On the Poisson relation for compact Lie groups

Intuition drawn from quantum mechanics and geometric optics raises the following long-standing question: Can the length spectrum of a closed Riemannian manifold be recovered from its Laplace spectrum? By demonstrating that the Poisson relation is an equality for a generic bi-invariant metric on a compact Lie group, we establish that the length spectrum of a generic bi-invariant metric on a compact Lie group can be recovered from its Laplace spectrum. Furthermore, we exhibit a substantial collection G of compact Lie groups—including those that are either tori, simple, simply connected, or products thereof—with the property that for each group U ∈ G the length spectrum of any bi-invariant metric g carried by U is encoded in the Laplace spectrum of g . The preceding statements are special cases of results concerning compact globally symmetric spaces for which the semi-simple part of the universal cover is split-rank. The manifolds considered herein join a short list of families of non-“bumpy” Riemannian manifolds for which the Poisson relation is known to be an equality.