Faithfulness of generalised Verma modules for Iwasawa algebras

TL;DR

This paper establishes the faithfulness of certain infinite-dimensional modules over Iwasawa algebras associated with split simple Lie algebras, providing new insights into their structure and prime ideals.

Contribution

It proves faithfulness of generalized Verma modules for Iwasawa algebras and extends results to all highest-weight modules in type A2, revealing prime ideal properties.

Findings

Faithfulness of generalized Verma modules for split simple Lie algebras.

Faithfulness of all infinite-dimensional highest-weight modules in type A2.

Prime ideals are annihilators of finite-dimensional simple modules in type A2.

Abstract

We prove faithfulness of infinite-dimensional generalised Verma modules for Iwasawa algebras corresponding to split simple Lie algebras with a Chevalley basis. We use this to prove faithfulness of all infinite-dimensional highest-weight modules in the case of type $A_{2}$. In this case we also show that all prime ideals of the corresponding Iwasawa algebras are annihilators of finite-dimensional simple modules.