Spectrally distinguishing symmetric spaces I

We prove that the irreducible symmetric space of complex structures on R2n (resp.
quaternionic structures on C2n) is spectrally unique within a 2-parameter (resp. 3-parameter)
family of homogeneous metrics on the underlying differentiable manifold. Such families are
strong candidates to contain all homogeneous metrics admitted on the corresponding manifolds.
The main tool in the proof is an explicit expression for the smallest positive eigenvalue of the
Laplace-Beltrami operator associated to each homogeneous metric involved. As a second conse-
quence of this expression, we prove that any non-symmetric Einstein metric in the homogeneous
families mentioned above are ν-unstable.