On the Poisson relation for compact Lie groups

TL;DR

This paper proves that for a broad class of compact Lie groups with bi-invariant metrics, the length spectrum can be recovered from the Laplace spectrum, confirming the Poisson relation as an equality in these cases.

Contribution

It establishes the Poisson relation as an equality for generic bi-invariant metrics on compact Lie groups, enabling spectral recovery of the length spectrum and group rank.

Findings

Poisson relation is an equality for generic bi-invariant metrics on compact Lie groups.

The Laplace spectrum encodes the length spectrum and rank for a wide class of Lie groups.

Results extend to compact globally symmetric spaces with split-rank universal covers.

Abstract

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? The Poisson relation states that for any closed Riemannian manifold $(M,g)$ the singular support of the trace of its wave group---a spectrally determined tempered distribution---is contained in the set consisting of $\pm \tau$, where $\tau$ is the length of a smoothly closed geodesic in $(M,g)$. Therefore, in cases where the Poisson relation is an equality, we obtain a method for retrieving the length spectrum of a manifold from its Laplace spectrum. The Poisson relation is known to be an equality for sufficiently "bumpy" Riemannian manifolds and there are no known counterexamples. We demonstrate that the Poisson relation is an equality for a compact Lie group equipped with a generic bi-invariant metric. Consequently, the length spectrum of a generic bi-invariant metric (and the rank of its underlying Lie group) can be recovered from its Laplace spectrum. Furthermore, we exhibit a substantial collection $\mathscr{G}$ of compact Lie groups---including those that are either tori, simple, simply-connected, or products thereof---with the property that for each group $U \in \mathscr{G}$ the Laplace spectrum of any bi-invariant metric $g$ carried by $U$ encodes the length spectrum of $g$ and the rank of $U$. 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.