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

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 $\mathrm{\Gamma }$ 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 $L^2(\mathrm{\Gamma }\mathrm{\backslash }G)$. 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.