Breuil's locally analytic socle conjecture for global modular Galois representations

Let $L$ be a finite extension of $\mathbb{Q}_p$, let $E$ be a coefficient field, and let $T(L)$ be the diagonal torus of $\operatorname{GL}_2(L)$. For a continuous very regular character $\chi$ of $T(L)$, write

$$I(\chi):=\operatorname{soc}\left(\operatorname{Ind}_{\overline{B}(L)}^{\operatorname{GL}_2(L)}\chi\right)^{\mathbb{Q}_p\text{-an}}.$$

Let $\rho_L$ be the local component of a global modular Galois representation, let $\widehat{\Pi}(\rho)$ be the associated admissible unitary Banach representation, and let $\delta_B$ be the modulus character of the upper triangular Borel subgroup. For subsets $J\subseteq\Sigma$ and $J\subseteq\widetilde{\Sigma}$, let $\delta_J^c$ and $\widetilde{\delta}_J^c$ denote the associated companion characters, and let $\chi^{\sharp}$ denote the twist defined in the surrounding notation.

Breuil's conjecture. For a character $\chi:T(L)\rightarrow E^{\times}$,

$$I(\chi)\hookrightarrow \widehat{\Pi}(\rho)^{\operatorname{an}}$$

if and only if $\chi=(\delta_J^c)^{\sharp}\delta_B^{-1}$ for some $J\subseteq\Sigma$, or $\chi=(\widetilde{\delta}_J^c)^{\sharp}\delta_B^{-1}$ for some $J\subseteq\widetilde{\Sigma}$.

This describes the locally analytic socle of the representation arising in the $p$-adic Langlands program. In the modular-curve case $L=\mathbb{Q}_p$, the conjecture was proved using $p$-adic comparison theorems and overconvergent modular forms; the general statement is not resolved by the supplied source.