The Grothendieck group of completed distribution algebras

TL;DR

This paper computes the Grothendieck group of completed distribution algebras on certain compact p-adic groups, revealing it as a free abelian group with rank tied to conjugacy classes coprime to p.

Contribution

It establishes an explicit isomorphism for the Grothendieck group of completed distribution algebras on compact p-adic groups, linking it to conjugacy classes and extending to continuous distributions.

Findings

Grothendieck group is isomorphic to Z^c where c counts conjugacy classes coprime to p.

The algebra of continuous distributions shares the same Grothendieck group.

Results hold for sufficiently large extensions K of Q_p.

Abstract

Let $G$ be a compact $p$-adic analytic group with no element of order $p$ and $H$ be its maximal uniform normal subgroup. Let $K$ be a finite extention of $\mathbb{Q}_p$. We show that the Grothendieck group of the completion of the algebra of locally analytic distributions on $G$ is isomorphic to $\mathbb{Z}^c$ where $c$ is the number of conjugacy classes in $G/H$ relative prime to $p$, provided that $K$ is big enough. In addition we will see the algebra $K[[G]]$ of continuous distributions on $G$ has the same Grothendieck group.