The Wave Trace Invariants of the Spectrum of the $G$-Invariant Laplacian

Given a compact boundaryless Riemannian manifold $Y$ on which a compact Lie group $G$ acts, there is always a metric on $Y$ such that the action is by isometries. Assuming $Y$ is equipped with such a metric, recall that the $G$-invariant Laplacian is the restriction of the ordinary Laplacian to the space of functions which are constant along the orbits of $G$. In this paper, the author analyzes the wave trace of the $G$-invariant Laplacian and shows that the singularities of this wave trace occur at the lengths of certain geodesic arcs–those that are orthogonal to the orbits and whose endpoints are related by the action of $G$. This defines a notion of the length spectrum for the group action. Further, using the deep connection between foliated manifolds and isometric group actions on manifolds, the asymptotics of these singularities are calculated for an arbitrary non-zero number in the length spectrum.