From étale $P_{+}$-representations to $G$-equivariant sheaves on $G/P$

Let $K/\mathbb Q_{p}$ be a finite extension with ring of integers $o$, let $G$ be a connected reductive split $\mathbb Q_{p}$-group of Borel subgroup $P=TN$ and let $α$ be a simple root of $T$ in $N$. We associate to a finitely generated module $D$ over the Fontaine ring over $o$ endowed with a semilinear étale action of the monoid $T_{+}$ (acting on the Fontaine ring via $α$), a $G(\mathbb Q_{p})$-equivariant sheaf of $o$-modules on the compact space $G(\mathbb Q_{p})/P(\mathbb Q_{p})$. Our construction generalizes the representation $D\boxtimes \mathbb P^{1}$ of $GL(2,\mathbb Q_{p})$ associated by Colmez to a $(φ,Γ)$-module $D$ endowed with a character of $\mathbb Q_{p}^{*}$.

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