The spectral function of a Riemannian orbifold

Hörmander’s theorem on the asymptotics of the spectral function of an elliptic operator is extended to the setting of compact Riemannian orbifolds. In contrast to the manifold case, the asymptotics depend on the isotropy type of the point at which the spectral function is computed. It is shown that “on average” the eigenfunctions of the operator are larger at singular points than at manifold points, by a factor of the order of the isotropy type. A sketch of a more direct approach to the wave trace formula on orbifolds is also given, obtaining results already shown separately by M. Sandoval and Y. Kordyukov in the setting of Riemmannian foliations.