On Infinitesimal $\tau$-Isospectrality of Locally Symmetric Spaces

TL;DR

This paper develops an infinitesimal version of the Matsushima-Murakami formula to analyze the spectra of locally symmetric spaces, providing new tools to study their infinitesimal $ au$-isospectrality and spectral relations between lattices.

Contribution

It introduces an infinitesimal formula relating automorphic forms and spectra, advancing the understanding of $ au$-isospectrality in locally symmetric spaces.

Findings

Infinitesimal Matsushima-Murakami formula derived.

Almost equality of spectra implies actual spectral equality.

New approach to study joint spectra of central operators.

Abstract

Let $(\tau, V_{\tau})$ be a finite dimensional representation of a maximal compact subgroup $K$ of a connected non-compact semisimple Lie group $G$, and let $\Gamma$ 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 $\tau$ and multiplicities of irreducible $\tau^\vee$-spherical spectra in $L^2(\Gamma \backslash G)$. 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 $\tau$-isospectrality. Along with this we prove that the almost equality of $\tau$-spherical spectra of two lattices assures the equality of their $\tau$-spherical spectra.