Number Theory – Condition on a Fontaine Laffaille module that specifies the image of the associated Galois representation

The establishment:

Let $ mathbb {F} $ to be a finite feature body $ p $ and $ W ( mathbb {F}) $ the Witt vector ring with residue field $ mathbb {F} $.
Recall that filtered Dieudonné & # 39; $ W ( mathbb {F}) $-module also known as Fontaine-Laffaille module is a $ W ( mathbb {F}) $-module equipped with a separate filtration, decreasing, exhaustive sub-modules $ {F ^ i M } $ and for each integer $ i $ a $ sigma $semilinear $ varphi ^ i = varphi_M ^ i: F ^ i M rightarrow M $. These cards are necessary to satisfy two conditions

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

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

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

The functor of Fontaine-Laffaille $ text {U}: text {MF} _ {tor} ^ {f,[0,p]} rightarrow text {Rep} _ {W ( mathbb {F})} ( text {G} _ { mathbb {Q} _p}) $
or $ text {Rep} _ {W ( mathbb {F})} ( text {G} _ { mathbb {Q} _p}) $ is the category of continuous $ W ( mathbb {F})[text{G}_{mathbb{Q}_p}]$ finite length modules $ W ( mathbb {F}) $-modules. It is a fundamental fact that if $ M $ has the structure of a free $ W ( mathbb {F}) $-module of rank $ n $ (more in detail, $ F ^ j M $ are all free $ W ( mathbb {F}) $-modules and maps $ varphi ^ j $ and $ W ( mathbb {F}) $-modules and can be seen as matrices with entries in $ W ( mathbb {F}) $) then $ rho_M: = text {U} (M) $ is a representation of Galois $ rho_M: text {G} _ { mathbb {Q} _p} rightarrow text {GL} _n (W ( mathbb {F})) $.

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

Comment: For classical groups like $ text {GSp} _ {2n} $ I know how to do that. It involves making use of the alternating form and the functoriality of $ U $.