The index of the pseudo-Riemannian Dirac operator as a transversally elliptic operator

Let (M,r) be a closed, space- and time-orientable, pseudo-Riemannian spin manifold and-let G be a compact group of orientation-preserving isometries on (M,r). If there are no isotropic directions transversal to the orbits of G, then the Dirac operator on (M,r) is transversally elliptic. In this paper we calculate its index.