Some remarks on real minimal nilpotent orbits and symmetric pairs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916410491404288 |
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| author | Okuda, Takayuki |
| author_facet | Okuda, Takayuki |
| contents | For a non-compact simple Lie algebra $\mathfrak{g}$ over $\mathbb{R}$, we denote by $\mathcal{O}^{\mathbb{C}}_{\min,\mathfrak{g}}$ the unique complex nilpotent orbit in $\mathfrak{g} \otimes_\mathbb{R} \mathbb{C}$ containing all minimal real nilpotent orbits in $\mathfrak{g}$. In this paper, we give a complete classification of symmetric pairs $(\mathfrak{g},\mathfrak{h})$ such that $\mathcal{O}^{\mathbb{C}}_{\min,\mathfrak{g}} \cap \mathfrak{g}^d = \emptyset$, where $\mathfrak{g}^d$ denotes the dual Lie algebra of $(\mathfrak{g},\mathfrak{h})$. Furthermore, for symmetric pairs $(G,H)$ with real simple Lie group $G$, we apply our classification to theorems given by T. Kobayashi [J. Lie Theory (2023)], and study bounded multiplicity properties of restrictions on $H$ of infinite-dimensional irreducible $G$-representations with minimum Gelfand--Kirillov dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00675 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some remarks on real minimal nilpotent orbits and symmetric pairs Okuda, Takayuki Representation Theory Primary 22E46, Secondary 22E45, 53C35, 32M15, 53C15, 57S30 For a non-compact simple Lie algebra $\mathfrak{g}$ over $\mathbb{R}$, we denote by $\mathcal{O}^{\mathbb{C}}_{\min,\mathfrak{g}}$ the unique complex nilpotent orbit in $\mathfrak{g} \otimes_\mathbb{R} \mathbb{C}$ containing all minimal real nilpotent orbits in $\mathfrak{g}$. In this paper, we give a complete classification of symmetric pairs $(\mathfrak{g},\mathfrak{h})$ such that $\mathcal{O}^{\mathbb{C}}_{\min,\mathfrak{g}} \cap \mathfrak{g}^d = \emptyset$, where $\mathfrak{g}^d$ denotes the dual Lie algebra of $(\mathfrak{g},\mathfrak{h})$. Furthermore, for symmetric pairs $(G,H)$ with real simple Lie group $G$, we apply our classification to theorems given by T. Kobayashi [J. Lie Theory (2023)], and study bounded multiplicity properties of restrictions on $H$ of infinite-dimensional irreducible $G$-representations with minimum Gelfand--Kirillov dimension. |
| title | Some remarks on real minimal nilpotent orbits and symmetric pairs |
| topic | Representation Theory Primary 22E46, Secondary 22E45, 53C35, 32M15, 53C15, 57S30 |
| url | https://arxiv.org/abs/2407.00675 |