The functor $\check{\mathcal{F}}^G_P$ at the level of $K_0$

TL;DR

This paper investigates a functor related to $p$-adic Lie groups and their representations, showing it induces injective maps on Grothendieck groups and exploring its interaction with translation functors.

Contribution

It proves that a specific functor from $ ext{D}(G)$-modules induces injective homomorphisms on Grothendieck groups and analyzes its interaction with translation functors.

Findings

Functor induces injective homomorphisms on Grothendieck groups.

Interaction with translation functors is clarified.

Provides new insights into the structure of $D(G)$-modules.

Abstract

Let $G$ be a $p$-adic Lie group with reductive Lie algebra $\mathfrak{g}$. Denote by $D(G)$ the locally analytic distribution algebra of $G$. Orlik-Strauch and Agrawal-Strauch have studied certain exact functors defined on various categories of $\mathfrak{g}$-representations with image in the category of locally analytic $G$-representations or $D(G)$-modules. In this paper we prove that for suitably defined categories of $D(G)$-modules, this functor gives rise to injective homomorphisms at the level of Grothendieck groups. We also explain how this functor interacts with translation functors at the level of Grothendieck groups.