Local-global compatibility for regular algebraic cuspidal automorphic representation when ℓ ≠ p

Original · EN

We prove the compatibility of local and global Langlands correspondences for GLₙ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne and Scholze. More precisely, let rₚ(π) denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation π of GLₙ(AF). We show that the restriction of rₚ(π) to the decomposition group of a place v p of F corresponds up to semisimplification to rec(πᵥ), the image of πᵥ under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of.rₚ(π)|GFᵥ is `more nilpotent' than the monodromy of rec(πᵥ).