Let $\rho_p$ be a 3-dimensional semi-stable representation of $\operatorname{Gal} (\overline{\mathbb{Q}_p} / \mathbb{Q}_p)$ with Hodge–Tate weights $(0, 1, 2)$ (up to shift) and such that $N^2 \neq 0$ on $D_\mathrm{st} (\rho_p)$. When $\rho_p$ comes from an automorphic representation $\pi$ of $G(\mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real field $F^+$ which is compact at infinite places and $\mathrm{GL}_3$ at $p$‑adic places), we show under mild genericity assumptions that the associated Hecke-isotypic subspaces of the Banach spaces of $p$‑adic automorphic forms on $G(\mathbb{A}^{\infty}_{F^+})$ of arbitrary fixed tame level contain (copies of) a unique admissible finite length locally analytic representation of $\mathrm{GL}_3 (\mathbb{Q}_p)$ of the form considered in [4] which only depends on and completely determines $\rho_p$.

中文翻译：

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$ \ mathrm {GL} _3（\ mathbb {Q} _p）$和本地-全局兼容性具有更高的$ \ mathcal {L} $不变量

假设$ \ rho_p $是$ \ operatorname {Gal}（\ overline {\ mathbb {Q} _p} / \ mathbb {Q} _p）$的3维半稳定表示，且其Hodge–Tate权重$（0， 1、2）$（最多移位），这样$ D_ \ mathrm {st}（\ rho_p）$上的$ N ^ 2 \ neq 0 $。当$ \ rho_p $来自$ G（\ mathbb {A} _ {F ^ +}）$$的自同构表示$ \ pi $时（对于整体实场$ F ^ + $上的a组$ G $在无穷大的地方是紧凑的，在$ p $-adic的地方是$ \ mathrm {GL} _3 $），我们在温和的一般性假设下证明了$ p $的Banach空间的相关Hecke-同型子空间-$任意固定驯服度的G（\ mathbb {A} ^ {\ infty} _ {F ^ +}）$包含（副本）唯一的$ \ mathrm {GL} _3（\ mathbb { [4]中考虑的形式的Q} _p）$仅取决于并完全确定$ \ rho_p $。