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

TL;DR

This paper develops a method to translate modules over pro-$p$ Iwahori-Hecke algebras for classical groups into $(, )$-modules, revealing a symmetry correspondence for supersingular modules.

Contribution

It explicitly constructs data for classical matrix groups to realize a functor linking Hecke modules to $(, )$-modules, extending prior abstract principles.

Findings

Identifies supersingular modules with symmetric $(, )$-modules.

Provides explicit data for classical groups to realize the functor.

Establishes a correspondence between algebraic modules and Galois-type representations.

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 $(C^{(\bullet)},\phi,\tau)$ an exact functor $M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$ from finite length ${\mathcal H}(G,I_0)$-modules to $(\varphi^r,\Gamma)$-modules. In the present paper we concretely work out such data $(C^{(\bullet)},\phi,\tau)$ 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 $(\varphi^r,\Gamma)$-modules satisfying a certain symmetry condition.