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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.02530 |
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Table of Contents:
- Given a real semisimple connected Lie group $G$ and a discrete subgroup $Γ< G$ we prove a precise connection between growth rates of the group $Γ$, polyhedral bounds on the joint spectrum of the ring of invariant differential operators, and the decay of matrix coefficients. In particular, this allows us to completely characterize temperedness of $L^2(Γ\backslash G)$ in terms of Quint's growth indicator function. As an application of our sharp polyhedral bounds we prove temperedness of $L^2(Γ\backslash G)$ for all Borel Anosov subgroups $Γ$ in higher rank Lie groups $G$ not locally isomorphic to $\mathfrak{sl}_3(\mathbb{K}),\mathbb{K}=\R,\C,\mathbb H,$ or $\mathfrak{e}_{6(-26)}$.