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The spectrum of the Laplace-Beltrami operator can be computed using Freudenthal's formula for normal homogeneous spaces $(G/K,g)$, i.e. the metric $g$ is the restriction of a biinvariant metric of $G$. We extend this formula to naturally reductive homogeneous spaces and provide naturally reductive realizations of deformations of normal homogeneous metrics along commuting subalgebras. This method can be applied to $3$-Sasaki manifolds together with their canonical deformation, which are positive $3$-$(\alpha,\delta)$-Sasaki manifolds. We obtain a formula to compute the spectrum of $3$-$(\alpha,\delta)$-Sasaki manifolds. Our approach has been inspired by Wilking's normal homogeneous realization of the Aloff-Wallach manifold $W^{1,1}=SU(3)/S^{1}$. We compute its full $3$-$(\alpha,\delta)$-Sasaki spectrum and related its spectrum to its geometry.

Spectrum of the Laplace-Beltrami Operator on Naturally Reductive Spaces