From pro-p Iwahori–Hecke modules to (φ,Γ)-modules, I

Abstract

Let $\mathfrak{o}$ be the ring of integers in a finite extension $K$ of ${\mathbb{Q}}_p$, and let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb{Q}}_p$, and let $T$ be a maximal split torus in $G$. Let $\mathcal{H}(G,I_0)$ be the pro-$p$ Iwahori–Hecke $\mathfrak{o}$-algebra. Given a semi-infinite reduced chamber gallery (alcove walk) $C^{(•)}$ in the $T$-stable apartment, a period $\varphi \in N\left(T\right)$ of $C^{(•)}$ of length $r$, and a homomorphism $\tau :{\mathbb{Z}}_p^{\times }\to T$ compatible with $\varphi$, we construct a functor from the category ${Mod}^{\mathrm{fin}}\left(\mathcal{H}\right(G,I_0\left)\right)$ of finite-length $\mathcal{H}(G,I_0)$-modules to étale $({\phi }^r,\Gamma )$-modules over Fontaine’s ring ${\mathcal{O}}_{\mathcal{E}}$. If $G={GL}_{d+1}\left({\mathbb{Q}}_p\right)$, then there are essentially two choices of $(C^{(•)},\varphi ,\tau )$ with $r=1$, both leading to a functor from ${Mod}^{\mathrm{fin}}\left(\mathcal{H}\right(G,I_0\left)\right)$ to étale $(\phi ,\Gamma )$-modules and hence to ${\mathrm{Gal}}_{{\mathbb{Q}}_p}$-representations. Both induce a bijection between the set of absolutely simple supersingular $\mathcal{H}(G,I_0){\otimes }_{\mathfrak{o}}k$-modules of dimension $d+1$ and the set of irreducible representations of ${\mathrm{Gal}}_{{\mathbb{Q}}_p}$ over $k$ of dimension $d+1$. We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of $G$ over $K$. For $d=1$, we recover Colmez’s functor (when restricted to $\mathfrak{o}$-torsion ${GL}_2\left({\mathbb{Q}}_p\right)$-representations generated by their pro-$p$ Iwahori invariants).