Unitary and Nonunitary Representations of the Heisenberg-Weyl Lie Algebra
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| Format: | Preprint |
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2026
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| author | Douglas, Andrew de Guise, Hubert Repka, Joe |
| author_facet | Douglas, Andrew de Guise, Hubert Repka, Joe |
| contents | We examine unitary and nonunitary representations of the Heisenberg-Weyl Lie algebra $\mathfrak{hw}_n$, with particular emphasis on tensor products of unitary representations and on indecomposable nonunitary representations. In the unitary setting, the irreducible representations with nontrivial central character are the Schrödinger representations, as classified by the Stone-von Neumann theorem. Although tensor products of these representations are considered in the literature, we give a detailed Lie-algebraic analysis and construct explicit unitary intertwining operators, including the case where the central characters sum to zero. In the nonunitary setting, we consider a natural realization of $\mathfrak{hw}_n$ as a subalgebra of the real symplectic Lie algebra $\mathfrak{sp}_{2n+2}(\mathbb R)$ and prove that every finite-dimensional complex irreducible representation of $\mathfrak{sp}_{2n+2}(\mathbb{R})$ remains indecomposable upon restriction to $\mathfrak{hw}_n$. This yields a large natural family of finite-dimensional, nonunitary indecomposable representations of $\mathfrak{hw}_n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_05782 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unitary and Nonunitary Representations of the Heisenberg-Weyl Lie Algebra Douglas, Andrew de Guise, Hubert Repka, Joe Representation Theory 17B10, 17B20, 17B30, 22E25, 22E27 We examine unitary and nonunitary representations of the Heisenberg-Weyl Lie algebra $\mathfrak{hw}_n$, with particular emphasis on tensor products of unitary representations and on indecomposable nonunitary representations. In the unitary setting, the irreducible representations with nontrivial central character are the Schrödinger representations, as classified by the Stone-von Neumann theorem. Although tensor products of these representations are considered in the literature, we give a detailed Lie-algebraic analysis and construct explicit unitary intertwining operators, including the case where the central characters sum to zero. In the nonunitary setting, we consider a natural realization of $\mathfrak{hw}_n$ as a subalgebra of the real symplectic Lie algebra $\mathfrak{sp}_{2n+2}(\mathbb R)$ and prove that every finite-dimensional complex irreducible representation of $\mathfrak{sp}_{2n+2}(\mathbb{R})$ remains indecomposable upon restriction to $\mathfrak{hw}_n$. This yields a large natural family of finite-dimensional, nonunitary indecomposable representations of $\mathfrak{hw}_n$. |
| title | Unitary and Nonunitary Representations of the Heisenberg-Weyl Lie Algebra |
| topic | Representation Theory 17B10, 17B20, 17B30, 22E25, 22E27 |
| url | https://arxiv.org/abs/2603.05782 |