From \'etale $P_{+}$-representations to $G$-equivariant sheaves on $G\mathrm{/}P$

Let $K\mathrm{/}{\mathbb{Q}}_p$ 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 $G({\mathbb{Q}}_p)\mathrm{/}P({\mathbb{Q}}_p)$. Our construction generalizes the representation $D⊠{\mathbb{P}}^1$ of $GL(2,{\mathbb{Q}}_p)$ associated by Colmez to a $(\phi ,\mathrm{\Gamma })$-module $D$ endowed with a character of ${\mathbb{Q}}_p^{\ast }$.