Classification of irreducible $(\mathfrak{g},\mathfrak{k})$-modules associated to the ideals of minimal nilpotent orbits for simple Lie groups of type $A$

TL;DR

This paper classifies prime primitive ideals and irreducible $(rak{g},rak{k})$-modules related to minimal nilpotent orbits in $rak{sl}(n,b C)$, providing a detailed understanding of their structure and decomposition.

Contribution

It provides a complete classification of prime primitive ideals and irreducible $(rak{g},rak{k})$-modules associated with minimal nilpotent orbits for type A Lie algebras.

Findings

Classification of prime primitive ideals with minimal nilpotent orbit closure

Explicit description of irreducible $(rak{g},rak{k})$-modules

Decomposition of modules as $rak{k}$-modules

Abstract

We classify completely prime primitive ideals whose associated varieties are the closure of the minimal nilpotent orbit of $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, and classify irreducible $(\mathfrak{g},\mathfrak{k})$-modules which have those ideals as annihilators. Moreover, we irreducibly decompose them as $\mathfrak{k}$-modules.