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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:

  1. $V$ is trianguline.
  2. $(W,\rho)^{ss}$ is totally reducible.
  3. $\pi$ 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".

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  • $\begingroup$ The formulation is imprecise about the coefficient field ($n$-dimensional over $\mathbf{Q}_p$ or $\overline{\mathbf{Q}}_p$ or...) and "totally reducible" should be defined (giving due attention to the coefficient field). $\endgroup$
    – Marguax
    Oct 13, 2013 at 19:23
  • $\begingroup$ Due attention given. $\endgroup$ Oct 14, 2013 at 11:21
  • $\begingroup$ Nice question. I presume you know Berger-Chenevier's paper and that you have checked its result are compatible with your conjecture? $\endgroup$
    – Joël
    Oct 14, 2013 at 13:44
  • $\begingroup$ David, so you have a precise reference for the case $n=2$? $\endgroup$
    – Joël
    Oct 20, 2013 at 17:09
  • $\begingroup$ Hi Joel, it's Theorem A.4 in this paper of Nakamura: arxiv.org/abs/0801.1230 $\endgroup$ Oct 20, 2013 at 20:43

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