On infinitesimal $\boldsymbol { \tau }$-isospectrality of locally symmetric spaces

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.