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For a continuous irreducible representation

$\rho: G_{\mathbb{Q}_p}\rightarrow GL_n(\overline{\mathbb{Q}_p})$,

is it possible for both $D_{cris}(\rho)$ and $D_{cris}(\chi\otimes\rho)$ to be nonzero, where $\chi$ is some non-crystalline character?

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Let me use Colmez' article "Representations triangulines" as a reference. Let $V$ be a repn which satisfies your condition.

By proposition 4.3, $V$ is trianguline. By proposition 4.10, the HT weight of $\chi$ has to be an integer. You can then assume that $\chi$ has finite order, and this implies that $V$ is potentially crystalline on an abelian extension of $Q_p$ (aka crystabelline).

Conversely, it seems likely that one can give examples of irreducible crystabelline representations $V$ such that $D_{cris}(V)$ and $D_{cris}(V \otimes \chi)$ are both nonzero, with $\chi$ of finite order (see the examples in 2.4 of Berger-Breuil's "Sur quelques representations potentiellement cristallines de $GL_2(Q_p)$").

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