Construction of Irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-Modules and Discretely Decomposable Restrictions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915604915552256 |
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| author | Kitagawa, Masatoshi |
| author_facet | Kitagawa, Masatoshi |
| contents | In this paper, we study the irreducibility of $\mathcal{U}(\mathfrak{g})^{G'}$-modules on the spaces of intertwining operators in the branching problem of reductive Lie algebras, and construct a family of finite-dimensional irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-modules using the Zuckerman derived functors. We provide criteria for the irreducibility of $\mathcal{U}(\mathfrak{g})^{G'}$-modules in the cases of generalized Verma modules, cohomologically induced modules, and discrete series representations. We treat only discrete decomposable restrictions with certain dominance conditions (quasi-abelian and in the good range). To describe the $\mathcal{U}(\mathfrak{g})^{G'}$-modules, we give branching laws of cohomologically induced modules using ones of generalized Verma modules when $K'$ acts on $K/L_K$ transitively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17125 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Construction of Irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-Modules and Discretely Decomposable Restrictions Kitagawa, Masatoshi Representation Theory In this paper, we study the irreducibility of $\mathcal{U}(\mathfrak{g})^{G'}$-modules on the spaces of intertwining operators in the branching problem of reductive Lie algebras, and construct a family of finite-dimensional irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-modules using the Zuckerman derived functors. We provide criteria for the irreducibility of $\mathcal{U}(\mathfrak{g})^{G'}$-modules in the cases of generalized Verma modules, cohomologically induced modules, and discrete series representations. We treat only discrete decomposable restrictions with certain dominance conditions (quasi-abelian and in the good range). To describe the $\mathcal{U}(\mathfrak{g})^{G'}$-modules, we give branching laws of cohomologically induced modules using ones of generalized Verma modules when $K'$ acts on $K/L_K$ transitively. |
| title | Construction of Irreducible $\mathcal{U}(\mathfrak{g})^{G'}$-Modules and Discretely Decomposable Restrictions |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2410.17125 |