The Setup:

Let $m\geq 1$ be an integer, $\mathbb{F}$ be a finite field of characteristic $p$ and $W(\mathbb{F})$ the ring of Witt-vectors with residue field $\mathbb{F}$ and $\sigma:W(\mathbb{F})\xrightarrow{\sim} W(\mathbb{F})$ a lift of the Frobenius $\bar{\sigma}:\mathbb{F}\xrightarrow{\sim}\mathbb{F}$. Recall that filtered Dieudonne' $W(\mathbb{F})$-module also known as a Fontaine-Laffaille module is a $W(\mathbb{F})$-module furnished with a decreasing, exhaustive, separated filtration of submodules $\{F^i M\}$ and for each integer $i$ a $\sigma$-semilinear map $\varphi^i=\varphi_M^i:F^i M\rightarrow M$. These maps are required to satisfy two conditions

  1. the following compatibility relation is satisfied $\varphi^{i+1}=p \varphi^i$,

  2. $\sum_i \varphi^i(F^i M)=M$.

Let $\text{MF}_{tor}^{f}$ denote the category of Fontaine-Laffaille modules $M$ with morphisms satisfying the conditions alluded to above. For $a<b$ let $\text{MF}_{tor}^{f,[a,b]}$ let the full subcategory of $\text{MF}_{tor}^{f}$ whose underlying modules $M$ satisfy $F^0 M=M$ and $F^p M=0$.

The Fontaine-Laffaille functor $\text{U}:\text{MF}_{tor}^{f,[0,p]}\rightarrow \text{Rep}_{W(\mathbb{F})}(\text{G}_{\mathbb{Q}_p})$ where $\text{Rep}_{W(\mathbb{F})}(\text{G}_{\mathbb{Q}_p})$ is the category of continuous $W(\mathbb{F})[\text{G}_{\mathbb{Q}_p}]$ modules that are finite-length $W(\mathbb{F})$-modules. It is a basic fact that if $M$ has the structure of a free $W(\mathbb{F})/p^m$-module of rank $n$ (in greater detail, $F^j M$ are all free $W(\mathbb{F})/p^m$-modules and the maps $\varphi^j$ and semilinear $W(\mathbb{F})$-module maps) then $\rho_M:=\text{U}(M)$ is a Galois representation $\rho_M:\text{G}_{\mathbb{Q}_p}\rightarrow \text{GL}_n(W(\mathbb{F})/p^m)$.

Question: Let $G\subset \text{GL}_n$ be an algebraic subgroup of $\text{GL}_n$ defined over $\mathbb{Q}$ (I'm mainly interested in the exceptional groups, like for instance $G_2$). What condition on the maps $\varphi^j_M$ ensures that the image of $\rho_M:=\text{U}(M)$ lies in $G(W(\mathbb{F})/p^m)$ so that it is a Galois representation $\rho_M:\text{G}_{\mathbb{Q}_p}\rightarrow G(W(\mathbb{F})/p^m )$.

Comment: For the classical groups like $\text{GSp}_{2n}$ this is done in section 2 of the paper of Clozel, Harris and Taylor "Automorphy of some $l$-adic lifts of Automorphic Mod $l$ Galois Representations". It involves making use of the alternating form and the functoriality of $U$.


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