Does the isotropic group of local galois representation have finite index?

Question

Let $\rho : G\to \mathrm{GL}_n(\mathbb Z_p)$ be a crystalline representation of $G=\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$. For any non zero element $a\in \mathbb{Z}_p^n$ , it spans a rank one $\mathbb{Z}_p$-submodule $L_a$. Now consider the subgroup $H$ of $G$, $H=\{g\in G| g(L_a)\subset L_a\}$. Does $H$ have finite index in $G$, i.e .$[G:H]< \infty$?

Answer 1

If $\rho$ is a representation with this property, then by taking $a$ to run through a basis of $\mathbf{Z}_p^n$ and intersecting the corresponding $H$'s, there must be a finite-index subgroup of $G$ whose image under $\rho$ is diagonal. Most crystalline representations will not have this property. There are even unramified representations that are not of this form, e.g. the one mapping Frobenius to $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.