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On Infinitesimal $τ$-Isospectrality of Locally Symmetric Spaces

Chandrasheel Bhagwat, Kaustabh Mondal, Gunja Sachdeva

Abstract

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.

On Infinitesimal $τ$-Isospectrality of Locally Symmetric Spaces

Abstract

Let be a finite dimensional representation of a maximal compact subgroup of a connected non-compact semisimple Lie group , and let be a uniform torsion-free lattice in . We obtain an infinitesimal version of the celebrated Matsushima-Murakami formula, which relates the dimension of the space of automorphic forms associated to and multiplicities of irreducible -spherical spectra in . 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 -isospectrality. Along with this we prove that the almost equality of -spherical spectra of two lattices assures the equality of their -spherical spectra.
Paper Structure (17 sections, 46 equations)

This paper contains 17 sections, 46 equations.

Table of Contents

  1. Introduction
  2. $\boldsymbol{\tau}$-Isospectrality and Representation Equivalence
  3. Infinitesimal $\boldsymbol{\tau}$-Isospectrality
  4. Literature Review and Methodology
  5. Preliminaries
  6. Basic setup
  7. Results on Harish-Chandra characters
  8. Isospectrality and representation equivalence
  9. A further refinement
  10. Selberg trace formula for $L^2(\Gamma \backslash G, V_{\tau})$
  11. Proof of the first main result Thm. \ref{['thm1.3.1']}
  12. Some preliminary results
  13. Proof of Theorem \ref{['thm1.3.1']}
  14. Proof of second main result Thm. \ref{['thm2']}
  15. Some observations
  16. ...and 2 more sections

Theorems & Definitions (9)

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