Classification of irreducible Harish-Chandra modules over extended Divergence-zero Lie algebras

TL;DR

This paper classifies irreducible Harish-Chandra modules over a specific divergence-zero Lie algebra, showing they are either cuspidal or generalized highest weight modules, and further characterizing the latter.

Contribution

It provides a complete classification of irreducible Harish-Chandra modules over the extended divergence-zero Lie algebra with nontrivial action, including their highest weight structure.

Findings

Every such module is either cuspidal or a generalized highest weight module.

Every irreducible generalized highest weight module is a highest weight module with respect to a suitable triangular decomposition.

The classification of irreducible Harish-Chandra modules over the algebra is achieved.

Abstract

Let $\mathcal{A}_n = \C[t_1^{\pm1}, t_2^{\pm1}, \ldots, t_n^{\pm1}]$, and let $\EuScript{D}_n$ denote the divergence-zero subalgebra of $\text{Der}\,(\mathcal{A}_n)$. In this paper, we classify irreducible Harish-Chandra modules over the extended divergence-zero Lie algebra $\EuScript{G}:=\EuScript{D}_n \ltimes \mathcal{A}_n$ with nontrivial $\mathcal{A}_n'$-action, where $\mathcal{A}'n= \oplus_{{\bf{m}} \in \Z^n\setminus \{\bf{0}\}} \C t^{\bf{m}}$. We prove that every such module is either cuspidal or a generalised highest weight module. We further prove that every irreducible generalised highest weight $\EuScript{G}$-module is an irreducible highest weight module with respect to a suitable triangular decomposition of $\EuScript{G}$. As a consequence, we obtain a classification of irreducible Harish-Chandra modules over $\EuScript{G}$ with nontrivial $\mathcal{A}_n'$-action.