Let $K/\mathbf{Q}_p$ be a finite extension, and let $V$ be an $n$-dimensional $\overline{\mathbf{Q}_p}$-vector space with a continuous action of $G_K$. Suppose $V$ is de Rham, so potentially semistable. Let $L/K$ be an extension over which $V$ becomes semistable, with maximal absolutely unramified subfield $L_0$. Choose an embedding $\tau:L_0 \to \overline{\mathbf{Q}_p}$ and set $W=(V \otimes_{L_0,\tau}\mathbf{B}_{\mathrm{st}})^{\mathrm{Gal}(\overline{K}/L)}$. A standard construction of Fontaine produces from $W$ a Weil-Deligne representation $\mathrm{WD}(V)=(W,\rho,N)$ independent of all choices. Let $\pi=\mathrm{rec}_K^{-1}(\mathrm{WD}(V))$ be its local Langlands correspondent. Here is a natural speculation:
The following conditions are equivalent:
- $V$ is trianguline.
- $(W,\rho)^{ss}$ is totally reducible.
- $\pi$ is a subquotient of a representation induced from the Borel subgroup of $\mathrm{GL}_n(K)$.
Is this considered anywhere in the literature? When $n=2$ this is a theorem of Nakamura, but things simplify a bit since in that case the third condition above is the same as "$\pi$ is not supercuspidal".