On Infinitesimal $τ$-Isospectrality of Locally Symmetric Spaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913800048869376 |
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| author | Bhagwat, Chandrasheel Mondal, Kaustabh Sachdeva, Gunja |
| author_facet | Bhagwat, Chandrasheel Mondal, Kaustabh Sachdeva, Gunja |
| contents | Let $(τ, V_τ)$ be a finite dimensional representation of a maximal compact subgroup $K$ of a connected non-compact semisimple Lie group $G$, and let $Γ$ be a uniform torsion-free lattice in $G$. We obtain an infinitesimal version of the celebrated Matsushima-Murakami formula, which relates the dimension of the space of automorphic forms associated to $τ$ and multiplicities of irreducible $τ^\vee$-spherical spectra in $L^2(Γ\backslash G)$. This result gives a promising tool to study the joint spectra of all central operators on the homogenous bundle associated to the locally symmetric space and hence its infinitesimal $τ$-isospectrality. Along with this we prove that the almost equality of $τ$-spherical spectra of two lattices assures the equality of their $τ$-spherical spectra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09847 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Infinitesimal $τ$-Isospectrality of Locally Symmetric Spaces Bhagwat, Chandrasheel Mondal, Kaustabh Sachdeva, Gunja Representation Theory 22E40, 22E45, 53C35 Let $(τ, V_τ)$ be a finite dimensional representation of a maximal compact subgroup $K$ of a connected non-compact semisimple Lie group $G$, and let $Γ$ be a uniform torsion-free lattice in $G$. We obtain an infinitesimal version of the celebrated Matsushima-Murakami formula, which relates the dimension of the space of automorphic forms associated to $τ$ and multiplicities of irreducible $τ^\vee$-spherical spectra in $L^2(Γ\backslash G)$. This result gives a promising tool to study the joint spectra of all central operators on the homogenous bundle associated to the locally symmetric space and hence its infinitesimal $τ$-isospectrality. Along with this we prove that the almost equality of $τ$-spherical spectra of two lattices assures the equality of their $τ$-spherical spectra. |
| title | On Infinitesimal $τ$-Isospectrality of Locally Symmetric Spaces |
| topic | Representation Theory 22E40, 22E45, 53C35 |
| url | https://arxiv.org/abs/2405.09847 |