Borel-de Siebenthal Positive Root Systems
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910916645224448 |
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| author | Paul, Pampa |
| author_facet | Paul, Pampa |
| contents | Let $G$ be a connected simple Lie group with finite centre, $K$ be a maximal compact subgroup of $G,$ and rank$(G)=$ rank$(K).$ Let $\frak{g}_0=$Lie$(G), \frak{k}_0=$Lie$(K) \subset \frak{g}_0, \frak{t}_0$ be a maximal abelian subalgebra of $\frak{k}_0, \frak{g}=\frak{g}_0^\mathbb{C}, \frak{k}=\frak{k}_0^\mathbb{C},$ and $\frak{h}=\frak{t}_0^\mathbb{C}.$ The existence of a Borel-de Siebenthal positive root system of $Δ(\frak{g}, \frak{h})$ is proved by Borel and de Siebenthal. In this article, we have determined all Borel-de Siebenthal positive root systems of $Δ(\frak{g}, \frak{h}),$ assuming the existence. As an application, we have determined the number of unitary equivalence classes of all Borel-de Siebenthal discrete series representations of $G$ (if $G/K$ is not Hermitian symmetric) with a fixed infinitesimal character. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_11099 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Borel-de Siebenthal Positive Root Systems Paul, Pampa Representation Theory 17B10, 17B20, 17B22, 17B25, 22E46 Let $G$ be a connected simple Lie group with finite centre, $K$ be a maximal compact subgroup of $G,$ and rank$(G)=$ rank$(K).$ Let $\frak{g}_0=$Lie$(G), \frak{k}_0=$Lie$(K) \subset \frak{g}_0, \frak{t}_0$ be a maximal abelian subalgebra of $\frak{k}_0, \frak{g}=\frak{g}_0^\mathbb{C}, \frak{k}=\frak{k}_0^\mathbb{C},$ and $\frak{h}=\frak{t}_0^\mathbb{C}.$ The existence of a Borel-de Siebenthal positive root system of $Δ(\frak{g}, \frak{h})$ is proved by Borel and de Siebenthal. In this article, we have determined all Borel-de Siebenthal positive root systems of $Δ(\frak{g}, \frak{h}),$ assuming the existence. As an application, we have determined the number of unitary equivalence classes of all Borel-de Siebenthal discrete series representations of $G$ (if $G/K$ is not Hermitian symmetric) with a fixed infinitesimal character. |
| title | Borel-de Siebenthal Positive Root Systems |
| topic | Representation Theory 17B10, 17B20, 17B22, 17B25, 22E46 |
| url | https://arxiv.org/abs/2309.11099 |