Towards the Derived Jacquet-Emerton Module Functor

TL;DR

This paper introduces a derived functor extension of the Jacquet-Emerton module, broadening the tools for analyzing admissible locally analytic representations of p-adic Lie groups.

Contribution

It defines a new $ ext{delta}$-functor $H^{ullet}J_{P}$ that extends the Jacquet-Emerton module to a derived setting for admissible locally analytic representations.

Findings

Defines a $ ext{delta}$-functor extending $J_P$

Establishes properties of the derived functor

Provides new tools for representation theory of p-adic groups

Abstract

Let $G$ be a $p$-adic Lie group associated to a connected reductive group over $\mathbb{Q}_{p}$. Let $P$ be a parabolic subgroup of $G$ and let $M$ be a Levi quotient of $P$. In this paper, we define a $\delta$-functor $H^{\star}J_{P}$ from the category of admissible locally analytic $G$-representations to the category of essentially admissible locally analytic $M$-representations that extends the Jacquet-Emerton module functor $J_{P}$ defined by Emerton.