Buzzard–Gee conjecture for Galois representations attached to automorphic forms

Let $F$ be a number field, $G$ a reductive group, $S$ a finite set of places, and $\bar{\rho}$ and $\rho$ denote the residual and integral Galois representations associated with Hecke data as in the source. Let $\rho:\Gamma_F\to{}^LG(k)$ be a continuous semisimple representation, and let $cal A^S_{\operatorname{ac}}(G,F)$ and $\mathcal{HE}^S_{\operatorname{coh}}(G,F)_{\overline{\mathbb F}_\ell}$ denote the indicated spaces of automorphic and Hecke-eigenvalue data.

Buzzard–Gee conjecture. The following are true. There exists a map

$$\operatorname{GLC}_{G,\overline{\mathbb Q}_\ell}:\mathcal A^S_{\operatorname{ac}}(G,F)\longrightarrow \mathcal G^S_{\operatorname{dR}}({}^LG,F)_{\overline{\mathbb Q}_\ell},\qquad \pi=\bigotimes'_v\pi_v\longmapsto\rho_{\pi,\iota},$$

such that $\iota\pi_v$ corresponds to $(\rho_{\pi,\iota}|_{\Gamma_{F_v}})^{\operatorname{ss}}$ via unramified local Langlands at every $v\notin S$. There also exists a map

$$\operatorname{GLC}_{G,\overline{\mathbb F}_\ell}:\mathcal{HE}^S_{\operatorname{coh}}(G,F)_{\overline{\mathbb F}_\ell}\longrightarrow\mathcal G^S({}^LG,F)_{\overline{\mathbb F}_\ell},$$

that assigns $frak m$ a representation $foverline\rho_{\mathfrak m}$ unramified outside $S$, with characteristic polynomials of Frobenius explicitly determined by the Hecke eigenvalues encoded in $frak m$ at every $v\notin S$.

This is a coarse form of the Buzzard–Gee conjecture; the general formulation should use $C$-groups. The analogous mod-$\ell$ statement was suggested by Ash, while the precise scope and refinements depend on the cohomological setting.