Generic equivariant simplicity of the spectrum of Laplace-type operators with finite symmetry

Abstract

We prove that for generic elements in three natural families of Laplace-type differential operators on smooth manifolds, which are equivariant with respect to a finite symmetry group, all the eigenspaces are irreducible representations. In other words, the spectrum of these operators is generically equivariantly simple. The three families of operators with symmetry are Schrödinger operators parametrized by the potential, going back to the work of Wigner, Laplace-Beltrami operators parametrized by invariant Riemannian metrics and conformal Laplacians parametrized by invariant metrics in a given conformal class. The latter two results extend prior work of Zelditch for certain free group actions and resolve, for finite group actions, a well-known open question raised by Guillemin and Yau. Our approach stems from methods developed by Wendl and Doan-Walpuski to address equivariant transversality in symplectic topology.