Inverse spectral geometry is the study of the relationship between the geometry of a Riemannian manifold and the spectrum of its associated Laplace operator. Motivated in part by considerations from quantum mechanics, it is a long-standing folk-conjecture that the spectrum of a manifold determines its length spectrum (i.e., the set consisting of the lengths of the closed geodesics). Using the trace formula of Duistermaat and Guillemin one can see that this conjecture is true for sufficiently ``bumpy'' Riemannian manifolds. However, our understanding of the conjecture in the homogeneous setting---where closed geodesics occur in large families---is rather incomplete. In this talk, we will demonstrate that the conjecture is true for compact simple Lie groups equipped with a bi-invariant metric and, more generally, compact irreducible symmetric spaces of splitting rank by showing that the so-called Poisson relation is an equality in this setting.