On the generic irreducibility of the spectrum of the Laplacian on homogeneous spaces.

Abstract: We discuss the generic irreducibility of Laplace eigenspaces on compact homogeneous spaces $M=G/K$ with $G$-invariant metrics. While Uhlenbeck’s theorem suggests that, for generic metrics, Laplace eigenvalues are simple, $G$-invariance forces multiplicities. Since each eigenspace carries a natural representation of $G$, the appropriate substitute is representation-theoretic simplicity: each eigenspace should be an irreducible $G$-module. Building on Schueth’s viewpoint for compact Lie groups with left-invariant metrics, we present the framework developed for homogeneous spaces, emphasizing the results of Petrecca--Röser and the remarks in de Oliveira--Marrocos on real versus complex irreducibility and on structural sources of eigenvalue collisions in higher rank. We conclude with a discrete analogue: generic spectra of weighted Laplacians on Cayley graphs.

differential geometrygeometric topologymetric geometry

Audience: researchers in the topic

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

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