Local-global compatibility

Let $F$ be a number field and let

\pi=\bigotimes_v'\pi_v

be an restricted tensor product defining an algebraic automorphic representation, and let $\rho_{\pi,\lambda}$ be an associated $\ell$ -adic Galois representation. Local–global compatibility at $v$ asserts that the restriction of $\rho_{\pi,\lambda}$ to a decomposition group at $v$ matches the local Langlands parameter of $\pi_v$ , after choosing an identification between complex and $\ell$ -adic coefficient fields and applying the stated normalization.

Away from ell

For $v\nmid\ell$ , the comparison for $\operatorname{GL}_n$ is commonly written

\operatorname{WD}\!\left( \rho_{\pi,\lambda}|_{\Gamma_{F_v}} \right)^{\mathrm{F\text{-}ss}} \cong \operatorname{rec}_{F_v}\!\left( \pi_v\otimes |\det|^{(1-n)/2} \right),

with the twist altered if a different local Langlands normalization is used. The superscript denotes Frobenius semisimplification: replace the Frobenius action in the Weil–Deligne representation by its semisimple part while retaining monodromy. Some theorems prove only ordinary semisimplification and therefore do not identify the monodromy operator.

At an unramified place, the statement reduces to equality of the Frobenius conjugacy class with the Satake parameter.

At places above ell

For $v\mid\ell$ , the comparison uses $p$ -adic Hodge theory: the Galois representation should be de Rham, its Hodge–Tate cocharacters should match the archimedean algebraic weights through the chosen embedding, and its potentially semistable Weil–Deligne parameter should match $\pi_v$ . This is a different and generally harder assertion than compatibility away from $\ell$ .

Status must be qualified

There is no single theorem giving full local–global compatibility for every reductive group and every algebraic automorphic representation. Strong theorems exist for broad regular algebraic cuspidal families on $\operatorname{GL}_n$ , with hypotheses depending on the number field, polarization, and place. A statement must specify whether it preserves monodromy and whether $v\mid\ell$ .

References

Ila Varma, “Local-global compatibility for regular algebraic cuspidal automorphic representations when $\ell\neq p$ ,” 2014. arXiv.
Lambert A'Campo, Bence Hevesi, Jack A. Thorne, and Dmitri Whitmore, “Local-global compatibility of automorphic Galois representations over CM fields at $p$ ,” 2026. arXiv.