Primitive ideals in rational, nilpotent Iwasawa algebras

TL;DR

This paper proves that in the setting of $p$-adic fields and nilpotent uniform pro-$p$ groups, all primitive ideals in the $K$-rational Iwasawa algebra are maximal and can be expressed in a standard form, using modules over affinoid envelopes.

Contribution

It establishes that all primitive ideals in the $K$-rational Iwasawa algebra of a nilpotent uniform pro-$p$ group are maximal and reducible to annihilators of modules over affinoid envelopes, providing a standard form.

Findings

Primitive ideals are maximal in the $K$-rational Iwasawa algebra.

Primitive ideals can be reduced to annihilators of modules over affinoid envelopes.

All primitive ideals can be expressed in a standard form.

Abstract

Given a $p$-adic field $K$ and a nilpotent uniform pro-$p$ group $G$, we prove that all primitive ideals in the $K$-rational Iwasawa algebra $KG$ are maximal, and can be reduced to a particular standard form. Setting $\mathcal{L}$ as the associated $\mathbb{Z}_p$-Lie algebra of $G$, our approach is to study the action of $KG$ on a Dixmier module $\widehat{D(\lambda)}$ over the affinoid envelope $\widehat{U(\mathcal{L})}_K$, and to prove that all primitive ideals can be reduced to annihilators of modules of this form.