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

TL;DR

This paper investigates the generic spectral properties of the Laplacian on homogeneous spaces, revealing how complex and quaternionic representations influence the spectrum's simplicity and symmetries.

Contribution

It demonstrates the impact of complex and quaternionic representations on the spectrum, introducing a $Q_8$-action that affects the spectral properties of the complex Laplacian.

Findings

Real $G$-simple spectrum for generic metrics on certain homogeneous spaces.

Complex representations induce a $Q_8$-action affecting spectral properties.

Symmetries in root systems relate different irreducible representations in higher rank symmetric spaces.

Abstract

Petrecca and R\"oser (2018, \cite{Petrecca2019}), and Schueth (2017, \cite{Schueth2017}) 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. Also we argue that for symmetric spaces on rank $\geq 2$ there are algebraic symmetries on the corresponding root systems which relates distinct irreducible representations on the same eigenspace.