\pi }_v)$.

28 pages, Comment welcome!

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