Classification of irreducible Harish-Chandra modules over extended Divergence-zero Lie algebras
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909014725492736 |
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| author | Mukherjee, Sudipta |
| author_facet | Mukherjee, Sudipta |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03985 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Classification of irreducible Harish-Chandra modules over extended Divergence-zero Lie algebras Mukherjee, Sudipta Representation Theory 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. |
| title | Classification of irreducible Harish-Chandra modules over extended Divergence-zero Lie algebras |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2605.03985 |