The wave equation, O(2, 2), and separation of variables on hyperboloids

We classify group-theoretically all separable coordinate systems for the eigenvalue equation of the Laplace-Beltrami operator on the hyperboloid [FORMULA OMITED], finding 71 orthogonal and 3 non-orthogonal systems. For a number of cases the explicit spectral resolutions are worked out. We show that our results have application to the problem of separation of variables for the wave equation and to harmonic analysis on the hyperboloid and the group manifold SL(2, R). In particular, most past studies of SL(2, R) have employed only 6 of the 74 coordinate systems in which the Casimir eigenvalue equation separates.