# Rigid Analytic Vectors in Locally Analytic Representations

## TL;DR

This paper explores the structure of rigid analytic vectors in locally analytic representations of uniform pro-p groups, establishing isomorphisms with distribution algebras and proving functor exactness.

## Contribution

It introduces a canonical isomorphism between the dual of rigid analytic vectors and Emerton's distribution algebra, and proves the exactness of the rigid analytic vector functor.

## Key findings

- Canonical isomorphism between dual of rigid analytic vectors and distribution algebra.
- Exactness of the functor assigning rigid analytic vectors to admissible representations.

## Abstract

Let $H$ be a uniform pro-$p$ group. Associated to $H$ are rigid analytic affinoid groups $\bbH_n$, and their "wide open" subgroups $\bbH_n^{\circ}$. Denote by $D^\la(H)= C^\la(H)'_b$ the locally analytic distribution algebra of $H$ and by $\DHnn$ Emerton's ring of $\bbH_n^{\circ}$-rigid analytic distributions on $H$. If $V$ is an admissible locally analytic representation of $H$, and if $V_{\bbH_n^\circ-\an}$ denotes the subspace of $\bbH_n^\circ$-rigid analytic vectors (with its intrinsic topology), then we show that the continuous dual of $V_{\bbH_n^\circ-\an}$ is canonically isomorphic to $\DHnn \ot_{D^\la(H)} V'$. From this we deduce the exactness of the functor $V \rightsquigarrow V_{\bbH_n^\circ-\an}$ on the category of admissible locally analytic representations of $H$.