From pro-$p$ Iwahori-Hecke modules to $(\phi ,\Gamma )$-modules I

Abstract

Let $\mathfrak{o}$ be the ring of integers in a finite extension $K$ of ${\mathbb{Q}}_p$, let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb{Q}}_p$, 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 semiinfinite reduced chamber gallery (alcove walk) $C^{(\bullet )}$ in the $T$-stable apartment, a period $\varphi \in N(T)$ of $C^{(\bullet )}$ of length $r$ and a homomorphism $\tau :{\mathbb{Z}}_p^{\times }\to T$ compatible with $\varphi$, we construct a functor from the category ${\mathrm{Mod}}^{\mathrm{fin}}(\mathcal{H}(G,I_0))$ of finite length $\mathcal{H}(G,I_0)$-modules to étale $({\phi }^r,\Gamma )$-modules over Fontaine's ring ${\mathcal{O}}_{\mathcal{E}}$. If $G={\mathrm{GL}}_{d+1}({\mathbb{Q}}_p)$ there are essentially two choices of ($C^{(\bullet )}$, $\varphi$, $\tau$) with $r=1$, both leading to a functor from ${\mathrm{Mod}}^{\mathrm{fin}}(\mathcal{H}(G,I_0))$ 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' functor (when restricted to $\mathfrak{o}$-torsion ${\mathrm{GL}}_2({\mathbb{Q}}_p)$-representations generated by their pro-$p$-Iwahori invariants)

References in corpus (6)

• Ordinary representations of G(Q_p) and fundamental algebraic representations
• From pro-$p$ Iwahori-Hecke modules to $(\phi ,\Gamma )$-modules I
• Modules universels de GL(3) sur un corps p-adique en caractéristique p
• Pro-p Iwahori-Hecke algebras are Gorenstein
• Locally unitary principal series representations of ${\mathrm{GL}}_{d+1}(F)$
• From pro-$p$ Iwahori-Hecke modules to $(\phi ,\Gamma )$-modules, II

Cited by in corpus (6)

• From pro-$p$ Iwahori-Hecke modules to $(\phi ,\Gamma )$-modules I
• Central elements in affine mod $p$ Hecke algebras via perverse ${\mathbb{F}}_p$-sheaves
• Perverse ${\mathbb{F}}_p$-sheaves on the affine Grassmannian
• Supersingular Hecke modules as Galois representations
• Geometrization of the Satake transform for mod $p$ Hecke algebras
• From pro-$p$ Iwahori-Hecke modules to $(\phi ,\Gamma )$-modules, II