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.
Faithfulness of highest-weight modules for Iwasawa algebras in type D
Faithfulness of simple 2-representations of $\mathfrak{sl}_2$
Representations of twisted current algebras
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