Generic irreducibility of Laplace eigenspaces with finite symmetry

I will report on a joint work in progress with Egor Morozov proving that for generic elements in several families of Laplace-type operators invariant under a finite group action, all eigenspaces are irreducible representations. In particular, for the case of Laplace-Beltrami operators, this provides a natural generalization of Uhlenbeck's result on the generic simplicity of the spectrum to the equivariant setting. Moreover, this extends previous work of Zelditch and solves the finite group case of a well-known question raised by Guillemin and Yau. For Schrödinger operators, our results rigorously underpin the notion of accidental degeneracy for certain quantum-mechanical systems with finite symmetry. Our approach involves modern methods of equivariant transversality which we extend to higher dimensions.