Spectrally distinguishing symmetric spaces I

We prove that the irreducible symmetric space of complex structures on $\mathbb R^{2n}$ (resp.\ quaternionic structures on $\mathbb C^{2n}$) 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 consequence of this expression, we prove that any non-symmetric Einstein metric in the homogeneous families mentioned above is $\nu$-unstable.