From \'etale 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 $\alpha$ 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 \'etale action of the monoid $T_{+} $ (acting on the Fontaine ring via $\alpha$), 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 $(\varphi,\Gamma)$-module $D$ endowed with a character of $\mathbb Q_{p}^{*}$.