On infinitesimal τ -isospectrality of locally symmetric spaces

Abstract

Let (τ,Vτ) 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∨ -spherical spectra in L2(Γ∖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.