On some properties of the functors ${\mathcal F}^G_P$ from Lie algebra to locally analytic representations

TL;DR

This paper investigates the properties of certain functors from Lie algebras to locally analytic representations for split reductive groups over p-adic fields, focusing on their faithfulness, projectivity, Ext-groups, and adjunctions.

Contribution

It provides a detailed analysis of the functors ${ m extbf{F}}^G_P$, exploring their categorical properties and relationships within the context of p-adic representation theory.

Findings

Analysis of faithfulness of the functors

Characterization of projective and injective objects

Descriptions of Ext-groups and adjunction formulas

Abstract

For a split reductive group $G$ over a finite extension $L$ of ${\mathbb Q}_p$, and a parabolic subgroup $P \subset G$ we examine functorial properties of the functors ${\mathcal F}^G_P$ introduced in \cite{OS2}. We discuss the aspects of faithfulness, projective and injective objects, Ext-groups and some kind of adjunction formulas.