On the Smooth Part Functor

TL;DR

This paper proves that all 1-cocycles in smooth representations of compact p-adic analytic groups are continuous, leading to automatic smoothness of extensions and vanishing of the first derived functor of the smooth part functor.

Contribution

It establishes the continuity of 1-cocycles and the vanishing of the first derived functor for smooth representations of compact p-adic groups, a novel result in representation theory.

Findings

All 1-cocycles are continuous in smooth representations.

Extensions of smooth representations are automatically smooth.

The first derived functor of the smooth part functor vanishes.

Abstract

Let $G$ be a compact $p$-adic analytic group and $k$ a field positive characteristic. We prove that for every smooth representation of $G$ on a $k$-vector space $V$, every 1-cocycle $G\to V$ is continuous. We deduce that the first derived functor of the smooth part functor vanishes on smooth representations. As a corollary, we obtain that extensions of smooth representations are automatically smooth.