Bibliographic Information: Bhagwat, C., Mondal, K., & Sachdeva, G. (2024). On Infinitesimal τ-Isospectrality of Locally Symmetric Spaces. arXiv:2405.09847v3 [math.RT].
Research Objective: This paper investigates the relationship between the spectra of locally symmetric spaces and the multiplicities of irreducible representations in the right regular representation of the associated Lie group. The authors aim to prove that almost-τ-representation equivalence implies τ-representation equivalence for uniform torsion-free lattices in non-compact symmetric spaces. Additionally, they seek to establish an infinitesimal version of the Matsushima-Murakami formula, connecting the dimension of spaces of automorphic forms to the multiplicities of irreducible representations.
Methodology: The authors employ the Selberg Trace Formula as the primary tool for their analysis. They utilize τ-equivariant test functions to focus on τ-spherical representations and leverage the properties of Harish-Chandra character distributions. The proof relies on constructing specific open sets in the Lie group and analyzing the vanishing of orbital integrals.
Key Findings:
Main Conclusions:
Significance: This research significantly contributes to the field of spectral geometry and the study of locally symmetric spaces. The results provide deeper insights into the interplay between spectral data and representation multiplicities, advancing the understanding of isospectrality and related concepts.
Limitations and Future Research: The paper focuses on uniform torsion-free lattices. Exploring similar relationships for non-uniform or non-torsion-free lattices could be a potential direction for future research. Additionally, investigating the connections between the infinitesimal Matsushima-Murakami formula and Langlands L-packets could offer further insights.
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by Chandrasheel... at arxiv.org 10-15-2024
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