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On Infinitesimal τ-Isospectrality of Locally Symmetric Spaces: Relating Representation Multiplicities and Spectral Data


Core Concepts
This mathematics research paper establishes a connection between the spectral data of locally symmetric spaces and the multiplicities of certain representations in the right regular representation of the underlying Lie group.
Abstract
  • 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:

    • The paper proves that almost-τ-representation equivalence implies τ-representation equivalence for uniform torsion-free lattices in non-compact symmetric spaces. This result implies that if the multiplicities of almost all τ-spherical representations agree for two lattices, then the multiplicities of all such representations must agree.
    • The authors derive an infinitesimal version of the Matsushima-Murakami formula, which relates the dimension of the space of automorphic forms associated with a representation τ to the multiplicities of irreducible τ∨-spherical representations in the right regular representation.
  • Main Conclusions:

    • The study establishes a strong link between the spectral data of locally symmetric spaces and the representation theory of the underlying Lie groups.
    • The infinitesimal Matsushima-Murakami formula provides a valuable tool for studying the spectral geometry of locally symmetric spaces and their isospectrality properties.
  • 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

https://arxiv.org/pdf/2405.09847.pdf
On Infinitesimal $\tau$-Isospectrality of Locally Symmetric Spaces

Deeper Inquiries

How do the results of this paper extend to the setting of non-uniform or non-torsion-free lattices?

Extending the results of this paper to non-uniform or non-torsion-free lattices presents significant challenges. Here's why: Continuous Spectrum: Non-uniform lattices lead to non-compact locally symmetric spaces, introducing a continuous spectrum in addition to the discrete spectrum considered in the paper. The Matsushima-Murakami formula, a cornerstone of the paper's approach, specifically relates the discrete spectrum to representation multiplicities. Dealing with the continuous spectrum would require different techniques and likely a more sophisticated version of the trace formula. Torsion Elements: Torsion elements in the lattice introduce singularities in the geometric side of the Selberg Trace Formula. The analysis of these orbital integrals for torsion elements is considerably more intricate. Convergence Issues: The convergence of the trace formula itself becomes more delicate in the non-compact setting. One often needs to impose additional growth conditions on the test functions and representations involved. Potential Approaches and Modifications: Reduction Theory: One might try to use reduction theory to decompose the locally symmetric space into simpler pieces, some of which might be compact. This could allow for a partial application of the existing techniques. Arthur-Selberg Trace Formula: For non-uniform lattices, the Arthur-Selberg Trace Formula, a more general and powerful version of the trace formula, would be necessary. This formula incorporates both the discrete and continuous spectra. Supercuspidal Representations: Focusing on the cuspidal or supercuspidal spectrum (representations with vanishing integrals along certain subgroups) might simplify the analysis, as these representations often exhibit more regular behavior.

Could there be a counterexample to the implication of almost-τ-representation equivalence to τ-representation equivalence in a more general setting beyond the assumptions made in the paper?

It is certainly conceivable that counterexamples to the implication of almost-τ-representation equivalence to τ-representation equivalence could exist in more general settings. Here are some scenarios where this implication might break down: Infinitely Many Exceptional Representations: The proof in the paper relies heavily on the finiteness of the set S of representations where the multiplicities differ. If we relax the "almost" condition to allow for infinitely many exceptions, the linear independence argument used in the proof no longer holds. Non-Linear Groups: The paper focuses on linear groups (groups that admit faithful finite-dimensional representations). The representation theory of non-linear groups can be significantly more complicated, and the techniques used in the paper might not readily generalize. Twists by Characters: It's possible that twisting one of the lattices by a non-trivial character of the group G could preserve almost-τ-representation equivalence without preserving full τ-representation equivalence. Constructing explicit counterexamples in these settings would be an interesting and challenging problem. It would likely require a deep understanding of the representation theory of the groups involved and the construction of lattices with very specific spectral properties.

How can the connection between the infinitesimal Matsushima-Murakami formula and Langlands L-packets be further explored to gain deeper insights into the spectral theory of locally symmetric spaces?

The connection between the infinitesimal Matsushima-Murakami formula and Langlands L-packets is intriguing and potentially very fruitful for deepening our understanding of the spectral theory of locally symmetric spaces. Here are some avenues for further exploration: L-packet Version of Matsushima-Murakami: As the paper mentions, formulating a variant of the Matsushima-Murakami formula directly in terms of L-packets would be a significant step. This would require understanding how the multiplicities of representations within an L-packet contribute to the dimensions of spaces of automorphic forms. Endoscopy and Transfer: Endoscopy provides a way to relate representations of different groups. Investigating how the infinitesimal Matsushima-Murakami formula behaves under endoscopic transfer could reveal hidden relationships between the spectral data of different locally symmetric spaces. Beyond the Discrete Spectrum: Exploring how the continuous spectrum of locally symmetric spaces interacts with L-packets is a major challenge. The Langlands program suggests that the continuous spectrum should also be organized into L-packets, but the precise nature of this organization is still an active area of research. Arithmetic Geometry: The Langlands program has deep connections to arithmetic geometry. Interpreting the infinitesimal Matsushima-Murakami formula in an arithmetic geometric setting, perhaps in terms of cohomology of Shimura varieties, could lead to new insights and connections. By pursuing these directions, we can hope to gain a more unified and comprehensive understanding of the spectral theory of locally symmetric spaces, connecting it to the broader framework of the Langlands program and its rich tapestry of ideas.
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