Spectral analysis on standard locally homogeneous spaces
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866916792724619264 |
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| author | Kassel, Fanny Kobayashi, Toshiyuki |
| author_facet | Kassel, Fanny Kobayashi, Toshiyuki |
| contents | Let $X=G/H$ be a reductive homogeneous space with $H$ noncompact, endowed with a $G$-invariant pseudo-Riemannian structure. Let $L$ be a reductive subgroup of $G$ acting properly on $X$ and $Γ$ a torsion-free discrete subgroup of $L$. Under the assumption that the complexification $X_{\mathbb C}$ is $L_{\mathbb C}$-spherical, we prove an explicit correspondence between spectral analysis on the standard locally homogeneous space $X_Γ=Γ\backslash X$ and on $Γ\backslash L$ via branching laws for the restriction to $L$ of irreducible representations of $G$. In particular, we prove that the pseudo-Riemannian Laplacian on $X_Γ$ is essentially self-adjoint, and that it admits an infinite point spectrum when $X_Γ$ is compact or $Γ\subset L$ is arithmetic. The proof builds on structural results for invariant differential operators on spherical homogeneous spaces with overgroups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1912_12601 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Spectral analysis on standard locally homogeneous spaces Kassel, Fanny Kobayashi, Toshiyuki Representation Theory Differential Geometry Spectral Theory 22E40, 22E46, 58J50, 11F72, 53C35 Let $X=G/H$ be a reductive homogeneous space with $H$ noncompact, endowed with a $G$-invariant pseudo-Riemannian structure. Let $L$ be a reductive subgroup of $G$ acting properly on $X$ and $Γ$ a torsion-free discrete subgroup of $L$. Under the assumption that the complexification $X_{\mathbb C}$ is $L_{\mathbb C}$-spherical, we prove an explicit correspondence between spectral analysis on the standard locally homogeneous space $X_Γ=Γ\backslash X$ and on $Γ\backslash L$ via branching laws for the restriction to $L$ of irreducible representations of $G$. In particular, we prove that the pseudo-Riemannian Laplacian on $X_Γ$ is essentially self-adjoint, and that it admits an infinite point spectrum when $X_Γ$ is compact or $Γ\subset L$ is arithmetic. The proof builds on structural results for invariant differential operators on spherical homogeneous spaces with overgroups. |
| title | Spectral analysis on standard locally homogeneous spaces |
| topic | Representation Theory Differential Geometry Spectral Theory 22E40, 22E46, 58J50, 11F72, 53C35 |
| url | https://arxiv.org/abs/1912.12601 |