Affinoid Dixmier modules and the deformed Dixmier-Moeglin equivalence

TL;DR

This paper investigates the structure of affinoid envelopes of Lie algebras over $Z_p$, introduces Dixmier modules in this context, and demonstrates their role in classifying primitive ideals and establishing a deformed Dixmier-Moeglin equivalence.

Contribution

It defines Dixmier modules over affinoid envelopes and proves their irreducibility for nilpotent Lie algebras, advancing the understanding of primitive ideals and the Dixmier-Moeglin equivalence in this setting.

Findings

Dixmier modules are generally irreducible for nilpotent Lie algebras.

Primitive ideals can be characterized via Dixmier modules.

The algebra satisfies a deformed version of the Dixmier-Moeglin equivalence.

Abstract

The affinoid envelope, $\widehat{U(\mathcal{L})}$ of a free, finitely generated $\mathbb{Z}_p$-Lie algebra $\mathcal{L}$ has proven to be useful within the representation theory of compact $p$-adic Lie groups. Our aim is to further understand the algebraic structure of $\widehat{U(\mathcal{L})}$, and to this end, we will define a Dixmier module over $\widehat{U(\mathcal{L})}$, and prove that this object is generally irreducible in case where $\mathcal{L}$ is nilpotent. Ultimately, we will prove that all primitive ideals in the affinoid envelope can be described in terms of the annihilators of Dixmier modules, and using this, we aim towards proving that these algebras satisfy a version of the classical Dixmier-Moeglin equivalence.