On Infinitesimally $\tau$-Isospectrality of Locally Symmetric Spaces
| Autor: | Bhagwat, Chandrasheel, Mondal, Kaustabh, Sachdeva, Gunja |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $(\tau, V_{\tau})$ be a finite dimensional representation of $K$, a maximal compact subgroup of a connected non-compact semisimple Lie group $G$ and let $\Gamma_i (i=1,2)$ be uniform torsion free lattices in $G$. We prove that if all but finitely many irreducible unitary $\tau$-spherical representations of $G$ occur with equal multiplicities in $L^2(\Gamma_1 \backslash G)$ and $L^2(\Gamma_2 \backslash G)$, then $L^2(\Gamma_1 \backslash G) \cong L^2(\Gamma_2 \backslash G)$ as representations of $G$. We introduce a notion of infinitesimally $\tau$-isospectrality and obtain a generalisation of the celebrated Matsushima-Murakami formula relating dimension of space of automorphic forms associated to $\tau$ and multiplicities of irreducible $\tau^\vee$-spherical representations in $L^2(\Gamma \backslash G)$. Comment: 22 pages, minor revisions |
| Databáze: | arXiv |
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