| Résume | Let $\mathfrak{X}_+$ be the Bruhat Tits halftree of $\textrm{Z}_2(\mathbb{Q}_p)$ on which the monoid $\left(\begin{array}{cc}\mathbb{Z}_p-\{0\} & \mathbb{Z}_p\\0&1\end{array}\right)$ acts. Let $G$ be a split reductive group over $\mathbb{Q}_p$ with connected center. Let $X$ be the semisimple Bruhat Tits building of $G$. The choice of a semiinfinite chamber gallery in an apartment of $X$ (satisfying some conditions) defines an embedding of $\mathfrak{X}_+$ into $X$ which is ``equivariant'' in a certain sense: It allows one to pull back $G$-equivariant coefficient systems $\mathcal{V}$ on $X$, satisfying a local smoothness condition, to equivariant coefficient systems on $\mathfrak{X}_+$.
Now let $\mathcal{H}$ be the pro-$p$-Iwahori Hecke algebra, with coefficients in a finite field $k$ of characteristic $p$, corresponding to a pro-$p$-Iwahori subgroup in $G$. A finite dimensional $\mathcal{H}$-module $M$ gives rise to a coefficient system $\mathcal{V}=\mathcal{V}_M$ on $X$ as above. Pulling it back to $\mathfrak{X}_+$ as indicated and then carrying out Colmez' construction we obtain a $(\varphi,\Gamma)$-module over $k$. Composing with Fontaine's equivalence, we obtain a functor from the category of finite dimensional $\mathcal{H}$-modules to the category of $\textrm{Gal}_{\mathbb{Q}_p}$-representations over $k$. If $G=\textrm{Z}_n(\mathbb{Q}_p)$ it induces a bijection between simple supersingular $n$-dimensional $\mathcal{H}$-modules and irreducible $n$-dimensional $\textrm{Gal}_{\mathbb{Q}_p}$-representations over $k$. |
