A note about the generic irreducibility of the spectrum of the Laplacian on homogeneous spaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915129778503680 |
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| author | de Oliveira, Diego S. Marrocos, Marcus A. M. |
| author_facet | de Oliveira, Diego S. Marrocos, Marcus A. M. |
| contents | Petrecca and Röser (2018, \cite{Petrecca2019}), and Schueth (2017, \cite{Schueth2017}) had shown that for a generic $G$-invariant metric $g$ on certain compact homogeneous spaces $M=G/K$ (including symmetric spaces of rank 1 and some Lie groups), the spectrum of the Laplace-Beltrami operator $Δ_g$ was real $G$-simple. The same is not true for the complex version of $Δ_g$ when there is a presence of representations of complex or quaternionic type. We show that these types of representations induces a $Q_8$-action that commutes with the Laplacian in such way that $G$-properties of the real version of the operator have to be understood as $(Q_8 \times G)$-properties on its corresponding complex version. Also we argue that for symmetric spaces on rank $\geq 2$ there are algebraic symmetries on the corresponding root systems which relates distinct irreducible representations on the same eigenspace. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18747 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note about the generic irreducibility of the spectrum of the Laplacian on homogeneous spaces de Oliveira, Diego S. Marrocos, Marcus A. M. Spectral Theory 35J05, 22C05, 22E46, 53C35 Petrecca and Röser (2018, \cite{Petrecca2019}), and Schueth (2017, \cite{Schueth2017}) had shown that for a generic $G$-invariant metric $g$ on certain compact homogeneous spaces $M=G/K$ (including symmetric spaces of rank 1 and some Lie groups), the spectrum of the Laplace-Beltrami operator $Δ_g$ was real $G$-simple. The same is not true for the complex version of $Δ_g$ when there is a presence of representations of complex or quaternionic type. We show that these types of representations induces a $Q_8$-action that commutes with the Laplacian in such way that $G$-properties of the real version of the operator have to be understood as $(Q_8 \times G)$-properties on its corresponding complex version. Also we argue that for symmetric spaces on rank $\geq 2$ there are algebraic symmetries on the corresponding root systems which relates distinct irreducible representations on the same eigenspace. |
| title | A note about the generic irreducibility of the spectrum of the Laplacian on homogeneous spaces |
| topic | Spectral Theory 35J05, 22C05, 22E46, 53C35 |
| url | https://arxiv.org/abs/2501.18747 |