A note about the generic irreducibility of the spectrum of the Laplacian on homogeneous spaces

Petrecca and Röser (Mathematische Zeitschrift 291:395–419, 2018) and Schueth (Ann Global Anal Geom 52:187–200, 2017) had shown that for a generic G-invariant metric g on certain compact homogeneous spaces \(M=G/K\) (including symmetric spaces of rank 1 and some Lie groups), the spectrum of the Laplace-Beltrami operator \(\Delta _g\) was real G-simple. The same is not true for the complex version of \(\Delta _g\) when there is a presence of representations of complex or quaternionic type. We show that these types of representations induces a \(Q_8\)-action that commutes with the Laplacian in such way that G-properties of the real version of the operator have to be understood as \((Q_8 \times G)\)-properties on its corresponding complex version.