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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2405.10382 |
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| _version_ | 1866929371517812736 |
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| author | Kitagawa, Masatoshi |
| author_facet | Kitagawa, Masatoshi |
| contents | In this paper, we deal with the $\mathcal{U}(\mathfrak{g})$-action on a $\mathfrak{g}$-module on which a larger algebra $\mathcal{A}$ acts irreducibly. Under a mild condition, we will show that the support of the $\mathcal{Z}(\mathfrak{g})$-action is a union of affine subspaces in the dual of a Cartan subalgebra modulo the Weyl group action. As a consequence, we propose a definition of a Cartan subalgebra for such a $\mathfrak{g}$-module.
The support of the $\mathcal{Z}(\mathfrak{g})$-module is an algebraic counterpart of the support of the measure in the irreducible decomposition of a unitary representation. This consideration is motivated by the theory of the discrete decomposability initiated by T. Kobayashi. Defining a Cartan subalgebra for a $\mathfrak{g}$-module is motivated by the study of I. Losev on Poisson $G$-varieties. These are related each other through the associated variety and the nilpotent orbit associated to a $\mathfrak{g}$-module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10382 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cartan subalgebras for restrictions of $\mathfrak{g}$-modules Kitagawa, Masatoshi Representation Theory In this paper, we deal with the $\mathcal{U}(\mathfrak{g})$-action on a $\mathfrak{g}$-module on which a larger algebra $\mathcal{A}$ acts irreducibly. Under a mild condition, we will show that the support of the $\mathcal{Z}(\mathfrak{g})$-action is a union of affine subspaces in the dual of a Cartan subalgebra modulo the Weyl group action. As a consequence, we propose a definition of a Cartan subalgebra for such a $\mathfrak{g}$-module. The support of the $\mathcal{Z}(\mathfrak{g})$-module is an algebraic counterpart of the support of the measure in the irreducible decomposition of a unitary representation. This consideration is motivated by the theory of the discrete decomposability initiated by T. Kobayashi. Defining a Cartan subalgebra for a $\mathfrak{g}$-module is motivated by the study of I. Losev on Poisson $G$-varieties. These are related each other through the associated variety and the nilpotent orbit associated to a $\mathfrak{g}$-module. |
| title | Cartan subalgebras for restrictions of $\mathfrak{g}$-modules |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2405.10382 |