From \'etale $P_{+}$-representations to $G$-equivariant sheaves on $G/P$

TL;DR

This paper constructs $G$-equivariant sheaves on $G/P$ from modules over Fontaine rings with semilinear étale actions, generalizing Colmez's work on $GL(2, Q_p)$ to broader reductive groups.

Contribution

It introduces a new method to associate $G$-equivariant sheaves to modules over Fontaine rings, extending Colmez's $GL(2, Q_p)$ representation theory to general reductive groups.

Findings

Generalizes Colmez's $GL(2, Q_p)$ construction

Provides a new link between Fontaine modules and sheaves on $G/P$

Establishes a framework for $G$-equivariant sheaves from Fontaine modules

Abstract

Let $K/\mathbb Q_{p}$ be a finite extension with ring of integers $o$, let $G$ be a connected reductive split $\mathbb Q_{p}$-group of Borel subgroup $P=TN$ and let $\alpha$ be a simple root of $T$ in $N$. We associate to a finitely generated module $D$ over the Fontaine ring over $o $ endowed with a semilinear \'etale action of the monoid $T_{+} $ (acting on the Fontaine ring via $\alpha$), a $G(\mathbb Q_{p})$-equivariant sheaf of $o$-modules on the compact space $G(\mathbb Q_{p})/P(\mathbb Q_{p})$. Our construction generalizes the representation $D\boxtimes \mathbb P^{1} $ of $ GL(2,\mathbb Q_{p})$ associated by Colmez to a $(\varphi,\Gamma)$-module $D$ endowed with a character of $\mathbb Q_{p}^{*}$.