Local-global compatibility conjecture for cspan class="math-inline"cspan class="math-inline"GL_3$\mathbb Q_p$c/spanc/spanc

Let $F$ be the number field in the setup, let $\rho:\operatorname{Gal}_F\to\operatorname{GL}_3(E)$ be continuous and absolutely irreducible, and assume it is unramified at the places of $D(U^p)$. Suppose

$$\widehat{S}(U^{\wp},W^{\wp})[\mathfrak m_\rho]^{\operatorname{lalg}}\ne 0,$$

$F^+_\wp\cong F_{\widetilde\wp}=\mathbb Q_p$, and $\rho_{\widetilde\wp}:=\rho|_{\operatorname{Gal}_{F_{\widetilde\wp}}}$ is semistable with $N^2\ne0$ on $D_{\operatorname{st}}(\rho_{\widetilde\wp})$. Assume also that $D_{\operatorname{rig}}(\rho_{\widetilde\wp})$ is sufficiently generic, and let $\Pi(\rho_{\widetilde\wp})$ be the associated locally analytic representation of $\operatorname{GL}_3(\mathbb Q_p)$, with

$$\Pi(\rho_{\widetilde\wp})^{\operatorname{lalg}}=\operatorname{soc}_{\operatorname{GL}_3(\mathbb Q_p)}\Pi(\rho_{\widetilde\wp}).$$

Local-global compatibility conjecture. The restriction morphism is bijective:

$$\operatorname{Hom}_{\operatorname{GL}_3(\mathbb Q_p)}\bigl(\Pi(\rho_{\widetilde\wp}),\widehat{S}(U^{\wp},W^{\wp})[\mathfrak m_\rho]\bigr)\xrightarrow{\sim}\operatorname{Hom}_{\operatorname{GL}_3(\mathbb Q_p)}\bigl(\Pi(\rho_{\widetilde\wp})^{\operatorname{lalg}},\widehat{S}(U^{\wp},W^{\wp})[\mathfrak m_\rho]\bigr).$$

This is the paper's main local-global compatibility conjecture: the full locally analytic representation attached to the local Galois representation should carry no more multiplicity information in completed cohomology than its locally algebraic socle. The conjecture concerns the semistable, nontrivial-monodromy, sufficiently generic case for $\operatorname{GL}_3(\mathbb Q_p)$.