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.