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

Abstract

Let $\mathfrak{o}$ be the ring of integers in a finite extension field of ${\mathbb{Q}}_p$, let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb{Q}}_p$, let $\mathcal{H}(G,I_0)$ be its pro-$p$-Iwahori Hecke $\mathfrak{o}$-algebra. In \cite{dfun} we introduced a general principle how to assign to a certain additionally chosen datum $(C^{(\bullet )},\varphi ,\tau )$ an exact functor $M\mapsto \mathbf{D}({\Theta }_{\ast }{\mathcal{V}}_M)$ from finite length $\mathcal{H}(G,I_0)$-modules to $({\phi }^r,\Gamma )$-modules. In the present paper we concretely work out such data $(C^{(\bullet )},\varphi ,\tau )$ for the classical matrix groups. We show that the corresponding functor identifies the set of (standard) supersingular $\mathcal{H}(G,I_0){\otimes }_{\mathfrak{o}}k$-modules with the set of $({\phi }^r,\Gamma )$-modules satisfying a certain symmetry condition.

International Mathematics Research Notices

References in corpus (3)

• Minimal length elements of extended affine Coxeter groups, II
• From pro-$p$ Iwahori-Hecke modules to $(\phi ,\Gamma )$-modules I
• Compatibility between Satake and Bernstein-type isomorphisms in characteristic p

Cited by in corpus (1)

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